GRE CAT Success 2004 (Peterson's Ultimate Gre Tool Kit)

GRE CAT SUCCESS T

2004

GRE CAT SUCCESS T

2004 Includes a vocabulary-building chapter by Merriam-Webster–with Greek and Latin roots, quizzes, and a practice exam

GRE is a registered trademark of Educational Testing Service (ETS). This book is not endorsed or approved by ETS.

About The Thomson Corporation and Peterson’s The Thomson Corporation, with 2002 revenues of US$7.8 billion, is a global leader in providing integrated information solutions to business and professional customers. The Corporation’s common shares are listed on the Toronto and New York stock exchanges (TSX: TOC; NYSE: TOC). Its learning businesses and brands serve the needs of individuals, learning institutions, corporations, and government agencies with products and services for both traditional and distributed learning. Peterson’s (www.petersons.com) is a leading provider of education information and advice, with books and online resources focusing on education search, test preparation, and financial aid. Its Web site offers searchable databases and interactive tools for contacting educational institutions, online practice tests and instruction, and planning tools for securing financial aid. Peterson’s serves 110 million education consumers annually.

GRE CAT Success is published with a CD. The CD will allow you to practice what you have learned using state-of-the-art computer adaptive software. The software was created by Cambridge Educational Services, 2720 River Road, Ste. 36, Des Plaines, IL 60018.

Editorial Development: American BookWorks Corporation Editorial Review: Joan Marie Rosebush

For more information, contact Peterson’s, 2000 Lenox Drive, Lawrenceville, NJ 08648; 800-338-3282; or find us on the World Wide Web at www.petersons.com/about. COPYRIGHT © 2003 Peterson’s, a division of Thomson Learning, Inc. Thomson Learning™ is a trademark used herein under license. Previous editions © 1997, 1998, 1999, 2000, 2002 “Merriam-Webster’s Roots to Word Mastery” Copyright © 2003 Merriam-Webster, Incorporated. ALL RIGHTS RESERVED. No part of this work covered by the copyright herein may be reproduced or used in any form or by any means—graphic, electronic, or mechanical, including photocopying, recording, taping, Web distribution, or information storage and retrieval systems—without the prior written permission of the publisher. For permission to use material from this text or product, contact us by Phone: 800-730-2214 Fax: 800-730-2215 Web: www.thomsonrights.com ISBN 0-7689-1230-X Printed in the United States of America 10

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Contents RED ALERT

Introduction to the GRE CAT . . . . . . . . . .

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DIAGNOSTIC TEST. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Analytical Writing Measure . . . . . . . . . . . . . . . . . . . . . Verbal Ability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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RED ALERT UNIT 1

RED ALERT UNIT 2 UNIT 3 UNIT 4

RED ALERT UNIT 5

RED ALERT UNIT 6

RED ALERT UNIT 7

RED ALERT UNIT 8

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GRE Analytical Writing Measure Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Analytical Writing Review. . . . . . . . . . . . . . . . . . . .

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Verbal Ability Strategies. . . . . . . . . . . . . .

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Sentence Completion Review. . . . . . . . . . . . . . . . . Analogy Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Antonym Review . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Reading Comprehension Strategies. . .

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Reading Comprehension Review . . . . . . . . . . . . .

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Why Study Vocabulary for the GRE? . .

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Merriam-Webster’s Roots to Word Mastery. . . .

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Quantitative Ability Strategies . . . . . . . .

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Mathematics Review . . . . . . . . . . . . . . . . . . . . . . . . .

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Quantitative Comparisons Strategies . .

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QUANTITATIVE COMPARISONS REVIEW . . . . . . . .

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Data Analysis Strategies. . . . . . . . . . . . . .

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Data Analysis Review . . . . . . . . . . . . . . . . . . . . . . . .

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CONTENTS

PRACTICE TEST 1 PRACTICE TEST 2

Analytical Writing Measure . . . . . . . . . Analytical Writing Measure . . . . . . . . .

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PRACTICE TEST 1 Verbal Ability. . . . . . . . . . . . . . . . . . . . . . PRACTICE TEST 2 Verbal Ability. . . . . . . . . . . . . . . . . . . . . .

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PRACTICE TEST 1 PRACTICE TEST 2

Quantitative Ability . . . . . . . . . . . . . . . . Quantitative Ability . . . . . . . . . . . . . . . .

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APPENDIX A The GRE CAT Success Math Review . . . . . APPENDIX B The GRE CAT Success Stress-Busting Guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . APPENDIX C Applying to Graduate School . . . . . . . . . . . APPENDIX D Writing a Good Personal Essay . . . . . . . . .

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APPENDICES

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GRE CAT Success

R E D A LERT INTRODUCTION TO THE GRE CAT The Graduate Record Examination (GRE) General Test is primarily a multiple-choice test that most graduate schools use for admission into their graduate programs that measures your verbal, quantitative, and writing skills. Fortunately, because most of this test is in a multiple-choice format, you can study for it by using this book and learning some of the “tricks-of-the trade” that have been developed by educators who have helped thousands of students prepare for this and similar exams. We’ll also show you how to ace the Analytical Writing section and provide sample essays to use as models. The Graduate Record Examination Program, which is administered by the Educational Testing Service (ETS), is offered year-round at hundreds of test centers around the world. You can schedule your appointment at a time that is convenient for you. You can also retake the test up to five times in one year but no more than once in any given month. ETS also offers Subject Tests in eight discipline areas (Biochemistry, Cell and Molecular Biology, Literature in English, Biology, Mathematics, Chemistry, Physics, Computer Science, and Psychology), each of which measures achievement in specific fields. If you are planning to take the General Test or any of the Subject Tests, you can obtain a registration packet and additional information about each test, by writing directly to: Graduate Record Examination Educational Testing Service P.O. Box 6000 Princeton, New Jersey 08541 http://www.gre.org Since 1999, the regular “pencil and paper” test for the Graduate Record Examination has been largely discontinued, and only the computerized version, known as the computer-adaptive test (CAT), is available in the United States and many other countries.

WHAT IS

A COMPUTER-ADAPTIVE TEST? A computer-adaptive test is—as the title says—adaptive. That means that each time you answer a question, the computer adjusts to your responses when determining which question to present next. If you answer the question correctly, you are presented with a question of increased difficulty. If you answer the question incorrectly, you will receive a question of lesser difficulty. For example, the first question in a section will be of moderate difficulty. If you answer it correctly, the computer adapts so that the next question is slightly more difficult. If your answer was incorrect, the next question will be somewhat easier. The computer will continue presenting questions based on your responses, with the goal of determining your ability level.

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It is important to understand that the questions at the beginning of a section affect your score more than those at the end. That’s because the early questions are used to determine your general ability level. Once the computer determines your general ability level, it presents questions to identify your specific ability level. As you progress farther into a section, it will be difficult to raise your score very much, even if you answer most items correctly. That’s because the later questions affect your score less, as they are used to pinpoint your exact score once the computer has identified your general ability level. Therefore, take as much time as you can afford to answer the early questions correctly. Your score on each section is based on the number of questions you answer correctly as well as the difficulty level of those questions. You will receive a score on each of the sections, whether or not you complete all of the questions. If you do not answer any of the questions in a section, a “No Score” will be reported. Because of the nature of this type of test, you must answer each question as it is presented to you before you get the next question. If you don’t answer the question, you can’t go forward. And once you answer a question, you can’t go back either, since the computer has already selected the next question, based on what you answered. Also, once you exit a section, you cannot go back. If you are not computer literate, don’t worry. You will only be required to have basic computer skills, and fortunately, you will have a tutorial prior to taking the exam, in order to demonstrate that you are capable of taking the test. Of course, it would be helpful to practice on a computer prior to taking the actual test, especially since the Analytical Writing portion of the exam will require typing skills. At the test center, you will have up to 4 hours for your test appointment, but only 2 hours and 15 minutes are allotted for the actual exam. So you will have enough time to take the tutorial and answer most of the other questions that you will receive, along with an ETS survey.

SCORING The scoring for the GRE CAT is similar to the scoring for the pencil-and-paper test. The number of correct answers is adjusted to the difficulty level of the questions you answered. As we mentioned earlier, this is why it is important to answer the first questions correctly, so that the difficulty level increases, as does your score. The final score incorporates the properties of the questions, how many questions you answered correctly, and the number of questions that you answered. Your score report will range from 200 to 800, will be separated into separate scores for each section, and will be accompanied by a percentile rank for those sections. One advantage of the CAT exam is that you will receive your verbal and quantitative scores as soon as you complete the test, if you wish. You can cancel the test at that time before getting your scores, if you think you did poorly. If you click on the “Test Quit” box on your screen, you will exit the test, and no scores will be reported—not even for those sections you have already completed. If you decide to see your scores, you will also receive a paper report in the mail within six weeks after taking the test. Another advantage of taking the CAT is that the test is offered year-round at hundreds of test centers around the world. You can schedule your appointment at a time that is convenient for you. You can also retake the test up to five times in one year but no more than once in any given month.

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GRE CAT Success

INTRODUCTION TO THE GRE CAT

ABOUT THIS BOOK This book has been written for you—the student—to help you prepare for this examination. It will provide you with the tools and the understanding of the overall test and guide you, step-by-step, through each of the major areas covered on this examination. There is a logical approach to this process and to the format of this book. Let’s take a closer look at this book in order to get a fuller understanding of how you can benefit from using it. Actually, we should start at the end. At the back of the book you’ll find a CD-ROM. This disk contains practice CATs. We suggest that you go through this book first, take the paper-and-pencil tests in the book, study the review sections, and then go to the CD to take the tests, which will simulate the actual GRE experience. First, let’s look at the basic make-up of the GRE. The test includes three major areas: Verbal (English), Quantitative (Mathematics), and Analytical Writing Assessment. You will also have one or two experimental sections that will be either Verbal or Quantitative and that do not count toward your final score. (ETS pretests questions to gather statistical information about them before using them on a real test.) Later on in this introductory section, we will cover these areas in more depth. However, it is important that you understand how the test is constructed. Within the three major sections, there are different types of questions, and unlike the old written test, these questions are integrated within a section. For example, in the old exam, you would first be asked seven Sentence Completion Questions, then nine Analogies, and so on. On the CAT, these questions are integrated within the Verbal section, and you won’t know what type of question you will be asked next. The Analytical Writing Measure was formerly a separate, independent test. However, as of October 2002, it replaced the Analytical Ability section. It is identical to the former GRE Writing Assessment test. We have prepared this book in order to give you practice answering the different questions as well as in-depth review material. Obviously, you won’t encounter the randomized and adaptive questions in the book, so instead, we focus here on comprehension. You must first understand what the different questions are that you will encounter, in order for you to do well on the actual test. The first part of the book is a Diagnostic Test. In actuality, we have presented three separate tests, one in each of the subject areas. Its purpose is to help you zero in on those specific topics that give you trouble. Armed with this knowledge, you can then focus your studying on those areas that need more review. The point is to save you time and effort, and it doesn’t make sense to study an entire course if you really only need to focus on one or two sections. Therefore, it is important in this studying process to take the Diagnostic Test, check your answers carefully (reading the explanations, if necessary), and then identify those areas that need additional work. In the second section of the book, each type of question is reviewed. Various strategies for attacking each type of question are presented. Following each set of strategies, there are also additional review questions in order for you to practice the material. But keep in mind that depending upon how you scored on your Diagnostic Test, you may not have to read this entire book. Following the review section are six complete Practice Tests—two in each of the three subject areas. Depending upon the amount of time you have to study prior to taking the actual GRE, you should try to take these tests under actual conditions, if possible. Score your tests and then check the explanations of those questions you answered incorrectly.

GRE CAT Success

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However you normally study, we strongly recommend that you try to follow one of the courses of study presented in the GRE Study Plans below. These plans offer approaches that will best use your available time. The plans are flexible, as any plan should be. After you take the Diagnostic Test, revise the study plan to fit your needs, weaknesses, and schedule. Let’s take a closer look at the components of the exam, in order to prepare you for what you will encounter—assuming this is the first time you’ve picked up a GRE review book. If you’ve already taken the test and are studying to take it again or you’ve used another book prior to this, you’re probably very familiar with the different types of questions that you will encounter. In that case, skip this chapter and move on to the Diagnostic Test and the review sections.

ANALYTICAL WRITING MEASURE As we mentioned earlier, this section replaces the Analytical Ability section that was on the previous GRE exams. The Analytical Writing Measure (AWM) is identical to the former GRE Writing Assessment test that was given separately from the GRE General Test. The AWM consists of two analytical writing tasks. The first 45-minute assignment is an Issue task. The second is an Argument task. The Issue task asks you to write from any perspective on a given opinion, and you’ll have a choice from two tasks randomly selected by the computer. The Argument task asks you to critique an argument by analyzing the issues presented, how logical it is, and so on. You are not asked to take sides.

VERBAL ABILITY There are four major question types in the Verbal Ability section: Analogies, Antonyms, Sentence Completions, and Reading Comprehension. There are a total of 30 questions in this section, as follows: 1. 2. 3. 4.

Sentence Completion—6 questions Analogies—7 questions Reading Comprehension—8 questions Antonyms—9 questions

Each Verbal Ability test is only 30 minutes, so time is of the essence. Throughout this book, we continually stress the idea of making time count, and one of the most important time-savers is to be intimately familiar with the directions for the questions. If you have to take the time to read them over again when you take the actual GRE, you’re losing time. Keep in mind that with 30 questions to be answered in 30 minutes, you have about 1 minute to answer each question. This portion of the GRE essentially is a test of vocabulary. The stronger your vocabulary, the easier it will be to answer the questions in the Antonyms, Sentence Completions, and Analogies. The Reading Comprehension test will measure your ability to understand reading passages, and a command of vocabulary will be useful here as well. The review units will give you some strong pointers and additional practice. There are several basic skills involved in this section, and in order to do well, you must learn those skills. Our inclusion of “Merriam-Webster’s Roots to Word Mastery” will help to improve your vocabulary.

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INTRODUCTION TO THE GRE CAT

QUANTITATIVE ABILITY (MATHEMATICS) The quantitative section of the GRE requires a basic understanding of fundamental mathematical concepts. You are asked to solve problems and to utilize mathematical reasoning. Fortunately, most of the mathematics on this test is high school level and should not be that difficult. Unfortunately, by the time you have reached your last year in college or have been out of college for some time and are getting ready to take the GRE, you have probably forgotten most of your high school math. We have provided you with a basic math review, covering all of the topics that will be included on this portion of the test. The mathematics section of the test includes three main areas and contains 28 questions to be 1 answered in 45 minutes—about 1 minutes per question. 2 1. Quantitative Comparisons—14 questions 2. Basic Mathematics—9 questions 3. Data Analysis—5 questions

Quantitative Comparisons The Quantitative Comparison questions require you to be able to reason quickly and accurately about two quantities provided. Thus, not only does this section require you to have mathematical ability, but it also requires a sense of logic and reasoning. The chapter on Quantitative Comparisons offers numerous fully explained examples as well as dozens of practice questions.

Basic Mathematics Basic Mathematics involves traditional computational skills and includes arithmetic, algebra, and geometry. Fortunately, even if your math skills are somewhat weak, you can develop strong questionanswering skills, that improve your chances of accurately narrowing your choices.

Data Analysis The Data Analysis section is a test of interpretation of charts, graphs, and tables. Much of the information is fairly clear, but many of the questions require you to analyze the material, select the data required, and then perform a variety of calculations. Don’t be misled by apparently easy answers. It’s likely that you’ll have to perform one or more mathematical operations in order to find the correct answer.

POINTERS There are a few things you should keep in mind when taking the GRE CAT—some that are generic tips and other that are CAT-specific. Although these may have appeared earlier in this chapter, they are important enough for you to read them again—and learn them. 1. You do not need to be computer literate in order to take the test. You will receive a tutorial before the exam, so that you are totally familiar with the computer, word processing, answering a test question, and using the mouse. 2. You must answer every question as it is presented to you. If you don’t answer a question and accept it at the time, you cannot get the next question. As we said earlier, take more time on the early questions, since they will count for more than those at the end of the test.

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3. Questions are not grouped by type within each section. Thus, you might find it disruptive to jump back and forth from one question type to another, but you should learn to develop your own method of dealing with this. 4. During the test, there is a time display that you can turn on or off. When there are 5 minutes remaining for a section, the time will automatically turn on and flash briefly to alert you. It will be helpful, though, to occasionally monitor how much time remains. 5. Use the process of elimination. One of the basic methods of answering multiple-choice questions is the process of elimination. Cross off the wrong answers and work toward the correct one. Eliminate those that are obviously incorrect. Select the one that strikes you as correct right from the start. The more choices you eliminate, the better your odds are for getting the correct answer. Now that you have a good idea of what the exam consists of and how it is presented, it’s time to begin studying. Try to pick a study plan that makes sense to you—it’s good discipline for test preparation. Then start by taking the Diagnostic Tests that follow. By the time you have completed all of the material in this book, you should be ready to score high on the actual GRE. Good luck!

GRE STUDY PLANS There are several ways to prepare for the GRE. We offer you these different study plans to help maximize your time and studying. The first is a 10-Week Plan that involves concentrated studying and a focus on the sample test results. The second is the 20-Week Plan, or Semester Plan, that is favored by schools. Finally, the Panic Plan is for those of you who have only a few weeks to prepare. Obviously, the more time you have to prepare, the easier it will be to review all of the material and find yourself somewhat more relaxed when taking the actual exam. These plans are not set in stone—feel free to modify them to suit your needs and your own study habits. But start immediately. The more you study and review the questions, the better your results will be.

THE 10-WEEK PLAN—2 LESSONS PER WEEK

Week 1

Lesson 1

Diagnostic Test Take the entire paper-and-pencil Diagnostic Test in one sitting. There are three sections: Verbal, Math, and Analytical Writing Assessment. Save the grading for Lesson 2.

Lesson 2 Diagnostic Answers Spend the time checking all of your answers and reading through the explanations. Although these first two lessons are an enormous amount of work, it is well worth it to be able to analyze your strengths and weaknesses at this point. It will enable you to select those subject areas that you should focus on and the areas in which to spend the most amount of time studying. Once you have determined the areas that need further study, amend this plan to suit your own needs. If you have done well on the Diagnostic Test, you might just want to take each of the Practice Tests separately or section by section, carefully checking your answers as you complete each portion of the test. www.petersons.com

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INTRODUCTION TO THE GRE CAT

Week 2

Lesson 1 Analytical Writing Measure Read through the first part of the Red Alert on Analytical Writing, focusing on the Issues section. Write outlines for several Issue essays. Write a response to the issue in Unit 1. Lesson 2

Analytical Writing Measure Read through the Red Alert part on Argument tasks. Write outlines for several Argument essays. You can select any of them. Write a response to the argument in Unit 1.

Week 3

Lesson 1 Verbal Study the Red Alert section for the Verbal test. Also, answer the questions in Units 2 (Sentence Completions), 3 (Analogies), and 4 (Antonyms). Lesson 2

Reading Comprehension and Vocabulary Read through the overview section for Reading Comprehension, and then answer the questions in Unit 5. Try to apply some of the Reading Comprehension strategies offered in the Red Alert. Begin to build your vocabulary with Merriam-Webster’s Roots to Word Mastery. You may find it useful to pace yourself through this review by returning to it over the remaining weeks.

Week 4

Lesson 1 Mathematics Review This is the major mathematics overview section. Begin with Quantitative Ability Strategies. It is in this Red Alert section that you can review all of the mathematics that will be covered on the test. In this first lesson, start with Arithmetic and read up to Decimals. Answer the questions that accompany each subsection. Lesson 2 Mathematics Review Read and study from Decimals up to Algebra. This is somewhat complicated material, especially if you have been away from mathematics for some time. By working through the accompanying problems, you will begin to get a feel again for the types of questions that you might encounter as well as help yourself to refresh your knowledge of the subject.

Week 5

Lesson 1

Mathematics Review Read from Algebra to Plane Geometry. Answer the questions and make sure you understand the answers before you move on to a new topic.

Lesson 2 Mathematics Review Complete the section on Plane Geometry. Study the review material and take the quiz.

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RED ALERT

Week 6

Lesson 1 Quantitative Comparisons This section of the GRE seems to present a lot more problems than many of the other sections on the test, perhaps because it involves as much reasoning as it does computational skills. Read through the Red Alert and answer the questions. Lesson 2 Mathematics Review This is the home stretch. Complete unit on Data Analysis. Now it’s time to move on to the practice tests and evaluate your progress.

Week 7

Lesson 1

Analytical Writing Measure Practice Test 1 This is a somewhat difficult portion of the test, unless you’re an excellent writer. Write the two essays that are presented here. Check your response against the sample response.

Lesson 2

Analytical Writing Measure Practice Test 2 Write the two essays that are presented here. Check your response against the sample response.

Week 8

Lesson 1

Verbal Ability Practice Tests Take the two tests and answer all of the questions you can, and then guess at those you don’t know. Circle those questions that you guessed at, so that you can zero in on those specific answers and so that you don’t delude yourself into thinking that you really knew those answers in the first place. There is a lot of work here, so it would make sense to break this and subsequent lessons into separate time periods during the day. It is more important to understand the type of question and to make sure you have memorized the directions for every part of the test. Check all of your answers. Keep track of those types of questions that are still giving you problems.

Lesson 2

Quantitative Ability Practice Tests Take the two Mathematics tests and answer all of the questions you can. Again, circle those questions that you guessed at, so that you can zero in on those specific answers. Try to break this lesson into two separate mini-lessons. Check all of your answers to all parts of the test.

Week 9

Lesson 1

GRE Diagnostic Test on CD Take the GRE CAT Diagnostic Test. Keep a written record of what types of questions gave you problems.

Lesson 2 GRE Untimed Practice Test on CD Take the GRE CAT Untimed Practice Test. Keep a written record of what types of questions gave you problems. www.petersons.com

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INTRODUCTION TO THE GRE CAT

Week 10 Lesson 1

GRE CAT Test on CD This is the computer-adaptive Practice Test. By now, you should be ready for this test-taking format. Carefully check your understanding of the answers. Take the test as many times as you need.

Lesson 2 Final Review It’s time to do a final review of your understanding of all of the parts of the tests. Try to analyze everything you’ve learned by using the book and taking the paper-and-pencil and the CAT tests. What’s left to review? Consult your notes and go back to the book to reread whatever you had trouble with.

THE 20-WEEK PLAN—1 Lesson Per Week If you have the luxury of time, the 20-Week Plan will enable you to better utilize your study time. You can spread out your plan into one lesson a week. This plan is ideal because you are not under any pressure and can take more time to review the material in the Red Alert chapters. You will also have enough time to go back and double-check the answers to those questions that might have given you problems. The basis for all test success is practice, practice, practice.

THE PANIC PLAN Not everyone has the time to study in the proper way for the GRE. School pressures may be great or your job may monopolize your time. You can’t do everything at once. With this in mind, perhaps we can offer a few helpful hints to get you through this period. 1. Read through the ETS test booklet and this GRE CAT Success book and memorize the directions. We’ve said it earlier, and it bears repeating. It’s a way of saving time when you take the actual test and of maximizing the time you have to work on the questions. 2. Read the introduction to this book. It will help you be prepared for the different types of questions you will encounter and give you an idea of how much time you will have on each section of the test. 3. If you don’t have time to take the CATs, take the paper-and-pencil diagnostic GRE test as well as the Practice Tests at the end of this book. By doing so, you will have had some practice answering the types of questions that will appear on the actual test. 4. Try to take as many of the GRE CATs as you can. It will be great practice, not only for understanding how the test works, but also for more computer experience. 5. Focus whatever time you have left on those specific areas of the test that gave you the most difficulty when you took the practice tests in this book. Whatever time you have before the exam, keep in mind that the more you practice on the actual question types that will appear on the exam, the better you will come to understand them, thereby improving your chances for a higher score.

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Diagnostic Test ANALYTICAL WRITING MEASURE Directions: Present your perspective on one of the issues below, using relevant reasons and/or examples to support your views. (Note: On page 27, a sample response is provided for the first issue only. While your essay will be quite different, compare it to the sample in terms of organization, grammar, and logic.) “Students would benefit if they worked in groups rather than working alone on major class projects.” “In today’s technological world, printed books are not as important as they once were. The computer has replaced the printed page.” Directions: Discuss how well reasoned you find this argument. Time—30 minutes. The following advertisement appeared in a big-city want-ad section: “In a dead-end job? Tired of being bossed around? Want the independent lifestyle of being in business for yourself? Then telemarketing is for you! Work in the privacy of your own home, inviting clients to take advantage of our tremendous super-values, money-savers, coupon specials, and one-time-only offers! No time clock! No quotas! No deadlines! Work from our huge list of potential customers, right at your own desk or easy chair (or stay in bed!). Starter kit, first client list, phone dialogue check-off list, tips on how to keep them interested, what to do about cranks and hang-ups—the whole package comes to your mailbox in five days! Send $29.99 for your passport to financial paradise! This is how you can start being in charge of your life.”

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VERBAL ABILITY SENTENCE COMPLETION Directions: Each of the following sentences has one or two blanks, indicating that something has been omitted. Beneath the sentence are five lettered words or sets of words. Choose the word or set of words for each blank that best fits the meaning of the sentence as a whole.

GRE CAT Success

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In a that (A) (B) (C) (D) (E)

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Attracted to ______ at celebrations, the ______ manager fortunately resided in the building where he attended parties. (A) charades. .aesthetic (B) canapés. .zany (C) potables. .bibulous (D) parsimony. .execrable (E) fustian. .stupefied

3.

______ by the ______ graduation requirements, the student diligently prepared for the upcoming examination. (A) Dissuaded. .negligible (B) Persuaded. .acrimonious (C) Undismayed. .marginal (D) Undaunted. .stringent (E) Aghast. .picayune

4.

Because of his ______ record of lying to police, the suspect was regarded as a(n) ______ criminal. (A) irascible. .disputatious (B) chaste. .incorrigible (C) circuitous. .insipid (D) inveterate. .habitual (E) crass. .impecunious

fit of ______, the ______ child, tired of waiting, whined to his mother he no longer wanted the baseball player’s autograph. exhaustion. .unruly pique. .petulant crying. .apathetic rage. .indifferent insouciance. .vexed

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In a child’s early years, he is ______, accepting instruction and quickly learning new skills; however, by adolescence, this same child is typically not as ______. (A) quixotic. .pragmatic (B) fulsome. .acquiescent (C) docile. .malleable (D) sybaritic. .facetious (E) resilient. .truant

6.

Since the terms of the proposed agreement between the vendors were ______, both businessmen were ______ and hesitant to sign. (A) benign. .edified (B) recondite. .stymied (C) vaunted. .nebulous (D) equivocal. .exigent (E) irrefutable. .fallow

ANTONYMS Directions: Each item below consists of a word printed in capital letters, followed by five lettered words or phrases. Choose the lettered word or phrase that is more nearly opposite in meaning from the word in capital letters. Since some of the questions require you to distinguish fine shades of meaning, be sure to consider all the choices before deciding which one is best.

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1.

ANATHEMA (A) boon (B) fear (C) curse (D) virulent (E) allergic

2.

TAWDRY (A) salacious (B) immune (C) prized (D) odious (E) gaudy

3.

ACERBIC (A) torrid (B) synthetic (C) indifferent (D) mellifluous (E) nefarious

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4.

PUSILLANIMOUS (A) valorous (B) obliterated (C) satiated (D) captious (E) abstruse

5.

SURFEIT (A) prosaic (B) paucity (C) succulence (D) sluggish (E) vacancy

6.

SEDITION (A) treason (B) ignorance (C) patriotism (D) perfidy (E) havoc

7.

REVERENCE (A) harbinger (B) presentiment (C) amulet (D) contempt (E) quotidian

8.

INCHOATE (A) galactic (B) opulent (C) plenary (D) verdant (E) neophyte

9.

SOPORIFIC (A) fatiguing (B) hypnotic (C) stimulating (D) innocuous (E) prodigious

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DIAGNOSTIC TEST

READING COMPREHENSION Directions: Each passage in this group is followed by questions based on the content. After reading a passage, choose the best answer to each question. Answer all questions following a passage on the basis of what is stated or implied in that passage.

PASSAGE 1 Line

5

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There is some evidence to suggest that Neanderthals were cannibalistic. The cave dwellers, who lived as long as 125,000 years ago, were an important link in the evolution of humankind. They had brains as large as modern man and developed a “culture” of their own that included the burying of their dead with perhaps a religious ceremonial aspect attached to the custom. In addition, they made jewelry-like ornaments that demonstrate a sense of creativity and aestheticism. However, in a recent report in Science magazine, there is evidence presented that shows that Neanderthals may have slaughtered some of their numbers and actually butchered them for the meat. Since Neanderthals were cave-dwellers, the evidence was discovered in one such cave near the Rhone River in France. Shockingly enough, human bones were found that bore the signs of deliberate butchering. The bones were from adults, teenagers, and even children of six or seven years of age and were 100,000 years old. They were found next to deer bones. Not unlike those bones, the human bones showed signs of slashes at the joints like the elbow, foot, and ankle, indicating that muscles and tendons were deliberately cut to facilitate the removal of “meat.” Flint could have been used for this purpose. Some bones had been smashed to remove their marrow, and skulls had been broken to remove the brains. Although this likelihood of cannibalism is substantiated by this ancient evidence, what cannot be proven is whether it was a regularly practiced custom among the Neanderthals to methodically slaughter their own kind or only practice cannibalism out of the necessity caused by famine. While many cultures throughout time and from around the world have placed a taboo on such practices, instances of cannibalism have occurred as tradition, religious ritual, or out of necessity in other places and eras. The “practice” of the Neanderthals may be one of the earliest precedence for such behavior and forever taint the image of early man as a primitive brute rather than the growing consensus that they, with a brain as large as contemporary man, were more like us than not. 1.

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The burying of the dead with an attendant religious ceremony would be a benchmark of cultural development because it is (A) typical of most species to acknowledge mortality and mourn the passing of their own kind. (B) a ritual that has been practiced historically by humans in many cultures throughout the world. (C) foreign to all species that have been observed in nature. (D) indicative of a species with the brain size of modern man. (E) atypical of cave dwellers from the same region and time in France.

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2.

As used in line 6 “aestheticism” most closely means (A) the ability to manipulate metals. (B) a capacity to polish gem stones. (C) an ability to evaluate the financial worth of objects. (D) an appreciation of beauty. (E) both (A) and (B).

3.

The (A) (B) (C) (D) (E)

4.

The main point of the article is that Neanderthals did practice cannibalism as (A) supported by evidence that was discovered. (B) part of their culture. (C) a necessity because of famine. (D) a religious ritual. (E) a regularly practiced custom or out of need.

evidence of cannibalism is supported by the fact that the bones were found near the bones of deer. completely lacking tendons, muscle, and flesh. scorched from being burned over an open fire. those of children and teenagers. cut at the joints.

PASSAGE 2 Line

5

10

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In a seemingly repeating cycle, two diet “fads” seem to follow one another; the Low-Carb (carbohydrate) approach versus the High-Carb method. Currently, the former is the one in vogue with millions of people across the America eating a high-protein, low-carbohydrate diet. So bring on the bacon, ham, eggs, cheese and any other high-fat food, but eliminate as many carbohydrates as possible like pasta, bread, fruit, soda, and high-sugar alcoholic beverages. The many versions of Low-Carb diets decrease carbohydrates, thereby, causing blood-sugar levels to fall. This causes the pancreas to produce less insulin. Insulin stimulates energy and without this resource, the body is forced to burn fat reserves to create needed energy. The result is a quick loss in weight. Conversely, when one eats carbohydrates they are reduced by enzymes into simple sugars. These sugars stimulate the pancreas to produce insulin, which allows sugars to enter tissue. Although cells use the sugar for energy, the excess sugars are stored as fat. Since many Americans, especially young people, have high-sugar diets, most of the excess is stored as fat. If blood levels can be dropped low enough, the body will burn this excess fat. While this mechanism is agreed upon by many scientists, there is some disagreement about how people lose weight on Low-Carb diets. Most people who write the diet books are not medical doctors and have come under criticism for not understanding the process of weight loss and the harmful effects of Low-Carb, high-protein diets. According to some doctors and scientists, the reason why people lose weight on these diets is that by reducing the ingestion of carbs, there is a corresponding reduction in caloric intake; therefore, people are simply consuming less calories. Similarly, since Americans traditionally have eaten so much sugar and sugar products, when the consumption of those are reduced, caloric intake is lessened, and it results in a weight loss. In addition to the misunderstandings regarding the process of weight loss, these

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DIAGNOSTIC TEST

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same doctors and scientists contend that a Low-Carb, high-protein diet can be harmful in a number of significant ways. The lack of fiber in the diet can cause constipation, weakness, nausea, as well as dehydration. Also, the high-protein diet is a great strain on the kidneys. These are major bad health effects compared to the additional side-effect of halitosis (bad breath). The important thing to remember about “diets” is that unless they involve a change of lifestyle that can be maintained for a lifetime, they are just a “quick fix” for short-term weight loss, not a solution to an ongoing problem. In addition, even some of these short-term solutions can affect health adversely, so when one considers a change in eating habits, consulting a doctor who is a specialist in this area would be beneficial. 1.

A Low-Carb diet affects the blood sugar level by (A) having enzymes reduce the blood sugar into simple sugars, which are more readably burned. (B) causing the pancreas to produce less insulin, thereby lowering the blood sugar level and the energy level. (C) stimulating the pancreas to produce more insulin to burn excess fat that has been stored. (D) allowing excess sugars to be secreted so they can’t be stored as excess fat in the cells of the body. (E) lowering it through lack of absorption of sugar by the cells in tissue.

2.

The passage implies that the debate between designers of these diets and doctors and scientists centers on the process of weight loss and adverse effects on health in regards to (A) whether excess fat is burned or there is simply less intake of calories. (B) the kinds of adverse health effects that may result. (C) which diet, Low-Carb or High-Carb, is more effective for weight loss. (D) both (B) and (C) (E) both (A) and (B)

3.

The (A) (B) (C) (D) (E)

main point of the article is that no one should try to alter their diet in an attempt to lose weight. doctors and scientists disagree with the writers of diet books. the mechanism by which Low-Carb diets work is debatable. reduction of caloric intake will result in weight loss. any change in diet must be one that can be maintained for a lifetime.

4.

The (A) (B) (C) (D) (E)

overall tone of the article can be described as informative and whimsical. adversarial but supportive. informative but cautionary. supportive but cautionary. adversarial and whimsical.

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ANALOGIES Directions: In each of the following questions, a related pair of words or phrases is followed by five lettered pairs of words or phrases. Select the lettered pair that best expresses a relationship similar to that expressed in the original pair.

GRE CAT Success

1.

AMICABLE : CHARISMA :: (A) perspicacious : perception (B) stylish : panache (C) ductile : refractory (D) prolix : taciturnity (E) turgid : drought

2.

DOG : CANINE :: (A) feline : cat (B) porcine : pig (C) fish : aquarium (D) vulture : vulpine (E) bear : ursine

3.

SCONCE : CANDLE :: (A) rack : framework (B) citadel : dragon (C) dormer : roof (D) well : ink (E) liter : grain

4.

TENACIOUS : HOLD :: (A) fortuitous : unlucky (B) wary : rash (C) tapering : invigorate (D) enervating : weaken (E) vacillating : vivify

5.

METICULOUS : MESSY :: (A) particular : finicky (B) military : precise (C) pacific : muddled (D) vitriolic : scathing (E) fastidious : slovenly

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DIAGNOSTIC TEST

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6.

TROWEL : MASON :: (A) artist : brush (B) spade : gardener (C) spatula : bartender (D) needle : farrier (E) decoupage : writer

7.

VERACITY : FALSEHOOD :: (A) submission : subjugation (B) truth : probity (C) recalcitrant : amiable (D) slake : thirst (E) carnage : accident

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MATHEMATICS

MATHEMATICS Directions: Each of the questions 1–14 consists of two quantities, one in Column A and one in Column B. You are to compare the two quantities and choose: (A) if the quantity in Column A is greater; (B) if the quantity in Column B is greater; (C) if the two quantities are equal; (D) if the relationship cannot be determined from the information given. Note: Since there are only four choices, NEVER MARK (E). Numbers: All numbers used are real numbers. Figures: Position of points, angles, regions, etc., can be assumed to be in the order shown; and angle measures can be assumed to be positive. Lines shown as straight can be assumed to be straight. Figures can be assumed to lie in a plane unless otherwise indicated. Figures that accompany questions are intended to provide information useful in answering the questions. However, unless a note states that a figure is drawn to scale, you should solve these problems NOT by estimating sizes by sight or by measurement, but by using your knowledge of mathematics.

Column A

Column B

A, B, C, D, and E are consecutive even integers 1.

2.

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A18

E

A number between 10 and 20

19

A number between 15 and 25

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DIAGNOSTIC TEST

Column A

Column B x57 y 5 22

3.

x2y

x2y

ABCD is a square AB 5 3

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4.

AC

5

5.

=17 1 =32

=49

z 2

3z

6.

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MATHEMATICS

Column A

Column B

The average of w, x, y, and z is 28, and w 1 x 5 56. 7.

w1x

y1z

x2 1 x 5 20 8.

5

x

a.c 9.

CB

AC

10.

0.81

=0.81

a is 4 times b. 11.

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a b

b a

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DIAGNOSTIC TEST

Column A

Column B (9)(27)(81) 5 3

12.

x

8

x

(p 1 q)3 5 64 p . 0, q , 0 13.

?p?

?q?

Angles A and B are complementary 14.

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12

x

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GRE CAT Success

MATHEMATICS

Directions: Each of the questions 15–28 has five answer choices. For each of these questions, select the best of the answer choices given.

GRE CAT Success

=25?

15.

What percent of 25 is (A) 20% (B) 25% 1 (C) 33 % 3 (D) 50% (E) 75%

16.

In the equation p 5 qr, if q is multiplied by 7 and r is divided by 7, then p is (A) multiplied by 7 (B) multiplied by 49 (C) divided by 7 (D) divided by 49 (E) left unchanged

17.

The price of a cassette deck was increased from $90 to $120. This represents what percent of increase in the price of the cassette deck? (A) 25% (B) 30% 1 (C) 33 % 3 (D) 75% 1 (E) 133 % 3

18.

How many minutes would it take a typist to complete a 300-word letter if 1 he can type 2 words in 10 seconds? 2 (A) 2 (B) 10 (C) 15 (D) 20 (E) 25

19.

What is the value of 2x3 2 y2 1 z if x 5 22, y 5 21, and z 5 3? (A) 213 (B) 26 (C) 24 (D) 10 (E) 12

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DIAGNOSTIC TEST

20.

In the figure above, PQRS is a parallelogram. If a 5 50, then what is the value of b? (A) (B) (C) (D) (E)

21.

Four managers at Shop-Well receive incentive bonuses. The total amount of money available for the four bonuses is $32,000. If Manager A gets $9,000, and Manager B gets $2,000 more than each of Managers C and D, what is the amount of Manager C’s bonus? (A) $5,000 (B) $6,000 (C) $7,000 (D) $7,500 (E) $8,000

22.

If WXYZ is a four-digit number that is divisible by 2 and by 5, what is the value of Z? (A) 0 (B) 2 (C) 4 (D) 5 (E) It cannot be determined.

23.

If xy Þ 0 and 15xy2 2 10xy3 5 0, what is the value of y? (A) (B) (C) (D) (E)

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25 50 65 70 100

2 3 3 2 2 3 6

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MATHEMATICS

Questions 24–28 refer to the following graphs.

GRE CAT Success

24.

What was the amount of student aid provided through federal loans during the 2000–2001 academic year? (A) $11,280,000 (B) $32,400,000 (C) $3,240,000,000 (D) $11,280,000,000 (E) $32,400,000,000

25.

By what percent did the amount of total aid awarded increase from the 1990–1991 academic year to the 2000–2001 academic year? (A) 40% (B) 50% (C) 60% (D) 150% (E) 250%

26.

Which of the following best describes the change in the amount of aid provided by state grants from the 1990–1991 academic year to the 2000– 2001 academic year? (A) It remained constant. (B) It decreased slightly. (C) It increased slightly. (D) It more than doubled. (E) It more than quadrupled.

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DIAGNOSTIC TEST

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27.

How much more grant money was awarded through other federal programs than through federal campus-based aid during the 1990–1991 academic year? (A) $0.36 billion (B) $0.48 billion (C) $3.6 billion (D) $4.8 billion (E) $48 billion

28.

By approximately what percent did the amount of money awarded through Federal Pell Grants increase from the 1990–1991 academic year to the 2000–2001 academic year? (A) 38% (B) 42% (C) 52% (D) 56% (E) 62%

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QUICK SCORE ANSWERS

Quick Score Answers Verbal Ability Sentence Completion 1. 2. 3. 4. 5. 6.

B C D D C B

Antonyms 1. 2. 3. 4. 5. 6. 7. 8. 9.

A C D A B C D C C

Reading Comprehension Passage 1 1. B 2. D 3. E 4. E Passage 2 1. B 2. E 3. E 4. C

Analogies 1. 2. 3. 4. 5. 6. 7.

B E D D E B C

Mathematics 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

C D B B A D C B D D A A A C

15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28.

A E C D D C C A B E D D B D

ANSWERS AND EXPLANATIONS

ANALYTICAL WRITING MEASURE Sample Response to First Issue—Score 5–6 The statement “Students would benefit if they worked in groups rather than working alone on major class projects” is a valid one for many reasons. In the era of technological advancement in classroom resources and equipment and an increasing multicultural environment in the classroom, it is especially important for students of any age to learn to utilize all available resources, especially their peers, when in an academic setting. In the following essay, I will explain three central reasons why students should, in fact, spend a sizeable amount of time working together in groups in the classroom, especially on large projects. First, students who begin major academic projects in groups or teams develop one of the most common and most serviceable skills for idea generation—brainstorming. Participating in brainstorming sessions allows students to learn to think creatively, to think quickly, and to interact appropriately with peers. Brainstorming is usually a flurry of ideas at first, then a redefining of those ideas, discussion of why an idea will or will not work, exclusion of ideas that the group believes will not work, and finally, a group decision that delineates the assigned project. Skills built by brainstorming are teamwork, listening, critical thinking, logic, compromise, communication, and organization. Second, working as a group to test ideas or points of view is beneficial. Knowledge of the assigned project, researching skills, and reporting of ideas are all essential tasks that major academic projects require. Students working within a group or team must use their individual knowledge, research, and reporting to enhance the group project. Personal gain is a secondary benefit to the responsibility to the team. Peer pressure compels individuals to bring only their best attributes to the team project. Knowing that each idea or issue

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DIAGNOSTIC TEST will be tested by the group encourages students to clearly and concisely communicate ideas and research to the team. Third, the skills discussed above allow students to practice proficiencies that will be useful in almost any career. Placing students in situations that emulate workplaces promotes positive reinforcement of appropriate communication skills, problem-solving facility, teamwork, compromise, and the organization and completion of both small and large tasks. Instead of reading about the workplace in a solitary assignment, students actually practice workplace skills with their group. This promotes successful activity on the job once a student has begun a career. Of course, there are benefits to individual research and learning as well. Research can be conducted at the student’s own pace, the student can choose projects that are of great interest to him or her, and individual accomplishment is certainly a self-esteem booster for most students. However, group projects challenge the student to discover ways to learn in different styles from what he or she is used to, thereby expanding the methods of learning. Also, working in groups helps students develop problem-solving skills, since not all students are working at the same pace with the same ideas, or in the same manner. Flexibility, empathy, and responsibility are all important aspects of group learning. For the reasons listed above, I believe that it is essential that students spend a sizeable amount of time working together in groups in the classroom. The benefits far outweigh the negative aspects, and teamwork in the classroom promotes involvement, skill building, and a sense of accomplishment.

VERBAL ABILITY Sentence Completion 1. The correct answer is (B). Pique means “irritation” or “resentment,” while petulant means “peevish.” The context clue in this sentence is “whined.” Choice (A), exhaustion and unruly, can describe a child’s condition; however, to say “a fit of exhaustion” is illogical. In choice (C), if the child were crying, then he would not be apathetic, or “without interest or feeling.” Choice (D) offers a similar contrast. Rage does not accompany indifference. Choice (E), insouciance, or “lighthearted nonchalance,” does not logically exist with vexed, which means “irritated.” 2. The correct answer is (C). In this sentence, “celebrations” and “parties” are context clues. Potables refer to “something suitable for drinking”; bibulous means “inclined to drink.” In choice (A), charades is a party game; aesthetic refers to “beauty.” In choice (B), canapés are cocktail food; zany means “goofy.” In choice (D), parsimony refers to “stinginess” while execrable means “detestable.” In choice (E), fustian is a “pretentious writing or speech” while stupefied means “astonished.” 3. The correct answer is (D). “Diligently prepared” suggests hard work. In choice (D), undaunted means “undiscouraged” while stringent means “very strict.” The original sentence, then, suggests that the student takes into account the strict requirements for graduation and works hard. In choice (A), dissuaded means “advised against,” while negligible means “insignificant.” This pair does not logically complete the sentence. Neither does the pair in choice (B). Persuaded has a positive connotation, but acrimonious, which means “caustic,” has a negative connotation. The choices in (C) begin well: undismayed is appropriate for a “diligent” student; however, marginal suggests the requirements are so easy, the student does not need to study much. In choice (E), aghast means “horrified,” while picayune means “trivial.” These words are paired illogically.

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ANSWERS AND EXPLANATIONS 4. The correct answer is (D). This kind of sentence can often be approached by inserting some words that you can think of that will logically fit the context; then see if your choices—or at least one of them—are among those offered. Choice (D) presents inveterate, meaning “chronic” or “confirmed”; habitual is a close synonym. The choices offered in the other items are not logically paired to complete these blanks. Choice (A), irascible, means “bad tempered” while disputatious means “controversial.” In choice (B), chaste means “pure,” while incorrigible means “unmanageable.” Choice (C) offers circuitous, which means “roundabout,” and insipid, which means “tasteless” or “dull.” Choice (E) presents crass, which means “gross” or “insensitive,” and impecunious, which means “penniless.” 5. The correct answer is (C). The context clues here are “child’s early years,” which contrast “adolescence.” Choice (C) offers docile, which means “easily taught,” and malleable, which means “adaptable” or “flexible.” Choice (A) offers quixotic, which means “imaginary” and “idealistic to an extreme degree”—like Don Quixote, from whose name the word is derived—and pragmatic, which means “practical.” In choice (B), fulsome means “disgusting,” while acquiescent means “agreeable.” In choice (D), sybaritic means “sensual” and facetious means “humorous.” Choice (E) offers resilient, which means “pliant,” and truant, which refers to someone who stays out of school without permission. 6. The correct answer is (B). The context clue is “hesitant.” Choice (B) presents recondite, which means “hard to understand,” and stymied, which means “perplexed” or “confused.” Choice (A) begins with benign, which means “harmless,” and edified, which means “educated.” Choice (C) offers vaunted, which means “boastful,” and nebulous, which means “vague.” Choice (D) has equivocal, which looks like a likely choice; it means “undecided,” but exigent means “demanding.” Choice (E) presents irrefutable, which means “undeniable,” and fallow, which means “idle” or “left unplanted.”

Antonyms 1. The correct answer is (A). Anathema means “curse.” Choice (A), boon, means “gift.” Notice that curse, the synonym for anathema, is one of the choices. Watch for this practice of including synonyms among the choices so that you won’t be misled. Choice (D), virulent, means “noxious, full of poison.” The other choices are irrelevant. 2. The correct answer is (C). Tawdry means “gaudy” or “cheap.” Choice (A), salacious, means “lascivious” or “lewd.” Choice (D), odious, means “hateful.” See again the synonym gaudy among the choices. 3. The correct answer is (D). Mellifluous means “honey-toned” or “sweetly flowing.” Acerbic means “bitter” or “harsh.” Choice (A), torrid, means “very hot” or “scorching.” Choice (E), nefarious, means “wicked.” 4. The correct answer is (A). Pusillanimous means “cowardly.” Valorous is “heroic.” Choice (B), obliterated, means “destroyed.” Choice (C), satiated, means “glutted.” Choice (D), captious, means “highly critical.” Choice (E), abstruse, means “difficult to comprehend.” 5. The correct answer is (B). Surfeit means “excess,” while paucity means “a lack of” or “dearth.” Choice (A), prosaic, means “ordinary.” Choice (C), succulence, means “juiciness.” Choice (D), sluggish, means “moving slowly.”

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DIAGNOSTIC TEST 6. The correct answer is (C). Sedition means “treason,” which is listed first among the choices. Choice (D), perfidy, is closely related to sedition; it means “treachery.” Choice (E), havoc, means “commotion” or “pandemonium.” 7. The correct answer is (D). Contempt means “disdain.” Choice (A), harbinger, is a “forerunner” or “hint of what is to come.” Choice (B), presentiment, also means “omen.” Choice (C), amulet, is a “fetish” or “magic charm.” Choice (E), quotidian, means “daily.” 8. The correct answer is (C), Plenary means “full” or “complete.” Inchoate means “imperfectly formed” or “incipient.” Choice (A), galactic, pertains to the galaxy; choice (E), neophyte, is a “beginner.” Choice (D), verdant, means “lush and green.” Choice (B), opulent, means “rich” or “wealthy.” 9. The correct answer is (C). Soporific means “sleep-inducing.” Choice (D), innocuous, means “harmless.” Choice (E), prodigious, means “gigantic.”

Reading Comprehension Passage 1 1. The correct answer is (B). Choice (B) establishes the burying of the dead and an attendant religious ceremony to commemorate it as a common practice among diverse cultures throughout the world. Choice (A) is incorrect because it is common knowledge that it is not typical behavior for most species to practice such a ritual. Choice (C) is incorrect because of the key word “all.” Often in multiple-choice questions, answers that state something categorically are not the right choice. Choice (C), therefore, makes too sweeping an assertion. Choice (D) is incorrect; although the Neanderthal had brains as large as modern man, there is no direct correlation between that fact and the burying of the dead with religious ceremony. Choice (E) is incorrect because although cannibalism may be the aberration in light of the new evidence discovered, burying the dead may or may not have been a common practice. 2. The correct answer is (D). The ornaments are not functional in nature but meant to be appreciated as an expression of creativity and for their beauty. Choices (A) and (B) are incorrect because although they fit the context of “jewelry” in the sense that some jewelry consists of metals and/or gem stones and the ability to manipulate those materials, there is no mention of such materials in the article. Choice (C) is incorrect because in the context of Neanderthal “culture,” one can infer that there is not a monetary structure to evaluate worth. Choice (E) is incorrect because both choices have been eliminated as stated above. 3. The correct answer is (E). It states in paragraph two that the cuts at the joints facilitated the removal of meat by cutting through tendons and muscle. Choice (A) is incorrect because it is not conclusive evidence just because the deer bones were a source of meat, too. Choice (B) is incorrect because the bones would lack all of that matter because of their age. Choice (C) is incorrect because nowhere in the article does it mention the bones being scorched. Choice (D) is incorrect because it implies that the young were eaten by the old; however, bones of adults were also found, and all showed signs of cannibalism.

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ANSWERS AND EXPLANATIONS 4. The correct answer is (E). Researchers can only speculate as to whether cannibalism among the Neanderthals was a regularly practiced custom or done out of need because of famine. While choices (A) and (B) are true, they are incomplete because they cannot definitively prove whether cannibalism was a custom or a necessity. Choice (C) is incorrect because it falsely states that famine was the cause of cannibalism; that relationship was not clearly established in the article. Choice (D) is incorrect because although Neanderthals perhaps had a religious observance of death, this is not always connected to the practice of cannibalism.

Passage 2 1. The correct answer is (B). In paragraph two, the process by which the blood sugar is lowered is described. Choice (A) is incorrect. Although it correctly states that the blood sugar level is reduced, the process is misstated. Choice (C) is incorrect because it mistakenly states that the pancreas increases the production of insulin to burn excess fat. Choice (D) is incorrect because nowhere in the article is there an explanation regarding secretions. In paragraph three, there is mention of the adverse affect of this diet on the kidneys, but not in relation to the secretion of sugars to induce weight loss. Choice (E) is incorrect. Although it correctly refers to the lowering of the blood sugar, it misstates the process. 2. The correct answer is (E). Choices (A) and (B) are incorrect because they only state one main discrepancy between the designers of the diets and the doctors and scientists. Choice (C) is incorrect because a Low-Carb diet is mentioned only briefly and is not compared to a High-Carb diet in terms of effectiveness. Choice (D) is incorrect. As a general “rule,” usually when there is a choice that combines choices, it is the correct choice. However, choice (D), although combining choices like choice (E), incorporates choice (C), which was already determined to be incorrect. 3. The correct answer is (E). Often, the main point of a piece of writing will be in the concluding paragraph as it is in this example. Choice (A) is incorrect. Although the article is cautionary in regards to “fad” diets, it is not adverse to a change in diet for better health. Choices (B), (C), and (D) are incorrect. Although they are statements that are made in the article, no one of them represents the main point. They provide the evidence that supports the main point. 4. The correct answer is (C). The first three paragraphs are informative and lead to the cautionary stance that is stated in the last paragraph. Choices (A) and (E) are incorrect because the term “whimsical” (fanciful, unpredictable) does not relate to this article at all. Choice (B) is incorrect because the article is adversarial but does not conclude by supporting a High-Carb diet. Choice (D) is incorrect because the tone is not supportive but cautionary.

Analogies 1. The correct answer is (B). A person who has charisma is amicable. In choice (B), the same relationship exists: a person who has panache, which means “verve,” is stylish. In choice (A), perspicacious means “keen” or “insightful”; someone with perception is perspicacious. However, notice that the order has been reversed in this item. Watch out for this technique. In choice (C), ductile means “easily molded,” which is the opposite of refractory, meaning “stubborn.” In choice (D), prolix means “wordy,” which is the opposite of taciturnity, meaning “silence.” In choice (E), turgid means “swollen,” while drought means “dry” or “without water.”

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DIAGNOSTIC TEST 2. The correct answer is (E). All of the choices offer animals; the word order is most important. Most of the choices offer an adjectival form based on Latin roots. Canine, for instance, is derived from the Latin canis for “dog.” Choice (E) reflects the same order: bear 1 adjective form ursine, which is derived from ursa, “bear.” Choice (A) has the right words; feline does refer to cats. However, the order is reversed here. The same is true for choice (B). Porcine means “pig-like,” but the order does not match that in the question. Choice (C) omits the adjectival form. Choice (D) offers the correct order of animal 1 adjectival; however, vulpine refers to foxes, not vultures. 3. The correct answer is (D). A sconce holds candles; a well can hold ink. The relationship in these two is the same. In choice (A), rack is a kind of framework. In choice (B), citadel is a fort; it does not hold a dragon. In choice (C), dormer is a characteristic of some roofs. In choice (E), liter is a type of measure for liquids, which grain clearly is not. 4. The correct answer is (D). Tenacious means “persistent” or “holding fast.” The pairs of words are then synonyms. Enervating means “weaken.” In choice (A), fortuitous, which means “lucky,” is the opposite of unlucky. In choice (B), wary, which means “cautious,” is the opposite of rash, which means “reckless.” In choice (C), tapering is “diminishing,” while invigorating means to “strengthen.” In choice (E), vacillating means “wavering,” while vivify means to “give life to.” 5. The correct answer is (E). The original pair, meticulous and messy, are antonyms. In choice (E), fastidious, which means “excessively attentive to details,” is the opposite of slovenly, which means “messy.” Choice (A) offers a pair of words that are antonyms, but the order is reversed. Among the other choices, pacific means “calm,” and vitriolic means “caustic,” which is synonymous with scathing. 6. The correct answer is (B). A trowel is a tool used by a mason, someone who lays bricks or stones; a spade or shovel is a tool used by a gardener. Choice (A) offers a craftsman with a tool, but the order is reversed. Choice (C) presents a cooking tool, a spatula, which is not used by a bartender. Choice (D) offers a needle, a tool not used by a farrier, someone who shoes horses. Finally, in choice (E), decoupage is a type of decoration using cutout pictures, gluing them to surfaces, and then varnishing them. A writer does not use this type of “tool.” 7. The correct answer is (C). The original pair, veracity and falsehood, are antonyms. Veracity means “truth.” Choice (C) presents a similar pair: recalcitrant means “rebellious” or “disobedient.” Amiable means “agreeable.” In choice (A), submission and subjugation are virtual synonyms. In choice (B), truth and probity are also synonyms. In choice (D), the relationship changes: slake means to “quench.” In choice (E), carnage is “massive bloodshed,” which does not necessarily occur in an accident; this pair is not opposite either.

MATHEMATICS 1. The correct answer is (C). Consecutive even integers differ by two. Thus, A 1 2 5 B, A 1 4 5 C, A 1 6 5 D, and, finally, A 1 8 5 E. 2. The correct answer is (D). There is no way to tell which number is bigger. For example, both numbers could be 16. Or, the number in Column A could be 19 and the number in Column B could be 16. Or, the number in Column A could be 11 and the number in Column B could be 24, etc. 3. The correct answer is (B). If x 5 7 and y 5 22, we have x2y 5 (7)2(22) 5 49(22) 5 298, while x 2 y 5 7 2 (22) 5 7 1 2 5 9.

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ANSWERS AND EXPLANATIONS 4. The correct answer is (B). Using the fact that the hypotenuse of an isosceles triangle is equal to the length of the legs times =2, we can determine that AC is equal to 3=2. Since =2 ' 1.4, AC is approximately 3(1.4) 5 4.2 , 5. 5. The correct answer is (A). Clearly the entry in Column B is equal to 7. To answer the question, all we need is a reasonable estimate of the value of the expression in Column A. Since =17 is a bit bigger than 4, and =32 is bigger than 5, the expression in Column A must be greater than 9. 6. The correct answer is (D). The entries in the two columns are equal if z 5 0. On the other hand, if z is, say, equal to 1, the entry in Column B is larger. 7. The correct answer is (C). We are given that the average of w, x, y, and z is 28. w1x1y1z 5 28. We also know that w 1 x is 56, so This means that 4 56 1 y 1 z 5 28. We can now manipulate this equation to determine the value of 4 y1 z. First, multiply both sides by 4. 56 1 y 1 z 5 112, or y 1 z 5 56. 8. The correct answer is (B). Begin by solving the given equation: x2 1 x 5 20 x2 1 x 2 20 5 0 ~x 1 5!~x 2 4! 5 0 Therefore, x 5 25 or 4. In either case, the value in Column B is larger. 9. The correct answer is (D). The fact that a . c tells us that CB . AB. However, we know nothing about the length of AC. 10. The correct answer is (B). Column B is equal to 0.9, which is larger than 0.81. Note that when an expression such as =0.81 is given, the principal (positive) square root is intended. If the equation x2 5 0.81 were given, x could equal either 0.9 or 20.9. 11. The correct answer is (A). The given information tells us that a 5 4b. Therefore, b 1 a 5 4, while 5 . b a 4 12. The correct answer is (A). Note that 9 5 32, 27 5 33, and 81 5 34. Thus, (9)(27)(81) 5 (32)(33)(34) 5 39, and this means that x 5 9. 13. The correct answer is (A). For (p 1 q)3 to be 64, which is positive, p 1 q must be positive. Since we are given that p is positive and q is negative, the absolute value of p must be greater than that of q. 14. The correct answer is (C). If angles A and B are complementary, angle C has to be a right angle. By the Pythagorean Theorem, AC 5 12. 15. The correct answer is (A). To begin,=25 5 5, so we need to find what percent 1 1 5 is of 25. Since 5 is of 25, and 5 20%, 5 is 20% of 25. 5 5 16. The correct answer is (E). Beginning with p 5 qr, multiply q by 7 and divide r by 7: r 7 5 ~qr! 5 qr. Thus, the equation would be left unchanged. ~7q! 7 7

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DIAGNOSTIC TEST 17. The correct answer is (C). To find the percent of increase, divide the amount of the increase by the original value, and express the result as a percent: Percent of increase 5

1 1 30 1 5 . The fraction expressed as a percent is 33 %. 90 3 3 3

18. The correct answer is (D). This problem can be completed by solving a proportion: 300 words 2.5 words 5 10 seconds x seconds 3000 5 1,200. Since it takes 1,200 2.5 seconds to type the letter dividing by 60, we get 20 minutes. Next, cross multiply: 2.5 x 5 3000. Therefore, x 5

19. The correct answer is (D). Substituting the given values into 2x3 2y2 1 z , we get: 2x3 2y2 1 z 5 5 5 5

2(22)3 2(21)2 1 3 2(28) 2 (1) 1 3 82113 10

20. The correct answer is (C). If a 5 50, the measure of angle QPS is 50°, since opposite angles in a parallelogram are equal. Since a straight angle has 180°, it must be true that b 1 b 1 50 5 180. Therefore, 2b 5 130, so b 5 65. 21. The correct answer is (C). After Manager A gets $9,000, there is $23,000 of bonus money left. Let C 5 the amount of Manager C’s bonus. Then, C 5 D, and B 5 C 1 $2,000. Therefore, since B 1 C 1 D 5 $23,000, we can see that (C 1 $2,000) 1 C 1 C 5 $23,000. 3C 1 $2,000 5 $23,000. 3C 5 $21,000 or C 5 $7,000. 22. The correct answer is (A). For a number to be divisible by 2, its last digit must be even, that is, either 0, 2, 4, 6, or 8. For a number to be divisible by 5, it must end in 0 or 5. Thus, the only way a number can be divisible by both 2 and 5 is if its last digit is 0. 23. The correct answer is (B). Begin by factoring the left-hand side of 15xy2 2 10xy3 5 0. The equation becomes: 5xy2~3 2 2y! 5 0. Since we are told that xy Þ 0, we know that neither x nor y is 0. Thus, 3 2 2y must equal 0. 3 2 2y 5 0 3 5 2y or 3 y5 2 24. The correct answer is (E). The amount of student aid provided through federal loans during the 2000–2001 academic year was 54% of the total amount of 60 billion. 54% of $60 billion 5 .54 3 $60 billion 5 $32.4 billion 5 $32,400,000,000

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ANSWERS AND EXPLANATIONS 25. The correct answer is (D). Total aid in the 1990–1991 academic year was $24 billion. Total aid in the 2000–2001 academic year was $60 billion. The percent of increase can be computed by dividing the increase by the original value and expressing the result as a percent. Percent of increase 5 36 4 24 5 1.5 5 150% 26. The correct answer is (D). The amount of aid provided by state grants in the 1990–1991 academic year was 6% of $24 billion 5 $1.44 billion. The amount of aid provided by state grants in the 2000–2001 academic year was 6% of $60 billion 5 $3.6 billion. Thus, the amount of aid more than doubled. 27. The correct answer is (B). The quickest way to solve this problem is to note that other federal programs contributed 2% more money than did federal campus-based aid. The total amount of aid was $24 billion. Thus, 2% of $24 billion 5 $.48 billion. 28. The correct answer is (D). The amount of money awarded through Pell Grants in the 1990–1991 academic year was 16% of $24 billion 5 $3.84 billion. The amount of money awarded through Pell Grants in the 2000–2001 academic year was 10% of $60 billion 5 $6.0 billion. The percent of increase was approximately (2.16 4 3.84) 3 100% ' 56%.

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R E D A LERT GRE ANALYTICAL WRITING MEASURE STRATEGIES The GRE Analytical Writing Measure (AWM) test became a required component of the GRE comprehensive examination starting in October 2002. The Analytical Writing Measure is designed to allow the candidate to demonstrate skills in logical reasoning and composition. These skills are becoming more and more important, not only in your graduate college experience, but also in the global workplace. By doing well on this section of the GRE test, you are demonstrating to potential graduate schools the following: • You can recognize and restate complex ideas. • You can analyze the structure of an argument. • You can advocate a point of view and support it with evidence. • You can construct a well-organized discussion of controversial topics. • You have control of the mechanics and stylistic elements of standard English. Essentially, the Analytical Writing Measure calls for two kinds of thinking: 1. Constructive: In the constructive portion of the test, called “the issue task,” you are asked to construct a strong, logical, well-supported argument in favor of or against a controversial issue of importance, one that can be argued on either side with equal success. 2. Analytical: In the analytical portion, called “the argument task,” you are asked to take apart the logical argument of another’s point of view, recognizing its components and assessing the success or failure of the argument’s support. The combined test calls for you to think clearly, creatively, logically, and methodically and to express your points of view and conclusions in strong, well-organized rhetoric.

THE ISSUE TASK Beginning with our first exposure to the topic, unknown to you before the test time, we can proceed in easy steps: 1. 2. 3. 4. 5.

Do I understand the topic’s balanced points of view? Does one side of the controversy seem on its face to be more valid? Can I build an argument by listing support in outline form in favor of my view? Can I refute the support elements of the opposing view? Can I express my view in strong English writing style, with appropriate mechanics?

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Sample Outline “Present Your Perspective on an Issue” Suppose your topic reads, “Science is a need for order; art is a rage for chaos.”

Step 1. Understanding Do you understand the statement? How would you paraphrase it? What element submits to agreement or disagreement? To comparison or contrast? In this example, what does “chaos” mean? Is the statement contrasting science and art as diametrically opposed? What is meant by “need” versus “rage”? Are “order” and “chaos” mutually exclusive?

Step 2. Reaction What is your first reaction to the statement? Does it “ring true”? Or does it “sound phony”? Do you feel it is overly simple? Too vague? Too specific? What examples spring to mind on each side of the question? How are the abstractions solidified by example and illustration? In this example, do you like the juxtaposition of “rage” and “chaos,” or does it disturb you? Do you have preconceived notions about the value or nonvalue of art? Do you admire scientific inquiry, or do you find it uncreative? Is Picasso a good example of “chaos”? Does the taxonomy of Mendeleev’s chart serve as “scientific order”? How about the asymmetrical planetary orbits? Could the universe’s “chaos” contradict the statement? While Escher’s drawings are “chaos,” are they also examples of “rage”?

Step 3. Commitment Commit to an agreement or disagreement stance. Does the preponderance of examples that come to mind support or refute the statement? Which side seems more supportable? What immediately apparent flaws are there in the opposite view? List the arguments on both sides, preparing to support your evidence and attack the opposing evidence. In this example, you decide that science does indeed seem to seek out and even demand order, even when that order must be forced on it. You think of the exceptions to the Mendeleev’s chart, those elements that do not really fit neatly into rows. You think of “chaos theory,” an attempt to give order even to the idea of non-order. You remember the “chaotic” lives of artists, their daring innovations, their powerful, even violent attacks on conventions and “rules.” You determine that artists do in fact direct their creative energies on an assault on order, and you are prepared to defend the statement. List the chaotic artists you can recall, and list all the failures of science to order the universe: Art

Science

Picasso Pollock Breughel, etc.

Missing quark Failure of unified field theory Mathematical enigmas, surds, etc.

Step 4. Defense Begin to write your essay. Start with an introduction, in which you set the tone (“Scientists would like us to think that everything is in place in the universe, but the artist is always there to remind us of the omnipresence of the impulse toward chaos. . . .”). Answer the counterarguments. Refute the

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opposing view. Anticipate how someone would argue the other side of the question, dismissing their contentions and defusing their examples. In this example, “If a scientist tries to suggest that chaos theory is anti-order, remind him or her of Mandelbaum’s fractals, which become orderly, predictable designs despite all efforts to randomize the results. Even Nature makes clouds, leaves, snow, and waves by orderly design.” “The French Academy’s strict rules automatically excluded some of the greatest artists of their time—Van Gogh, Renoir, Matisse, Modigliani.” End the first paragraph with your thesis statement: “Clearly, science seeks order, while art gives voice to the energy of chaos in all its forms.” Gradually unfold your argument, support by support, illustrating each support with strong, visual examples: “Mathematics, the language of science, is the numbering of everything that exists; even impossible ideas such as pi are subjected to a number, even if it goes on infinitely. When a notion is mathematically absurd, we still give it a neat, orderly name: surd.” “If the rage for chaos ever had a signature, it is in the violent, physical exertions of throwing paint arbitrarily on canvas that made Pollock’s work absolutely unique in art history.” Conclude your essay by recapping your view, summarizing your arguments, and ending with a convincing, specific statement that “clinches” your argument. In this example: “Clearly, then, the underlying difference between science and art is the struggle between a belief in a Grand Design reflected in the benign neatness of ‘everything in its place’ and a grim but honest belief that individuality, newness, creativity, invention, and chaos are the ultimate definitions of humankind, represented by the artist in the studio, staring at the blank canvas, about to make something, whether or not the universe has a pigeonhole for it.”

Step 5.

Composition

Write your essay, and review your mechanics, especially sentence structure, removing weak verb constructions, strengthening clichés with fresh rewordings, and editing out extraneous verbiage, wheel-spinning, and the like.

GENERAL DIRECTIONS

FOR THE ISSUE

TASK

Directions: You will have 45 minutes to plan, outline, and compose an essay that presents your perspective on one of several topics. You may not choose your own topic on which to write. The topic chosen will take the form of a quotation, stating or suggesting a controversy, with two, or several, points of view available for discussion. You need not agree with the quotation; rather, you are to find a supportable thesis in favor of or opposed to the implications of the topic. Personal observations, your education, or your general knowledge on a subject can be the basis for your approach. University/college faculty members will read and evaluate your essay based on the following: • • • •

Considerations of the complexities and implications of the issue Organization, development, and expression of your ideas Your ability to support your ideas with reason, example, and good sense Your demonstrated control of the conventions of standard written English

First, think carefully about the issue itself, and plan your writing approach. Think logically, and organize your ideas as they develop. Begin your writing when you have ordered your ideas in some logical fashion. Leave time to reread and revise your essay, including re-ordering of your argumentive supports for added strength.

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SAMPLE TOPICS

FOR THE ISSUE

TASK

Directions: Present your perspective on the issue below, using relevant reasons and/or examples to support your views. “Students with physical disabilities should be given special classes in special classrooms.” “The death penalty acts as a preventative to violent crime in others.” “Same-sex marriages do not meet the requirements of a true marriage.” “A person with a gun is less likely to be the victim of a crime.” “An individual’s right to choose must be protected by the government.” “Modern art is a trick played on a gullible public hungry for anything new.” “America is not good at preserving its architectural heritage.” “Poverty is a necessary ingredient of the free-enterprise system.”

THE ARGUMENT TASK To take an argument apart, you can follow these steps: 1. 2. 3. 4. 5.

Can I examine an argumentive claim, separate its thesis statement from its support claims, assess the logical value of each support claim, recognize argumentive fallacies embedded in the rhetoric, and express my assessment of the argument’s strengths and weaknesses?

Your task here is to ask, “How well reasoned is this argument? Is the thesis statement clearly articulated? Does the reader follow the supports in an organized way? Has the writer used measured, reasonable examples to illustrate those supports? Has the writer avoided logical fallacies, such as generalization, post hoc, argument by analogy, and the like? Has the arguer convinced through strong evidentiary expression?” 1. 2. 3. 4. 5.

Do I understand the topic’s balanced but controversial issue? Do I recognize a clear thesis statement in the opening paragraph? Can I list in outline form the evidence presented in favor of that view? Does the argument contain fallacies, false statistics, or illogical rhetoric? Am I convinced or unconvinced by the argument’s structural support?

Sample Outline “Analyze an Argument” Suppose your essay argues in favor of stricter controls on illegal immigration into the U.S. from Mexico. It cites statistics (calling them “startling” and “overwhelming”), gives anecdotal evidence of the failure of the present system, and calls for a physical barrier between the countries, reinforced by an increase in border patrol personnel.

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Step 1 Do you understand the argument’s thesis? Is it clearly stated in the opening paragraph? Does the rhetoric contain the sense of “should” or “it is necessary to”? Step 2 How does the author line up the support elements? Is there a number in the thesis statement (such as “for the following six reasons . . .”)? Can you assign a number to each support element? Are they arranged in ascending or descending order of power to convince or in some other logical order? Or are they arbitrary or haphazardly arranged? Step 3 What is the tone of the support material? Reasoned, logical, with qualifying statements of comparison (such as “by and large,” “the trend is . . . ,” “it would make sense that . . . ,” etc.) or is it slanted, impassioned, emotional (such as “disease-ridden peons,” “lazy, corrupt border guards,” “deadly economic parasites,” etc.)? Step 4 Is each separate support convincing in its own right, or do logical fallacies insert themselves in the support (such as “We let Mexican lettuce workers unionize, and now look at the soaring price of food”)? Does each element have a logical “weight” to it? Step 5 Is the conclusion warranted by the evidence? Has the author given the reader enough facts and reasoning to actually convince?

GENERAL DIRECTIONS

FOR THE

ARGUMENT TASK

Within a 30-minute time limit, you will be called upon to read a short essay supporting one specific point of view. Your task is as follows: • Find its thesis statement. • Order its support statements. • Evaluate the strength of the general argument and support statements made. • Examine the rhetorical devices and linguistic choices. • Find possible argumentative fallacies. • Determine the validity of the argument made. Most importantly, you are asked to consider the logical soundness of the argument rather than to agree or disagree with the position it presents. You will then write an analytical essay presenting your findings. Begin your writing when you have ordered your ideas in some logical fashion. Leave time to reread and revise your essay for sense and for mechanics.

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SAMPLE TOPICS

FOR THE

ARGUMENT TASK

Directions: Discuss how well reasoned you find this argument. 1. The following appeared in a national political magazine article: “Rather than constantly fighting against the strict interpretations of the Supreme Court regarding the outmoded and dated wording of the U.S. Constitution, Congress should call for a Third Constitutional Convention, whose agenda would be to redesign the Constitution around the exigencies and consequences of 200 years of change. No one should be ‘bearing arms’ based on the single phrase in the 2nd Amendment, nor should criminals live a life of comfort and ease because chunky peanut butter is ‘cruel and unusual punishment.’ ” 2. The following appeared in a magazine subscribed to by the large-vehicle and excavation equipment industry: “We’re advocating replacing the term ‘strip mining’ with ‘improvement mining,’ because new methods of replenishing the land, planting grasses, and in some cases even landscaping the terrain have actually benefited the region and its citizens. Parks, walking paths, open spaces uncluttered by the haphazard contours of Mother Nature now grace many rural areas, after the mighty but environment-friendly tools of modern industry have been hauled away on low-boys. Communities should thank us.” 3. The following appeared in a nationally syndicated op-ed column of several large-city newspapers: “Any vestigial objections to an expanded military presence, and the budget to support it, disappear in the face of these recent bioterrorist threats. Until there is a well-armed, well-trained soldier at every post office in this country, there will be no such thing as homeland security, and those who oppose spending the money can take the responsibility when anthrax becomes the Bubonic Plague of the twenty-first century.” 4. A recent automobile newsletter, sent to national car dealers, said the following: “We are predicting that the internal combustion engine will be the next victim of OPEC’s near-monopoly. The American public, fed up with the wildly fluctuating price of fossil fuel, a price held hostage at every international political flare-up, will turn to hydrogen fuel cells by 2010. The small-car mentality, blind to safety issues, drives the market for these two-seat scooters, stripped of comfort, but high on feisty American autonomy and independence.” 5. The following appeared on the syllabus of a college English class: “Attendance is required to make this kind of instruction effective. We understand the complexities, contingencies, and emergencies of your everyday modern life and expect you to prioritize your tasks according to your own values. In this sense, all absences are excused absences; that is, excused by yourself. However, if your college education is important, you should give attendance a very high priority, because absences, especially contiguous absences, interrupt the developmental flow of the instructional material and reduce the student’s ability to focus on the topics, a focus necessary to build the cumulative reading/writing skills necessary to complete the course. Rule of thumb: More than three absences will probably affect your grade indirectly.”

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6. The following letter to the editor appeared in a mountain state newspaper: “There is a certain dignity to the proposition that Montana should enforce its own sovereignty. By the strength of its preservation of the original principles of the Founding Fathers, by its pioneering spirit, but its belief in the power of the individual to get into and out of his own scrapes without the condescending intervention of a bureaucratic hegemony 2000 miles away, and by the isolation, beauty, and sheer precipitous fact of its symbolic mountain terrain, Montana no more feels obligated to identify with the corruption, sloth, and self-indulgence of the rest of the so-called United States.” 7. The following appeared in a letter to a U.S. senator from one of his constituents: “Please don’t vote for the upcoming bill giving government subsidies to the elderly for their medications. My family has been in the pharmacy business for three generations, and my uncle now runs a small chain of pharmacies in our community. He says the subsidies would ruin his business, because the government checks never come on time, and the elderly customers he has now will go to the big pharmacy in town, which can afford to wait for those government checks. Who is there to help the small-business person when things get tough? Certainly not the federal government. My uncle’s business will go under if he starts to lose all his best customers.” 8. The following appeared in a journal on criminal justice: “Prisoners’ rights regarding conjugal visits in the penitentiary are jeopardized by the smallmindedness of ultra-conservative wardens and the dominantly white male boards of administration who hire them. To them, normal healthy relations of any kind are antithetical to the unwritten law of every penitentiary: Make the inmates’ lives as miserable and unproductive as possible, even when common decency and the natural impulse toward human companionship dictate to the contrary.

TEST-TAKING STRATEGIES Writing and analyzing arguments can be an overwhelming and daunting task, unless you understand the simple outline steps toward breaking down a complex argument into its component parts. The following checklist can be used with sample essays in practice and can help you plan your strategies for the actual test: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

Simplify the argument. Rephrase the argument. Make sure you know the definition of all the words. Find and number the supports. List examples from both sides of an argument. Mentally argue the other side to find weaknesses. Circle “loaded” words with rhetorical connotations. Identify fallacies. Write your draft. Revise your draft.

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Unit 1 ANALYTICAL WRITING REVIEW Directions: Present your perspective on the issue below, using relevant reasons and/or examples to support your views. Time—45 minutes (Note: In the actual GRE, you will be given a choice of two issues.) “Financial gain should be the most important factor in choosing a career.” Directions: Discuss how well reasoned you find this argument. The following appeared in the editorial section of a local newspaper: “In the first four years that Jones-Harrison has served as President of the country of Salmantir, the consumer spending has decreased and the unemployment rate has increased. Two businesses have closed for each new business that has opened. Under Bentley, who served for four years before Jones-Harrison, the unemployment rate decreased and the consumer spending increased. Clearly, the residents of Salmantir would be best served if they voted Jones-Harrison out of office and reelected Bentley.”

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ANALYTICAL WRITING REVIEW

ANALYTICAL WRITING REVIEW—ISSUE There is no right or wrong answer to a GRE essay topic—only a strong response or a weak one. Your point of view and approach to an issue or argument will differ from those of other test-takers, and therefore your essays will differ from the ones given below. We provide these poor responses, notes for improvement, and then high-level responses to serve as models for how you might construct a response to the given topics. When you review your own essay or have someone else review it, look for clear construction, logic, and good grammar, and follow the strategies found in the previous pages.

Response of Score 3 “Financial gain should be the most important factor in choosing a career.” Financial gain is an extremely important factor to consider in choosing a career path. Without adequate monetary income, it would be very difficult to maintain appropriate shelter, nourishing food, and reliable transportation. However, in the essay that follows, I will argue that ample financial gain, while especially important, should not be the most imperative factor when deciding how to spend your professional life. In a capitalistic society, there is a certain percentage of people who believe that financial gain should be the foremost consideration when deciding a career path. Because most sectors of the United States economy spent 15 years (from the mid-eighties to the end of the 1990’s) in tremendous fiscal growth, a bull market, and exponential explosion in personal finances, those individuals who strove for and created their own wealth only to see it plummet in the past 24 months could argue that securing adequate finances, if not wealth, should be the most important factor in determining a career path. These individuals created, then maintained, a lifestyle that required substantial income. Beautiful homes, exotic vacations, luxury vehicles, Ivy League educations, nightly socializing, and lavish entertainment cost tens of thousands of dollars each year. When the stock market began its swift decline in October of 2000, many of these individuals lost sizeable amounts of money and were forced to condense spending on non-essential items. Closure of large corporations forced an increase in unemployment of white-collar jobs. Political and commercial scandals pulled even more money out the economy. Due to these hardships, the individuals who have been forced to live more conservatively than they are use to could argue that it takes a significant amount of money to fund a “moderate” lifestyle. Their skewed definition of a “moderate” lifestyle includes the homes, vacations, luxury vehicles, Ivy League educations, socializing, and entertainment. Because they grew accustomed to that level of lifestyle, they considered it “moderate”, or normal. Now that they have had to cut back, they feel the loss of status and material goods could be salvaged if only there was a way to produce more income. Alternately, individuals who chose a career based more upon their interests, skills, talents, and even their “callings” rather than finances have not had to make

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such drastic adjustments as the economy has fluctuated. They have lived more conservatively overall, and have not felt the pressure of substantially decreased income. Also, because they chose occupations that were fulfilling both personally and professionally, they tend to have higher levels of job satisfaction and a desire to continue to further to develop skills in their field. Statistics have shown that the happier a person is at his or her job, the happier his or her overall life is. Of course, financial security is also necessary for a comfortable life. Therefore, career choices should be made first upon skills, interests, and talents, and secondarily upon financial security. Both are necessary for a well rounded and fulfilling life.

Notes for Improvement While this essay is appropriate in overall length and contains an adequate analysis of the topic, there are areas in the essay where the writer tries to convey a point and writes around the point without getting directly to it. For example, when the writer discusses why the wealthier sector of society feels economic distress more acutely than other sectors, he or she could make the point more succinctly. Coherence is compromised with long sentences and ideas that, although valid, are not as well organized as they could be. This writer has also focused upon current economic conditions a little too heavily and made the assumption that the wealthy suffer more in economic downturns than do other sectors of the population. While this discussion may be related to his or her point, it is laboriously presented and takes up too much of the discussion. More importantly, it is largely irrelevant to the main point. Cutting down on run-on sentences would help this writer clarify and condense, as would limiting the use of unnecessary adjectives. And, while the writing is generally acceptable, there are errors in grammar and syntax that detract from its effectiveness. The example below represents a score of 5 or 6 and is more concisely presented, substantially more cohesive, and grammatically correct.

Response of Score 5–6 Financial gain is an important consideration when choosing a career. Without adequate income, it is difficult to maintain shelter, food, and transportation. In this essay, however, I will argue that financial gain, while important, should not be the most important factor when deciding how to spend your professional life. People who choose occupations that are closely aligned with their interests, talents, and beliefs will spend their work time engaged in more fulfilling activity. Since one is likely to spend at least half of one’s waking weekday hours at work, fulfillment there is crucial to one’s happiness. Numerous studies have shown that the happier a person is at work, the happier he or she is in other aspects of life. If a person is fulfilled by the work itself, that sense of fulfillment is an experience that can easily be renewed each day simply by going to work. But what of people focused solely on financial gain? When they suffer a financial decline, their source of satisfaction may not be so easily renewed. Throughout the

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1990s, many people accumulated unprecedented wealth. They may not have enjoyed their work—they may even have short-changed their nonwork lives to accumulate money—but at least they had the satisfaction of seeing their portfolios grow. As their wealth increased, they may have leveraged it to borrow money for a more expensive car or home. Then the bear market of 2000–2003 struck. Much of their financial gain—the very thing that made their work worthwhile— evaporated and left behind no sense of fulfillment or satisfaction. During this same time period, the people who chose personally fulfilling work still had access to it, while people who chose financial gain alone were left holding decimated balance statements and perhaps crushing debt. For these people, renewal of their sense of fulfillment would take much longer than driving to work the next day. For these reasons, financial gain should not be the most important factor in choosing a career. It is an end-product of work rather than integral to the daily experience of work. The person who enjoys work for itself day by day is more likely to experience fulfillment and better positioned to renew that sense of fulfillment in financially difficult times.

ANALYTICAL WRITING REVIEW—ARGUMENT The following appeared in the editorial section of a local newspaper: “In the first four years that Jones-Harrison has served as President of the country of Salmantir, the consumer spending has decreased and the unemployment rate has increased. Two businesses have closed for each new business that has opened. Under Bentley, who served for four years before Jones-Harrison, the unemployment rate decreased and the consumer spending increased. Clearly, the residents of Salmantir would be best served if they voted Jones-Harrison out of office and reelected Bentley.”

Sample Response—Score 2 This argument makes several good points, but it is not strong enough to be completely believable. Examples and statistics are given but not everyone would agree with the writer of this argument, just because there are facts presented. Just because more businesses have closed than opened, doesn’t mean Jones-Harrison has done a bad job. Also, it is possible that when Bentley was President, the general economy is better or there were more jobs available; And it isn’t the fault of a President if the consumer spending decreases. Overall, Jones-Harrison may not want to run for reelection, and Bentley may not want to by President again in the country. Basically, the writer doesn’t give enough information to tell. Before an election could occur, condition could change for the better, and the people might be very happy with Jones-Harrison as President after all in the position.

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Notes for Improvement There are several areas where the analysis of this argument could be made more coherent. First, the analysis is too brief to represent a thorough identification of claims, exploration of alternative explanations, and a discussion of the line of reasoning. The organization of thoughts and ideas presented by the writer of the analysis is unclear and follows no logical arrangement. Second, this analysis does not delve into why the argument is effective or not. Rather, it states reasons why the events may or may not have occurred and injects the analyzer’s opinions into the analysis. A successful analysis steers clear of personal opinion or taking one side over the other. The analysis would have been more effective if this writer had acknowledged the claims and given evidence to support the claims or reasons why the claims could not be substantiated. Third, this analysis does list alternatives to the argument: Jones-Harrison may not want to run for reelection, Bentley may not want to be President again, and before an election could occur, conditions could change for the better and the people might be happy with Jones-Harrison as President after all. But, the alternatives are to the actions of the participants in the argument, not to the validity of the argument itself. Remember to analyze the argument, not the actions presented in the argument. The writer of the analysis does point out some logical inconsistencies (“Just because more businesses have closed than opened, doesn’t mean Jones-Harrison has done a bad job” and “Also, it is possible that when Bentley was President, the general economy is better or there were more jobs available”). But he or she does not develop or organize them or present them in a grammatically correct manner. The essay also contains dangling modifiers, standard punctuation errors, and subject–verb agreement errors. These errors strongly detract from the cohesiveness of the analysis. There is also a lack of organization and coherence in the structure of the sentences and paragraphs. The lack of transitions makes the analysis difficult to follow. With the linguistic weaknesses in the analysis (simple sentences, inexpressive language, grammatical errors) and the failure of the writer to construct a critique based on logical analysis, this essay would score a 2. The essay below implements the points just discussed and demonstrates how to raise the score to a 5 or 6.

Sample Response—Score 5–6 This argument is presented clearly and backs up several of its claims with facts, and the writer is passionate about his or her issue. However, the argument isn’t strong enough to be convincing. Examples are given and a conclusion is drawn, but the items offered as evidence are assumptions without justification, and the ideas and facts offered as evidence can be disputed with simple logic. The essay that follows will demonstrate the weak points in the argument, as well as suggest ways to clarify and strengthen the argument. Each of the four sentences in the argument contains an underlying assumption that can be easily and logically challenged. In the first sentence, the implication is that President Jones-Harrison is solely responsible for a decline in consumer

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spending and an increase in unemployment in Salmantir. A President does not exclusively control all of the economic factors that affect consumer spending and employment. The general condition of the national economy; the growth or decline of industry in the area; and local and national leadership other than that of the President are just a few of the factors that influence consumer spending and the stability of a workforce. The second sentence states that more businesses have closed than opened under President Jones-Harrison. There is no mention of the types of businesses, the demographics of the country, the strength of the general economy, the availability of funds for new business owners, or the conditions of the businesses prior to Jones-Harrison’s taking office. Again, the assumption that Jones-Harrison is solely responsible is a fallacious one. The third sentence implies that Bentley controlled consumer spending and employment better than Jones-Harrison has, but Bentley could not have controlled these things any more than Jones-Harrison can. The general economy may have been stronger, nonpresidential leadership may have been more proactive, and any number of other reasons could explain the superior consumer spending and employment under Bentley. The final sentence boldly states that the residents of Salmantir would be better off replacing Jones-Harrison with Bentley. Jones-Harrison may not want to run for reelection, however, and Bentley may not be interested in seeking public office again. The factors listed in the paragraphs above will most likely not change with a change in president, and if they do change, it will be a slow change based on numerous factors other than just a change in presidential leadership. Overall, this argument would have been more convincing if, in discussing the employment and consumer spending situation of Salmantir, factors other than presidential leadership were discussed. The line of reasoning in the argument is not based on reasoned, logical, or supported claims. The argument for one presidential candidate over another should be based upon a complete understanding of the issues of the community and the political, professional, and personal stance of each candidate. Ill-founded confidence in a candidate could lead to national turmoil.

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R E D A LERT VERBAL ABILITY STRATEGIES To succeed on this part of the GRE, you need to become familiar with the types of questions that the test contains. These include vocabulary questions dealing with sentence completion, analogies, reading passages, and antonyms. The section that follows offers explanation, strategies, and practice with each type of question. Working through this information should help you improve your scores on the examination. Each review contains sample questions with explanations of what skills and techniques should be used for success. Working your way carefully through each section, you will increase your understanding of the kinds of questions as you strengthen your skills.

SENTENCE COMPLETIONS In a sentence completion question, one or more words have been removed. You are required to supply the missing word(s) that will best complete a sentence. These questions demand skill in figuring out meanings from context. Choose words that BEST fit the meaning of the sentence. In order to handle this type of question, you should first read the sentence as you see it without trying to fill in the word(s). After reading, consider the MAIN IDEA of the sentence and THEN read the choices. Remember, BOTH words must fit into the meaning of the sentence; therefore, read your choice into the sentence supplying and evaluating BOTH words. Example Choose words that best fit the meaning of the sentence: The zoology students sat quietly in their observation post; they were pleasantly surprised to observe, over the course of two days, a band of gorillas build a ______ camp each night. This always followed a day of ______ for the berries and leaves that constitute their diet. (A) (B) (C) (D) (E)

solid..trading sturdy..roaming interesting..seeking makeshift..foraging circular..farming

Your knowledge of the meanings of words and the ability to use those words appropriately within a given context will help you answer sentence completion questions. In addition, each sentence provides key words, specific examples, or an overall logic that helps direct you to the correct answer, regardless of your knowledge of the subject. The following strategies listed are also useful.

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Rules 1. Relationships: As you read the sentence, note key words that show relationships. For example, but, although, however, and on the other hand indicate contrasting ideas. And, another, and the same denote similarity. Therefore, as a result, consequently, since, and because signify a cause–effect relationship. In the example, followed indicates a time relationship. 2. Grammar and Logic: Eliminate any choices that make no sense or that are grammatically incorrect. Choice (C) cannot be correct because the first blank requires a word beginning with a consonant. Choice (E) cannot be correct because farming does not apply to gorillas or their food. 3. Both blanks: Be sure that your choice of answer offers words that fit both blanks logically. Often only one of a pair may seem a sensible choice. Read through both words in each possible answer because both words must make sense. For example, in choice (A), solid logically could be used to complete the sentence; however, trading—a human activity— does not fit logically into the context of the sentence. Also, if two choices still seem to be possibly correct answers, examine the choice of vocabulary carefully to determine any nuances of meaning. Choices (B) and (D) both offer words that could be used to complete the sentence; however, since the camp is remade each night, it is probably makeshift rather than sturdy. Also, while the gorillas may be said to be roaming for food, foraging is a more specific and suitable word because it means “searching for food.” Example I attend the local college games, especially the one with our arch rival, State College. This year was extremely tough for us. State led throughout the game; but, after the ______ of a strong rally late in the ball game, we really thought we had a great chance of winning. Therefore, we were doubly ______ when our team lost. (A) (B) (C) (D) (E)

lack. .surprised threat. .amused dispute. .annoyed excitement. .disappointed skill. .doubtful

Using the aforementioned clues and procedures, select the answer you think is best. Rule 1 The key words in the sentence that help you determine this answer are strong, rally, and lost. You can determine that by looking at the entire selection to see what its intent is. Rule 2 indicates that choice (A) is not possible because it would make sense ONLY if the team had won. Rule 3 indicates that choice (B) cannot be correct. While the word threat seems reasonable, the word amused does not. Applying this rule also helps you make the right selection of choice (D) by pointing out that in the context of strong, rally, and lost, logically this one is the correct choice. Choices (C) and (E) offer words whose meanings are incorrect in the context of the sentence; therefore, rule 3 applies to them as well.

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Example Traditionally, countries with ______ borders requiring ______ must maintain a large army and support it by imposing taxes. (A) (B) (C) (D) (E)

historic. .markers vulnerable. .defense vague. .exploration unwanted. .elimination contested. .estimation

Now, by applying the three rules again, which choice did you make? Let’s look at choice (A) first. While historic will work in the sentence, markers does not because it makes no sense. A country does NOT employ an army to maintain its markers; therefore, rule 3 fits here. Choice (B) offers two words that are logical options, so rule 3 applies again. However, you must be sure to read all of the possible choices before you select an answer. Choices (C) and (D) present options that are NOT logical. A vague border would not require exploration; an unwanted border does not require an army to eliminate the border. Choice (E) makes no sense at all. Therefore, rule 2 applies to all three of these selections.

ANALOGIES An analogy question presents two words that are related in some way, and it requires you to first discover the relationship, then find another pair of words that is related in the same way. Note the following example: ADVERTISING : SELLING :: (A) (B) (C) (D) (E)

reporting : informing training : helping discovering : exploring marketing : research creating : destroying

To answer analogy questions, use the following strategies: 1. First, determine the relationship between the first pair of words and state that relationship in the form of a complete sentence: “Advertising is a means of selling products to an audience.” 2. Then find a pair of words in the answer choices that can be substituted for the original pair: “Reporting is a means of informing an audience.” None of the other choices expresses quite the same relationship. Although you can say, “Training is a means of helping an audience,” the context is much more general. Choice (A) is the best. 3. You may ask, “What if I do not know the definition of some word? How do I make a choice?” It is essential that you establish a logical connection between the original pair of words. If you are unsure of the definition of either, you must consider whether you are willing to risk a guess. The question may be one you choose to skip at this point and perhaps return to if you have time. Remember as well that your score is calculated by penalizing you one fourth of a point for each incorrect answer. Omitting a question neither reduces nor adds to your score.

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Types of Relationships in Analogy Questions The following table illustrates some of the most common types of relationships you will encounter in analogy questions: Type of Analogy

Example

Action of Object

PLAY : CLARINET :: incise : knife

Cause to Effect

SUN : SUNBURN :: overeating : indigestion

Item to Category

IGUANA : REPTILE :: cat : mammal

Object to Its Function

PENCIL : WRITING :: tractor : plowing

Object to Its Material

CURTAINS : CLOTH :: windows : glass

Part to Whole

PAGE : BOOK :: limb : tree

Time Sequence

RECENT : CURRENT :: antique : obsolete

Word to Antonym

ASSIST : HINDER :: enthrall : bore

Word to Synonym

PROVISIONS : SUPPLIES:: portent : omen

Worker and Creation

ARTIST : SKETCH :: composer : etude

Worker and Workplace

CHEF : KITCHEN :: judge : courtroom

Word and Word Derived from

ACT : ACTION :: image : imagine

Now, using the two previously described procedures and the preceding table, look at these examples. Example 1 MNEMONIC : MEMORY :: (A) (B) (C) (D) (E)

trousers : speech glasses : vision earmuffs : movement blinders : hearing glove : hand

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The correct answer is (B). Now, consider the relationship between the words MNEMONIC : MEMORY. A mnemonic device helps one to remember. Choices (A), (C), (D), and (E) cannot be logical answers because none of these offers the same relationship. Speech has no relationship to trousers. Earmuffs have no relationship to movement, and blinders have no relationship to hearing. While a glove covers a hand, it does not help to produce a hand. Choice (B) is correct because glasses are designed to aid vision or to help one to see. The relationship is identical to that of the original pair; it is a Cause to Effect relationship. Example 2 WAGGISH : LAUGHS :: (A) (B) (C) (D) (E)

risible : yawns bilious : smiles lachrymose : tears ribald : moans frown : grin

The correct answer is (C). Again, using the previously described procedures, you can determine that a waggish remark is designed to produce laughs. Looking at choices (A), (B), (D), and (E), you can see that they are incorrect because they do not produce the same relationship. Risible means laughable, while bilious refers to a yellowish coloration of the skin, and ribald pertains to coarse, offensive humor. Choices (A) and (E) are incorrect because the relationship in each is Word to Antonym, while choices (B) and (D) are wrong because the relationship in each is not Cause to Effect. Example 3 PHILIPPIC : VITUPERATIVE :: (A) (B) (C) (D) (E)

liturgy : ribald encomium : complimentary harangue : restrained paean : scurrilous anecdote : story

The correct answer is (B). The relationship is Word to Synonym since a philippic is a kind of speech that is, by definition, vituperative or scathing. Choices (A), (C), and (D) have the relationship of Word to Antonym. Choice (B), on the other hand, is correct because an encomium is a kind of speech that is, by definition, complimentary. Choice (E) is incorrect since the relationship is Item to Category.

ANTONYMS Antonym questions require you to use your vocabulary skills as well as to develop relationships and thought processes. Antonym questions provide a single word and ask you to select from a list of words the ONE that is most opposite in meaning.

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Example 1 APATHY (A) (B) (C) (D) (E)

indifference wrath zeal expression bewilderment

Several strategies can help you to answer antonym questions: 1. Try to define the meaning of the question word yourself before reviewing the choices. Remember to look for a word that is most nearly opposite in meaning. Be careful NOT to choose a synonym, such as indifference in the preceding example. 2. If no answer is immediately apparent, eliminate the obviously incorrect choices. Keep in mind that many words have more than one meaning. If no choice seems to have the opposite meaning, think of other meanings for the question word. 3. Choose the word that is most nearly opposite in meaning. In the preceding example, both wrath and zeal indicate strong emotional involvement. However, wrath refers to intense anger, while zeal refers to intense enthusiasm. The correct answer, therefore, would be choice (C), zeal. Example 2 WILT (A) (B) (C) (D) (E)

prevent drain expose revive stick

The correct answer is (D). Using the procedure already described, consider the meaning of wilt. You probably thought of a flower as it begins to suffer from a loss of water, or because of extreme heat. You recognize that the wilting flower is about to die. So, what you need is an opposite. Neither choices (A), (B), (C), nor (E) denotes the same relationship. Indeed, prevent, choice (A), seems to ward off the wilting condition. Choices (B) and (E) make no sense. Choice (C) might be the CAUSE of the wilting, but it would not be the opposite of wilt. Therefore, the answer is choice (D), revive. The word revive means to bring back to life or to stop the wilting. The meaning is opposite. Example 3 PREMEDITATED (A) (B) (C) (D) (E)

spontaneous conclusive disruptive vindictive strenuous

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The correct answer is (A). Using the procedure, what do you discover? Premeditated means to plan or set out to accomplish. One might even say that premeditated is planned. So, we need an antonym that indicates no planning at all. Choice (A) is correct. Spontaneous indicates an action that is NOT planned or premeditated. Choice (B) at first seems a possible choice; however, when we think about it, conclusive means either closing or decisive. Premeditated is NOT opening or tentative; therefore, the answer is not correct. Choice (C) indicates a loss of control that has nothing to do with our word. Choice (D) evokes the meaning of a grudge or an attempt to make up for something that has already happened. Premeditated activities are not always vindictive. Choice (E) means using great amounts of energy. Premeditated activities are not necessarily strenuous. Do you see how the reasoning process works? Now, let us try one more. Example 4 BREADTH (A) (B) (C) (D) (E)

rarity mobility complexity narrowness roughness

The correct answer is (D). Remembering the procedure, did you decide that breadth means broad or wide? Good, that is correct! Now, look at the answers. Choice (A) cannot be correct because something broad does not have to be common, which would be necessary for rarity to be the opposite. In addition, choice (B) cannot be correct because something that is broad is not always stationary; therefore, mobility as the opposite would not be true either. Choice (C) cannot be correct because something broad or wide does not have to be simple; therefore, the opposite cannot be complex. Choice (E) cannot be correct because something broad does not have to be smooth, which would make roughness an antonym. You see that something that has breadth MAY be the opposite of all of these things, but none of them is required for the definition to fit. Choice (D) is correct because something that has breadth cannot be narrow. Now it’s time to practice what you have learned in this section. Following are three review sections: Sentence Completions, Analogies, and Antonyms. Go through each section, answering all of the questions and then carefully checking your answers. If you are still having problems with any one section, come back to this chapter and reread it.

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Unit 2 SENTENCE COMPLETION REVIEW Directions: Each sentence below has one or two blanks, each blank indicating that something has been omitted. Beneath the sentence are five lettered words or sets of words. Choose a set of words for each pair of blanks that best fits the meaning of the sentence as a whole. 1.

Philosophers tell us that one’s lifetime is ______ when considered from the viewpoint of ______ making humans appear much less important than they think in the grand scheme of things. (A) laudatory. .prestidigitation (B) jaded. .youth (C) ephemeral. .eternity (D) superfluous. .transience (E) gauche. .theology

2.

The primitive emotions of love and hate, even though extreme opposites, are found in varying degrees even in the most ______ and ____ person, according to sociologists. (A) brackish. .mature (B) sylvan. .intellectual (C) celestial. .civilized (D) beneficent. .stable (E) defunct. .healthy

3.

When surveying the rule of the elderly king, we could only conclude that as he neared his ______ he became a(n) ______ ruler, which was obvious by his inattention to some matters. (A) pinnacle. .blatant (B) dotage. .effete (C) prime. .voluble (D) euphony. .dissident (E) prerogative. .covert

4.

Surveying the college course guide, we could conclude that ______ is a phase of the study of ______. (A) nihilism. .gynecology (B) hypertension. .etymology (C) recidivism. .criminology (D) altruism. .paleontology (E) hallucination. .chivalry

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5.

A refugee may be forced to ______ allegiance to his former country and ______ all of his former friends in order to work in a new country. (A) fabricate. .garble (B) fetch. .extradite (C) fluctuate. .expurgate (D) abjure. .forsake (E) lacerate. .occlude

6.

Some experts think that the origin of schizophrenia is ______; others believe it is ______. (A) contiguous. .environmental (B) congenital. .environmental (C) congenital. .deleterious (D) contagious. .pathological (E) exogenous. .celestial

7.

Even though we had heard that Professor Smith of the English Department taught an easy class, we knew that ______ and ______ are usually studied by those who enjoy the language. (A) liturgy. .pantheism (B) philology. .etymology (C) prosody. .ubiquity (D) tautology. .simony (E) raillery. .verity

8.

When I am ______, I am also ______, I explained to my friends who wondered at my long face. (A) scintillating. .verbose (B) quiescent. .succinct (C) lugubrious. .lachrymose (D) reviled. .providential (E) providential. .rubicund

9.

One of the things we learned in health class is that when eating, it is important to ______ thoroughly in order for proper ______ to occur. (A) rankle. .temerity (B) mitigate. .digestion (C) transmute. .veneration (D) query. .progeny (E) masticate. .digestion

10.

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Compelled by my professor to attend a lecture by an aging former teacher, I found the lecture was full of ______, and, as I had suspected and dreaded, it became most ______. (A) clichés. .bromidic (B) gabble. .blatant (C) foibles. .bombastic (D) histrionics. .insidious (E) metaphors. .laconic

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11.

After introducing two of my friends, I learned that the introduction was a disaster because her ______ immediately led her to suspect his ______ in discussing his life experiences. (A) philology. .valiant (B) rancor. .secular (C) vigilance. .petulance (D) perspicacity. .fraudulence (E) vagary. .indolent

12.

My friends were absolutely amazed when attending a religious convocation where the ______ outbursts of the congregation were ignored by the ______. (A) heretical. .indigent (B) heinous. .indolent (C) profane. .ecclesiastic (D) ebullient. .commissary (E) flagrant. .exodus

13.

After ruining her dress, I would have preferred her most biting ______ to the ______ looks she directed my way. (A) euphemisms. .consummate (B) anodynes. .feckless (C) diatribes. .reproachful (D) effigies. .refulgent (E) histrionics. .penitent

14.

During the fearful storm, the people in its path ______ God for divine ______. (A) importuned. .intervention (B) imputed. .favors (C) expiated. .revelation (D) deprecated. .power (E) immortalized. .gifts

15.

After studying psychology for a quarter, I can see that my friend is a ______ because he is always ______ favors from others. (A) sycophant. .currying (B) benediction. .eliciting (C) brigand. .flouting (D) facade. .brandishing (E) tryst. .avowing

16.

Many of my peers have turned to religion, realizing that the ______ in the church was a sign of ______ rather than money-hungry leaders. (A) tithe. .redress (B) windfall. .sacrilege (C) skeptic. .predilection (D) wraith. .piety (E) schism. .sedition

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17.

After the burglarizing of my home, I overheard the detective remark to the police officer that apparently the thief had moved in a ______, ______ manner. (A) sensuous. .tangible (B) furtive. .surreptitious (C) phlegmatic. .probing (D) moribund. .menial (E) ostentatious. .patrician

18.

During our commencement, the student body president delivered the ______, which had a ______ effect on the audience. (A) martinet. .pernicious (B) patrimony. .depraved (C) salutatory. .bracing (D) elixir. .blatant (E) cudgel. .brusque

19.

Returning home for vacation, I learned that my mother’s new medicine had made her extremely ______ and ______. (A) articulate. .copious (B) doltish. .overt (C) autocratic. .congruent (D) torpid. .phlegmatic (E) ludicrous. .remiss

20.

When I interviewed for a journalist’s position, I was told that often the editor was very ______; he made numerous ______. (A) sedentary. .rifts (B) fastidious. .emendations (C) saline. .parables (D) maudlin. .orifices (E) onerous. .idylls

21.

My erratic brother gives us all kinds of problems; his occasional ______ ______ are frightening to the family members. (A) spurious. .tacks (B) transitory. .oblations (C) turgid. .zephyrs (D) sporadic. .fulminations (E) perfidious. .nosegays

22.

When listening to nursery rhymes, my daughter likes the part in which the ______ witch uses a tiny doll as a ______. (A) ductile. .missal (B) eviscerated. .derelict (C) exacting. .crux (D) malevolent. .fetish (E) doughty. .doxology

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UNIT 2

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23.

When my Criminal Justice class observed a courtroom proceeding, we watched while one accused was examined; the ______ heard the testimony and ______ the man to jail. (A) iconoclast. .condoled (B) bourgeois. .denuded (C) doggerel. .eulogized (D) consort. .imbibed (E) arbiter. .remanded

24.

My friend’s uncle is a member of Alcoholics Anonymous; his ______ led to an ______ liver disease. (A) dipsomania. .ineluctable (B) avarice. .auspicious (C) volition. .unctuous (D) sojourn. .audacious (E) tableau. .incipient

25.

Can you believe that I won the photo contest with the ______ of the two pictures that was most ______, even though I only tried to hang them where there were nails! (A) sophistry. .hallow (B) juxtaposition. .esthetic (C) trappings. .emaciated (D) pseudonym. .facile (E) corollary. .extraneous

26.

In Biology class, we learned about animal families, so that I was able to understand that members of the ______ family are ______ and why my farmer uncle grows grass. (A) bovine. .herbivorous (B) anthropoid. .adamant (C) conduit. .corpulent (D) congenital. .incarnadine (E) heretic. .chivalrous

27.

When my sister got her first job, her novice ______ led to a ridiculous ______. (A) collusion. .consanguinity (B) synthesis. .cordovan (C) colophon. .temerity (D) ineptitude. .imbroglio (E) chauvinism. .quirk

28.

In Meteorology, we learned that the ______ around the moon is a(n) ______ sign. (A) parody. .audacious (B) oblation. .heretic (C) hiatus. .onerous (D) corona. .auspicious (E) dregs. .organic

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SENTENCE COMPLETION REVIEW

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29.

During the time of Teddy Roosevelt, soldiers in battle might see the ______ troop appear over the ______. (A) equestrian. .butte (B) albino. .heyday (C) exorbitant. .pendant (D) diabolic. .ventricle (E) incendiary. .rhesus

30.

After a year of hard work in the metropolitan rush, as a relief from ______ pressures, many plan to ______ on their vacation. (A) inveterate. .pique (B) urban. .rusticate (C) pent. .prate (D) neolithic. .venerate (E) laconic. .slake

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UNIT 2

Quick Score Answers 1. 2. 3. 4. 5.

C D B C D

6. 7. 8. 9. 10.

B B C E A

11. 12. 13. 14. 15.

D C C A A

16. 17. 18. 19. 20.

A B C D B

21. 22. 23. 24. 25.

D D E A B

26. 27. 28. 29. 30.

A D D A B

ANSWERS AND EXPLANATIONS

SENTENCE COMPLETION REVIEW 1. The correct answer is (C). The key words are less important. A good word would be “unimportant.” That eliminates choices (A), (D), and (E). The trigger for the second blank is grand scheme of things. A good word choice would be “lifetime.” That eliminates choice (B). 2. The correct answer is (D). The clue in the sentence is emotions . . . are found. The trigger words are even in the most. A good word for a person who would have both qualities would be “good.” That eliminates choices (A), (B), and (E). Since celestial means “heavenly,” it cannot apply to a person. This eliminates choice (C). 3. The correct answer is (B). The key word here is elderly. A good word would be “senile.” This eliminates choices (A) and (E). The trigger for the second blank is inattention. A good word would be “tired.” That eliminates choices (C) and (D). 4. The correct answer is (C). It is the only choice that correctly describes a phase of an academic subject. 5. The correct answer is (D). The key word is refugee. A good word for what a refugee must do is “surrender.” That eliminates choices (A), (C), and (E). A good word for another act for a refugee would be “forgo.” That eliminates choice (B). 6. The correct answer is (B). The key words are some experts think. A good word for what some would believe is “genetic.” That would eliminate choices (A) and (E). The trigger for the second blank is others believe. A good word for the opposite of what is in the first blank would be “acquired.” That eliminates choices (C) and (D). 7. The correct answer is (B). The key word is English. A good word for a characteristic of an English teacher is “scholar.” That eliminates choices (A), (D), and (E). The trigger for the second blank is language. That eliminates choice (C). 8. The correct answer is (C). The key words are long face. A good word would be “sad.” That eliminates choices (A), (D), and (E). Another good word is “tearful.” That eliminates choice (B). 9. The correct answer is (E). The key word is eating. A good word is “chew.” That would eliminate choices (A), (B), and (D). A good word for the result of chewing is “utilize.” That eliminates choice (C). 10. The correct answer is (A). The key word for the second blank is dreaded. A good choice would be “boring.” That eliminates choices (B), (D), and (E). The trigger word for the first blank is aging. A good choice would be “overused phrases.” That eliminates choice (C).

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SENTENCE COMPLETION REVIEW 11. The correct answer is (D). The key word is suspect. A good word would be “distrust.” That eliminates choices (A) and (E). The trigger for the second blank is discussing his life experiences. A good word would be “honesty.” That eliminates choices (B) and (C). 12. The correct answer is (C). The key words for the second blank are church leader. A good word would be “priest.” That eliminates choices (A), (B), and (E). The trigger for the first blank is outbursts. A good word would be “irreligious.” That eliminates choice (D). 13. The correct answer is (C). The key words are would have preferred. A good word is “cursing.” That eliminates choices (A), (B), and (D). The trigger for the second blank is looks. A good word is “blaming.” That eliminates choice (E). 14. The correct answer is (A). The key words are fearful storm. A good word would be “prayed.” That eliminates choices (C), (D), and (E). The trigger for the second blank is divine. A good word would be “assistance.” That eliminates choice (B). 15. The correct answer is (A). The key words for the second blank are favors from others. A good word would be “begging.” That eliminates choices (C), (D), and (E). For the first blank, key words are studying psychology, which would lead one to observe behavior. A good word would be “flatterer.” That eliminates choice (B). 16. The correct answer is (A). The key word is money-hungry. A good word would be “begging.” That eliminates choices (C), (D), and (E). The trigger words for the second blank are turned to. A good word would be “good.” That eliminates choice (B). 17. The correct answer is (B). The key word is burglarizing. Two synonyms are needed, as indicated by the comma between the blanks. A good word would be “sneaky.” That eliminates choices (A), (C), and (E). Another good word for such behavior would be “careful.” That eliminates choice (D). 18. The correct answer is (C). The key word is commencement. A good word for a speech delivered on such an occasion is “address.” That eliminates choices (A), (B), and (E). The trigger for the second blank is effect. A good word would be “energizing.” That eliminates choice (D). 19. The correct answer is (D). The key here is that we need synonyms, as indicated by the pairing divided by “and.” A good word for the result of medication is “drowsy.” This eliminates choices (A), (C), and (E). Another good word is “sleepy.” This eliminates choice (B). 20. The correct answer is (B). The key word here is editor. A good word for the work of an editor is “exacting.” That eliminates choices (A), (C), and (D). The trigger for the second blank is numerous. A good word would be “corrections.” This eliminates choice (E). 21. The correct answer is (D). The key words are erratic and problems. We need an adjective and a noun, as indicated by the lack of a comma between the two words. A good word choice would be “infrequent.” That eliminates choices (A) and (E). The trigger for the second blank is frightening. A good word would be “displays.” That eliminates choices (B) and (C). 22. The correct answer is (D). The key words are nursery rhymes. A good word for a witch in such a work would be “evil.” That eliminates choices (A), (C), and (E). The trigger for the second blank is doll. A good word would be an “effigy.” That eliminates choice (B).

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UNIT 2 23. The correct answer is (E). The key word is courtroom. A good word for one who hears in a courtroom is “judge.” That eliminates choices (B), (C), and (D). The trigger words for the second blank are to jail. Good word choices would be “sent back.” That eliminates choice (A). 24. The correct answer is (A). The clue here is Alcoholics Anonymous. A good word would be “alcoholism.” That eliminates choices (B), (D), and (E). The trigger for the second blank is liver disease. A good word would be “cirrhosis.” That eliminates choice (C). 25. The correct answer is (B). The clue here is hang them. A good word would be “arrangement.” That eliminates choices (A), (D), and (E). The trigger for the second blank requires a result. A good word would be “interesting.” That eliminates choice (C). 26. The correct answer is (A). The key word for the second blank is grass. That eliminates choices (B), (C), and (E). The clue for the first blank is animal. A good word would be “cow.” That eliminates choice (D). 27. The correct answer is (D). The key words are first job and novice. A characteristic of a beginner would be “lack of training.” That eliminates choices (B), (C), and (E). The trigger for the second blank is ridiculous. A good word would be “confusion.” That eliminates choice (A). 28. The correct answer is (D). The key words here are around the moon. A good word would be “circle.” That eliminates choices (A), (B), and (E). The trigger for the second blank is sign. A good word would be “beautiful.” That eliminates choice (C). 29. The correct answer is (A). The key words here are Teddy Roosevelt, whom we revere as our president, who rode with the cavalry. A good word would be “horse.” That eliminates choices (C), (D), and (E). The trigger for the second word is over. A good word would be “rise.” That eliminates choice (B). 30. The correct answer is (B). The key words are metropolitan rush. A good word would be “city.” That eliminates choices (C), (D), and (E). The trigger for the second blank is vacation. A good word would be “country.” That eliminates choice (A).

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Unit 3 ANALOGY REVIEW Directions: In each of the following questions, a related pair of words or phrases is followed by five lettered pairs of words or phrases. Select the lettered pair that best expresses a relationship similar to that expressed in the original pair. 1.

ASKEW : STRAIGHT :: (A) smooth : soft (B) rough : smooth (C) tall : high (D) rough : tough (E) often : frequent

2.

NEEDLE : SEW :: (A) pencil : paper (B) radio : electricity (C) picture : color (D) towel : dry (E) book : cover

3.

FAST : HUNGER :: (A) camp : fire (B) jog : fatigue (C) sing : voice (D) tight : choke (E) play : win

4.

APPETIZER : DESSERT :: (A) hat : shoes (B) right : left (C) rug : carpet (D) introduction : epilogue (E) step : stair

5.

CANDLE : WICK :: (A) hammer : nail (B) light : bulb (C) oven : fire (D) bicycle : ride (E) drill : bit

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6.

BARREL : VIAL :: (A) honey : milk (B) shovel : hoe (C) key : door (D) pit : peach (E) volume : monograph

7.

ARROW : ROCKET :: (A) bow : gun (B) Ford : car (C) sand : glass (D) tent : camp (E) stagecoach : jet

8.

IGLOO : ESKIMO :: (A) house : man (B) tree : bark (C) cabin : hunter (D) tent : camping (E) tepee : Indian

9.

CHURCH : STATE :: (A) confusion : adaptation (B) priest : officer (C) time : minutes (D) team : player (E) breeze : sunshine

10.

APPLE : PIE :: (A) dentist : teeth (B) milk : cake (C) sin : evil (D) flour : bread (E) eat : salad

11.

PRESIDENT : NATION :: (A) snake : bite (B) frog : swim (C) mayor : city (D) student : college (E) land : human

12.

TEPID : BOILING :: (A) car : tire (B) fast : long (C) charcoal : flame (D) cool : freezing (E) light : dark

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ANALOGY REVIEW

GRE CAT Success

13.

CATERPILLAR : BUTTERFLY :: (A) star : flag (B) face : head (C) mountain : field (D) tadpole : frog (E) animal : fur

14.

NORTH AMERICA : CANADA :: (A) land : lake (B) valley : hill (C) wolf : dog (D) Europe : Ireland (E) New York : Rochester

15.

RAIN : DROP :: (A) milk : bucket (B) ice : skid (C) water : icicle (D) snow : flake (E) pudding : bowl

16.

ANTWERP : BELGIUM :: (A) Russia : France (B) Italy : Sicily (C) Lima : Peru (D) New York : New York City (E) England : Ireland

17.

BLADE : SKATE :: (A) car : gas (B) chair : leg (C) table : knife (D) wheel : bike (E) bowl : soup

18.

PEDIATRICS : GERIATRICS :: (A) sociology : anthropology (B) medicine : law (C) history : biology (D) obstetrics : thanatology (E) geology : chemistry

19.

BOOKKEEPER : ACCOUNTANT :: (A) player : coach (B) farmer : cowboy (C) senator : congressman (D) typist : secretary (E) janitor : engineer

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UNIT 3

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20.

LAWYER : JUDGE :: (A) message : messenger (B) capitalist : interest (C) mother : daughter (D) reporter : editor (E) lieutenant : army

21.

RAISIN : GRAPE :: (A) bread : toast (B) orange : kumquat (C) prune : plum (D) apple : berry (E) wash : hang

22.

STARVATION : FAMINE :: (A) energy : resistance (B) ship : harbor (C) dissect : join (D) disease : epidemic (E) surgeon : operation

23.

COURT : JUSTICE :: (A) camp : counselor (B) hospital : health (C) school : books (D) palace : royal (E) airport : hangar

24.

RUNG : LADDER :: (A) buckle : belt (B) logs : cabin (C) step : stairway (D) bricks : concrete (E) limbs : body

25.

AFTERNOON : DUSK :: (A) spring : fall (B) 2 p.m. : 5 p.m. (C) Sunday : Friday (D) light : dark (E) sun : moon

26.

VIBRATION : SOUND :: (A) ring : bell (B) staff : stick (C) courage : strength (D) dust : chalk (E) gravity : pull

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ANALOGY REVIEW

GRE CAT Success

27.

HYDROGEN : GAS :: (A) oxygen : breathe (B) gold : jewelry (C) mercury : liquid (D) plant : grow (E) steel : solid

28.

BOMB : TARGET :: (A) aim : miss (B) train : station (C) brow : forehead (D) ball : throw (E) vest : suit

29.

ATOM : MOLECULE :: (A) toothpick : tooth (B) ocean : lake (C) shrub : tree (D) raspberry : apple (E) star : galaxy

30.

SPOTS : MEASLES :: (A) clamp : hold (B) fire : flames (C) dollar : penny (D) trail : path (E) swellings : mumps

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UNIT 3

Quick Score Answers 1. 2. 3. 4. 5.

B D B D E

6. 7. 8. 9. 10.

E E E B D

11. 12. 13. 14. 15.

C D D D D

16. 17. 18. 19. 20.

C D D D D

21. 22. 23. 24. 25.

C D B C B

26. 27. 28. 29. 30.

E C B E E

ANSWERS AND EXPLANATIONS

ANALOGY REVIEW 1. The correct answer is (B). The words are antonyms. Only choice (B) contains antonyms; therefore, the best answer is choice (B). Choice (A) is incorrect; smooth and soft are similar and not opposites. Choice (C) is incorrect for the same reason; tall and high are similar. Choice (D) is incorrect; rough and tough are rhyming words and slightly similar, and therefore incorrect. Choice (E) is incorrect; often and frequent are synonyms. 2. The correct answer is (D). The relationship is Object to Its Function. A needle is used by a person to sew; a towel is used by a person to dry, therefore choice (D) is correct. Choice (A) is incorrect; pencil and paper do have a relationship; however, if paper is a noun, the relationship is not Object to Its Function. If it is a verb, there is no relationship because a pencil is not used to paper a wall. Choice (B) is incorrect; a radio uses electricity. Choice (C) is incorrect; a picture might have color. Choice (E) is incorrect; a book has a cover. In choices (A), (B), (C), and (E), the relationship is not that of Object to Its Function. 3. The correct answer is (B). Fasting causes hunger; jogging causes fatigue. The relationship is Cause to Effect. Choice (A) is incorrect; a camp does not cause a fire. The camp may have a fire as a part of its program, but it does not cause the fire. Choice (C) is incorrect; sing does not cause voice. A voice is needed to sing, but will not necessarily cause singing. Choice (D) is incorrect; something tight may or may not cause one to choke. Choice (E) is incorrect; playing does not cause winning. One may wish that it did, but it does not. Only choice (B) has the proper Cause to Effect relationship. 4. The correct answer is (D). Both sets list a beginning and an end—an appetizer begins the meal, a dessert ends the meal; an introduction begins a speech, an epilogue ends the speech. Choice (A) is incorrect; while a hat is on the head and shoes are on the feet, this is not a relationship of beginning and ending. Choices (B), (C), and (E) are incorrect. The relationship of right and left is Opposites; the same is true of rug and carpet, and of step and stair. 5. The correct answer is (E). A candle cannot serve its purpose without a wick; similarly, a drill cannot serve its purpose without a bit. The relationship is Part to Whole. Choice (A) is incorrect; a hammer pounds a nail, but the nail is not part of the hammer. Choice (B) is incorrect; a bulb may give light but light is not part of the bulb. Choice (C) is incorrect; there may be fire in an oven, but fire is not part of the oven. Choice (D) is incorrect; a bicycle is used to ride but requires a third party to accomplish it.

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ANALOGY REVIEW 6. The correct answer is (E). A barrel is a large container; a vial is a small container. A volume is a large book; a monograph is a small one. The relationship is Opposites. Choice (A) is incorrect; honey and milk are natural products but have no relationship as opposites. Choice (B) is incorrect; shovel and hoe are both objects used for basically the same purpose and are not opposites. Choice (C) is incorrect; a key is used to open a door; it is not the opposite of a door. Choice (D) is incorrect; a pit is a part of a peach and therefore not an opposite. 7. The correct answer is (E). A rocket is a modern projectile; an arrow is an archaic one. A jet is a modern means of transportation; a stagecoach is an archaic one. The relationship is that of old and new, or Opposites. Choice (A) is incorrect; the bow and the gun are both weapons but do not have the old/new relationship. Choice (B) is incorrect; a Ford is a make of car, and without the proper relationship. Choice (C) is incorrect; sand is used to make glass, but it is not the opposite of glass. Choice (D) is incorrect; a tent is used in camping, which does not have the old/new relationship. Only choice (E) has the same old/new relationship. 8. The correct answer is (E). An igloo is the precise name for an Eskimo house, just as a tepee is the precise name for an Indian house; the relationship is Synonyms. While choice (A) is similar, it is not precise; it is too general; mankind does live in a house, but the meaning is not precise. Choice (B) is incorrect; a tree is covered with bark, but the two are not synonymous. Choice (C) is incorrect; the relationship is imprecise; a hunter might use a cabin, but these words are not synonyms. Choice (D) is incorrect; a tent is used for camping, but it is not a synonym for camping. 9. The correct answer is (B). A church and the state are organizations in a society. The head of a church is a priest; the head of a state is an officer; the relationship is Parts of a Whole. Choice (A) is incorrect; confusion might cause adaptation; but it is not a part of adaptation. Choice (C) is incorrect; time is measured in minutes but the minutes are parts of an hour, not time. Choice (D) is incorrect; a team is made up of players, but the two are not part of the same whole. Choice (E) is incorrect; breeze and sunshine are both attributes of weather, but the relationship is not as specific as choice (B). 10. The correct answer is (D). Apples are made into a pie; flour is made into bread. The relationship is Object to Outcome. Choice (A) is incorrect; a dentist works on teeth but does not make them. Choice (B) is incorrect; milk is sometimes used in making a cake, but the function is not as specific. Choice (C) is incorrect; sin is not made into evil; sin and evil are synonymous. Choice (E) is incorrect; one might eat a salad, but the relationship is different, because it has a verb and a noun, rather than two nouns. 11. The correct answer is (C). The head of a nation is a president; the head of a city is a mayor; the relationship is Part to Whole. Choice (A) is incorrect; a snake may bite; however, the snake is not head of the bite. Choice (B) is incorrect; a frog may swim but is not head of the swim. Choice (D) is incorrect; a student may attend college, but a student is not the head of the college. Choice (E) is incorrect; a human may use the land, but is not the head of it. 12. The correct answer is (D). Tepid is moderate: boiling is extreme. Similarly, cool is moderate; freezing is extreme. The words are Opposites. Choice (A) is incorrect; a car has a tire, but the two words are not opposites. Choice (B) is incorrect; fast and long do not have opposite connotations. Choice (C) is incorrect; charcoal is ignited by a flame, but these words are not opposites. Choice (E) is incorrect; while light and dark are opposites, there is not the specific relationship of temperature, as in choice (D).

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UNIT 3 13. The correct answer is (D). A caterpillar becomes a butterfly; a tadpole becomes a frog. The relationship is Object to Outcome. Choice (A) is incorrect; the flag may have a star, but a star will not become a flag. Choice (B) is incorrect; a face will not become a head; the face is part of the head. Choice (C) is incorrect; a mountain may contain a field, but it will not become a field. Choice (E) is incorrect; an animal will not become fur but is covered in fur. Only choice (D) has the correct relationship. 14. The correct answer is (D). North America is the continent where the country of Canada is found. Europe is the continent where the country of Ireland is found. The relationship is Part to Whole. Choice (A) is incorrect; a land is not necessarily where a lake is found; the relationship is nonspecific. Choice (B) is incorrect; a valley is not necessarily where a hill is found. Choice (C) is incorrect; a wolf is not where a dog is found: Choice (E) is incorrect; while New York is where Rochester is found, the specific relationship is continent to country, not state to city. 15. The correct answer is (D). Rain falls in drops; snow falls in flakes. The relationship is Whole to Part. Choice (A) is incorrect; milk does not come in a bucket. Choice (B) is incorrect; ice may relate to skid, but ice is not a part of a skid. Choice (C) is incorrect; water may become an icicle, but an icicle is not a part of water. The relationship is imprecise. Choice (E) is incorrect; pudding may be served in a bowl, but the relationship is not Whole to Part. Only choice (D) has the appropriate relationship. 16. The correct answer is (C). Antwerp is a city in Belgium; Lima is a city in Peru. Both Belgium and Peru are countries. Choice (A) is incorrect; Russia and France are both countries; however, there is not the relationship of City to Country. Choice (B) is incorrect; Italy and Sicily are both countries. Choice (D) is incorrect; New York is a state, and New York City is a city, but the relationship we seek is city to country. Choice (E) is incorrect; England is a country, as is Ireland. Only choice (C) has the proper relationship of city to country. 17. The correct answer is (D). A blade on a skate touches the ground; a wheel on a bike touches the ground. Choice (A) is incorrect; a car uses gas, but the relationship is not precise. Choice (B) is incorrect; a chair has a leg on which to stand, but the relationship is Object to Part, rather than Part to Object. Choice (C) is incorrect; a table may have a knife on it, but the knife is not a part of the table. Choice (E) is incorrect; a bowl may contain soup, but soup is not part of a bowl. 18. The correct answer is (D). Pediatrics deals with children; geriatrics deals with the aged. Similarly, obstetrics deals with birth, and thanatology deals with death. The relationship is Opposites. Choices (A), (B), (C), and (E) are incorrect; sociology and anthropology are both the study of man; medicine and law are both courses of professional study; history and biology are both courses of study; geology and chemistry are both courses of study in science; however, none has the same relationship as choice (D). 19. The correct answer is (D). A bookkeeper is a novice accountant as a typist is a novice secretary. The relationship is Cause to Effect. Choice (A) is incorrect; a player needs a coach and may one day become a coach, but a coach does not have to be a player. Choice (B) is incorrect; a farmer is not a novice cowboy. Choice (C) is incorrect; a senator is not a novice congressman. Choice (E) is not correct; a janitor is not a novice engineer.

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ANALOGY REVIEW 20. The correct answer is (D). A lawyer may become a judge; a reporter may become an editor. The relationship is the ultimate of accomplishments for the professional. Choice (A) is incorrect; a messenger carries a message; but there is no relationship. Choice (B) is incorrect; a capitalist may collect interest; but there is not the same relationship. Choice (C) is incorrect; a mother will not necessarily become a daughter; the reverse may be true. She is her mother’s daughter. The relationship is not the same. Choice (E) is incorrect; a lieutenant does not become an army; a lieutenant is part of an army. 21. The correct answer is (C). A raisin is a dried grape; a prune is a dried plum. The relationship is Object to Outcome. Choice (A) is incorrect; while bread may become toast, it does not represent the process of drying. Choice (B) is incorrect; an orange does not dry into a kumquat. Choice (D) is incorrect; an apple does not become a berry; there is no relationship. Choice (E) is incorrect; one may hang the wash to dry, but the relationship is not the same. 22. The correct answer is (D). Starvation is associated with famine as disease is associated with an epidemic. The relationship is Object to Outcome. Choice (A) is incorrect; energy may be associated with resistance but is not the outcome. Choice (B) is incorrect; a ship will enter a harbor but will not result in the harbor. Choice (C) is incorrect; dissect means to take apart; which is the opposite of join. The relationship is Opposites. Choice (E) is incorrect; a surgeon will perform an operation but will not cause it. The surgeon is not an object. 23. The correct answer is (B). One seeks justice in court and health in a hospital. The relationship is Cause to Effect. Choice (A) is incorrect; a camp will not seek a counselor in the same capacity of cause/effect. Choice (C) is incorrect; books are used in a school, but there is not the cause-and-effect relationship. Choice (D) is incorrect; a palace is royal, but there is no seeking of effect. Choice (E) is incorrect; an airport may contain a hangar, but it is not Cause to Effect. 24. The correct answer is (C). A rung is part of a ladder as a step is part of a stairway. They are both used to go up and down. Choice (A) is incorrect; a belt may contain a buckle, but it is not used in the same way. Choice (B) is incorrect; logs make a cabin, but the comparison is not used similarly. Choice (D) is incorrect; bricks and concrete are used in construction, but without the precise relationship. Choice (E) is incorrect; limbs are part of a body, but the usage is different. 25. The correct answer is (B). Afternoon is to dusk as 2 p.m. is to 5 p.m. 2 p.m. is usually afternoon; 5 p.m. can be dusk. The relationship is synonymous. Choice (A) is incorrect; spring and fall are seasons that occur with different manifestations. Choice (C) is incorrect; Sunday and Friday are days of the week, but without relationship. Choice (D) is incorrect; light and dark are opposites. Choice (E) is incorrect; sun and moon are both planets, but without the relationship of afternoon and dusk. 26. The correct answer is (E). A vibration causes sound; gravity causes a pull. The relationship is Object to Effect. Choice (A) appears to be similar; however, a ring does not cause a bell. Choice (B) is incorrect; a staff is a stick; the relationship is synonymous. Choice (C) is incorrect; courage and strength are similar, but not with the relationship of vibration and sound. Choice (D) is incorrect; dust comes from chalk but does not cause chalk. Choice (E) has the same relationship.

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UNIT 3 27. The correct answer is (C). Hydrogen is a gas; mercury is a liquid. Both hydrogen and mercury are elements. The relationship is Part to Whole. Choice (A) is incorrect; we breathe oxygen, but the relationship is not the same. Choice (B) is incorrect; we use gold to make jewelry but the relationship is not the same. Choice (D) is incorrect; a plant will grow, but there is not a Part-to-Whole relationship. Choice (E) is incorrect; steel is solid; but the relationship is not Part to Whole. 28. The correct answer is (B). A bomb travels to a target as a train travels to a station. The relationship is Object to Outcome. Choice (A) is incorrect; aim and miss have a negative relationship. Choice (C) is incorrect; brow and forehead are synonyms. Choice (D) is incorrect; while we may throw a ball, the destination is not mentioned. Choice (E) is incorrect; a vest is part of a suit. Only choice (B) has the proper relationship. 29. The correct answer is (E). An atom is a part of a molecule; a star is part of a galaxy. The relationship is Part to Whole. Choice (A) is incorrect; a toothpick is not part of a tooth. Choice (B) is incorrect; an ocean is not a part of a lake—both are bodies of water. Choice (C) is incorrect; a shrub is not a part of a tree. Choice (D) is incorrect; a raspberry is not a part of an apple. 30. The correct answer is (E). One of the symptoms of measles is spots. One of the symptoms of mumps is swellings. The relationship is Effect (the symptom) to Cause (the disease). Choice (A) is incorrect; a clamp is not a symptom of holding; a clamp will hold. Choice (B) is incorrect; fire and flames are synonyms. Choice (C) is incorrect; a dollar is not a symptom of a penny. Choice (D) is incorrect; trail and path are synonyms.

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Unit 4 ANTONYM REVIEW Directions: Each question below consists of a word printed in capital letters, followed by five lettered words or phrases. Choose the lettered word or phrase that is most nearly opposite in meaning to the word in capital letters. Since some of the questions require you to distinguish fine shades of meaning, be sure to consider all the choices before deciding which one is best. 1.

INTREPID (A) scurrilous (B) pusillanimous (C) propitious (D) mettlesome (E) militant

2.

EFFRONTERY (A) timidity (B) palpable (C) raillery (D) libel (E) forensic

3.

TURBULENT (A) quiescent (B) cursory (C) extol (D) gyrate (E) imbibe

4.

INEXORABLE (A) surreptitious (B) tractable (C) jaded (D) iconoclast (E) garish

5.

PEREMPTORY (A) glaucous (B) docile (C) extricate (D) panegyric (E) mnemonics 77

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6.

SAGACIOUS (A) trepidation (B) perspicuity (C) frugal (D) garish (E) ignorant

7.

TRUNCATE (A) enlarge (B) extrude (C) intrepid (D) pique (E) vacillate

8.

UNCOUTH (A) urbane (B) travail (C) sentient (D) prevaricate (E) maladroit

9.

ZEAL (A) flail (B) impute (C) ignoble (D) affable (E) indifference

10.

EMPYREAN (A) amenity (B) corpulent (C) exonerate (D) hellish (E) indolent

11.

JUDICIOUS (A) incongruous (B) poignant (C) imprudent (D) volition (E) syncopate

12.

VOCIFERATE (A) turgid (B) listen (C) resurgent (D) rapacity (E) vilify

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ANTONYM REVIEW

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13.

ABJURE (A) venerate (B) maintain (C) transpire (D) obdurate (E) lacerate

14.

RECALCITRANT (A) submissive (B) paroxysm (C) cryptic (D) exhort (E) divert

15.

OBDURATION (A) exogenous (B) approbation (C) decry (D) covetous (E) deference

16.

ENCOMIUM (A) censure (B) invoke (C) sequence (D) coherence (E) paradox

17.

FLUCTUATE (A) magnate (B) canter (C) inflate (D) spin (E) stabilize

18.

MITIGATE (A) lie (B) correct (C) increase (D) remark (E) integrate

19.

ANOMALOUS (A) audacious (B) congruous (C) obsolete (D) ominous (E) chronicle

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20.

TIMOROUS (A) daring (B) rigorous (C) perceptive (D) frugal (E) unctuous

21.

LACERATE (A) mend (B) tolerate (C) profligate (D) accept (E) masticate

22.

CIRCUMSPECT (A) negligent (B) fortuitous (C) delude (D) repressive (E) extrinsic

23.

FIASCO (A) production (B) gamut (C) analysis (D) success (E) allegory

24.

ETIOLATE (A) relegate (B) facilitate (C) clean (D) stain (E) discuss

25.

HERETIC (A) exorbitant (B) verbal (C) orthodox (D) clerical (E) stoic

26.

PREDILECTION (A) seclusion (B) limpid (C) repulsion (D) anachronism (E) gibe

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27.

LACONIC (A) cogent (B) voluble (C) prodigal (D) dulcet (E) acme

28.

MENDACIOUS (A) honest (B) adroit (C) theological (D) vituperative (E) harsh

29.

ANTEDILUVIAN (A) foible (B) modern (C) affable (D) pragmatic (E) foment

30.

MOTLEY (A) vermilion (B) malaise (C) aphorism (D) fecund (E) homogeneous

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UNIT 4

Quick Score Answers 1. 2. 3. 4. 5.

B A A B B

6. 7. 8. 9. 10.

E A A E D

11. 12. 13. 14. 15.

C B B A E

16. 17. 18. 19. 20.

A E C B A

21. 22. 23. 24. 25.

A A D D C

26. 27. 28. 29. 30.

C B A B E

ANSWERS AND EXPLANATIONS

ANTONYM REVIEW 1. The correct answer is (B). Intrepid means valorous and heroic; pusillanimous means mean-spirited or cowardly, and is therefore the opposite. Choice (A) is incorrect; while close in meaning, scurrilous means abusive language. Choice (C) is incorrect; propitious, meaning favorable circumstances, is close in meaning to intrepid; therefore, it is not an opposite. The same is true for choices (D) and (E), both of which mean helpful or brave. 2. The correct answer is (A). Effrontery means audacity, boldness; timidity means shy or awkward. The two words are opposites, and therefore antonyms. Choice (B) is incorrect because it means touchable and has no relationship to effrontery. Choice (C) is incorrect; good-natured has only a vague relationship with effrontery. Choice (D) is incorrect because it means a false statement, which is not the opposite of effrontery. Choice (E) means used in a court of law and has no relationship with effrontery. 3. The correct answer is (A). Turbulent means disorderly or unruly; quiescent means quiet, still, resting, or inactive, the opposite of turbulent. Choice (B) is incorrect; cursory means scant, which is not the opposite of turbulent. Choice (C) is incorrect; meaning praise. Choice (D) means to revolve, which has only a scant relationship to turbulent. Choice (E) is incorrect; meaning to drink, there is no relationship. Choice (A) is closest to being an exact opposite. 4. The correct answer is (B). Inexorable means uncompromising or rigid; tractable means yielding or docile. Choice (A) is incorrect; surreptitious means stealthy, which has no relationship to inexorable. Choice (C) means to become dull from overwork or overuse, which is not the opposite of inexorable. Choice (D) is incorrect; an iconoclast is one who destroys religious symbols. Choice (E) is incorrect; garish means gaudy. 5. The correct answer is (B). Peremptory means authoritative or positive; docile means complacent or changeable. Choice (A) is incorrect; glaucous means pale or bluish gray, which has no relationship to peremptory. Choice (C) means release, and has no relationship to peremptory. Choice (D), which means elaborate praise, while seemingly a bit of an opposite to peremptory, actually has no relationship. Choice (E) is incorrect; mnemonics means a memory aid, which has nothing to do with peremptory.

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ANTONYM REVIEW 6. The correct answer is choice (E). Sagacious means rational or knowledgeable; ignorant means unknowing. The two are opposites, and therefore antonyms. Choice (A) is incorrect; trepidation means dread and is not an opposite. Choice (B), perspicuity, means lucidity, which is similar to sagacious. Choice (C) means miserly, which has no relationship. Choice (D), garish, means gaudy, which, again, has no relationship. 7. The correct answer is (A). Truncate means amputate or shorten; enlarge means to grow or lengthen. Choice (B) is incorrect; extrude means to push or throw out, which, while similar to enlarge, is not as close to being the opposite of truncate as is enlarge. Choice (C) means valorous and heroic, which has no relationship to truncate. Choice (D), pique, means to irk or annoy, which has no relationship. Choice (E), vacillate, is incorrect because it means to switch back and forth; this word has no connection with truncate. 8. The correct answer is (A). Uncouth means barbarous or crude; urbane means proper and sophisticated. Choice (B) is incorrect; travail means toil and has no relationship at all. Choice (C), sentient, is incorrect because it means conscious, and has no relationship. Choice (D), prevaricate, mean falsehood, and while prevaricate may be uncouth, it is not the opposite. Choice (E) is incorrect; maladroit means inept and is close in meaning to uncouth and therefore incorrect. 9. The correct answer is (E). Zeal means enthusiasm or keenness; indifference means cool or without interest. Choice (A) is incorrect; flail means to beat, which might be done with zeal, but is not the opposite. Choice (B), impute, is incorrect, meaning to attribute, which is not an opposite. Choice (C) is incorrect; ignoble means common or ordinary, and is not related to zeal at all. Choice (D) means cordial or friendly. Only choice (E) is opposite. 10. The correct answer is (D). Empyrean means heavenly or celestial; hellish means acting with wickedness. Choice (A) is incorrect; amenity means an agreeable or pleasing manner, which is similar in meaning to empyrean and therefore not an antonym. Choice (B) means obese, which has no relationship to empyrean. Choice (C) means to free from blame, which is not an antonym. Choice (E) means disinclined or lazy, which has no relationship to empyrean. Only choice (D) is the opposite. 11. The correct answer is (C). Judicious means discreet or sensible; imprudent means without care or thinking. Choice (A) is incorrect; incongruous means incompatible, which is slightly the opposite, but not the exact opposite. Choice (B) is incorrect; poignant means sharp, which is not the opposite of sensible. Choice (D) means conscious choice, and is not the opposite of judicious. Choice (E) is a term used in music, which has no relationship to judicious. 12. The correct answer is (B). Vociferate means to bellow or shout; listen means to pay attention or be silent. Choice (A) is incorrect; turgid means ornate language, which might describe one who is vociferating, but it is not the opposite. Choice (C) is incorrect; resurgent means to return or come back, and has no relationship. Choice (D) is incorrect; it means greedy, and is not the opposite of vociferate. Choice (E) is incorrect; vilify means to slander, which might be done in a vociferous way, but is not the opposite. Only choice (B) is correct.

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UNIT 4 13. The correct answer is (B). Abjure means to forswear or repudiate; maintain means to keep or hold fast, which is opposite in meaning. Choice (A) is incorrect; venerate means to adore, which has no relationship. Choice (C), transpire, means to become known or occur, which has a similarity as an opposite, but it is not directly opposite of abjure. Choice (D), obdurate, means stubborn, which is not the best choice. Choice (E) means to tear out roughly, which has no relationship. 14. The correct answer is (A). Recalcitrant means obstinate or rebellious; submissive means to give in and be obedient. Choice (B) means a sudden outburst, which recalcitrant might describe, but paroxysm is not the opposite. Choice (C) means secret or puzzling, which has no relationship. Choice (D) means to appeal urgently, which might be done in a recalcitrant manner, but it is not the opposite. Choice (E) means to turn aside, which might be the result of recalcitrant behavior, but is not the opposite of it. 15. The correct answer is (E). Obduration means condemnation or reproof; deference means courtesy and kindness. Choice (A), exogenous, is incorrect; the meaning is out-of-body origin, which has nothing to do with obduration. Choice (B), approbation, is incorrect; the meaning is warm praise, which is similar to an opposite, but not the most opposite. Choice (C) is incorrect; decry means to put down and disapprove, which is similar to obduration and therefore not opposite. Choice (D) is incorrect; covetous means to want what others have and has no relationship to obduration. Only choice (E) is correct. 16. The correct answer is (A). Encomium means adulation or praise; censure means criticism or castigation. Choice (B) is incorrect; invoke means to ask or pray for, and is not the opposite. Choice (C) is incorrect; sequence means logical order, and has no relationship with encomium. Choice (D) is incorrect; coherence means the quality of being understandable; an encomium might be delivered in a coherent manner; the meanings are not opposite. Choice (E) means seemingly contradictory but actually true, which is not the opposite of encomium. 17. The correct answer is (E). Fluctuate means undulate or move; stabilize means to make steady or to maintain a given level. Choice (A) is incorrect; magnate means a powerful person; while such a person might fluctuate, the meaning is not opposite. Choice (B) is incorrect; canter is a gait of a horse, which might occur in a fluctuating manner, but is not the opposite. Choice (C) means to enlarge, which is not the opposite of fluctuate. Choice (D) is incorrect; spin means to turn, which might be done in a fluctuating manner. Only choice (E) is the direct opposite. 18. The correct answer is (C). Mitigate means to lessen or ameliorate; increase means to grow. Choices (A), (B), and (D) are incorrect; lie, choice (A), means a falsehood, which might be spoken to mitigate a problem; correct (B) means proper and provable, which might mitigate a situation; remark choice (D) is a statement that might mitigate something; however, none is an opposite. Choice (E) is incorrect; integrate means to bring together, which is not opposite. 19. The correct answer is (B). Anomalous means abnormal or inconsistent; congruous means appropriate or fitting. Choice (A) is incorrect; audacious means bold or daring, which is slightly similar to anomalous but not the opposite. Choice (C), absolute, means out of date and might be slightly similar. Choice (D), ominous; means threatening, which is definitely not the opposite. Choice (E), chronicle, means a narrative, which has no relationship with anomalous.

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ANTONYM REVIEW 20. The correct answer is (A). Timorous means afraid or faint-hearted; daring means brave. Choice (B) is incorrect; rigorous means harsh and a rigorous act, which might make one timorous, is not the opposite. Choice (C) is incorrect; perceptive means to know quickly, which is not the opposite of timorous. Choice (D) is incorrect; frugal means stingy or miserly, but a frugal person is not the opposite of a timorous one. Choice (E) is incorrect; unctuous means oily or excessively smooth manner. Only choice (A) is the opposite. 21. The correct answer is (A). Lacerate means to rend or sever; mend means to repair or make new. Choice (B) is incorrect; tolerate means to endure, which is similar to mend, but not the opposite of lacerate. Choice (C), profligate, means wildly extravagant, which is not the opposite of lacerate. Choice (D) is incorrect; accept means to receive, which has no relationship with lacerate. Choice (E), masticate, means to chew, which may be done in a lacerating manner, but it is therefore not the opposite. 22. The correct answer is (A). Circumspect means careful or watchful; negligent means careless. Choice (B) is incorrect; fortuitous means by accident or chance, which is similar to negligent but not the most nearly opposite of circumspect. Choice (C) is incorrect; delude means to deceive, which is not the opposite. Choice(D) is incorrect; repressive means keeping under control, which is not the opposite of circumspect. Choice (E), extrinsic, means not essential, which is not the opposite. Only choice (A) is the most nearly opposite. 23. The correct answer is (D). Fiasco means flop or failure; success means victory or achievement. Choice (A) is incorrect; production means a creation, which might become a fiasco, but it is not the most nearly opposite. Choice (B) is incorrect; gamut means the range; a fiasco might cover a gamut but is not the opposite. Choice (C) is incorrect; analysis means separating parts from a whole, which may or may not be a fiasco, but the word is not opposite. Choice (E) is incorrect; allegory is a story with a literal and a symbolic meaning, and it is not the opposite of fiasco. 24. The correct answer is (D). Etiolate means to blanch or whiten; stain means to soil or blacken. Choice (A) is incorrect; relegate means to send or consign. Choice (B) is incorrect; facilitate means to make easier; there is no relationship with etiolate. Choice (C) is incorrect; clean is similar to etiolate. Choice (E) means to talk, which has no relationship. 25. The correct answer is (C). Heretic means schismatic or nonconforming; orthodoxy means conforming to a religious doctrine. Choice (A) is incorrect; exorbitant means exceeding all bounds, which might describe heretic. Choice (B) is incorrect; verbal means oral. Choice (D) is incorrect; clerical means of or relating to the clergy, which might describe the opposite of heretic, but it is not most nearly opposite. Choice (E) is incorrect; stoic means unaffected by joy or pleasure, which might describe a heretic, but is certainly not opposite. Only choice (C) is most directly opposite. 26. The correct answer is (C). Predilection means inclination or liking; repulsion means disinclined or not to one’s liking. Choices (A), (B), (D), and (E) are incorrect; seclusion, choice (A), means isolation; limpid, choice (B), means clearness; anachronism, choice (D), means chronologically out of order; and a gibe, choice (E), is a taunting remark—all of these might be one’s predilection or repulsion, but none of them is an opposite of predilection.

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UNIT 4 27. The correct answer is (B). Laconic means brief or terse; voluble means readily flowing of speech or lengthy conversation. Choice (A) is incorrect; cogent means appealing to the intellect by clear presentation, which might be done in a laconic manner, but it is not the opposite. Choice (C) is incorrect; prodigal means extravagant or wasteful, which is similar to the opposite but not most nearly opposite. Choice (D) is incorrect; dulcet is sweetly pleasing, which might be in a laconic manner if we stretch it, but which is definitely not opposite. Choice (E) is incorrect; acme is the pinnacle or top and has no relationship to laconic. 28. The correct answer is (A). Mendacious means deceitful or dishonest; honest is the opposite. Choice (B) is incorrect; adroit means skillful and adept, which has no relationship to mendacious. Choice (C) is incorrect; theological means relating to religious study, which could be described as honest, but it is not the opposite of mendacious. Choice (D) is incorrect; vituperative means bitter, which could describe a mendacious person, but not an honest one. Choice (E) is incorrect; harsh means coarse or rough, which might describe someone who is mendacious, but is not the opposite. 29. The correct answer is (B). Antediluvian means ancient or hoary; modern means new and fresh. Choice (A) is incorrect; foible means minor weakness, which might be described by either antediluvian or modern, but is not the opposite of antediluvian. Choice (C), affable, means cordial or friendly, which might be similarly descriptive, but is not the opposite. Choice (D), pragmatic, means in a practical manner, which is not directly opposite. Choice (E) is incorrect; foment means incite or promote, which would not be the opposite. 30. The correct answer is (E). Motley means mixed or variegated; homogeneous means same type or without mixture. Choice (A) is incorrect; vermilion is the bright red color that might be made from a motley group of colors and therefore not opposite. Choice (B) is incorrect; malaise is a vague feeling, which is not the opposite. Choice (C) is incorrect; an aphorism is a short saying about life, which would not be the opposite. Choice (D) is incorrect; fecund means productive and is not the opposite.

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R E D A LERT READING COMPREHENSION STRATEGIES The GRE CAT requires that you read various kinds of passages and then show how well you understand them by answering reading comprehension questions. The passages may be selections from a work of literature; persuasive arguments for or against some idea; or passages about the biological or physical sciences, the humanities, or the social sciences. There are basically four types of questions concerning the reading comprehension selections: 1. Comprehension: Questions that test your comprehension require you to understand the material given in the reading selection. Remember, the material contained will NOT assess your prior knowledge of the subject but will assess your ability to understand the material covered by the selection under consideration. 2. Evaluation: Questions that test your ability to evaluate require you to look at the material of the reading selection and determine how reliable or accurate that information is in view of the selection presented. Again, this material will NOT assess your prior knowledge but will present facts and conclusions that you must evaluate for reliability. 3. Application: Questions that test your ability to apply require you to read the material presented and to then apply or use that material in a different situation. Often this involves situations in which a fact is presented and the question deals with the opposite of that fact. This means that you must apply what is presented and draw a conclusion regarding the opposite effect. 4. Incorporation of new information: Questions that test your ability to deal with new information presented in the light of that which is familiar require you to reevaluate the conclusions you have drawn in light of new facts or observations. Often this type of selection will present a “tried and true” situation about which you may or may not have prior knowledge. After this presentation, the selection will introduce new findings or additional pertinent information that may change—dramatically or slightly—the outcomes or conclusions. The following guidelines should help you answer Reading Comprehension questions: 1. You will see only the passage and the first question, so old paper-and-pencil techniques of reading all of the questions first don’t work. Thus, the first task is to read the first few sentences of the passage to get an idea of the topic. 2. Read the first and final sentences of each paragraph, and skip through the rest of it quickly to see what else you might notice.

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3. Read the final few sentences of the passage. 4. Read all the answer choices. Base your answers solely on material that is actually found in the passage. Do not choose an answer simply because it is a true fact in your own experience or because you hold an opinion about the answer. 5. Do not be misled by answers that are partially correct, or whose scope is too narrow or too broad for the content of the passage. Watch, too, for answers that contradict or distort the facts in the passage and avoid these. 6. Be very careful not to choose an answer that concerns a side effect, precursor, or outcome of the material contained in the selection. Remember, your answer choices must be based on the material you have read.

EXPECT QUESTIONS

ON THE

FOLLOWING

The Main Idea of the Passage Questions concerning the main idea of a reading selection test your ability to read for the theme or main idea that is the SUBJECT of the selection. Some main ideas impart a moral or teaching; others deal with cause and effect; still others deal with the presentation of a situation.

Specific Details in the Passage Questions about specific details are simply a matter of reading for meaning. Often, the answers are verbatim in the material presented. Reading for detail requires that you focus on what you are reading.

Inferences or Conclusions Based on the Passage Remember, “inference” means deriving logical conclusions from premises known or assumed to be true, or the act of reasoning from factual knowledge or evidence. With that in mind, these questions require you to understand the logic of the author’s statements and decide what is or is not reasonable.

The Meaning of Words in the Passage Remember, this material does not require previous knowledge on your part; therefore, there may well be words included with which you are not familiar and may have never seen or used in your real life. Often, these words are placed in the selection deliberately in order to test your ability to evaluate them within a specific context. This means that when the question deals with a word whose meaning you do not readily know, you must determine its meaning by looking at the sentence, or perhaps the paragraph, in which it is used. Evaluate the material presented around the word, and often you can determine its meaning. Another way is to look at the root of the word by eliminating the affixes and looking at the base.

The Mood or Tone of the Passage The mood or tone of the passage, or other evidence of the writer’s attitude toward the subject is often quite simply the author’s purpose for writing.

Specific Techniques Used by the Writer of the Passage Many of the passages you will read will have the writer presenting an argument on a given subject. Argumentation is a technique. At times the author may relate an anecdotal account of an actual

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READING COMPREHENSION STRATEGIES

happening; this is narrative, or telling a story. Much of the material presented for your comprehension is expository in technique. Exposition is explaining and elaborating. Often this technique is a “how to” organization. Another of the more popular techniques is description, which is a technique used frequently. This simply presents material in which the writer describes a situation, scene, or emotion.

The Writer’s Logic, Organization, or Message Questions on the writer’s techniques require you to determine the author’s purpose, argument, or reasoning. Now, let us look at some examples of the questions you may encounter. First, here is a brief article. Read this using the techniques just presented. NOTE: Articles on the actual test will be longer. Sample Line

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10

The United States and other Western industrial countries may face a period of “jobless growth” in the 1980s, even if President Carter and other nations’ leaders succeed in their declared aim of expanding business investment and ending the world recession. This is the warning that an increasing number of economists, officials, and business people are giving Western governments as they prepare for the Bonn economic summit meeting, to be held this month. It reflects fears that any upturn in business spending, stimulated by the summit meeting, will merely accelerate the present trend toward replacing human workers with sophisticated new machinery instead of creating additional jobs. “The evidence that we have is suggesting increasingly that the employmentdisplacing effects of automation, anticipated for the 1950s, are now beginning to arrive on a serious scale in the 1970s,” concludes an unpublished report by the Organization for Economic Cooperation and Development, which monitors the economic progress of Western nations.

1.

An ironic economic prediction contained in this article is that (A) unpublished statements can have a great effect. (B) an upturn in business spending may lead to great unemployment. (C) economic experts frequently know little about their subject. (D) a world recession seems inevitable. (E) economic recovery is a worldwide problem.

2.

It is apparent that statements alluded to in this selection stem from thoughts expressed prior to (A) a meeting of the Common Market nations. (B) a conference of the United Nations. (C) an international economic summit meeting. (D) a disarmament conference. (E) no particular meeting.

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3.

It would seem that the disappointing effects of automation indicated here (A) were impossible to predict. (B) took the economic community by surprise. (C) are largely to be discounted. (D) cannot be avoided. (E) had been anticipated more than twenty years ago.

Let’s Check Your Answers 1. The correct answer is (B). This is where you must use your ability to infer. The irony involved lies, of course, in the fact that an upturn in business—generally looked upon as a favorable development—may actually have an ADVERSE effect in at least one important area of the economy. Choice (A) is incorrect. There is nothing ironic about this fact; in addition, it is not a major point of the article. NOTE: Here you are utilizing the ability to discard ideas, no matter how true you know them to be, because they are NOT a major part of the article. You are also using your ability to discern the main idea. Choice (C) is incorrect. While this does indeed have an ironic tinge, it is NOT indicated as a matter of discussion in the article. NOTE: Again, you are discarding information that is NOT a part of the discussion. Choice (D) is incorrect. First, there is nothing ironic about this; second, the trend indicated by the article is in the opposite direction. Again, you are using your ability to discern the main idea or focus of the piece and to determine or infer as well. Choice (E) is incorrect. This is simply a matter of fact, without any ironic tinge. In addition, it is not a major aspect of the matter under discussion in this particular article. Again, you are using your ability to determine the focus or main idea. 2. The correct answer is (C). The topic sentence of the second paragraph substantiates this statement: “. . . as they prepare for the Bonn economic summit meeting . . .” Here you are drawing information directly from the material presented. You have read carefully and can therefore go directly to the material questioned. Choices (A), (B), (D), and (E) are all incorrect. These answers are presented in an irrelevant manner in order to encourage you to read for detail. While they sound reasonable, there is no correlation at all to what you have read. You must use your ability to focus on what you are reading and thereby be able to differentiate the material presented from that which “sounds good.” 3. The correct answer is (E). The final paragraph indicates that these effects “anticipated for the 1950s” are now beginning to arrive. Here you are inferring or drawing a conclusion, but, as well, you are using your ability to discern the main idea. You are also determining cause and effect and are extending the meaning presented by the material. Choice (A) is incorrect. This is contrary to the point made in the concluding paragraph. Here is an example of material that, while “sounding good,” is an answer that is contrary to the material presented. Choice (B) is incorrect. Again, there is every indication in the article that this was not so. Choice (C) is incorrect. On the contrary, these effects are being taken quite seriously. Here you are using your ability to read for meaning and to determine cause and effect. Choice (D) is incorrect. There is no indication of a conclusion supporting this thought. Remember, you must read carefully enough to discern what material IS supported by the details of the selection. Now, with these ideas in mind, take the following sample test, which incorporates many of the types of information described above. Follow the procedure above.

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Sample The passage below is followed by questions based on its content. Answer the questions following the passage on the basis of what is stated or implied about the planet Mars in the passage. Line

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Mars revolves around the Sun in 687 Earth days, which is equivalent to 23 Earth months. The axis of Mars’s rotation is tipped at a 25-degree angle from the plane of its orbit, nearly the same as the Earth’s tilt, which is about 23 degrees. Because the tilt causes the seasons, we know that Mars goes through a year with four seasons just as the Earth does. From the Earth, we have long watched the effect of the seasons on Mars. In the Martian winter, in a given hemisphere, there is a polar ice cap. As the Martian spring comes to the northern hemisphere, for example, the north polar cap shrinks, and material in the planet’s more temperate zones darkens. The surface of Mars is always mainly reddish, with darker gray areas that, from the Earth, appear blue green. In the spring, the darker regions spread. Half a Martian year later, the same process happens in the southern hemisphere. One possible explanation for these changes is biological: Martian vegetation could be blooming or spreading in the spring. However, there are other explanations. The theory that currently seems most reasonable is that each year during the northern hemisphere springtime, a dust storm starts, with winds that reach velocities as high as hundreds of kilometers per hour. Fine, light-colored dust is blown from slopes, exposing dark areas underneath. If the dust were composed of certain kinds of materials, such as limonite, the reddish color would be explained. It can be inferred that one characteristic of limonite is its (A) (B) (C) (D) (E)

2.

According to the author, seasonal variations on Mars are a direct result of the (A) (B) (C) (D) (E)

3.

reddish color. blue-green color. ability to change colors. ability to support rich vegetation. tendency to concentrate into a hard surface.

proximity of the planet to the Sun. proximity of the planet to the Earth. presence of ice caps at the poles of the planet. tilt of the planet’s rotational axis. length of time required by the planet to revolve around the Sun.

It can be inferred that, as spring arrives in the southern hemisphere of Mars, which of the following is also occurring? (A) The northern polar cap is increasing in size. (B) The axis of rotation is tipping at a greater angle. (C) A dust storm is ending in the southern hemisphere. (D) The material in the northern temperate zones is darkening. (E) Vegetation in the southern temperate zones is decaying.

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Let’s Check Your Answers 1. The correct answer is (A). The final sentence of the article contains this information. 2. The correct answer is (D). This is explained in paragraph 1, the very beginning of the article. 3. The correct answer is (A). This information can be inferred from paragraph 2 where the statement is made that “. . . the north polar cap shrinks . . .” Therefore, in the opposite season, the polar cap would increase.

As you check your answers, be especially aware of the other choices. In some cases, the other choices actually contain words from the article; however, the sequence is different or the inference is incorrect. Following this overview is a section with several sample passages for your review. Read through the passages and answer the questions. Check your answers carefully, and try to analyze those types of questions that give you difficulty. If necessary, reread this overview for clarification.

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Unit 5 READING COMPREHENSION REVIEW Directions: Each passage in this group is followed by questions based on its content. After reading a passage, choose the best answer to each question. Answer all questions following a passage on the basis of what is stated or implied in that passage.

Passage 1 Line

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On April 1, 1865, a party of Union soldiers surprised a lone Confederate scout in a clearing off the White Oak Road, a few miles outside Petersburg, Virginia. The end was coming quickly for the South. Already that day, Major General Philip Sheridan’s cavalry had captured the vital road junction to the west of Five Forks, and the Army of the Potomac was poised to sever the last railroad links to beleaguered Petersburg. Confederate General Robert E. Lee’s men, holding the town, faced the prospect of having to fight the enemy in the open once more after months of static siege warfare. The Union patrol grew confident when its three members spotted a ragged Southerner in the open. The soldiers called for him to surrender. But their self-assurance evaporated when he not only failed to drop his weapon, but swung it up to cover the Yankees. Immediately, they recognized his piece as a Spencer repeating rifle. Surprised and intimidated by his possession of the arm, the Federals meekly threw down their rifle-muskets and raised their hands. Aware of the weapon’s capacity for discharging explosives in quick succession, they surrendered, intimidated by the southerner’s possession of the superior weapon. Private Berry Benson, a member of an elite unit of South Carolina sharpshooters, marched his captives off to the rear. Only he knew that his Spencer’s magazine was empty. He had run out of cartridges for the captured weapon just the day before, expending his meager supply of forty rounds in beating back a Federal attack. Benson’s minor exploit went unnoticed amid the disaster that was engulfing the Confederacy that spring, but it symbolized both the valor and thwarted ingenuity of his nation’s war effort. The story of the confederate Spencer lends a unique footnote to the history of a struggle waged largely by a series of gallant Southern makeshifts. The arm that Private Benson carried toward Appomattox reflected the plight of a resourceful imagination stifled by technological inferiority. Outgunned or not, the Southern troops were still formidable fighters, and inevitably some of the repeaters fell into their hands. As early as October 1863,

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Union Colonel John T. Wilder’s famed “Lightning Brigade” was chagrined to report guerrillas had seized a supply wagon loaded with Spencers and 4,000 rounds of ammunition near Decherd, Alabama. 1.

Which of the following statements is true, according to the selection? (A) The weaponry of the North was superior to that of the South. (B) The weaponry of the South was superior to that of the North. (C) Northern soldiers knew that they were winning and could surrender this small battle without it affecting the outcome of the war. (D) The war did not depend upon one soldier. (E) Southern soldiers were braver than Northern soldiers.

2.

In comparison with the other weapons used in the Civil War, the Spencer rifle was (A) superior in its capacity to withstand excessive use. (B) easy for the soldiers to carry. (C) simple to load and operate. (D) powerful in its discharge of explosives. (E) all of the above.

3.

Evidence that the war was drawing to a close was apparent because (A) more Southern soldiers had been killed in battle than Northern soldiers. (B) the Northern soldiers had moved into and occupied the capital of the South. (C) Northern generals were better trained than Southern generals. (D) Northern soldiers had captured a key intersection and were in command of the railroad. (E) Southern generals had surrendered.

4.

Private Berry Benson was an elite soldier with possession of special knowledge that (A) there were others of his number waiting to ambush the Union patrol. (B) the war was almost over and a Union patrol would not shoot a lone soldier. (C) he would rather be on the Union side than the side of the South. (D) his weapon was empty of ammunition, but he knew its superiority and therefore perpetuated the ruse of its power. (E) the Union soldiers had unloaded guns.

5.

The sentence “The arm that Private Benson carried toward Appomattox reflected the plight of a resourceful imagination stifled by technological inferiority” means that (A) the North was inferior to the South. (B) the South had more ingenuity than the North. (C) the South was reduced to using whatever faculties were available, including confiscated weapons. (D) Spencer rifles were in great demand as technological advances. (E) all of the above.

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6.

On the previous day, Private Benson had been instrumental in (A) fending off a Union attack. (B) stealing a weapon. (C) surrendering to the enemy. (D) saving his unit from further attack. (E) marching into a slaughter.

7.

According to the article, the fact that the Southern soldier was in possession of a Northern weapon was (A) unexplainable, since the South was on the verge of losing the entire war. (B) evidence of a traitor in the midst of the Northern armies. (C) Colonel Wilder’s idea of the gross inferiority of the training of Northern soldiers. (D) not an isolated incident since an entire cache of weapons had been seized with their ammunition. (E) a total surprise to the entire Northern army.

Passage 2 Line

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Black holes, when imagined, are unimaginable. But popular culture got used to them anyway. Black holes are the stars of movies, the heroes of books, the byword for all kinds of bad risks. They are overfamiliar and all but cliché. Luckily, astronomers are not bored yet. In the last few years, they have found increasing evidence of black holes both in our galaxy and outside it. These days, what’s most unbelievable about black holes is that they seem to be real. For certain stars, black holes are the afterlife. Stars the size of our sun spend their lives burning fuel and radiating light, balancing the radiation’s push outward against gravity’s pull inward. As a star runs out of fuel, gravity begins to win. The star condenses and shrinks smaller and smaller until gravity’s pull is again balanced, this time by the force that keeps electrons from crowding too close together. The star, now called a white dwarf, shines for a while, then gradually cools and dims. In stars with masses more than eight times the sun’s, gravity is correspondingly stronger. These stars die with a bang in supernova explosions, which blow away much of the star’s mass. If what remains is less than three solar masses, gravity jams the negatively charged electrons and the positively charged protons together. The opposite charges neutralize each other, and the remnant core, now composed entirely of neutrons, is called a neutron star. It has shrunk to about ten miles in diameter. Matter this compact “beggars description,” says Jeffrey McClintock, an astronomer at the Harvard-Smithsonian Center for astrophysics in Massachusetts. If the Great Lakes were made this compact, they would fit into a bathroom sink. “ ‘Compact’ is the word we like to use,” McClintock adds, “because ‘dense’ doesn’t even cover it.” Neutron stars shine when they’re formed, most brightly in X rays; they also have magnetic fields that can send out crisp pulses of radio waves.

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In stars with masses forty times the sun’s, gravity is strong enough to make the unthinkable happen. These stars also die violently. If the remaining core is bigger than a neutron star—that is, greater than three solar masses—it condenses to nothingness, or near enough to make no difference. Physicists call this point a singularity and tend not to talk about it because they have no clue as to what happens to matter at these densities. “It most likely goes unstable,” says McClintock. “Does it exist anymore? I don’t know. It’s basically out the window. The elementary particles themselves are torn into fragments whose nature is not known and cannot be guessed.” Scientists do know that matter at these densities loses all properties except for mass, rotation, and charge. Says McClintock: “The trees out there, those pearls, the computer—any property they have, once in the black hole—they don’t have anymore.” The physicists’ phrase is “black holes have no hair.” “That means black holes don’t have you-name-it, just-list-it,” says McClintock. “Nothing nothing nothing nothing nothing.” 8.

Which of the following statements bears out an allusion to the relationship between gravity and the black hole? (A) Gravity has no relationship to the black hole at all. (B) The black hole is a byproduct of gravity’s effect on the universe. (C) A star’s loss of gravity allows the black hole to provide an afterlife. (D) Gravity increases the likelihood that black holes will cease to exist. (E) A black hole and gravity are the same thing.

9.

The (A) (B) (C) (D)

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logical connection between stars and the black hole is that for some, the black hole is the remnant or finality of the star. all stars sink into the black hole and disappear for eternity. the black hole is the birthplace of all stars. bright stars exist alongside the black hole, which contributes to their brightness. (E) there is no logical connection. Astronomers and the public have differing views toward black holes. Specifically the (A) astronomers see the phenomenon as a scientific wonder, and the public sees it as a hoax. (B) public is bored with the black hole as a subject of media exposure and literature, while the astronomers are eagerly pursuing the study of the black hole. (C) astronomer sees the black hole as a fluke, while the public believes that there is a signal for the end of the world in it. (D) public wants to visit the black holes, while the astronomers fear what may be lurking there. (E) public wants to know more about black holes, but the astronomer is not willing to share.

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11.

Which of the following statements is true, according to this selection? (A) The core of the black hole is a magnetic mass more than eight times the sun’s. (B) Neutron stars have more of a likelihood of sinking into the black hole than do the supernova stars. (C) Stars larger than our sun usually explode at their death, unlike stars, which are smaller and less bright. (D) The density of the magnetic field determines the destiny of the star. (E) The compact density of the exploded star is easily explainable in terms that a layperson can understand.

12.

The author uses the comparison to the Great Lakes in order to (A) show the differences in sizes of stars. (B) illustrate the effect of explosions in outer space upon matter existing there. (C) allow the novice to see that outer space defines our understanding. (D) illustrate the significance of density of matter when it is reduced to compactness. (E) create the illusion of an earthly equivalent to outer space’s matter.

13.

In comparison with the sun, some stars (A) are larger in size. (B) are smaller in size. (C) have a gravity pull that is significantly larger. (D) have a gravity pull that is significantly smaller. (E) all of the above

14.

The “death” of a star has long been a subject of interest to mankind, and especially to astronomers. One might infer from this article that (A) stars die different deaths, depending upon their size and gravity. (B) all stars die after a period of time and form black holes. (C) many stars explode violently, while others fade slowly. (D) the death of a star is determined by the presence of matter in outer space. (E) outer space is littered with dead stars.

15.

One (A) (B) (C) (D) (E)

should conclude from this reading that the word mass has to do with what is left after a star explodes. the core or center of black hole. the actual physical makeup of the star. the remnants of a dead star. the central ingredient for a black hole.

16.

The (A) (B) (C) (D) (E)

author uses the word singularity as an indicator for the density of the mass within a black hole. degree of compact gravity contained within a supernova. condensation to near nothingness of large masses. afterlife of certain stars forming a black hole. fuel required to assure the continued life of a star.

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17.

The conclusion that one may reach from this selection is that (A) black holes are formed by the collision of stars in outer space. (B) scientists have decided not to investigate black holes because of a lack of information. (C) physicists exploring black holes find unexplainable conditions. (D) black holes change the identity of their contents to nothing. (E) all of the above

Passage 3 Line

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Teachers and librarians need to be aware of the emotional, intellectual, and physical changes that young adults experience, and they need to give serious thought to how they can best accommodate such changes. Growing bodies need movement and exercise, but not just in ways that emphasize competition. Because they are adjusting to their new bodies and a whole host of new intellectual and emotional challenges, teenagers are especially self-conscious and need the reassurance that comes from achieving success and knowing that their accomplishments are admired by others. However, the typical teenage lifestyle is already filled with so much competition that it would be wise to plan activities in which there are more winners than losers; for example, publishing newsletters with many student-written book reviews, displaying student artwork, and sponsoring science fiction, fantasy, or other special-interest book discussion clubs. A variety of small clubs can provide multiple opportunities for leadership, as well as for practice in successful group dynamics. Making friends is extremely important to teenagers, and many shy students need the security of some kind of organization with a supportive adult barely visible in the background. In these activities, it is important to remember that young teens have short attention spans. A variety of activities should be organized so that participants can remain active as long as they want and then go on to something else without feeling guilty and without letting the other participants down. This does not mean that adults must accept irresponsibility. On the contrary, they can help students acquire a sense of commitment by planning for roles that are within their capabilities and their attention spans and by having clearly stated rules. Teenagers need limitations, but they also need the opportunity to help establish what these limits and expectations will be. Adults also need to realize that the goal of most adolescents is to leave childhood behind as they move into adulthood. This has implications for whether libraries treat young adult services as a branch of the children’s or the adults’ department. Few teenagers are going to want to sit on small children’s chairs or compete with nine- and ten-year-olds when they pick books off the shelves. Neither are they going to be attracted to books that use the word children or picture preteens on the covers. Young adults want a wide variety of informational books about aspects of their lives that are new; for example, the physical development of their bodies, the new freedom they have to associate mainly with peers instead of family, and the added responsibilities they feel in deciding what kinds of adult roles they will fit.

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18.

Which of the following statements accurately reflects the view of the author? (A) The reading material available in libraries and schools meets the emotional, intellectual, and physical changes for students. (B) Reading material meets the needs of life changes for teens. (C) Librarians direct students to the material that is appropriate. (D) Teachers are ready to assist students with reading material. (E) Young adults need to have the option for reading material that speaks to the needs of their developing physical and emotional makeup.

19.

It is to be inferred from the passage that (A) authors must be ready to write books for teens. (B) students respond to reading material that uses their lives as a background. (C) literature must be found to speak to the specific needs of changing students. (D) children’s literature is appropriate for adolescents. (E) students are eager to read.

20.

As compared with children’s literature, adolescent literature (A) deals with the emotional needs. (B) concerns itself with intellectual changes. (C) approaches the physical needs. (D) has topics that interest adolescent students. (E) all of the above

21.

This selection makes the point that meeting the needs of adolescent students often requires incorporating (A) some type of organization that incorporates adult support. (B) supervised reading programs. (C) a strict academic environment. (D) adult supervision of social programs. (E) competitive activities.

22.

The (A) (B) (C) (D) (E)

23.

One (A) (B) (C)

particular recommendation of this article is that children and adolescents need to be separated. the needs of adolescents are greater than those of children. libraries and classrooms are constructed for all students. the needs of changing adolescents must be accommodated. services are readily available to meet the needs of all children.

would conclude from this reading that there is a great market for authors of adolescent literature. libraries and classrooms need restructuring. the provision of appropriate reading material for adolescents can be helpful to their maturation. (D) role models are difficult to find for today’s students. (E) activities for students should provide a high profile for writing.

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24.

The main idea of this article is the (A) need for having clubs for students that will help them to compete. (B) reality that student activities can help to provide a non-threatening environment for youth. (C) environment for learning is set by furnishings. (D) implication that teachers and librarians should be aware of ways to assist young adults in coping with life’s changes. (E) students have great needs that are not being met.

25.

According to this selection, the primary desire of young adults is for literature that will (A) let them see themselves in a favorable light. (B) provide them with a pattern to follow. (C) give exciting looks into the future as an adult. (D) allow separation from the family unit. (E) provide information about moving from childhood to adulthood.

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One by one, it seems, American values are being restored. First, there was Liberty in New York, then Freedom in Washington. The statues, that is, both stalwart women worn by a century or more of exposure to the elements. Liberty received her face-lift right on her pedestal in New York Harbor in the mid-1980s, but before Freedom could be cleaned and repaired in 1993, the 19.5-foot, 7.5-ton bronze figure had to be removed from the top of the U.S. Capitol. Enter a large orange helicopter named Bubba. One morning in May 1993, a Sikorsky S-64F rose quickly from the Capitol grounds and hovered above the dome. With the aid of a gyro-coupled flight control system, pilot Max Evans held that spot in the sky while a rigging crew on the dome attached four dangling cables to a framework of bars and nylon straps that supported the statue. With the connections secure, the helicopter’s hoist began lifting, threading the statue through the scaffolding that had been erected around it. The statue swayed slightly, but did not twist on the short trip down. The company that operates Bubba has a suspension system that ensures that the load turns only with the helicopter. The Skycrane has three pilots: two facing forward, one aft. The aft pilot sits in a glass-enclosed booth, looking directly at the hoist. The aircraft is under his control as the cables are attached to the payload and the lift begins. Once all obstacles are cleared, the pilots in front take over. A cheer rose from hundreds of onlookers as the helicopter lowered the statue to the ground and workers bolted it to a metal base constructed on the Capitol’s east Plaza, the statue’s temporary home. The three pilots, wearing tan jumpsuits, stood by their helicopter and received some rare public adulation. “We were in Columbus last week on a much more difficult job,” one of them said with a grin. “We lifted a transformer through a narrow opening and carried it a block, and no one said a thing.”

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This was the first time Freedom had been moved since 1863, when pieces of the statue were first hoisted atop the dome by steam engine and bolted together. It took less than 10 minutes to lift the statue off the dome. Freedom was returned to its original position after the restoration. 26.

The (A) (B) (C)

27.

Which of the following statements accurately portrays the main idea of this selection? (A) The methods for moving a statue are complicated. (B) The Sikorsky S-64F helicopter is an efficient machine. (C) Moving a historical statue is similar to moving a transformer. (D) When dealing with matters of history, all care must be taken. (E) The statues Liberty and Freedom had to be refurbished.

28.

The mention of the crew and its part in the operation is included for the purpose of (A) showing the humanity of the operation. (B) illustrating the involvement of humanity in a technological operation. (C) depicting the statues as “human” as well. (D) showing how insufficient technology is, in that it requires human operation. (E) confusing the reader.

29.

As compared with Liberty, Freedom was (A) a gift from American schoolchildren. (B) restored on the ground. (C) given a long overdue face-lift. (D) restored with technology. (E) made of marble.

30.

The (A) (B) (C) (D) (E)

author implies that statues should be refurbished often. using a helicopter for such a delicate operation is inefficient. today’s technology allows major tasks to be performed in an efficient manner when viewed in a historical perspective. (D) helicopters are made for a variety of tasks. (E) our nation’s historical statues must receive care that will allow them to extend into the next century.

specially equipped helicopter included a special control system. a hoisting system. a special suspension system. a specially trained crew. all of the above.

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Quick Score Answers 1. 2. 3. 4. 5.

A D D D C

6. 7. 8. 9. 10.

A D C A B

11. 12. 13. 14. 15.

C D E A C

16. 17. 18. 19. 20.

C D E B E

21. 22. 23. 24. 25.

A D C D E

26. 27. 28. 29. 30.

C B B B E

ANSWERS AND EXPLANATIONS

PASSAGE 1 1. The correct answer is (A). The Southern soldier’s possession of a Spencer rifle “surprised and intimidated” the Northern soldiers, who threw down their weapons. They recognized the weapon as one of theirs (“the captured weapon;” line 19). The rifle is also described as a superior weapon. Choice (B) is incorrect. It is the opposite of the thrust of the selection. Choice (C) is incorrect. Although the war was coming to an end, the Northern soldiers were prepared to confront the Southern soldier until they saw his weapon. Choice (D) is incorrect. There is no evidence to draw this conclusion. Choice (E) is incorrect. Although the Southern soldier bravely covered the Northern soldiers with his empty weapon, this is not an indicator of the bravery of the South or the lack of bravery of the North. 2. The correct answer is (D). In the second paragraph, we learn that the Union soldiers, “aware of the weapon’s capacity for discharging explosives in quick succession,” surrendered. Choice (A) is incorrect; that the weapon was “sturdy” means that it would withstand use is not mentioned in this selection. Choice (B) is incorrect; there is no evidence in the selection that the Spencer was the premier small weapon. A small weapon is usually a pistol, not a rifle. Choice (C) is incorrect; there is no indication that the weapon was “simple to operate and load.” Thus, choice (E) cannot be a correct choice. 3. The correct answer is (D). The author states that “. . . cavalry had captured the vital road junction to the west . . . and the Army was poised to sever the last railroad link.” Choice (A) is incorrect; there is no direct evidence of this fact in the selection. Choice (B) is incorrect; there is no mention of the capital of the South. Choice (C) is incorrect; the training of generals is not mentioned, nor is there an allusion to it. Choice (E) is incorrect; there is no evidence of the surrender of Southern generals. 4. The correct answer is (D). The author states that “only he [Benson] knew that his Spencer’s magazine was empty.” Choice (A) is incorrect; there is no mention of the remainder of Benson’s South Carolina unit. Choice (B) is incorrect; while the author does opine that the war is about to end, there is no indication but that the Union patrol would take the lone prisoner. Choice (C) is incorrect; Benson’s allegiance to the South is never questioned. Choice (E) is incorrect; though it is not mentioned, the Union guns were obviously loaded. Benson’s gun was not loaded.

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READING COMPREHENSION REVIEW 5. The correct answer is (C). The Southern soldier had a captured weapon, and even though it was not loaded, he used its known superiority to his advantage. Choice (A) is incorrect. There is no evidence to indicate this fact. Choice (B) is incorrect; this one soldier was a lone example and not indicative of the entire Southern armies. Choice (D) is incorrect; while the guns were important, that is not the meaning of the sentence. Choice (E) is incorrect because choices (A), (B), and (D) are not true statements. 6. The correct answer is (A). The author states that Benson had used his weapon to “beat back a Federal attack.” Choice (B) is incorrect; Benson had not stolen the weapon, but had captured it at some previous time, not necessarily the previous day. Choices (C) and (D) are incorrect; there is no evidence that Benson surrendered or saved his unit. Choice (E) is incorrect; there is no evidence at all of a slaughter or of Benson marching into one. 7. The correct answer is (D). This is stated in the last paragraph of the selection. Choice (A) is incorrect; although it was true that the South was about to lose, the possession of the weapon had no connection with that fact. Choice (B) is incorrect; there is no evidence of treason in the article. Choice (C) is incorrect; Colonel Wilder had reported the capture of weapons in Alabama. Choice (E) is incorrect; while the patrol was surprised, there is no evidence that they represented the entire Union army.

PASSAGE 2 8. The correct answer is (C). The author states that a star running out of fuel is the victim of gravity and that the black hole is the star’s afterlife. Choice (A) is incorrect because the entire article deals with the reality of gravity as a part of the phenomenon of the black hole. Choice (B) is incorrect; there is no information to support this statement. Choice (D) is incorrect; gravity actually works with the black hole and would not add to the likelihood of the demise. Choice (E) is incorrect; gravity contributes to the function of the black hole but is not the same thing. 9. The correct answer is (A). The author states that for some stars the black hole is the final resting place or “afterlife.” Choice (B) is incorrect because there is no mention of eternity; therefore, there is no information to support this theory. Choice (C) is incorrect; the birthplace of stars is not mentioned. Choice (D) is incorrect; there is no evidence that stars shine brighter adjacent to a black hole. Choice (E) is incorrect because choice (A) provides the connection. 10. The correct answer is (B). In the opening paragraph, the author makes the statement that using the black hole as a subject for literature and film has bored the public, while the astronomer is still discovering truths. Choice (A) is incorrect; there is no evidence presented that the public considers the black hole a hoax. Choice (C) is incorrect; there is no evidence that astronomers view the black hole as a fluke, but rather as an interesting subject. Choices (D) and (E) are incorrect; there is no supporting evidence in the selection. 11. The correct answer is (C). The third paragraph of the article begins with a sentence that states this premise. Choice (A) is incorrect; there is no evidence to support this. Choice (B) is incorrect; in fact, the neutron star is often the by-product of the supernova explosion. Choice (D) is incorrect; this is a false premise. Choice (E) is incorrect; the opposite is true, according to McClintock, when he says it “beggars description.”

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UNIT 5 12. The correct answer is (D). The illustration is intended to show that something of enormous size when reduced as significantly as this article is discussing becomes a unit we can easily recognize—a bathroom sink. Choice (A) is incorrect; there is no attempt in this article to divide the different sizes of stars. Choice (B) is incorrect; at first glance this would seem correct—however, the term “matter existing there” renders this a false statement. Choice (C) is incorrect; again, at first glance this appears correct—however, remember the need to separate what we think we know from what is in the article. The illustration is used to assure that we DO understand. Choice (E) is incorrect; there is nothing in the article to support this statement. 13. The correct answer is (E). There is evidence in the article for all of the answers. Choice (A) is correct; the first sentence of the third paragraph renders this true. Choice (B) is correct; paragraph 2 provides the evidence for this inference. Choices (C) and (D) are correct; paragraph 3 provides information for this conclusion. 14. The correct answer is (A). The evidence presented in this article allows the reader to infer that there are differences, and that these differences lie in the composition of the star. Choice (B) is incorrect; this is a commonly held theory of laypeople— however, you have been cautioned not to select what you think to be true, but to base your choice on the material presented. There is no support for this theory in this article. Choice (C) is incorrect; there is evidence to support the first part of this premise, but not the last part; therefore, the answer is incorrect. Choices (D) and (E) are incorrect; there is no evidence to support either of these theories. 15. The correct answer is (C). Throughout the article, the author refers to the physical being of the stars as a “mass,” and the condensation of that physical property by gravity as a “mass.” Choice (A) is incorrect; after the violent death of a star, there are fragments only. Choice (B) is incorrect; the core or center is a “singularity.” Choice (D) is incorrect; the remnants of a dead star vary in designation from fragments to singularity. Choice (E) is incorrect. There is no evidence in this article to the central ingredient for a black hole. 16. The correct answer is (C). The author states that physicists prefer not to discuss the central condensation but to refer to it as “singularity” because they have no clue as to what happens. Choice (A) is incorrect; there is no evidence of a name for this condition. Choice (B) is incorrect; a supernova is a very bright star. Choice (D) is incorrect; the afterlife of such a star is not discussed in this article. Choice (E) is incorrect; there is no discussion of “fuel.” 17. The correct answer is (D). In the last paragraph, the author explains that property is changed to nothing. Choice (A) is incorrect; there is no reference to the collision of stars in this selection. Choice (B) is incorrect; the opposite is true. McClintock indicates that the investigation is ongoing. Choice (C) is incorrect; while there is an indication that the singularity condition of the black hole is puzzling, the explanation is that the condensation is so infinitesimal as to defy description. Choice (E) is incorrect, because choice (D) is correct.

PASSAGE 3 18. The correct answer is (E). Choice (A) is incorrect; the first sentence of the selection makes the point that the material SHOULD meet the needs. Choice (B) is incorrect; the point is clearly made that the material SHOULD meet these needs. Choice (C) is incorrect; again, the point is made that this SHOULD happen. Choice (D) is incorrect; teachers SHOULD be ready to do this.

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READING COMPREHENSION REVIEW 19. The correct answer is (B). The author makes the point that literature speaking to the needs of teens is preferred. Choice (A) is incorrect; there is no mention of authors in the selection. Choice (C) is incorrect; while there is mention of the availability of such material, there is no mention that it MUST be found. Choice (D) is incorrect; the point is made that children’s literature is NOT appropriate for adolescents. Choice (E) is incorrect; while this may or may not be a true statement, there is no mention of this in the selection. 20. The correct answer is (E). The author makes the point that the literature that interests adolescents deals with the emotional needs—choice (A); concerns itself with intellectual changes—choice (B); approaches the physical needs—choice (C); and has topics that interest adolescent students—choice (D). Therefore, choice (E) is correct. 21. The correct answer is (A). The author of the selection makes the point that students need “security . . . with a supportive adult barely visible in the background.” Choice (B) is incorrect; there is no mention of supervised reading programs. Choice (C) is incorrect; the author makes no mention of a strict academic environment. Choice (D) is incorrect; the author indicates that librarians and teachers should assist, but there is no requirement for adult supervision of social programs. Choice (E) is incorrect; competitive activities are mentioned as being one of the negatives for teens. 22. The correct answer is (D). The author makes the direct statement and supports it with a variety of examples that support the fact that the needs of changing adolescents—emotional, physical, and intellectual—must be accommodated. Choice (A) is incorrect. The author does not state a need for separation but for accommodation. Choice (B) is incorrect; while there may be some evidence that adolescent needs are greater, the author does not make this as a point. Choice (C) is incorrect; the author clearly states that having adolescents utilize furniture made for children in libraries and classrooms is unsatisfactory. Choice (E) is incorrect; the author’s point is that all services are NOT readily available to meet the needs of all children. 23. The correct answer is (C). This selection infers that the provision of appropriate reading material for adolescents can be helpful to their maturation. Choice (A) is incorrect; while there may be a great market for authors of adolescent literature, this point is not made in this selection. Choice (B) is incorrect; while this may be a true statement, it is not a part of this selection. Choice (D) is incorrect; again, this may be true—however, there is no mention of role models in this selection. Choice (E) is incorrect; the point is made that activities for students should provide more winners than losers, but there is no indication of a necessary high profile. 24. The correct answer is (D). The topic sentence of the first paragraph gives this indication. Choice (A) is incorrect; discussed in the passage, it is not the entire concept of the selection. Choice (B) is incorrect; there is no mention of a nonthreatening environment. Choice (C) is incorrect; there is a point that furnishings may be unpleasant when designed for children but used by adolescents, but this is not the main idea of the selection. Choice (E) is incorrect; while this may be a true statement, it is not the main idea of this selection. 25. The correct answer is (E). In the last paragraph of the selection, this point is made. Choice (A) is incorrect; there is no mention of this in the selection. Choice (B) is incorrect; while the inference is that students need a guide, there is no indicator of the need for a “pattern.” Choice (C) is incorrect; there is no mention of looking ahead. Choice (D) is incorrect; the point is made that young adults are associating mainly with peers and not family, but there is no indication of a “separation.”

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PASSAGE 4 26. The correct answer is (C). The comparison between the steam engine that placed the statue there and the fact that it took less than 10 minutes to hoist it by helicopter implies that modern technology, when viewed in a historical perspective, is far more efficient than technology in the past. Choice (A) is incorrect; there is no indicator that statues should be refurbished often. Choice (B) is incorrect; indeed, the use of the efficient helicopter is applauded. Choice (D) is incorrect; while this is a true statement, it is not the author’s inference. Choice (E) is not correct; although probably true, it is not the author’s inference. 27. The correct answer is (B). The author extols the virtues of the Sikorsky S-64F helicopter. Choice (A) is incorrect; the author makes the movements of the helicopter seem simple. Choice (C) is incorrect; the mention of the transformer is an aside. Choice (D) is incorrect; although perhaps true, this is not the main idea. Choice (E) is incorrect; again, although true, this is not the main idea of this selection. 28. The correct answer is (B). Putting the crew in and including its comments involves humanity. Choice (A) is incorrect; while this may be a truth, it is not the author’s intent. Choice (C) is incorrect; there is no effort to “humanize” the statues. Choice (D) is incorrect; there is no effort on the author’s part to demean technology. Choice (E) is incorrect; the author makes no effort to confuse the reader. 29. The correct answer is (B). The author states that Liberty was kept on her pedestal for her face-lift, while Freedom was brought to the ground. Choice (A) is incorrect; there is no mention of this fact. Choice (C) is incorrect; while perhaps true, this is not stated in the selection. Choice (D) is incorrect; this is not discussed with relation to Liberty. Choice (E) is incorrect; there is no mention of the composition of Freedom. 30. The correct answer is (E). All answers are correct and are spelled out in the selection. There is a gyro-coupled flight control system, which makes choice (A) correct. There is a third pilot to handle the hoisting system, which makes choice (B) correct. The special suspension system is mentioned, making choice (C) correct. The crew and its work is mentioned, making choice (D) correct. Therefore, choice (E) is the correct choice.

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R E D A LERT WHY STUDY VOCABULARY FOR THE GRE? If you’re a native speaker of English, you already know thousands of words. (The average person has a working vocabulary of over 10,000 words—and is probably capable of at least recognizing thousands more.) After four years of college, you probably have an extensive vocabulary of words drawn from many fields of study, to say nothing of the words you hear, see, and use in everyday life. Is it really necessary for you to study vocabulary in preparation for the GRE? For most people, the answer is yes. The test-makers consider vocabulary so important that they test it in several ways on the GRE. 1. As you know, the verbal sections of the exam include antonym questions, which require you to pick a word whose meaning is the opposite of some other word. In Unit 4, we provided you with a number of hints and strategies for tackling these items effectively. Then there are indirect and hidden vocabulary questions—of which there are plenty. 2. Your ability to fully understand reading comprehension passages will often turn on your knowledge of vocabulary. The broader, more varied, and more accurate your vocabulary knowledge, the better your chances of answering the questions that cover these passages quickly and correctly. 3. Analogy questions obviously depend to a large extent on vocabulary. It’s difficult—though not impossible, as we discussed in Unit 3—to decipher the analogy relationships unless you understand the words that are involved. One typical group of analogy items includes the words proficiency, embellish, carping, reclusive, tactile, criterion, intransigent, and strenuous, among others. (How many of these can you define?) You don’t have to throw up your hands in despair if an analogy item contains a word or two you don’t know; there’s more than one way to skin that cat. But the process will be a lot easier and faster if you know most of the words used, or at least have a nodding acquaintance with them. 4. The better your vocabulary knowledge, the easier you’ll find it to understand the sentence completion items (which are, in effect, mini-reading passages, each one sentence long). Even an occasional math item is made a little more complicated by the use of a challenging vocabulary word. 5. Your performance on the Analytical Writing Measure will be aided by vocabulary knowledge that is both broad and deep: broad, in the sense that you have a relatively large and varied pool of words to draw upon; deep, in the sense that your understanding of individual words is accurate and sophisticated. The words you use in your essays will have a significant impact on the grades you receive. Reliance on rudimentary or narrow vocabulary makes you sound “less smart”; so does misusing words.

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THE SIX BEST VOCABULARY-BUILDING TIPS FOR THE GRE

STUDY VOCABULARY DAILY There are some topics you can easily cram. Vocabulary isn’t one of them. Words generally stick in the mind not the first or second time you learn them but the fourth or fifth time. Try to begin your vocabulary study several weeks before the exam. Take fifteen or twenty minutes a day to learn new words. Periodically review all the words you’ve previously studied; quiz yourself, or have a friend quiz you. This simple regimen can enable you to learn several hundred new words before you take the GRE.

LEARN

A

FEW WORDS

AT A

TIME

Don’t try to gobble dozens of words in one sitting. They’re likely to blur into an indistinguishable mass. Instead, pick a reasonable quantity—say, ten to fifteen words—and study them in some depth. Learn the definition of each word; examine the sample sentence provided in the word list; learn the related words; and try writing a couple of sentences of your own that include the word. Refer to your own dictionary for further information if you like.

LEARN WORDS

IN

FAMILIES

Language is a living thing. Words are used by humans, innately creative beings who constantly twist, reshape, invent, and recombine words. (Think of the jargon of your favorite sport or hobby, or the new language currently blossoming in cyberspace, for some examples.) As a result, most words belong to families in which related ideas are expressed through related words. This makes it possible to learn several words each time you learn one.

USE

THE

WORDS YOU LEARN

Make a deliberate effort to include the new words you’re learning in your daily speech and writing. It will impress people (professors, bosses, friends and enemies) and it will help solidify your memory of the words and their meanings. Maybe you’ve heard this tip about meeting new people: if you use a new acquaintance’s name several times, you’re likely never to forget it. The same is true with new words: use them, and you won’t lose them.

CREATE YOUR OWN WORD LIST Get into the habit of reading a little every day with your dictionary nearby. When you encounter a new word in a newspaper, magazine, or book, look it up. Then jot down the new word, its definition, and the sentence in which you encountered it in a notebook set aside for this purpose. Review your vocabulary notebook periodically—say, once a week. Use the words you learn this way. Your notebook will reflect the kinds of things you read and the kinds of words you find most difficult. And the fact that you’ve taken the time and made the effort to write down the words and their meanings will help to fix them in your memory. Chances are good that you’ll encounter a few words from your vocabulary notebook on the exam.

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Unit 6 MERRIAM-WEBSTER’S ROOTS TO WORD MASTERY If you’re like many students today preparing for the GRE, you probably have never taken a course in Latin, which means you may never have learned how most English words came to be based on words from older languages. And you may never have realized how the study of word roots can lead to a much larger vocabulary than you now have. Studying and mastering vocabulary words can certainly improve your GRE Verbal and Analytical Writing scores. So to maximize your chances of scoring high on your test, this unit will set you on the path to learning a broad range of vocabulary words. You’ll learn 50 of the Greek and Latin roots that form the foundation of most of the words in the English language as well as 150 English words based on those roots. Many of these 150 words will actually lead you to several more words each. By learning the word credible, you’ll also understand credibly and credibility the next time you hear them; by learning gratify, you’ll also learn gratifying and gratification; and by learning theology, you’ll understand theological, theologically, and theologian when you run across them. So learning the roots and words in this chapter will help you to learn thousands of words. Ancient Greek and Latin have been the sources of most words in the English language. (The third-biggest source is the family of Germanic languages.) And not just of the older words: Almost the entire English vocabulary was created long after the fall of the Roman empire, and it continues to expand to this day. Of the new words that are constantly being invented, the majority—especially those in the sciences, where most new words are introduced—are still based on Greek and Latin roots. Even new buzzwords that you think appear out of nowhere may be Greek or Latin in origin. For instance, morph is a short form of metamorphose, which comes almost straight from Latin; def is short for definitely, which is also based on Latin; hype is probably short for hyperbole, which comes straight from Greek; and rad is short for radical, which comes from the Latin radix—which actually means “root”! While root study is very valuable, be cautious when you begin exploring it. A portion of a word may resemble a root only by coincidence. For example, the word center doesn’t have anything to do with the root cent (meaning “hundred”), and the words interest and interminable don’t have anything to do with the root

For more vocabulary-building exercises, visit Merriam-Webster’s Web site at www.m-w.com.

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inter (meaning “between”). It may take time to recognize which words actually contain the roots you think you see in them. Another problem is that not every root you think you’ve identified will necessarily be the right one. For example, ped may mean either “foot” or “child,” and liber may mean either “book” or “free.” A third problem is that many common roots are too short to recognize or change their spelling in a confusing way from word to word. So even though perception, deceive, recipe, capture, and receipt can all be traced to the same Latin root, the root changes form so much—cip, cept, cap, etc.—that root study probably won’t help the student looking for a memory aid. Similarly, when the Latin word ad (meaning “to” or “toward”) becomes a prefix, it usually changes to ac-, ad-, af-, ag-, am-, an-, or some other form, so the student can rarely recognize it. In addition, the meanings of some roots can change from word to word. So even though the cip-cept-cap root means “grasp,” “seize,” or “take,” it may seem to change its meaning completely when combined with a prefix (per-, de-, etc.). As long as you are aware of such difficulties, root study is an excellent way to learn English vocabulary (not to mention the vocabularies of Spanish, French, Italian, and Portuguese, all of which are based on Latin). In fact, it’s the only method of vocabulary acquisition that relies on broadly useful memory aids. Without it, vocabulary study consists of nothing but trying to memorize unrelated words one by one by one. So from here on, it’s up to you. The more fun you can have learning your new vocabulary, the better you’ll do. And it can be fun. For one thing, the results are instantaneous—you can show off your new knowledge any time you want. And you’ll almost feel your mind expanding as your vocabulary expands. This is why people talk about the “power” of a large vocabulary; you’ll soon realize your mental capacities are actually becoming more powerful with every new word. Take every opportunity to use the words you’re learning; the most effective way to keep a new word alive in your vocabulary is to use it regularly. Look and listen for the new words you’ve learned—you’ll be surprised to find yourself running into them often, especially if you’ve also begun reading more demanding books and magazines in your leisure time. Challenge your friends with them, even if just in a joking way. Make up games to test yourself, maybe using homemade flashcards. And don’t stop acquiring new vocabulary words after you’ve mastered this unit. Whenever you’re reading, look for roots in the new words you keep encountering and try to guess each word’s meaning before looking it up in a dictionary (which you should try to keep close at hand). Once you’ve acquired the habit, you’ll be astonished at how quickly your vocabulary will grow.

INSTRUCTIONS We introduce you to 50 of the most useful Greek and Latin roots (omitting the prefixes and suffixes that almost everyone knows—anti-, co-, de-, -ism, mis-, non-, un-, vice-, etc.). We call these roots useful because they are common and also because they nearly always keep their meaning in an obvious way when they appear in an English word. So if you encounter an unfamiliar word on your test, these roots may be the key to making an educated guess as to its meaning. www.petersons.com

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Each root is discussed in a short paragraph. Each paragraph is followed by three vocabulary words derived from the root. For each word, we provide the pronunciation, the definition, and a sentence showing how the word might actually be used in writing or conversation. You’ll be quizzed after every 15 words, and finally you’ll be tested on every one of the 150 words. (All answers are given at the end of the unit.) These tests will ensure that the words and roots become permanently fixed in your memory, just as if you’d been drilled on them in class. For further study on your own, near the end of the chapter we list an additional 50 useful roots, along with three English words based on each one.

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50 ROOTS TO SUCCESS Pronunciation Guide: \@\ abut \@r\ further \a\ ash \a¯\ ace \ä\ mop, mar \au ˙ \ out \ch\ chin \e\ bet \e¯\ easy \g\ go \i\ hit \ı¯\ ice \j\ job \Î\ sing \o ¯ \ go \o ˙ \ law \o ˙ i\ boy \th\ thin \th\ the \ü\ loot \u ˙ \ foot \y\ yet \zh\ vision agr Beginning Latin students traditionally learn the word agricola, meaning “farmer,” in their very first class. Though most of us tend to think of the Romans as soldiers, senators, and citizens of the city of Rome, most inhabitants of the empire were actually farmers. We see the root today in words such as agriculture. agronomy \@-'grä-n@-me¯\ A branch of agriculture dealing with field-crop production and soil management. • The poor country was in dire need of an agronomy team to introduce its farmers to new crops and techniques. agrochemical \µa-gro ¯ -'ke-mi-k@l\ An agricultural chemical (such as an herbicide or an insecticide). • The river’s pollution was easily traced to the runoff of agrochemicals from the cornfields. agrarian \@-'grer-e¯-@n\ Of or relating to fields, lands, or farmers, or characteristic of farming life. • The team of sharply dressed lawyers seemed nervous and awkward in this agrarian landscape of silos and feed stores. ante Ante means “before”; its opposite, post, means “after.” Both almost always appear as prefixes (that is, at the beginnings of words). Ante is easy to confuse with anti, meaning “against.” Antebellum means “before the war,” and we often speak of the antebellum South—that is, the South before the Civil War, not the “antiwar” South. antedate \'an-ti-µda¯t\ 1: To date as of a date prior to that of execution. 2: To precede in time. • It appeared that Crowley had antedated his check to the contractors, helping them evade taxes for work done in the new year. antecedent \µan-t@-'se¯-d@nt\ Prior, preceding. • As Mrs. Perkins told it, the scuffle had started spontaneously, and any antecedent events involving her rowdy son had been forgotten. anterior \an-'tir-e¯-@r\ Situated before or toward the front. • Dr. Singh was going on about anterior and posterior knee pain, but in her agony Karen could hardly remember a word.

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anthro The Latin anthro means “man” or “mankind.” Thus, in English we call the study of mankind anthropology. Anthro is very close to the Greek and Latin andro, which shows up in such words as android. anthropoid \'an-thr@-µpoid\ Any of several large, tailless apes. • The anthropoids—chimpanzees, bonobos, gorillas, orangutans, and gibbons— had diverged from the human evolutionary line by 5 million years ago. misanthrope \'mi-s@n-µthro ¯ p\ A person who hates or distrusts mankind. • Over the years she had retreated into an increasingly bitter solitude, and her former friends now dismissed her as a misanthrope. philanthropy \f@-'lan-thr@-pe¯\ Active effort to promote human welfare. • His philanthropy was so welcome that no one cared to inquire how he’d come by his fortune. aqu The Greek and Latin root aqu- refers to water. The ancient world regarded all matter as made up of four elements—earth, air, fire, and water. Today, the root is found in such familiar words as aquarium, aquatic, and aquamarine. aquaculture \'ä-kw@-µk@l-ch@r\ The cultivation of the natural produce of water, such as fish or shellfish. • Having grown hugely, the aquaculture industry now produces 30 percent of the world’s seafood. aquifer \'a-kw@-f@r\ A water-bearing stratum of rock, sand, or gravel. • The vast Ogallala aquifer, which irrigates most of the Great Plains, is monitored constantly to ensure that it isn’t dangerously depleted. Aquarius \@-'kwar-e¯-@s\ 1: A constellation south of Pegasus pictured as a man pouring water. 2: The 11th sign of the zodiac in astrology. • Many believe that the great Age of Aquarius began in 1962; others believe it commenced in 2000 or hasn’t yet begun. arti This root comes from the Latin word for “skill.” Art could also mean simply “cleverness,” and we still describe a clever solution as artful. Until recent centuries, almost no one made a real distinction between skilled craftsmanship and what we would now call art. So the words artistic and artificial turn out to be very closely related. artifice \'är-t@-f@s\ 1: Clever skill. 2: A clever trick. • She was stunned to find she’d been deceived by a masterpiece of artifice—the lifelike figure of a seated man talking on the phone, a lit cigarette in his right hand. artifact \'är-ti-µfakt\ A usually simple object, such as a tool or ornament, made by human workmanship. • Among the artifacts carried by the 5,000-year-old Iceman was a fur quiver with fourteen arrow shafts. artisan \'ar-t@-z@n\ A skilled worker or craftsperson. • Ducking down an alley, he weaved quickly through the artisans hawking their wares of handworked brass and leather.

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QUIZ 1 Answers appear at the end of this unit.

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1.

Carnegie spread his ________________ more widely than any previous American, building almost 1,700 libraries.

2.

A long list of __________________s—mainly herbicides and pesticides— were identified as health threats.

3.

News of the cave’s discovery soon leaked out, and local youths were soon plundering it of its Indian ________________s.

4.

Stalin moved swiftly to uproot Russia’s _______________ traditions and substitute his new vision of collectivized agriculture.

5.

They had drilled down 85 feet before they struck the ______________ and water bubbled to the surface.

6.

The first X-ray image, labeled “________________,” showed a frontal view of her heart.

7.

George Washington Carver, a hero of American ________________, transformed Southern agriculture through his research into the peanut.

8.

The throne itself, its surface glittering with ornaments, was the most extravagant example of the sculptor’s ________________.

9.

In his lecture on “The ________________ Causes of the Irish Famine,” he expressed wonder at rural Ireland’s absolute dependency on the potato by 1840.

10.

Before the development of ________________, the Atlantic salmon was threatened by overfishing.

11.

Her brother, always suspicious and unfriendly, was by now a genuine ________________, who left his phone unplugged and refused all invitations.

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12.

Any contracts that ________________ the new law by five years or more will remain in effect.

13.

The man resembled an ________________, with powerful sloping shoulders and arms that seemed to brush the ground.

14.

A young boy pouring water into the basin below reminded her of the astrological symbol of ________________.

15.

All the handcrafts turned out to be the work of a large family of ________________s.

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bene In Latin, bene means “well”; its near-opposite, mal, means “bad” or “poorly.” Both usually appear at the beginnings of words. We may hope to use this root often to list benefits and describe beneficial activities. benediction \µbe-n@-'dik-sh@n\ The pronouncement of a blessing, especially at the close of a worship service. • The restless children raced out to the church picnic immediately after the benediction. beneficent \b@-'ne-f@-s@nt\ Doing or producing good; especially performing acts of kindness or charity. • Even the busy and poor willingly contribute to organizations recognized as beneficent. benefactor \'be-n@-µfak-t@r\ A person or group that confers aid, such as a charitable donation. • Construction of the new playground had been funded by a generous benefactor. bio Bio comes from the Greek word for “life.” Thus, biology means the study of all living forms and life processes, and biotechnology uses the knowledge gained through biology. Antibiotics fight off bacteria, which are life forms, but not viruses, which may not be. bionic \bı¯-'ä-nik\ Having normal biological ability enhanced by electronic or mechanical devices. • A 1970s TV series featuring “the Bionic Woman” sparked interest in robotics. biopsy \'bı¯-µäp-se¯\ The removal and examination of tissue, cells, or fluids from the living body. • Until the biopsy results came back, there was no way to tell if the lump was cancerous. symbiosis \µsim-be¯-'o ¯ -s@s\ The intimate living together of two dissimilar organisms, especially when mutually beneficial. • In a display of symbiosis, the bird stands on the crocodile’s teeth and pecks leeches off its gums. chron This root comes from the Greek word for “time.” A chronicle records the events of a particular time. Chronometry is the measuring of time, which can be done with a chronometer, a timepiece more accurate than an ordinary watch or clock. chronic \'krä-nik\ Marked by long duration or frequent recurrence; habitual. • Her roommate was a chronic complainer, who started off every day grumbling about something new. anachronism \@-'na-kr@-µni-z@m\ 1: The error of placing a person or thing in the wrong period. 2: One that is out of its own time. • After the collapse of the Soviet Union, some analysts felt that NATO was an anachronism. chronology \kr@-'nä-l@-je¯\ An arrangement of events in the order of their occurrence. • Keeping a journal throughout her trip gave Joan an accurate record of its chronology afterward.

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circum Circum means “around” in Latin. So to circumnavigate is “to navigate around,” often around the world, and circumference means the “distance around” a circle or other object. A circumstance is a fact or event accompanying (“standing around”) another. circumvent \µs@r-k@m-'vent\ To evade or defeat, especially by trickery or deception. • During Prohibition, many citizens found ways to circumvent the laws against alcohol. circumspect \µs@r-k@m-'spekt\ Careful to consider all circumstances and consequences; cautious; prudent. • Unlike his impulsive twin brother, Claude was sober, circumspect, and thoughtful. circumstantial \µs@r-k@m-'stan-sh@l\ 1: Describing evidence based on inference, not directly observed facts. 2: Incidental. • The fact that he was gone all night was only circumstantial evidence, but still extremely important. cosm Cosm comes from the Greek word meaning “order.” Ultimate order, for the Greeks, related to the universe and the worlds within it, so cosmos for us means the universe. A cosmonaut was a space traveler from the former Soviet Union. cosmopolitan \µkäz-m@-'pä-l@-t@n\ International in outlook; sophisticated; worldly. • The cosmopolitan actress Audrey Hepburn was born in Belgium and educated in England but won fame in America. cosmology \käz-'mä-l@-je¯\ 1: A branch of astronomy dealing with the origin and structure of the universe. 2: A theory that describes the nature of the universe. • New Age philosophies and science fiction suggest a variety of possible cosmologies. microcosm \'mı¯-kr@-µkä-z@m\ An individual or community thought of as a miniature world or universe. • Early thinkers saw the whole human world as a microcosm of the universe, which was considered the macrocosm.

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QUIZ 2 Answers appear at the end of this unit.

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1.

In ant–aphid ________________, the aphids are protected by the ants, who “milk” them for their honeydew.

2.

A ________________ witch could end a drought by casting a spell to bring rain.

3.

The diner’s hours depended on such ________________ factors as whether the cook’s car had gotten repossessed.

4.

Phenomena such as time warps and black holes made theoretical ________________ the strangest subject in the curriculum.

5.

Church members were surprised by the closing ________________, “May God deny you peace, but grant you love.”

6.

Neuroscientists believe they will soon have developed a complete ________________ ear.

7.

Identifying a suspicious tumor almost always calls for a ________________ procedure.

8.

The children’s clinic was built soon after a significant gift by a single ________________.

9.

Both candidates had managed to ________________ campaign finance laws through fraud.

10.

Measles and flu are acute illnesses, while asthma and diabetes are ________________ conditions.

11.

Shakespeare’s Macbeth, set in the eleventh century, contains such ________________s as a reference to clocks.

12.

A detailed ________________ of the actions of company executives from April to July revealed some suspicious patterns.

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13.

Office life, with all its dramas and secrets, seemed to her a ________________ of the world outside.

14.

When we have only flimsy evidence, we should be ________________ in our opinions.

15.

With its international nightlife and a multitude of languages spoken on its beaches, Martinique is a ________________ island.

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cred This root comes from credere, the Latin verb meaning “to believe.” Thus something incredible is almost unbelievable. We have a good credit rating when institutions believe in our ability to repay a loan, and we carry credentials so that others will believe we are who we say we are. credence \'kre¯-d@ns\ Mental acceptance as true or real; belief. • Giving credence to gossip—or even to corporate financial reports these days—is risky. credible \'kre-d@-b@l\ Trustworthy; believable. • The defense team doubted that the ex-convict would make a credible witness. creed \'kre¯d\ A statement of the essential beliefs of a religious faith. • The Nicene Creed of A.D. 381 excluded Christian beliefs considered incorrect. dis In Latin, dis means “apart.” In English, its meanings have increased to include “do the opposite of” (as in disobey), “deprive of” (as in disillusion), “exclude or expel from” (disbar), “the opposite or absence of” (disaffection), and “not” (disagreeable). disarming \di-'sär-miÎ\ Reducing hostility or criticism; ingratiating. • Their ambassador to the United Nations has a disarming manner but a cunning mind. disburse \dis-'b@rs\ To pay out; distribute. • The World Bank agreed to disburse $20 million to Bolivia in recognition of its economic reforms. discredit \dis-'kre-d@t\ 1: To cause disbelief in the accuracy or authority of. 2: To disgrace. • Lawyers with the states suing the tobacco company sought to discredit testimony of its chief witness. dyna The Greek root dyna means “to be able” or “to have power.” Dynamite has enough power to blow up the hardest granite bedrock. A dynamic person or group is powerful and energetic. A dynamometer measures mechanical force, which is measured in dynes. dynamo \'dı¯-n@-µmo ¯ \ 1: A power generator. 2: A forceful, energetic person. • The early dynamo was a mysterious mechanism for many, who saw no relation between steam and electric current. dynasty \'dı¯-n@-ste¯\ 1: A line of rulers from the same family. 2: A powerful group or family that maintains its position for a long time. • After the Mongols and before the Manchus, the Ming dynasty provided China a very stable era. hydrodynamic \µhı¯-dro ¯ -dı¯-'na-mik\ Of or relating to the motion of fluids and the forces acting on moving bodies immersed in fluids. • Water temperature, resistance, and depth are among the hydrodynamic aspects of rowing.

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dys In Greek, dys means “bad” or “difficult.” As a prefix in English, it has the additional meanings “abnormal” and “impaired.” Dyspnea is difficult or labored breathing. Dyspepsia is indigestion (or ill humor). A dysfunctional family is one that functions badly. dyslexia \dis-'lek-se¯-@\ A disturbance of the ability to read or use language. • Dyslexia is regarded as the most widespread of the learning disabilities. dysentery \'di-s@n-µter-e¯\ An infectious intestinal disease with abdominal pain and severe diarrhea. • Considering the poor sanitation, travelers were not surprised to find dysentery widespread. dystrophy \'dis-tr@-fe¯\ A disorder involving wasting away of muscular tissue. • The telethon raises over $50 million a year to battle muscular dystrophy and related diseases. epi Coming from the Greek, this root means various things, particularly “on” and “over.” An epicenter is the part of the earth’s surface directly over the focus of an earthquake. The epidermis is the outer layer of the skin, overlying the inner “dermis.” An epitaph is an inscription upon a tomb in memory of the person buried there. epithet \'e-p@-µthet\ A characterizing and often abusive word or phrase. • Classic epithets used by Homer include “rosy-fingered dawn” and “Zeus, the cloud-gatherer.” epigraph \'e-p@-µgraf\ 1: An engraved inscription. 2: A quotation set at the beginning of a literary work to suggest its theme. • Chapter 5, describing the great battle, bears the Shakespearean epigraph “Let slip the dogs of war.” epilogue \'e-p@-µlo ˙ g\ A concluding section, especially to a literary or musical work. • Not until the novel’s epilogue do we realize that all the characters were based on the author’s family.

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QUIZ 3 Answers appear at the end of this unit.

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1.

Most early Christian ________________s developed around the act of baptism, where adult candidates proclaimed their faith.

2.

With his ________________ smile and quiet humor, he charms even the wariest clients.

3.

Amoebic ________________ is not just traveler’s diarrhea—it is contracted by people who live in unclean conditions, too.

4.

The dictator scornfully attempted to ________________ the proceedings at his war crimes trial.

5.

New reports lent ________________ to the captive’s story that the enemy had fled.

6.

When students with undiagnosed ________________ go on to higher education, their coping mechanisms often fall apart.

7.

Henry Ford founded a ________________; his great-grandson is now the company’s chairman.

8.

The listing of ________________s included 40,000 reports of inscriptions found on Roman ruins.

9.

Katie’s research focused on the ________________ drag of small sea kayaks.

10.

Relief agencies explored how best to ________________ funds and food to the disaster victims.

11.

At the close of Shakespeare’s The Tempest, Prospero speaks the wise ________________.

12.

Lou Gehrig’s disease is one of about forty diseases in the area of muscular ________________.

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13.

The shoplifter hurled obscene ________________s at the guard conducting her to the office.

14.

Her story is hardly ________________, since she’s already changed the facts twice.

15.

Mayor Fiorello La Guardia of New York was considered a ________________ in an already dynamic city.

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extra This root, from Latin, places words “outside” or “beyond” their usual or routine territory. Extraterrestrial events take place “beyond” the Earth. Something extravagant, such as an extravaganza, goes beyond the limits of moderation. And extra is itself a word, a shortening of extraordinary, “beyond the ordinary.” extrapolate \ik-'stra-p@-µla¯t\ To project (known data) into an unknown area to arrive at knowledge of the unknown area. • Her department pored over export-import data endlessly in order to extrapolate present trade trends and predict the future. extrovert \'ek-str@-µv@rt\ An outgoing, sociable, unreserved person. • Linda’s boss is an extrovert, always happiest in a roomful of people. extraneous \ek-'stra¯-ne¯-@s\ Not forming a vital part; irrelevant. • Coaching in diving and dance often seeks to reduce extraneous movements. fid Fid comes from fides, the Latin word for “faith.” An infidel is someone who lacks a particular kind of religious faith. An affidavit is a sworn statement, a statement you can have faith in. Something that’s bona fide is in “good faith”—absolutely genuine, the real deal. fiduciary \f@-'dü-she¯-µer-e¯\ 1: Involving a confidence or trust. 2: Held or holding in trust for another. • Corporate directors have often forgotten their fiduciary responsibility to their companies’ stockholders. confidante \'kän-f@-µdänt\ A person to whom secrets are entrusted, especially a woman. • The famed advice columnist Ann Landers was in many ways America’s confidante. fidelity \f@-'de-l@-te¯\ 1: The quality or state of being faithful. 2: Accuracy, as in sound reproduction. • Harriet’s comment left Lisa wondering about her husband’s fidelity. geo From the Greek word for “earth,” geo almost always appears as a prefix. Geography describes the Earth’s surface; geology deals with its history. We measure the Earth—and relationships of its points, lines, angles, surfaces, and solids—using geometry. geopolitical \µje¯-o ¯ -p@-'li-ti-k@l\ Combining geographic and political factors such as economics and population spread, usually with reference to a state. • Any invasion might trigger a series of geopolitical consequences, including the fall of other governments. geosynchronous \µje¯-o ¯ -'siÎ-kr@-n@s\ Having an orbit such that its position is fixed with respect to the Earth. • Satellites in geosynchronous orbits are usually positioned over the equator. geothermal \µje¯-o ¯-'th@r-m@l\ Of, relating to, or using the heat of the Earth’s interior. • Geothermal energy technology is most developed in areas of volcanic activity.

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graph This root is taken from the Greek word meaning “to write.” Something graphic is “vividly described.” Graphology is the study of handwriting. A graph is a diagram representing changes in something that varies. But graph, or graphy, actually most often appears at the end of a word. spectrography \spek-'trä-gr@-fe¯\ The dispersing of radiation (such as electromagnetic radiation or sound waves) into a spectrum to be photographed or mapped. • Spectrography can determine what elements stars are made of and how fast they are moving. seismograph \'sı¯z-m@-µgraf\ An apparatus for measuring and recording earthquakerelated vibrations. • Only recently have seismographs been enabling earthquake predictions that actually save lives. topography \t@-'pä-gr@-fe¯\ 1: The detailed mapping of geographical areas showing their elevations and natural and manmade features. 2: The contours of a geographical surface. • Watching for the next El Niño, NASA monitors ocean surface topography from space for clues. grat This root comes from gratus, the Latin word meaning “pleasing, welcome, or agreeable,” or from gratia, meaning “grace, agreeableness, or pleasantness.” A meal that is served graciously will be received with gratitude by grateful diners, unless they want to risk being called ingrates. gratify \'gra-t@-µfı¯\ 1: To be a source of pleasure or satisfaction. 2: To give in to; indulge or satisfy. • The victim’s family was gratified by the guilty verdict in the murder trial. ingratiate \in-'gra¯-she¯-µa¯t\ To gain favor by deliberate effort. • Backers of the proposed new mall sought to ingratiate themselves with community leaders. gratuitous \gr@-'tü-@-t@s\ Uncalled for; unwarranted. • Luckily for Linda and all concerned, her gratuitous and offensive remark was not recorded.

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QUIZ 4 Answers appear at the end of this unit.

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1.

An ________________ may assume that introverts are odd and antisocial.

2.

To be named a child’s guardian is to enter an important ________________ relationship.

3.

Broadcast journalists’ microphones now reduce surrounding ________________ noise to a whisper.

4.

He kept the embarrassing details a secret from everyone but Kendra, his longtime ________________.

5.

The growth of telecommunications is causing a rapid increase in the number of _________________ satellites.

6.

Her scheme to ________________ herself with the president began with freshly baked cookies.

7.

________________ energy usually derives from the heat associated with young volcanic systems.

8.

CAT scans and ________________ are being used to analyze old bones from the Southwest.

9.

After polling 840 well-chosen Americans, the firm ________________s its results to the entire country.

10.

The stock-market chart looked as if it had been produced by a ________________ set on the San Andreas fault.

11.

Chief Justice John Marshall called for an oath of ________________ to the Constitution.

12.

Haters of junk e-mail formed NAGS—Netizens Against ________________ Spamming.

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13.

The map detailed the region’s ________________, indicating the approximate altitude of every square foot of land.

14.

He hoped the award would ________________ her without swelling her head.

15.

Foster was devoted to national politics, but had no interest in wider ________________ issues.

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hydr Hydr flows from the Greek word for “water.” Hydrotherapy uses water for healing physical infirmities. Water may spout from a hydrant. “Water” can also be found in the lovely flower called hydrangea: its seed capsules resemble ancient Greek water vessels. hydraulic \hı¯-'dro ˙ -lik\ 1: Operated or moved by water. 2: Operated by the resistance offered or the pressure transmitted when a quantity of liquid is forced through a small opening or tube. • The hydraulic brake system used in automobiles is a multiple piston system. dehydrate \de¯-'hı¯-µdra¯t\ 1: To remove water from. 2: To lose liquid. • To minimize weight on the challenging trail, the hikers packed dehydrated fruits and vegetables. hydroelectric \µhı¯-dro ¯-i-'lek-trik\ Of or relating to production of electricity by waterpower. • Hydroelectric power sounded clean and renewable, but some asked about its social and environmental impact. hyper This Greek prefix means “above and beyond it all.” To be hypercritical or hypersensitive is to be critical or sensitive beyond what is normal. To hyperextend means to extend a joint (such as a knee or elbow) beyond its usual limits. Clicking on a hyperlink may take you beyond the Web site where you found it. hyperbole \hı¯-'p@r-b@-le¯\ Extravagant exaggeration. • The article called him the college’s most popular professor, which even he thought was hyperbole. hypertension \µhı¯-p@r-'ten-sh@n\ The condition accompanying high blood pressure. • Hypertension ran in Rachel’s family and seemed to be linked to her relatives’ heart attacks. hyperventilate \µhı¯-p@r-'ven-t@l-µa¯t\ To breathe rapidly and deeply. • Competitive short-distance runners hyperventilate briefly before running. hypo Coming from Greek, hypo as a prefix can mean “under” or “below normal.” A hypocrite says or does one thing while thinking or feeling something entirely different underneath. Many hypo- words are medical. A hypodermic needle injects medication under the skin. Hypotension, or low blood pressure, can be just as unhealthy as hypertension. hypochondriac \µhı¯-p@-'kän-dre¯-'ak\ A person depressed in mind or spirits because of imaginary physical ailments. • My grandmother is a hypochondriac; every time she hears about a new disease on the news, she thinks she has caught it. hypothetical \µhı¯-p@-'the-ti-k@l\ Involving an assumption made for the sake of argument. • The dating service provides hypothetical questions designed to predict success or failure. hypothermia \µhı¯-po ¯ -'th@r-me¯-@\ Subnormal body temperature. • Confusion and slurred speech are signs of hypothermia, a silent killer in all seasons.

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inter This prefix is the Latin word meaning “between or among.” Someone who interferes comes between two people; a player who intercepts a pass comes between the ball and its intended receiver. An intermission is a break between acts of a play. An international event takes place between or among nations. intercede \µin-t@r-'se¯d\ 1: To act between parties as a mediator. 2: To plead on another’s behalf. • The bishop prayed, asking Mother Mary to intercede for us. interdict \'in-t@r-µdikt\ To destroy, cut off, or damage. • U.S. Kosovo Force soldiers sought to interdict weapons at the Serbian and Albanian borders. interface \'in-t@r-µfa¯s\ 1: A surface forming a common boundary between two bodies, spaces, or phases. 2: The place where independent systems meet and act on each other. • Long before the computer age, the auto dashboard was designed as a man2machine interface. jur Jur comes from the Latin verb jurare, “to swear or take an oath,” and the noun juris, “right” or “law.” A jury, made up of jurors, makes judgments based on the law. A personal injury caused by another person is “not right.” perjury \'p@r-j@-re¯\ Violation of an oath to tell the truth; lying under oath. • Lying to a TV reporter is one thing; perjury before a Senate committee is another. jurisprudence \µju ˙ r-@s-'prü-d@ns\ 1: A system of laws. 2: The science or philosophy of law. • Juliana’s heroes were the crusaders of 20th-century jurisprudence, especially Thurgood Marshall. abjure \ab-'ju ˙ r\ 1: To give up, renounce, recant. 2: To abstain from. • To the prison counselor, the three conspirators always solemnly abjured a future life of crime.

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QUIZ 5 Answers appear at the end of this unit.

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1.

The novelist Lord Archer was found guilty of ________________ for lying during his libel suit.

2.

“As a _______________ example,” she said, “let’s suppose it were the other way around.”

3.

Jared led the team up the river to visit the principal ________________ power plant in the region.

4.

In the thinner air near the mountain top, the climbers began to ________________.

5.

________________ technology uses fluid to give bulldozers and cranes their great power.

6.

There are often no warning signs before ________________ triggers a stroke, heart attack, or heart failure.

7.

By 19 he was a ________________, calling his mother daily about some new ache or sniffle.

8.

They urged the UN Secretary-General to ________________ in the bloody Middle East conflict.

9.

By then her face was caked with ice and ________________ had caused her heart to stop.

10.

In Web site design, a user-friendly ________________ is essential.

11.

Accepting the peace prize, Hume again stressed the need to ________________ violence.

12.

To preserve fruits, we learned how to can and freeze and even ________________ them.

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13.

Feminist ________________ is a philosophy of law based on the political, economic, and social equality of the sexes.

14.

Jason’s mom said he read thick books and took quantities of notes, but this was surely ________________.

15.

Once the enriched uranium left the lab, there would be no chance to ________________ it.

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mal Mal, from the Latin, means “bad.” Malodorous things smell bad. A malefactor is someone guilty of bad deeds. A malady is a disease or disorder. Malnutrition is faulty or inadequate nutrition. Dismal means particularly bad. malevolent \m@-'le-v@-l@nt\ Having, showing, or arising from intense ill will, spite, or hatred. • Bookstores report that children still like stories with hairy beasts and malevolent aliens. malign \m@-'lı¯n\ To speak evil of; defame. • Amanda didn’t wish to malign her neighbors, but the late-night partying had to stop. malpractice \µmal-'prak-t@s\ An abandonment of professional duty or a failure of professional skill that results in injury, loss, or damage. • The soaring cost of malpractice insurance forced many doctors into early retirement. mar From the Latin word mare, meaning “sea,” mar brings its salty tang to English in words like marine, “having to do with the sea,” and submarine, “under the sea.” It also forms part of such place names as Del Mar (“of the sea”), California. Aquamarine is the color of clear seawater in sunlight. maritime \'mar-@-µtı¯m\ Of or relating to the sea, navigation, or commerce of the sea. • She achieved a national practice in maritime law, specializing in ship insurance. marina \m@-'re¯-n@\ A dock or basin providing secure moorings for pleasure boats. • Florida has marinas all along its coast to meet the needs of watercraft from enormous yachts to flimsy sailboats. mariner \'mar-@-n@r\ A sailor. • Ann was haunted by some lines about the old mariner in Coleridge’s famous poem. morph This form comes from the Greek word for “shape.” It appears in anthropomorphic, meaning “having human form.” And morph is itself a new English word; by morphing, filmmakers can alter photographic images or shapes digitally, transforming them in astonishing ways. amorphous \@-'mo ˙ r-f@s\ Shapeless; formless. • The sculptor swiftly molded an amorphous lump of clay into a rough human shape. metamorphosis \µme-t@-'mo ˙ r-f@-s@s\ 1: A change in physical form or substance. 2: A fundamental change in form and often habits of an animal as part of the transformation of a larva into an adult. • Day by day we watched the gradual metamorphosis of the tadpoles into frogs. morphology \mo ˙r-'fä-l@-je¯\ A branch of biology dealing with the form and structure of organisms. • As an example, she mentioned the morphology of whales, whose fins evolved from legs.

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mort / mori These roots come from the Latin noun mors (and its related form mortis), meaning “death.” A mortuary is a place where dead bodies are kept until burial, and a mortician prepares corpses for burial or cremation. Memento mori, a Latin phrase used in English, means “a reminder of death,” such as a skull. moribund \'mo ˙ r-@-b@nd\ 1: Dying or approaching death. 2: Inactive or becoming outmoded. • Evidence of the sagging industrial economy could be seen in the moribund factories and towns. mortify \'mo ˙ r-t@-µfı¯\ 1: To subdue or deaden (the body) with self-discipline or self-inflicted pain. 2: To embarrass greatly; humiliate. • The parents’ attempts to act youthful mortified their kids, who almost died of embarrassment when their friends were around. mortality \mo ˙ r-'ta-l@-te¯\ 1: The state of being subject to death. 2: The proportion of deaths to population. • The preacher takes every occasion to remind us of our mortality, as does the insurance agent. mut Mut comes from the Latin mutare, “to change.” Science-fiction movies often focus on weird mutations, changes in normal people or animals that lead to death, destruction, or comedy. A governor may commute or change a prison sentence; a person commuting between cities “exchanges” one location for another. permutation \µp@r-myu ˙ -'ta¯-sh@n\ 1: The changing of the order of a set of objects. 2: An ordering of a set of objects. • The letters A, B, and C have six possible permutations: ABC, ACB, BAC, BCA, CAB, and CBA. immutable \i-'myü-t@-b@l\ Unchangeable or unchanging. • The physical world is governed by the immutable laws of nature. transmute \trans-'myüt\ To change in shape, appearance, or nature, especially for the better; transform. • A meek person may dream of being transmuted into a tyrant, or a poor person into a rich one.

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UNIT 6

QUIZ 6 Answers appear at the end of this unit.

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1.

The ________________ at Hyannis has over 180 slips for deep-draft sailboats, motorboats, and yachts.

2.

________________ suits are being filed today against even fine doctors who have made no errors.

3.

The monarch’s transformation from caterpillar to butterfly represents a dramatic ________________.

4.

Computer users were warned about a ________________ virus hiding in e-Christmas cards.

5.

The fabled dream of the alchemist was to ________________ lead into gold.

6.

The store’s nautical antiques and pond yachts should interest the armchair ________________.

7.

It would ________________ her if she ever heard herself described as “middle-aged.”

8.

Al Capp’s Shmoo was an ________________ blob-like creature who sometimes helped his friends solve mysteries.

9.

The moment he left the party, she started to ________________ him mercilessly.

10.

In terms of ________________, bats’ wings are skeletal hands with very long fingers, webbed with membranes.

11.

The day the first CD appeared in the stores, the vinyl LP was ________________.

12.

The number of different ways eight people can line up in a row provides a nice illustration of ________________s.

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13.

Detailed ________________ records are kept by the National Center for Health Statistics.

14.

The National ________________ Museum was displaying personal possessions of the Bounty mutineers.

15.

In an ever-changing world, people hunger for standards and qualities that are ________________.

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neo Old as its Greek source, neo means “new.” Neon was a new gas when found in 1898. A neoconservative is a liberal who has become a conservative. A neophyte is a new convert, or a beginner. And a neologism is a new word. neoclassical \µne¯-o ¯-'kla-si-k@l\ Of or relating to a revival or adaptation of the style of classical antiquity. • Neoclassical paintings are dignified and restrained, and they often radiate a noble spirit. Neolithic \µne¯-@-'li-thik\ Of or relating to the latest period of the Stone Age, characterized by polished stone implements. • Doctors have asked how the life spans of the Neolithic farmers compared with those of earlier hunter-gatherers. neoplasm \'ne¯-@-µpla-z@m\ A new growth of tissue serving no useful purpose in the body; tumor. • Using digital X rays, the dentist examined Tom’s gums for neoplasms and cysts. omni This comes from the Latin prefix meaning “all.” Thus an omnidirectional antenna will draw in stations from all directions. Something omnipresent is thought to be present at all places and at all times. An omnivorous animal might eat almost everything. Some companies apparently meaning to be everything to their customers name themselves simply “Omni.” omnibus \'äm-ni-b@s\ Of, relating to, or providing for many things at once. • The Senate’s omnibus bill includes money for everything from snail research to new bombers. omnipotent \äm-'ni-p@-t@nt\ Having unlimited authority or influence; almighty. • The question arises, If God is good and omnipotent, why do bad things happen? omniscient \äm-'ni-sh@nt\ Having infinite awareness, understanding, insight, or knowledge. • His stories usually have an omniscient narrator, who reveals the thoughts of all the characters. ortho Ortho comes from orthos, the Greek word for “straight,” “right,” or “true.” Orthotics is a therapy that straightens out the stance or posture of the body by providing artificial support for weak joints or muscles. Orthograde animals, such as humans, walk with their bodies in an upright position. Orthography is correct spelling. orthodox \'o ˙ r-th@-µdäks\ 1: Holding established beliefs, especially in religion. 2: Conforming to established rules or traditions; conventional. • Gerald preferred orthodox, mainstream cancer treatments to untested alternative therapies. orthopedist \µo ˙ r-th@-'pe¯-dist\ A medical specialist concerned with correcting or preventing skeletal deformities. • A local orthopedist eventually managed to correct the child’s spinal curvature. orthodontic \µo ˙r-th@-'dän-tik\ Pertaining to irregularities of the teeth and their correction. • As much as she dreaded braces, Jennifer knew the time had come for orthodontic work.

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pan Directly from Greek, pan means “all”; as a prefix in English it can also mean “completely,” “whole,” or “general.” A panoramic view is a complete view in every direction. Pantheism is the worship of all gods. A pandemic outbreak of a disease will affect an exceptionally high proportion of the population, though probably not literally “all” people. panacea \µpa-n@-'se¯-@\ A remedy for all ills or difficulties; a cure-all. • Educational reform is sometimes seen as the panacea for society’s problems. panoply \'pa-n@-ple¯\ 1: A magnificent or impressive array. 2: A display of all appropriate accessory items. • The full panoply of a royal wedding was a thrilling sight for millions. pantheon \'pan-the¯-µän\ 1: The gods of a people. 2: A group of illustrious people. • Even during Dickens’s lifetime, the critics had admitted him into the literary pantheon. phon This Greek root means “sound” or “voice.” It shows up in such words as telephone (“far sound”), microphone (“small sound”), and xylophone (“wood sound”). Phonics teaches reading by focusing on the sounds of letter groups. A phonograph is an instrument for reproducing sounds. cacophony \ka-'kä-f@-ne¯\ Harsh or discordant sound. • According to his grandfather, popular music since Bing Crosby had been nothing but cacophony. phonetic \f@-'ne-tik\ Relating to or representing the sounds of the spoken language. • Some schools teach reading by the phonetic method, linking sounds with letters. polyphonic \'pä-le¯-'fä-nik\ Of or relating to music in which two or more independent melodies are sung or played against each other in harmony. • Children singing “Three Blind Mice” are performing the simplest kind of polyphonic music.

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UNIT 6

QUIZ 7 Answers appear at the end of this unit.

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1.

Some saw the antidepressant drug Prozac as a psychological ________________.

2.

Prehistory, the period of no written records, included the ________________ and Bronze Ages.

3.

Suzanne, age 16, said if she were ________________ for a day, she would bring about world peace and save the rainforest.

4.

Adventurous young people often challenge ________________ religious belief systems.

5.

The ________________ Trade and Competitiveness Act touched on many aspects of labor, global commerce, and regulation.

6.

The conductor chose a balanced program by composers from the musical ________________.

7.

A ________________ alphabet was developed by NATO to be understandable by all allies in the heat of battle.

8.

The checkup produced one cause for concern: a small ________________ on the bile duct.

9.

My ________________ traces all our lower back problems to the time when the first humans stood erect.

10.

Sandra’s new sequencer could take complex ________________ music and convert it into written notation.

11.

Even several university degrees and eyes in the back of your head do not make you ____________.

12.

Looking down the long row of ________________ buildings, we almost thought we were in ancient Rome.

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13.

His mouth was a disaster area, and his crooked rows of teeth had never had a minute of ________________ attention.

14.

Out over the ocean, the winter sky spread a brilliant ________________ of stars.

15.

The kids who liked producing the most outrageous music soon were styling themselves the “________________ Club.”

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photo Coming from the Greek word for “light,” photo enlightens us in words like photography, which is the use of light to create an image on film or paper. A photocopy is a printed copy made by light on an electrically charged surface. A photogenic person is one highly suitable for being photographed. photon \'fo ¯ -µtän\ A tiny particle or bundle of radiant energy. • Star Trek’s photon torpedoes destroy their targets with intense radiation in the X-ray range. photosynthesis \µfo ¯ -to ¯ -'sin-th@-s@s\ The process by which green plants use light to produce organic matter from carbon dioxide and water. • Sagebrush is a hardy plant that can carry on photosynthesis at very low temperatures. photoelectric \µfo ¯ -to ¯ -i-'lek-trik\ Relating to an electrical effect from the interaction of light with matter. • Photoelectric cells would trigger the yard lights when they sensed motion. post Post comes from a Latin word meaning “after” or “behind.” A postscript is a note that comes after an otherwise completed letter, usually as an afterthought. Postpartum refers to the period just after childbirth and all of its related concerns. To postdate a check is to give it a date after the date when it was written. posterior \pä-'stir-e¯-@r\ Situated behind or on the back; rear. • A posterior view of the animal revealed unusual coloring and an extremely long tail. posthumous \'päs-ch@-m@s\ Following or happening after one’s death. • The late singer achieved posthumous success when her film became a huge hit. postmortem \µpo ¯ st-'mo ˙ r-t@m\ 1: Occurring after death. 2: Following the event. • In 1999 the institute had issued a postmortem report on the Bosnian war, “NATO’s Empty Victory.” pre One of the most common of all English prefixes, pre comes from prae, the Latin word meaning “before” or “in front of.” A TV program precedes another by coming on before it. You predict an event by saying it will happen before it does. A person who presumes to know assumes something before having all the facts. precocious \pri-'ko ¯-sh@s\ Showing mature qualities at an unusually early age. • Some thought the sitcom’s precocious child star was cute; others thought she was a show-off. prerequisite \pre¯-'re-kw@-z@t\ An action, event, or object required in advance to achieve a goal. • Certain courses were prerequisites for majoring in engineering at the university. predisposed \µpre¯-di-'spo ¯zd\ Influenced in advance or made persuadable. • The commissioner was predisposed to vote for the project since its developer had given his campaign a large contribution.

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prim Prim comes from primus, the Latin word for “first.” A prime minister is the chief minister of a ruler or state. Something primary is first in time, rank, or importance. Something primitive seems to be in an early stage of development. primal \'prı¯-m@l\ 1: Original or primitive. 2: First in importance. • Much of civilization seems designed to disguise or soften the rawness of our primal urges. primordial \prı¯-'mo ˙ r-de¯-@l\ Existing in or from the very beginning. • He assumed his ancestors emerged from the primordial ooze, and not as gods. primate \'prı¯-ma¯t\ A member of an order of mammals that includes humans, apes, and monkeys. • Do we have anything important to learn about human behavior from our cousins the primates? rect This root comes directly from the Latin word rectus, meaning “straight” or “right.” A rectangle is a four-sided figure whose straight sides meet at right angles. To correct something is to make it right. To stand erect is to stand straight. rectitude \'rek-t@-µtüd\ Correctness in judgment; moral integrity. • The school superintendent wasn’t popular, but no one could question his fairness and rectitude. rectify \'rek-t@-µfı¯\ To make or set right; correct. • Problems with the Bowl Championship Series were rectified by a simple four-team playoff. rectilinear \µrek-t@-'li-ne¯-@r\ Characterized by straight lines. • In its rectilinear structure, the sculpture reflects the surrounding office buildings.

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UNIT 6

QUIZ 8 Answers appear at the end of this unit.

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1.

Hamstrings, deltoids, and gluteus maximus are human muscles on the _______________ side.

2.

The lighting engineering firm offered ________________ sensors for many uses.

3.

The West Point students’ reputation for ________________ was badly damaged by the cheating scandal.

4.

Average parents of specially gifted or ________________ children face unusual challenges.

5.

With the dead man now proven innocent, his relatives sought a ________________ pardon.

6.

The power of lasers results from a focused concentration of ________________s.

7.

For many, retreating to a rough-hewn home in the woods seems to satisfy a ________________ urge.

8.

Some children may be ________________ to asthma by their genes.

9.

Green plants don’t graze, hunt, or shop; they make food by using sunlight through ________________.

10.

The association called on Congress to ________________ the unfairness of health care funding.

11.

During the ________________ exam, the medical examiner discovered a mysterious blackening of the liver tissue.

12.

Her study of baboons earned Gloria a fellowship to the ________________ research center.

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13.

Over ten billion years ago, the Milky Way was just a giant ________________ gas cloud.

14.

Simple ________________ designs with bold vertical and horizontal lines dominated the hotel’s decor.

15.

Detailed knowledge of psychology is not a ________________ for interviewing of job applicants.

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retro Retro means “back,” “behind,” or “backward” in Latin. Retro itself is a fairly new word in English, meaning “nostalgically old-fashioned,” usually when describing styles or fashions. To retrogress is to go back to an earlier and usually worse state. A retrograde action is a backward or reverse action. retroactive \µre-tro ¯ -'ak-tiv\ Intended to apply or take effect at a date in the past. • The fact that the tax hike was retroactive was what annoyed the public the most. retrofit \'re-tro ¯ -µfit\ To furnish something with new or modified parts or equipment. • Owners were offered “fast-track” permits to retrofit their homes against earthquakes. retrospective \µre-tr@-'spek-tiv\ Of or relating to surveying the past. • Excitement grew in anticipation of the rare retrospective exhibition of Avedon’s photographs. scrib / scrip These roots come from the Latin verb scribere, “to write.” A script is written matter, such as lines for a play. Scriptures are sacred writings. Scribble means to write or draw carelessly. A written work that hasn’t been published is a manuscript. circumscribe \'s@r-k@m-µskrı¯b\ To limit the range or activity of. • Various laws have circumscribed the freedom of labor unions to strike and organize. inscribe \in-'skrı¯b\ 1: To write, engrave, or print. 2: To dedicate (a book) to someone. • As Mike turned to leave, the store clerk offered to inscribe the diamond ring free. proscribe \pro ¯ -'skrı¯b\ 1: To prohibit. 2: To condemn or forbid as harmful. • If the doctor proscribes certain foods, you’d better not eat them. sub Sub means “under,” as in subway, submarine, and substandard. A subject is a person who is under the authority of another. Subconscious activity exists in the mind just under the level of awareness. To subdue is to bring under control. subjugate \'s@b-ji-µga¯t\ To bring under control; conquer; subdue. • Bringing criminal charges against reporters seemed a government attempt to subjugate the media. subliminal \s@-'bli-m@-n@l\ Not quite strong enough to be sensed or perceived consciously. • Worried parents claimed that some songs contained disturbing subliminal messages. subversive \s@b-'v@r-siv\ 1: Tending to overthrow or undermine by working secretly from within. 2: Tending to corrupt someone or something by weakening loyalty, morals, or faith. • In the 1950s the nation became alarmed that subversive communists were lurking everywhere.

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syn From the Greek word meaning “with” or “together with,” syn as a prefix in English can also mean “at the same time.” Thus synesthesia is the remarkable awareness of another sense (such as color) at the same time as the one being stimulated (such as sound). Synergy is the useful “working together” of distinct elements. Syntax is about how words are put together. synthesis \'sin-th@-s@s\ The combination of parts or elements into a whole. • Chemical analysis separates a substance into its elements; chemical synthesis combines elements to produce something new. synopsis \s@-'näp-s@s\ A condensed statement or outline. • Having read the synopsis, Bill did not feel a need to read the full report. syndrome \'sin-µdro ¯ m\ A group of signs and symptoms that occur together and characterize a particular abnormality. • Sufferers from chronic fatigue syndrome fought for a decade to have their symptoms recognized as a specific illness. tele Tele comes from the Greek word for “far off”; in English its basic meaning is “distant” or “at a distance.” A telescope looks at faraway objects. A telephoto lens on a camera magnifies distant objects for a photograph. A television allows us to watch things taking place far away (or sometimes not far enough away). teleological \µte-le¯-@-'lä-ji-k@l\ Relating to design, purpose, or cause, especially in nature. • The traditional teleological argument claims that humans are so remarkable that only God could have designed them. telepathic \µte-l@-'pa-thik\ Communicating from one mind to another without known sensory means. • Suzanne never considered herself telepathic, but she awoke with a start when her brother died at 2:00 a.m. 3,000 miles away. telemetry \t@-'le-m@-tre¯\ The transmission, especially by radio, of measurements made by automatic instruments to a distant station. • Satellite telemetry allowed the tracking of this year’s great caribou migration.

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UNIT 6

QUIZ 9 Answers appear at the end of this unit.

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1.

Highly responsive to each other’s actions, the twins at times seemed almost ________________.

2.

The catalog featuring vintage dinnerware of the 1940s through the 1970s was really a ________________ display of modern design.

3.

As a semi-invalid, she led a ________________d life, rarely venturing beyond her garden.

4.

Approval of the pay increase was confirmed, ________________ to January 1st.

5.

Did an early experiment in ________________ advertising at a movie theater result in increased popcorn sales?

6.

A cherub helped an angel ________________ words so beautiful they fell like roses from her feather pen.

7.

Her new album seemed to be a ________________ of country and world music.

8.

The scary part was when the ________________ failed and the astronauts vanished from the screens.

9.

Once in power, the mullahs proceeded to ________________ the Westernized women of Tehran.

10.

Smoking is now ________________d in many U.S. medical and restaurant settings.

11.

A Hollywood-based Web site offers a helpful ________________ of the plots of hundreds of films.

12.

The mayor hoped to ________________ the vehicles to increase the mobility of the disabled.

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13.

The new special police unit was entrusted with intelligence gathering and monitoring _________________ activities.

14.

Schizophrenia is a ________________ related to a variety of causative factors.

15.

The claim that a gopher’s cheek pouches are intended for carrying food is, to zoologists, a ________________ statement.

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terr This root was dug up from the Latin terra, “earth.” Terra firma is a Latin phrase that means “firm ground,” as opposed to the swaying seas. A terrace is a leveled area along a sloping hill; territory is a specific piece of land. A terrier, literally an “earth dog,” was originally used by hunters to dig for small game. subterranean \µs@b-t@-'ra¯-ne¯-@n\ Underground. • The region, it was believed, was home to subterranean beings that emerged from their burrows only at night. terrestrial \t@-'res-tre¯-@l\ 1: Having to do with the earth or its inhabitants. 2: Having to do with land as distinct from air or water. • Unlike frogs, most toads are terrestrial, entering the water only to lay their eggs. terrain \t@-'ra¯n\ The surface features of an area of land. • Mountain unicycling proved especially challenging over such rough terrain. therm Still warm from centuries of use, therm comes from the Greek word meaning “heat.” A thermometer measures heat; a thermostat makes sure it stays at the same level. A rising body of warm air, used by hawks and sailplanes, is called a thermal. thermal \'th@r-m@l\ 1: Of, relating to, or caused by heat. 2: Designed to prevent loss of body heat. • Thermal vents on the ocean floor release steam as hot as 600°. thermodynamic \µth@r-mo ¯ -dı¯-'na-mik\ Of or relating to the physics of heat. • A chemical’s thermodynamic properties indicate how it will behave at various temperatures. thermonuclear \µth@r-mo ¯ -'nü-kle¯-@r\ Of or relating to changes in the nucleus of atoms of low atomic weight brought about by very high temperatures. • In those days thermonuclear devices were being proposed for such uses as excavating canals. trans This root comes across from Latin to indicate movement “through, across, or beyond” something. A translation carries the meaning across languages. A TV signal is transmitted or “sent through” the air (or a cable) to your set. Public transportation carries you across a distance, though you may need to transfer from one bus or subway across to another. transient \'tran-sh@nt\ 1: Passing through a place and staying only briefly. 2: Of brief duration. • Tristan’s inn in Vermont attracted transient tourists to come to gaze at the autumn foliage. transcendent \tran-'sen-d@nt\ 1: Exceeding usual limits; surpassing. 2: Beyond comprehension. • The symphony’s hushed ending, with the solo violin melody trailing off into silence, is almost transcendent. transfusion \trans-'fyü-zh@n\ 1: The process of diffusing into or through. 2: The process of moving (as of blood) into a vein. • Travelers needing blood transfusions have usually suffered severe accidents.

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uni Uni comes from the Latin word for “one.” A uniform is clothing of one design. A united group has one opinion or forms one unit. A unitard is a one-piece combination leotard and tights, very good for skating, skiing, dancing—or riding a one-wheeled unicycle. unicameral \µyü-ni-'kam-r@l\ Having a single legislative house or chamber. • Passing new laws was comparatively quick and easy in the unicameral government. unilateral \µyü-ni-'la-t@-r@l\ Having, affecting, or done by one side only. • Russia’s unilateral withdrawal from Afghanistan, in return for nothing, astonished the world. unison \'yü-n@-s@n\ 1: Sameness of musical pitch. 2: A state of harmonious agreement; accord. • Unable to read music well enough to harmonize, the village choir sang only in unison. viv Viv comes from vivere, the Latin verb meaning “to live or be alive.” A vivid memory is a lively one. A survivor has lived through something terrible. A revival brings something back to life—whether an old film, interest in a long-dead novelist, or the religious faith of a group. vivacious \v@-'va¯-sh@s\ Lively, sprightly. • For the cheerleading squad, Sheri chose the most outgoing, energetic, and vivacious candidates. vivisection \µvi-v@-'sek-sh@n\ Experimental operation on a living animal. • The firm reluctantly agreed to avoid research involving vivisection in favor of alternative methods. convivial \k@n-'viv-y@l\ 1: Enjoying companionship in feasting and drinking. 2: Festive. • Alberta was known for hosting relaxed and convivial gatherings, where the wine flowed freely.

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UNIT 6

QUIZ 10 Answers appear at the end of this unit.

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1.

At the height of the Cold War, some Americans began digging ________________ fallout shelters.

2.

The ________________ properties of metals affect technologies we don’t think of as heat-related.

3.

Forty-nine states have bicameral legislatures; only Nebraska’s is _______________.

4.

Any chemical reaction that produces heat is a ________________ reaction.

5.

Over such rugged ________________, mules were the only hope for transporting needed supplies.

6.

It’s a noisy, _________________ crowd that gathers at McSorley’s Restaurant after 5:00.

7.

His bright idea turned out to be a ________________ one, and he had soon moved on to something new.

8.

He returned from the backpacking trip energized as if he’d been given a ______________ of new blood.

9.

Detonating a ________________ bomb requires temperatures exceeding a million degrees Fahrenheit.

10.

While singing in parts is difficult, singing modern compositions for ________________ voices has challenges of its own.

11.

Jessie was so ________________ that she livened up every party she ever attended.

12.

She emerged from the concert hall in a daze, feeling she had undergone a ________________ experience.

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13.

Alabama has over 500 species of marine mollusks, and many ________________ mollusks as well.

14.

Animal lovers of every stripe wrote in, claiming that ________________ had little scientific merit.

15.

After failed negotiations with its neighbors, Iran announced a ________________ decision to develop its own oil wells in the Caspian Sea.

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REVIEW TEST Fill in each blank in the sentences on the following pages with one of the following words. Answers appear at the end of this unit. abjure agrarian agrochemical agronomy amorphous anachronism antecedent antedate anterior anthropoid aquaculture Aquarius aquifer artifact artifice artisan benediction benefactor beneficent bionic biopsy cacophony chronic chronology circumscribe circumspect circumstantial circumvent confidante convivial cosmology cosmopolitan credence credible creed dehydrate disarming disburse

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discredit dynamo dynasty dysentery dyslexia dystrophy epigraph epilogue epithet extraneous extrapolate extrovert fidelity fiduciary geopolitical geosynchronous geothermal gratify gratuitous hydraulic hydrodynamic hydroelectric hyperbole hypertension hyperventilate hypochondriac hypothermia hypothetical immutable ingratiate inscribe intercede interdict interface jurisprudence malevolent malign malpractice

marina mariner maritime metamorphosis microcosm misanthrope moribund morphology mortality mortify neoclassical Neolithic neoplasm omnibus omnipotent omniscient orthodontic orthodox orthopedist panacea panoply pantheon perjury permutation philanthropy phonetic photoelectric photon photosynthesis polyphonic posterior posthumous postmortem precocious predisposed prerequisite primal

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primate primordial proscribe rectify rectilinear rectitude retroactive retrofit retrospective seismograph spectrography subjugate subliminal subterranean subversive symbiosis syndrome synopsis synthesis telemetry teleological telepathic terrain terrestrial thermal thermodynamic thermonuclear topography transcendent transfusion transient transmute unicameral unilateral unison vivacious vivisection

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1.

Keith ________________d the novel “To Melissa, my only muse and inspiration.”

2.

After the first trial, Collins was called to answer charges of ________________ and evidence tampering.

3.

Is it ________________ to say that an eagle’s wings were “designed” for soaring?

4.

Rafael’s clumsy attempt to ________________ the contract led to his arrest for fraud.

5.

Some interactive games let players achieve virtual destruction worse than that of a ________________ bomb.

6.

After his divorce, his legal practice shrank and a ________________ suit almost bankrupted him.

7.

________________ apes resemble humans in that they lack tails and walk semi-erect.

8.

The “facts” on the “Astounding Facts” Web site turned out not to be very ________________.

9.

________________ runoff is blamed for creating a huge “dead zone” in the Gulf of Mexico.

10.

Most people picture ________________s as underground lakes rather than as expanses of soaked gravel.

11.

The Water-Carrier, ________________, is an old constellation carved in stones of the Babylonian Empire.

12.

The formal gardens were showplaces of ________________, with every tree and shrub shaped by human hands.

13.

Aaron’s fossil hunting in Alaska led to his unearthing of unusual ancient ________________s.

14.

The prison’s star inmate, he had undergone a ________________ from hardened criminal to contributing citizen.

15.

The blonde Evita was seen by Argentina’s poor as a ________________ angel dispensing charity.

16.

Paleolithic hunters, with their tools of chipped stone, gave way to ________________ farmers, with their polished stone tools.

17.

What anonymous ________________ had contributed $500,000 to the medical fund?

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18.

Her aunt, previously blind, could now recognize faces with her new ________________ eye.

19.

For years Carol had managed her ________________ heart condition through careful diet and exercise.

20.

Using a telephone in a play set in 1765 is an obvious ________________.

21.

Veterinarians have often relied on ________________ examinations in diagnosing disease.

22.

A newly hired 22-year-old had easily managed to ________________ the computer security system.

23.

The sunny and ________________ Doris Day started out as a jazz singer in the 1940s.

24.

When asked about Russia’s own success fighting corruption, the official quickly became ________________.

25.

The club was chic and ________________, and everyone seemed to have a French or German accent.

26.

Rural Maine is home to many ________________s: woodworkers, potters, weavers, and the like.

27.

The findings of Copernicus and Galileo proposed nothing less than a new ________________.

28.

Some building codes require ________________ sensors, which are quick to detect smoky fires.

29.

The ________________ provided needed services after an exhausting day on choppy seas.

30.

The game of Monopoly seems to present a ________________ of the world of real-estate dealing.

31.

Scientists often refer to the ocean’s surface as the ocean-atmosphere ________________.

32.

A new find lends ________________ to the claim that the first Americans came from Europe.

33.

She was skeptical about what lay behind his smooth and ________________ manner.

34.

Society often depends on ________________ to fill the gaps left by government spending.

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35.

The new president turned out to be a ferociously energetic human ________________.

36.

Each winter, outdoor adventure groups often publicize the best ways to avoid frostbite and ________________.

37.

She was always reading about alternative therapies, but her doctor was as ________________ as they come.

38.

Muscular ________________ is actually a family of disorders that causes muscle degeneration.

39.

A passage from Othello appeared as the ________________ of the longawaited report.

40.

The author’s ________________ listed the adventurers’ whereabouts five years after their rescue.

41.

She told him his concerns were ________________ and he should stick to the subject at hand.

42.

A data recorder and transmitters and receivers formed part of the satellite’s ________________ system.

43.

A prominent political ________________, the Kennedy family has seen many of its members elected to office.

44.

He was a drifter, hardly the kind of person for a ________________ responsibility such as executor of a will.

45.

“Attack ads” attempt to ________________ political candidates, often with half-truths and lies.

46.

Whether the coup succeeded or failed depended on the ________________ of the general’s soldiers.

47.

The clinic, in Canada’s far north, serves a ________________ Inuit population not likely to return for regular checkups.

48.

Could ________________ tensions in faraway Asia actually affect the national elections?

49.

Study of the sun’s magnetic fields requires ________________ to reveal the solar spectrum.

50.

For the America’s Cup yachts, the keel by itself presents complex ________________ problems.

51.

Severe stomach distress ruined their trip, and it turned out they both had ________________.

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52.

The Princeton Earth Physics Project tracks earthquakes using a network of ________________s.

53.

There were arguments about how best to ________________ financial aid following the disaster.

54.

Sandra knew it would ________________ her husband if she wore the necklace he’d given her.

55.

The fetus had gotten turned around into the ________________ position, which can make the birthing process difficult.

56.

Much modern ________________ still is practiced in small mom-and-pop fishpond operations.

57.

Those lurching virtual-reality thrill rides are powered by ________________, pressurized-fluid technology.

58.

Nick, the family ________________, played touch football, organized reunions, and was his company’s top salesman.

59.

Astonishing examples of cooperative living between species appear as ________________.

60.

Having quit smoking, he was told he must now adopt a strict diet for his ________________.

61.

The meteorite, billions of years old, offered clues about the ________________ solar system.

62.

Best-selling novelists can sell a book idea to their publishers with nothing but a short ________________.

63.

The Garden of Eden is the biblical vision of a ________________ paradise.

64.

Melanie’s problems with spelling and math were finally traced to ________________.

65.

Having entered the virtual body, the doctor may test a range of ________________ drug interactions.

66.

The response from the opposition was full of ________________ insults and slurs.

67.

A United Nations force was asked to ________________ on behalf of both combatants and restore peace.

68.

With public-relations help, the Nigerians hoped to ________________ reports of genocide during the Biafran war.

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69.

As a context for discussion, Abu handed out a detailed ________________ of Muslim history.

70.

At halftime, the ________________ mass of band musicians abruptly snapped into a tight formation.

71.

The greatest Supreme Court justices could often be called philosophers of ________________.

72.

The awesome ________________ of the procession made the hometown parade seem like a coronation.

73.

Fascinated by plant breeding, Heather began to enroll in ________________ courses.

74.

After a terrible two-month binge, he solemnly ________________d alcohol forever.

75.

That nasty remark was her first hint that her beloved Alex had a ________________ streak.

76.

Two ________________ power plants were being built near the volcano’s base.

77.

Following the hijackings, the airline was forced to ________________ its jets with new cockpit doors.

78.

To celebrate its seagoing history, the port city established a ________________ museum.

79.

A specialist in vertebrate ________________, he usually explained skeleton structures in terms of evolution.

80.

Looser laws in Canada may make it harder for the U.S. to ________________ drug trafficking.

81.

Devout medieval Christians sought to “________________ the flesh”—to reduce their sensitivity to hunger, cold, and discomfort.

82.

Shuffling the deck ensures that the cards will be dealt in almost infinite ________________s.

83.

The foes of freedom have tried to suppress books, films, and songs, calling them ________________.

84.

The congregation then recites the ________________, a concise statement of Christian beliefs.

85.

Many ________________ sculptures from the 1780s could be mistaken for works from ancient Rome.

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86.

It would take a heartfelt apology to ________________ the situation.

87.

Before taking Tree Physiology, you must have completed such ________________s as Forest Botany.

88.

The Senate finally threw everything together into a single ________________ bill.

89.

Though there had been no eyewitnesses, the ________________ evidence was enough to convict him.

90.

After a time, the supposed illnesses of a ________________ no longer attract the sympathy of friends.

91.

The investigation was focusing on a 48-year-old man—a single, unemployed loner and ________________.

92.

One need not be ________________ to write an encyclopedia, but it would help.

93.

The company produces a sports guard, dentures, braces, and other ________________ appliances.

94.

Lawyers argued that the state constitution ________________d the legislature’s power in cases like this one.

95.

Having failed to form a coalition, the president began to consider taking ________________ action.

96.

Apollo and Dionysus were two of the most widely worshiped gods in the Greek ________________.

97.

New to New York, Carol couldn’t fall asleep with the ________________ of street sounds.

98.

With her fortune declining, even Lady Armstrong could see that the great estate was ________________.

99.

To prevent moisture-related spoilage, Gretchen said we could ________________ some foods.

100.

The lawyers had argued that her injuries were actually ________________ to the accident.

101.

With a liver ________________, a small piece of tissue can be examined for signs of disease.

102.

Several simultaneous melodies combined to form a rich texture of ________________ sound.

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103.

The river’s ________________ dams block downstream movement of large wood, disturbing aquatic habitats.

104.

The verdict awarded her complete survivor’s benefits, ________________ to the date of her husband’s death.

105.

She would later claim that she had ________________d her grief into the songs that made her famous.

106.

The university awarded ________________ degrees to seniors killed in the crash.

107.

Sandra tried using ________________ motivational tapes while sleeping to improve her attitude.

108.

________________ reading ability in children may not be matched by advanced writing skills.

109.

Lottman’s film was an odd ________________ of ancient myth, film noir, and alternative comics.

110.

Gazing into a campfire at night, we feel a ________________ connection with our prehistoric ancestors.

111.

Any system that turns heat into mechanical energy represents a ________________ process.

112.

A salty ________________ may throw a tub of sea jargon at you to expose your ignorance.

113.

It required all his charm to ________________ himself with the power brokers.

114.

Several Roman emperors, convinced that they were ________________, declared themselves gods.

115.

Sitting up straight at the table isn’t necessarily an outward sign of moral ________________.

116.

The U.S. Geographic Survey has modeled and mapped the entire American ________________.

117.

The center compiled data on illness and ________________ from blood diseases.

118.

The region’s ________________ is dramatic, with sheer cliffs descending to parched plains.

119.

Hilda’s sleek, ________________ designs featured sharp clean lines and squared corners.

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120.

His CAT scan revealed a large ________________, but it turned out to be harmless.

121.

With remarkable skill and patience, the ________________ had restored Tyler’s spine by surgical means.

122.

The ________________ exhibit on Project Apollo began with its birth in 1961.

123.

Brett’s doctors called on his close relatives to donate blood for the ________________.

124.

Even in war there are rules and norms of behavior that ________________ the worst offenses.

125.

The Founding Fathers rejected the idea of a ________________ legislature, favoring a House and Senate to balance each other.

126.

Wherever the great khan’s army marched, it would conquer and ________________ the local tribes.

127.

From these incomplete statistics we can easily ________________ the complete data.

128.

People with this rare ________________ are smart and mentally retarded at the same time.

129.

Some groups argue that ________________ is a barbaric and unjustified form of animal cruelty.

130.

Audrey claims to have ________________ communication with her pet ferrets while she’s at the office.

131.

Many have wondered if some murderers were biologically ________________ to kill.

132.

In 1880 a traveling salesman might have tried to sell you a single ________________ for everything from mumps to arthritis.

133.

Prehistoric peoples in harsh climates often lived in caves or even ________________ dwellings.

134.

He delivered his praise as solemnly as a priest’s ________________.

135.

________________ orbits are ideal for maintaining contact with a specific location on Earth.

136.

We distinguish between outright lies on the one hand and mere ________________ on the other.

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137.

These get-togethers start out quietly but always become ________________, and sometimes even rowdy.

138.

She emitted the kind of radiant energy that isn’t measured in ________________s.

139.

Giant tubeworms live on the ocean floor near ________________ vents spouting scalding water.

140.

The broken jawbone was clearly visible in the X-ray image that showed an ________________ view of his skull.

141.

The mob outside the ________________ research center called for an end to tests on monkeys.

142.

After awakening from her coma, she recounted a ________________ experience of light and bliss.

143.

Phyllis used the ________________ approach with her first-graders, sounding out syllables one by one.

144.

The student complaint involved the alleged yelling of racial ________________s.

145.

The new Web site, called “________________.com,” is “for those who like to tell and those who like to listen.”

146.

It’s common, but also dangerous, for freedivers to ________________ in order to stay underwater longer.

147.

The rebellious workers began chanting in ________________, “No Contract, No Work!”

148.

Economic growth in poor countries often depends on ________________ reform and rural development.

149.

In green plants, light energy is converted into chemical energy during ________________.

150.

Alicia kept a fixed and ________________ order to her household, especially in the sock drawers.

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UNIT 6

50 MORE ROOTS The roots and derived words in the table below are intended for further study. Learn the meanings of any of the words you are unfamiliar with (perhaps by drilling yourself with homemade flash cards), and try using each of them in sentences. Try to think of other terms that use each of the roots in the left-hand column. aud (“hear”)

auditor

audition

auditory

aut/auto (“same, self”)

automaton

autonomy

autocratic

bell (“war”)

bellicose

belligerent

antebellum

bi (“two”)

bipartisan

binary

bipolar

carn (“flesh”)

carnage

incarnation

carnal

cata (“down”)

catalyst

catacomb

catatonic

cent (“hundred”)

centenary

centigrade

centimeter

cid (“kill”)

genocide

infanticide

fungicide

corp (“body”)

corporal

corpulent

corporeal

crac/crat (“power”)

bureaucrat

aristocracy

autocrat

crypt/cryph (“hidden”)

cryptic

apocryphal

crypt

culp (“guilt”)

culpable

exculpate

mea culpa

cur (“care”)

curator

sinecure

curative

dec (“ten”)

decathlon

decimate

decibel

demo (“people”)

demotic

endemic

demographic

dict (“speak”)

diction

edict

indict

domin (“lord”)

domineer

predominant

dominion

duct (“lead”)

abduct

duct

induct

ego (“I”)

alter ego

egocentric

egoist

equi (“equal”)

equivocal

equity

equilibrium

eu (“good”)

euphemism

euphoria

euthanasia

flu (“flow”)

influx

confluence

fluent

grad (“step”)

degradation

gradient

gradation

grav (“heavy”)

grave

gravitate

gravitas

hemi/demi/semi (“half”)

hemiplegic

semiconductor

demigod

homo (“same”)

homogeneous

homogenize

homologous

later (“side”)

bilateral

collateral

unilateral

medi (“middle”)

mediate

intermediary

median

mono (“single”)

monotone

monologue

monotheism

neuro (“nerve”)

neurology

neuron

neurotransmitter

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50 MORE ROOTS

(continued)

nom (“name”)

misnomer

nomenclature

nominal

patr/pater (“father”)

patriarch

patrimony

patrician

pun/pen (“punish”)

punitive

impunity

penal

peri (“around”)

peripheral

peripatetic

perimeter

phob (“fear”)

agoraphobia

xenophobia

acrophobia

plac (“please”)

placate

implacable

placebo

popul (“people”)

populist

populace

depopulate

proto (“first”)

protocol

protagonist

prototype

quadr (“four”)

quadrennial

quadriplegic

quadruped

sacr/sanct (“holy”)

sanctify

sacrosanct

sanctuary

simil/simul (“like”)

simile

simulate

assimilate

son (“sound”)

sonority

sonata

sonic

super/supra (“above”)

superannuated

superimpose

superfluous

the/theo (“god”)

theocracy

monotheism

theology

topo (“place”)

topical

topographical

utopia

tri (“three”)

trilogy

trinity

trimester

turb (“confused”)

perturb

turbid

turbine

ver/veri (“true”)

aver

veracity

veritable

verb (“word”)

verbiage

verbose

proverb

vert (“turn”)

subvert

revert

avert

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UNIT 6

ANSWER KEY

QUIZ 1 1. 2. 3. 4. 5.

philanthropy agrochemical artifact agrarian aquifer

6. 7. 8. 9. 10.

anterior agronomy artifice antecedent aquaculture

11. 12. 13. 14. 15.

misanthrope antedate anthropoid Aquarius artisan

6. 7. 8. 9. 10.

bionic biopsy benefactor circumvent chronic

11. 12. 13. 14. 15.

anachronism chronology microcosm circumspect cosmopolitan

6. 7. 8. 9. 10.

dyslexia dynasty epigraph hydrodynamic disburse

11. 12. 13. 14. 15.

epilogue dystrophy epithet credible dynamo

6. 7. 8. 9. 10.

ingratiate geothermal spectrography extrapolate seismograph

11. 12. 13. 14. 15.

fidelity gratuitous topography gratify geopolitical

6. 7. 8. 9. 10.

hypertension hypochondriac intercede hypothermia interface

11. 12. 13. 14. 15.

abjure dehydrate jurisprudence hyperbole interdict

6. 7. 8. 9. 10.

mariner mortify amorphous malign morphology

11. 12. 13. 14. 15.

moribund permutation mortality maritime immutable

QUIZ 2 1. 2. 3. 4. 5.

symbiosis beneficent circumstantial cosmology benediction

QUIZ 3 1. 2. 3. 4. 5.

creed disarming dysentery discredit credence

QUIZ 4 1. 2. 3. 4. 5.

extrovert fiduciary extraneous confidante geosynchronous

QUIZ 5 1. 2. 3. 4. 5.

perjury hypothetical hydroelectric hyperventilate hydraulic

QUIZ 6 1. 2. 3. 4. 5.

marina malpractice metamorphosis malevolent transmute

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QUIZ 7 1. 2. 3. 4. 5.

panacea Neolithic omnipotent orthodox omnibus

6. 7. 8. 9. 10.

pantheon phonetic neoplasm orthopedist polyphonic

11. 12. 13. 14. 15.

omniscient neoclassical orthodontic panoply cacophony

6. 7. 8. 9. 10.

photon primal predisposed photosynthesis rectify

11. 12. 13. 14. 15.

postmortem primate amorphous rectilinear prerequisite

6. 7. 8. 9. 10.

inscribe synthesis telemetry subjugate proscribe

11. 12. 13. 14. 15.

synopsis retrofit subversive syndrome teleological

6. 7. 8. 9. 10.

convivial transient transfusion thermonuclear unison

11. 12. 13. 14. 15.

vivacious transcendent terrestrial vivisection unilateral

QUIZ 8 1. 2. 3. 4. 5.

posterior photoelectric rectitude precocious posthumous

QUIZ 9 1. 2. 3. 4. 5.

telepathic retrospective circumscribe retroactive subliminal

QUIZ 10 1. 2. 3. 4. 5.

subterranean thermal unicameral thermodynamic terrain

REVIEW TEST 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

inscribe perjury teleological antedate thermonuclear malpractice anthropoid credible agrochemical aquifer Aquarius artifice artifact metamorphosis beneficent

GRE CAT Success

16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30.

Neolithic benefactor bionic chronic anachronism postmortem circumvent vivacious circumspect cosmopolitan artisan cosmology photoelectric marina microcosm 165

31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45.

interface credence disarming philanthropy dynamo hypothermia orthodox dystrophy epigraph epilogue extraneous telemetry dynasty fiduciary malign

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UNIT 6

46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80.

fidelity transient geopolitical spectrography hydrodynamic dysentery seismograph disburse gratify posterior aquaculture hydraulic extrovert symbiosis hypertension primordial synopsis terrestrial dyslexia hypothetical gratuitous intercede discredit chronology amorphous jurisprudence panoply agronomy abjure malevolent geothermal retrofit maritime morphology interdict

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81. 82. 83. 84. 85. 86. 87. 88. 89. 90. 91. 92. 93. 94. 95. 96. 97. 98. 99. 100. 101. 102. 103. 104. 105. 106. 107. 108. 109. 110. 111. 112. 113. 114. 115.

mortify permutation subversive creed neoclassical rectify prerequisite omnibus circumstantial hypochondriac misanthrope omniscient orthodontic circumscribe unilateral pantheon cacophony moribund dehydrate antecedent biopsy polyphonic hydroelectric retroactive transmute posthumous subliminal precocious synthesis primal thermodynamic mariner ingratiate omnipotent rectitude

166

116. 117. 118. 119. 120. 121. 122. 123. 124. 125. 126. 127. 128. 129. 130. 131. 132. 133. 134. 135. 136. 137. 138. 139. 140. 141. 142. 143. 144. 145. 146. 147. 148. 149. 150.

terrain mortality topography rectilinear neoplasm orthopedist retrospective transfusion proscribe unicameral subjugate extrapolate syndrome vivisection telepathic predisposed panacea subterranean benediction geosynchronous hyperbole convivial photon thermal anterior primate transcendent phonetic epithet confidante hyperventilate unison agrarian photosynthesis immutable

GRE CAT Success

R E D A LERT QUANTITATIVE ABILITY STRATEGIES The GRE CAT test combines three different types of questions: Quantitative Comparisons, Problem Solving, and Data Interpretation. There are a total of 28 questions in this section, and you will have 45 minutes in which to complete this section. The following is the breakdown of question types. Quantitative Comparison Problem Solving (multiple-choice) Data Interpretation (tables, charts, graphs)

14 questions 9 questions 5 questions

All of the questions in this section are based on the mathematics that is usually covered in high school math classes—arithmetic, algebra, and geometry. There are two different types of arithmetic questions that will appear on the GRE—one that asks you to perform a computation (add the fractions, multiply the decimals, manipulate the percents), and one that asks you to solve a word problem. Similarly, there will be algebraic computation problems (solve the equation, factor the expression, manipulate the square roots), as well as algebraic word problems. As far as the geometry problems are concerned, you will only be asked to solve problems by working with geometric properties. You will not need to create proofs or state definitions. In the following pages, you will find a thorough review of all of the mathematics covered on the GRE CAT. Prior to that, hints and strategies for the Quantitative Ability sections are given, as well as a discussion of the Quantitative Comparison and Data Interpretation formats. Read these sections carefully, and remember what you have read when you begin to work the practice tests.

HINTS

AND

STRATEGIES

FOR

QUANTITATIVE ABILITY QUESTIONS

1. If you are not able to answer a question in one or two minutes (at the most), take a guess. If you don’t answer a question, you can’t move on. Wasting time with a question may mean you won’t have time to work on subsequent ones. Keep in mind that as you answer and complete a question, the next one will be slightly more difficult if you answered it correctly, and a little easier if you answered it incorrectly. 2. Do not waste any time doing computations that are not necessary. Remember that one of the choices must be the correct one. Estimate as much as you possibly can as you try to determine which of the answers must be correct. 3. Be particularly careful when answering Quantitative Comparison questions, as these are the only questions on the entire GRE for which there are four answer choices instead of the usual five. Never mark (E) as the answer for a Quantitative Comparison question.

RED

167

ALERT

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4. Be careful (especially when solving geometry problems) to express your answer in the same units of measure as the multiple-choice answers. 5. All fractions that appear as the answers to questions will be expressed in reduced form. Therefore, if you solve a problem and obtain a fraction as the answer, this fraction must be reduced before you will find it among the multiple-choice answers. Similarly, all square root answers must be expressed in reduced form. In geometrical problems involving p, look at the answer choices to determine if you are supposed to leave the answer in terms of p or 22 use the approximate value . 7 6. Of course, you are not permitted to use a calculator to perform your computations. This means that you should brush up on the rules for multiplying and dividing numbers with decimals, etc. However, the problems are, in general, designed not to include messy computations. If you ever find yourself thinking “I wish I had a calculator to help me with this problem,” look at the problem again carefully. There is probably an easier way to do it that you may have missed. 7. If the answer you obtain doesn’t match one of the choices given, it might still be right. Try to write it in a different form, and then see if it matches. For example, the answer 2x 1 3x can also be written as x(x 1 3). 8. Make sure to answer the question that is being asked. Sometimes people get a problem wrong because, after finding the value of x, they choose that value as the answer, when the problem was actually asking for the value of x 1 2. 9. If you are stuck, try looking at the multiple-choice answers. Since one of them has to be right, the answers may give you some idea of how to proceed. 10. If you have no idea how to answer a question, be sure to make your best guess before moving on. Since there is no penalty on the GRE for a wrong answer, never leave a question unanswered. In the following sections, all of the principles of mathematics that you need to know for the GRE are reviewed. There are also numerous problems for you to solve. If you have difficulty with any of them, first check the answers carefully, and then review the material again. Finally, try to answer the questions without using the answers.

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Unit 7 MATHEMATICS REVIEW ARITHMETIC

WHOLE NUMBERS Definitions The set of numbers {1, 2, 3, 4, . . .} is called the set of counting numbers and/or natural numbers, and/or sometimes the set of positive integers. (The notation, { }, means “set” or collection, and the three dots after the number 4 indicate that the list continues without end.) Zero is usually not considered one of the counting numbers. Together, the counting numbers and zero make up the set of whole numbers. Place Value Whole numbers are expressed in a system of tens, called the decimal system. Ten digits—0, 1, 2, 3, 4, 5, 6, 7, 8, and 9—are used. Each digit differs not only in face value but also in place value, depending on where it stands in the number. Example 1 237 means: (2 { 100) 1 (3 { 10) 1 (7 { l) The digit 2 has face value 2 but place value of 200. Example 2 35,412 can be written as: (3 { 10,000) 1 (5 { 1000) 1 (4 { 100) 1 (1 { 10) 1 (2 { 1) The digit in the last place on the right is said to be in the units or ones place; the digit to the left of that in the tens place; the next digit to the left of that in the hundreds place; and so on.

Odd and Even Numbers A whole number is even if it is divisible by 2; it is odd if it is not divisible by 2. Zero is thus an even number. Example 2, 4, 6, 8, and 320 are even numbers; 3, 7, 9, 21, and 45 are odd numbers.

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Prime Numbers The positive integer p is said to be a prime number (or simply a prime) if p 5 1, or the only positive divisors of p are itself and 1. The positive integer 1 is called a unit. The first ten primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29. All other positive integers that are neither 1 nor prime are composite numbers. Composite numbers can be factored, that is, expressed as products of their divisors or factors; for example, 56 5 7 { 8 5 7 { 4 { 2. In particular, composite numbers can be expressed as products of their prime factors in just one way (except for order). To factor a composite number into its prime factors, proceed as follows. First try to divide the number by the prime number 2. If this is successful, continue to divide by 2 until an odd number is obtained. Then attempt to divide the last quotient by the prime number 3 and by 3 again, as many times as possible. Then move on to dividing by the prime number 5 and other successive primes until a prime quotient is obtained. Express the original number as a product of all its prime divisors. Example Find the prime factors of 210. 2 )210 3 )105 5 ) 35 7 Therefore: 210 5 2 { 3 { 5 { 7 (written in any order) and 210 is an integer multiple of 2, of 3, of 5, and of 7.

Consecutive Whole Numbers Numbers are consecutive if each number is the successor of the number that precedes it. In a consecutive series of whole numbers, an odd number is always followed by an even number and an even number by an odd. If three consecutive whole numbers are given, either two of them are odd and one is even or two are even and one is odd. Example 1 7, 8, 9, 10, and 11 are consecutive whole numbers. Example 2 8, 10, 12, and 14 are consecutive even numbers. Example 3 21, 23, 25, and 27 are consecutive odd numbers.

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MATHEMATICS REVIEW

Example 4 21, 23, and 27 are not consecutive odd numbers because 25 is missing.

The Number Line A useful method of representing numbers geometrically makes it easier to understand numbers. It is called the number line. Draw a horizontal line, considered to extend without end in both directions. Select some point on the line and label it with the number 0. This point is called the origin. Choose some convenient distance as a unit of length. Take the point on the number line that lies one unit to the right of the origin and label it with the number 1. The point on the number line that is one unit to the right of 1 is labeled 2, and so on. In this way, every whole number is associated with one point on the line, but it is not true that every point on the line represents a whole number.

Ordering of Whole Numbers On the number line the point representing 8 lies to the right of the point representing 5, and we say 8 . 5 (read “8 is greater than 5”). One can also say 5 , 8 (“5 is less than 8”). For any two whole numbers a and b, there are always three possibilities: a , b,

a 5 b,

or

a.b

If a 5 b, the points representing the numbers a and b coincide on the number line.

Operations with Whole Numbers The basic operations on whole numbers are addition (1), subtraction (2), multiplication ({ or 3), and division (4). These are all binary operations—that is, one works with two numbers at a time in order to get a unique answer. The operations of addition and multiplication on whole numbers are said to be closed because the answer in each case is also a whole number. The operations of subtraction and division on whole numbers are not closed because the unique answer is not necessarily a member of the set of whole numbers. Examples 31457 4 { 3 5 12 2 2 5 5 23 3485

GRE CAT Success

3 8

a whole number a whole number not a whole number not a whole number

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UNIT 7

Addition If addition is a binary operation, how are three numbers—say, 3, 4, and 8—added? One way is to write: (3 1 4) 1 8 5 7 1 8 5 15 Another way is to write: 3 1 (4 1 8) 5 3 1 12 5 15 The parentheses merely group the numbers together. The fact that the same answer, 15, is obtained either way illustrates the associative property of addition: (r 1 s) 1 t 5 r 1 (s 1 t) The order in which whole numbers are added is immaterial—that is, 3 1 4 5 4 1 3. This principle is called the commutative property of addition. Most people use this property without realizing it when they add a column of numbers from the top down and then check their result by beginning over again from the bottom. (Even though there may be a long column of numbers, only two numbers are added at a time.) If 0 is added to any whole number, the whole number is unchanged. Zero is called the identity element for addition.

Subtraction Subtraction is the inverse of addition. The order in which the numbers are written is important; there is no commutative property for subtraction. 423Þ324 The Þ is read “not equal.”

Multiplication Multiplication is a commutative operation: 43 { 73 5 73 { 43 The result or answer in a multiplication problem is called the product. If a number is multiplied by 1, the number is unchanged; the identity element for multiplication is 1. Zero times any number is 0: 42 { 0 5 0 Multiplication can be expressed with several different symbols: 9 { 7 { 3 5 9 3 7 3 3 5 9(7)(3) Besides being commutative, multiplication is associative: (9 { 7) { 3 5 63 { 3 5 189 and 9 { (7 { 3) 5 9 { 21 5 189

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A number can be quickly multiplied by 10 by adding a zero at the right of the number. Similarly, a number can be multiplied by 100 by adding two zeros at the right: 38 { 10 5 380 and 100 { 76 5 7600

Division Division is the inverse of multiplication. It is not commutative: 844Þ448 The parts of a division example are named as follows: quotient divisorq dividend If a number is divided by 1, the quotient is the original number. Division by 0 is not defined (has no meaning). Zero divided by any number other than 0 is 0: 0 4 56 5 0

Divisors and Multiples The whole number b divides the whole number a if there exists a whole number k such that a 5 bk. The whole number a is then said to be an integer multiple of b, and b is called a divisor (or factor) of a. Example 1 3 divides 15 because 15 5 3 { 5. Thus, 3 is a divisor of 15 (and so is 5), and 15 is an integer multiple of 3 (and of 5). Example 2 3 does not divide 8 because 8 Þ 3k for a whole number k. Example 3 Divisors of 28 are 1, 2, 4, 7, 14, and 28. Example 4 Multiples of 3 are 3, 6, 9, 12, 15, . . .

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UNIT 7

QUIZ

WHOLE NUMBER PROBLEMS 1.

What is the prime factorization of 78?

2.

What are the divisors of 56?

3.

Which property is illustrated by the following statement? (3 1 5) 1 8 5 3 1 (5 1 8)

4.

Which property is illustrated by the following statement? (5 { 7) { 3 5 (7 { 5) { 3

5.

Find the first five multiples of 7. SOLUTIONS

1.

78 5 2 { 39 5 2 { 3 { 13

2.

The divisors of 56 are 1, 2, 4, 7, 8, 14, 28, 56

3.

The Associative Property of Addition

4.

The Commutative Property of Multiplication

5.

7, 14, 21, 28, 35

FRACTIONS Definitions a (or a/b) is called a fraction. The b upper part, a, is called the numerator, and the lower part, b, is called the denominator. The denominator indicates into how many parts something is divided, and the numerator tells how many of these parts are taken. A fraction indicates division:

If a and b are whole numbers and b Þ 0, the symbol

7 5 8q7 8 If the numerator of a fraction is 0, the value of the fraction is 0. If the denominator of a fraction is 0, the fraction is not defined (has no meaning): 0 50 17

17 not defined (has no meaning) 0

If the denominator of a fraction is 1, the value of the fraction is the same as the numerator: 18 5 18 1 If the numerator and denominator are the same number, the value of the fraction is 1: 7 51 7 www.petersons.com

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MATHEMATICS REVIEW

Equivalent Fractions Fractions that represent the same number are said to be equivalent. If m is a countm3a a m a a a 5 because 5 1 and 1 3 5 ing number and is a fraction, then: b m3b b m b b Example 2 4 6 8 5 5 5 3 6 9 12 These fractions are all equivalent.

Inequality of Fractions If two fractions are not equivalent, one is smaller than the other. The ideas of “less than” and “greater than” were previously defined and used for whole numbers. For the fractions

a c and : b b

a c , if a , c b b That is, if two fractions have the same denominator, the one with the smaller numerator has the smaller value. If two fractions have different denominators, find a common denominator by multiplying one denominator by the other. Then use the common denominator to compare numerators. Example 5 4 Which is smaller, or ? 8 7 8 { 7 5 56 5 common denominator 5 7 35 3 5 8 7 56

4 8 32 3 5 7 8 56

Since 32 , 35, 32 35 4 5 , and , 56 56 7 8

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UNIT 7

Reducing to Lowest Terms The principle that m3a a 5 m3b b can be particularly useful in reducing fractions to lowest terms. Fractions are expressed in lowest terms when the numerator and denominator have no common factor except 1. To reduce a fraction to an equivalent fraction in lowest terms, express the numerator and denominator as products of their prime factors. Each time a prime appears in the numerator over the same prime in the denomip nator, , substitute its equal value, 1. p Example Reduce

30 to an equivalent fraction in lowest terms. 42

30 2 {3 { 5 5 5 5 51{1{ 5 42 2 { 3 { 7 7 7 In practice, this can be done even more quickly by dividing the numerator and denominator by any number, prime or not, which will divide both evenly. Repeat this process until there is no prime factor remaining that is common to both numerator and denominator: 30 15 5 5 5 42 21 7

PROPER FRACTIONS, IMPROPER FRACTIONS,

AND

MIXED NUMBERS

Definitions A proper fraction is a fraction whose numerator is smaller than its denominator. Proper fractions always have a value less than 1: 3 4

5 8

121 132

0 1

An improper fraction is a fraction with the numerator equal to or greater than the denominator. Improper fractions always have a value equal to or greater than 1: 3 2

17 17

9 1

15 14

A mixed number is a number composed of a whole number and a proper fraction. It is always greater than 1 in value: 7 3 8

1 5 4

3 11 14

7 7 The symbol 3 means 3 1 and is read “three and seven-eighths.” 8 8

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MATHEMATICS REVIEW

To Change a Mixed Number into an Improper Fraction Multiply the denominator by the whole number and add this product to the numerator. Use the sum so obtained as the new numerator, and keep the original denominator. Example 4 Write 9 as an improper fraction. 11 9

~11 3 9! 1 4 99 1 4 103 4 5 5 5 11 11 11 11

Note: In any calculations with mixed numbers, first change the mixed numbers to improper fractions.

To Change an Improper Fraction into a Mixed Number Divide the numerator by the denominator. The result is the whole-number part of the mixed number. If there is a remainder in the division process, because the division does not come out evenly, put the remainder over the denominator (divisor). This gives the fractional part of the mixed number: 6 20 5 3q20 3 18

2 56 3

2 remainder

MULTIPLICATION Proper and Improper Fractions Multiply the two numerators and then multiply the two denominators. If the numerator obtained is larger than the denominator, divide the numerator of the resulting fraction by its denominator: 3 15 45 3 5 8 11 88

3 22 66 10 3 5 51 8 7 56 56

Multiplication of fractions is commutative. Three or more fractions are multiplied in the same way; two numerators are done at a time and the result multiplied by the next numerator. The product in the multiplication of fractions is usually expressed in lowest terms.

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UNIT 7

Canceling In multiplying fractions, if any of the numerators and denominators have a common divisor (factor), divide each of them by this common factor and the value of the fraction remains the same. This process is called canceling or cancellation. Example 9 10 27 90 3 5? 18 300 2 100 27 90 27 9 3 5 3 18 300 18 30 9 1 27 9 5 3 18 30 2 10 5

931 9 5 2 3 10 20

Divide second fraction by

10 10

Cancel: 18 and 9 each divisible by 9; 27 and 30 each divisible by 3 Multiply numerators; multiply denominators

Another method: 3 3 27 9 333 9 3 5 5 18 30 2 3 10 20 2 10

Cancel: 27 and 18 have common factor 9; 9 and 30 have common factor 3

Note: Canceling can take place only between a numerator and a denominator, in the same or a different fraction, never between two numerators or between two denominators.

Mixed Numbers Mixed numbers should be changed to improper fractions before multiplying. Then multiply as described above. Example To multiply 4 5 33 7 8 5 change 3 to an improper fraction: 8 5 ~8 3 3! 1 5 24 1 5 29 5 5 3 5 8 8 8 8

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MATHEMATICS REVIEW

Multiply 1 4 29 29 3 5 7 8 14 2 The answer can be left in this form or changed to a mixed number: 2

1 14

Fractions with Whole Numbers Write the whole number as a fraction with a denominator of 1 and then multiply: 3 3 7 21 1 375 3 5 55 4 4 1 4 4 Note: When any fraction is multiplied by 1, its value remains unchanged. When any fraction is multiplied by 0, the product is 0.

DIVISION Reciprocals Division of fractions involves reciprocals. One fraction is the reciprocal of another if the product of the fractions is 1. Example 1 3 4 and are reciprocals since 4 3 1 1 3 4 131 3 5 51 4 3 131 1 1 Example 2 1 and 3 are reciprocals since 3 1 1 3 3 51 3 1 1 To find the reciprocal of a fraction, interchange the numerator and denominator— that is, invert the fraction, or turn it upside down.

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UNIT 7

Proper and Improper Fractions Multiply the first fraction (dividend) by the reciprocal of the second fraction (divisor). Reduce by cancellation if possible. If you wish to, change the answer to a mixed number when possible: Example 9 4 9 7 4 5 3 2 7 2 4 The reciprocal of 5

4 7 4 7 is because 3 5 1 7 4 7 4

63 8

7 57 8

Mixed Numbers and/or Whole Numbers Both mixed numbers and whole numbers must first be changed to equivalent improper fractions. Then proceed as described above. Note: If a fraction or a mixed number is divided by 1, its value is unchanged. Division of a fraction or a mixed number by 0 is not defined. If a fraction is divided by itself or an equivalent fraction, the quotient is 1: 19 19 19 7 4 5 3 7 7 7 19

Reciprocal of

19 7 is 7 19

513151

ADDITION Fractions can be added only if their denominators are the same (called the common denominator). Add the numerators; the denominator remains the same. Reduce the sum to the lowest terms: 3 2 1 31211 6 3 1 1 5 5 5 8 8 8 8 8 4

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MATHEMATICS REVIEW

When the fractions have different denominators, you must find a common denominator. One way of doing this is to find the product of the different denominators. Example 5 1 1 5? 6 4 A common denominator is 6 { 4 5 24. 5 4 20 3 5 6 4 24

and

1 6 6 3 5 4 6 24

5 1 20 6 1 5 1 6 4 24 24 5

20 1 6 24

5

26 24

5

13 12

1 51 12

Least-Common Denominator A denominator can often be found that is smaller than the product of the different denominators. If the denominator of each fraction will divide into such a number evenly and it is the smallest such number, it is called the least (or lowest) common denominator, abbreviated as LCD. Finding a least-common denominator may make it unnecessary to reduce the answer and enables one to work with smaller numbers. There are two common methods. First Method

By inspection 5 1 1 5? 6 4 LCD 5 12 because 12 is the smallest number into which 6 and 4 divide evenly. Therefore:

GRE CAT Success

12 4 6 5 2

multiply

5 2 10 3 5 6 2 12

12 4 4 5 3

multiply

1 3 3 3 5 4 3 12

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UNIT 7

Then: 5 1 10 3 1 5 1 6 4 12 12 5

13 12

1 51 12 Second Method

By Factoring This method can be used when the LCD is not recognized by inspection. Factor each denominator into its prime factors. The LCD is the product of the highest power of each separate factor, where power refers to the number of times a factor occurs. Example 5 1 1 5? 6 4 Factoring denominators gives: 652{3

452{2

and

LCD 5 2 { 2 { 3 5 12 Convert to LCD: 5 2 10 3 5 6 2 12

1 3 3 5 12 4 3

5 1 10 3 1 5 1 6 4 12 12 5

13 12

1 51 12 The denominators 4 and 6 factor into 2 { 2 and 2 { 3, respectively. Although the factor 2 appears three times, its power is 22 from factoring 4. The factor 3 appears once, so its power is 31. Therefore, the LCD as a product of the highest power of each separate factor is 2 3 2 3 3.

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MATHEMATICS REVIEW

The factoring method of adding fractions can be extended to three or more fractions. Example 1 3 1 1 1 5? 4 8 12 Factoring denominators gives: 452{2

852{2{2

12 5 2 { 2 { 3

LCD 5 2 { 2 { 2 { 3 5 24 Convert to LCD: 1 6 6 3 5 4 6 24

3 3 9 3 5 8 3 24

1 2 2 3 5 12 2 24

1 3 1 6 9 2 1 1 5 1 1 4 8 12 24 24 24 5

61912 24

5

17 24

Addition of Mixed Numbers Change any mixed numbers to fractions. If the fractions have the same denominator, add the numerators. If the fractions have different denominators, find the LCD of the several denominators and then add numerators. Reduce the answer if possible. Write the answer as a mixed number if you wish. Example 2 1 2 2 15 11 5? 3 2 9 Factoring denominators gives: 353

252

953{3

LCD 5 2 { 3 { 3 5 18

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UNIT 7

Convert to LCD: 8 6 48 3 5 3 6 18

11 9 99 3 5 2 9 18

11 2 22 3 5 9 2 18

1 2 8 11 11 2 1 2 15 11 5 1 3 2 9 3 2 9 5

48 99 22 1 1 18 18 18

5

48 1 99 1 22 18

5

169 7 59 18 18

SUBTRACTION Fractions can be subtracted only if the denominators are the same. If the denominators are the same, find the difference between the numerators. The denominator remains unchanged. Example 19 2 2 5? 3 3 5

19 2 2 3

5

17 3

2 55 3 When fractions have different denominators, find equivalent fractions with a common denominator, and then subtract numerators. Example 7 3 2 5? 8 4 Factoring denominators gives: 852{2{2

452{2

LCD 5 2 { 2 { 2 58

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MATHEMATICS REVIEW

Convert to LCD: 7 7 5 8 8

3 2 6 3 5 4 2 8

7 3 7 6 2 5 2 8 4 8 8 726 5 8 5

1 8

Mixed Numbers To subtract mixed numbers, change each mixed number to a fraction. Find the LCD for the fractions. Write each fraction as an equivalent fraction whose denominator is the common denominator. Find the difference between the numerators. Example 3 5 3 22 5? 8 6 LCD 5 24 3 5 27 17 3 22 5 2 8 6 8 6 5

81 68 2 24 24

5

13 24

If zero is subtracted from a fraction, the result is the original fraction: 3 3 0 3 205 2 5 4 4 4 4

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UNIT 7

QUIZ

FRACTION PROBLEMS In the following problems, perform the indicated operations and reduce the answers to lowest terms. 1. 2. 3. 4. 5. 6.

5 4 3 12 15 1 3 4 2 8 5 2 1 12 3 2 5 2 3 11 1 4 3 3 3 5 1 4 7 22 5 3 SOLUTIONS

1.

2.

3. 4.

5.

6.

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1 1 5 4 5 4 1 3 5 3 5 12 15 12 15 9 3 3 4 1 3 1 8 1 8 4 4 5 3 5 3 5 2 8 2 3 2 3 3 1 5 2 5 8 13 1 1 5 1 5 51 12 3 12 12 12 12 2 5 22 15 7 2 5 2 5 3 11 33 33 33 2 2 1 4 10 4 10 4 8 3 5 3 5 52 3 3 5 3 5 3 5 3 5 3 3 1 4 1 39 7 117 35 82 7 7 22 5 2 5 2 5 55 5 3 5 3 15 15 15 15

186

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MATHEMATICS REVIEW

DECIMALS Earlier, we stated that whole numbers are expressed in a system of tens, or the decimal system, using the digits from 0 to 9. This system can be extended to fractions by using a period called a decimal point. The digits after a decimal point form a decimal fraction. Decimal fractions are smaller than 1—for example, .3, .37, .372, and .105. The first position to the right of the decimal point is called the tenths’ place, since the digit in that position tells how many tenths there are. The second digit to the right of the decimal point is in the hundredths’ place. The third digit to the right of the decimal point is in the thousandths’ place, and so on. Example 1 .3 is a decimal fraction that means 33

1 3 5 10 10

read “three-tenths.”

Example 2 The decimal fraction of .37 means 33

1 1 10 1 173 533 173 10 100 100 100 5

30 7 37 1 5 100 100 100

read “thirty-seven hundredths.”

Example 3 The decimal fraction .372 means 300 70 2 372 1 1 5 1000 1000 1000 1000 read “three hundred seventy-two thousandths.” Whole numbers have an understood (unwritten) decimal point to the right of the last digit (i.e., 4 5 4.0). Decimal fractions can be combined with whole numbers to make decimals—for example, 3.246, 10.85, and 4.7. Note: Adding zeros to the right of a decimal after the last digit does not change the value of the decimal.

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UNIT 7

Rounding Off Sometimes a decimal is expressed with more digits than desired. As the number of digits to the right of the decimal point increases, the number increases in accuracy, but a high degree of accuracy is not always needed. Then, the number can be “rounded off” to a certain decimal place. To round off, identify the place to be rounded off. If the digit to the right of it is 0, 1, 2, 3, or 4, the round-off place digit remains the same. If the digit to the right is 5, 6, 7, 8, or 9, add 1 to the round-off place digit. Example 1 Round off .6384 to the nearest thousandth. The digit in the thousandths’ place is 8. The digit to the right in the ten-thousandths’ place is 4, so the 8 stays the same. Example 2 .6386 rounded to the nearest thousandth is .639, rounded to the nearest hundredth is .64, and rounded to the nearest tenth is .6. After a decimal fraction has been rounded off to a particular decimal place, all the digits to the right of that place will be 0. Note: Rounding off whole numbers can be done by a similar method. It is less common but is sometimes used to get approximate answers quickly. Example 3 Round 32,756 to the nearest hundred. This means, to find the multiple of 100 that is nearest the given number. The number in the hundreds’ place is 7. The number immediately to the right is 5, so 32,756 rounds to 32,800.

DECIMALS

AND

FRACTIONS

Changing a Decimal to a Fraction Place the digits to the right of the decimal point over the value of the place in which the last digit appears and reduce if possible. The whole number remains the same. Example Change 2.14 to a fraction or mixed number. Observe that 4 is the last digit and is in the hundredths’ place. .14 5

14 7 5 100 50

Therefore: 7 2.14 5 2 50

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MATHEMATICS REVIEW

Changing a Fraction to a Decimal Divide the numerator of the fraction by the denominator. First put a decimal point followed by zeros to the right of the number in the numerator. Add and divide until there is no remainder. The decimal point in the quotient is aligned directly above the decimal point in the dividend. Example Change

3 to a decimal. 8

Divide .375 8q3.000 24 60 56 40 40 When the division does not terminate with a 0 remainder, two courses are possible. First Method

Divide to three decimal places. Example 5 Change to a decimal. 6 .833 6q5.000 48 20 18 20 18 2 The 3 in the quotient will be repeated indefinitely. It is called an infinite decimal and is written .833 . . .

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UNIT 7

Second Method

Divide until there are two decimal places in the quotient and then write the remainder over the divisor. Example 5 Change to a decimal. 6 .83 1 6q5.00 5 .83 3 48 20 18 2

ADDITION Addition of decimals is both commutative and associative. Decimals are simpler to add than fractions. Place the decimals in a column with the decimal points aligned under each other. Add in the usual way. The decimal point of the answer is also aligned under the other decimal points. Example 43 1 2.73 1 .9 1 3.01 5 ? 43. 2.73 .9 3.01 49.64

SUBTRACTION For subtraction, the decimal points must be aligned under each other. Add zeros to the right of the decimal point if desired. Subtract as with whole numbers. Examples 21.567 2 9.4 12.167

21.567 2 9.48 12.087

39.00 2 17.48 21.52

MULTIPLICATION Multiplication of decimals is commutative and associative: 5.39 3 .04 5 .04 3 5.39 (.7 3 .02) 3 .1 5 .7 { (.02 3 .1)

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MATHEMATICS REVIEW

Multiply the decimals as if they were whole numbers. The total number of decimal places in the product is the sum of the number of places (to the right of the decimal point) in all of the numbers multiplied. Example 8.64 3 .003 5 ? 8.64 3 .003 .02592

2 13 5

places to right of decimal point places to right of decimal point places to right of decimal point

A zero had to be added to the left of the product before writing the decimal point to ensure that there would be five decimal places in the product. Note: To multiply a decimal by 10, simply move the decimal point one place to the right; to multiply by 100, move the decimal point two places to the right.

DIVISION To divide one decimal (the dividend) by another (the divisor), move the decimal point in the divisor as many places as necessary to the right to make the divisor a whole number. Then move the decimal point in the dividend (expressed or understood) a corresponding number of places, adding zeros if necessary. Then divide as with whole numbers. The decimal point in the quotient is placed above the decimal point in the dividend after the decimal point has been moved. Example Divide 7.6 by .32. 23.75 .32q7.60 5 32q760.00 64 120 96 240 224 160 160 Note: “Divide 7.6 by .32” can be written as

7.6 . If this fraction is multiplied by .32

100 an equivalent fraction is obtained with a whole number in the denominator: 100 7.6 100 760 3 5 .32 100 32 Moving the decimal point two places to the right in both divisor and dividend is equivalent to multiplying each number by 100.

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UNIT 7

Special Cases If the dividend has a decimal point and the divisor does not, divide as with whole numbers and place the decimal point of the quotient above the decimal point in the divisor. If both dividend and divisor are whole numbers but the quotient is a decimal, place a decimal point after the last digit of the dividend and add zeros as necessary to get the required degree of accuracy. (See Changing a Fraction to a Decimal, page 189). Note: To divide any number by 10, simply move its decimal point (understood to be after the last digit for a whole number) one place to the left; to divide by 100, move the decimal point two places to the left; and so on.

Percents Percents, like fractions and decimals, are ways of expressing parts of whole numbers, as 93%, 50%, and 22.4%. Percents are expressions of hundredths—that is, of fractions whose denominator is 100. The symbol for percent is “%.” Example 25% 5 twenty-five hundredths 5

25 1 5 100 4

The word percent means per hundred. Its main use is in comparing fractions with equal denominators of 100.

RELATIONSHIP

WITH

FRACTIONS

AND

DECIMALS

Changing a Percent to a Decimal Divide the percent by 100 and drop the symbol for percent. Add zeros to the left when necessary: 30% 5 .30

1% 5 .01

Remember that the short method of dividing by 100 is to move the decimal point two places to the left.

Changing a Decimal to a Percent Multiply the decimal by 100 by moving the decimal point two places to the right, and add the symbol for percent: .375 5 37.5%

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.001 5 .1%

192

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MATHEMATICS REVIEW

QUIZ

DECIMAL PROBLEMS 1.

Change the following decimals into fractions, and reduce. a. b.

2.

1.16 15.05

Change the following fractions into decimals. a. b.

3 8 2 3

In the following problems, perform the indicated operations. 3.

3.762 1 23.43

4.

1.368 2 .559

5.

8.7 3 .8

6.

.045 4 .5 SOLUTIONS

1.

a. b.

2.

a.

b.

3.

GRE CAT Success

8 4 16 51 51 1.16 5 1 100 50 25 5 1 15.05 5 15 5 15 100 20 .375 3 5 8q3.000 8 24 60 256 40 .666 . . . 2 5 3q2.00 3 18 20 218 20

3.762 123.43 27.192

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UNIT 7

4.

1.368 2.559 .809

5.

8.7 3.8 6.96

6.

0.09 .5.q0.0.45 \

\

Changing a Percent to a Fraction Drop the percent sign. Write the number as a numerator over a denominator of 100. If the numerator has a decimal point, move the decimal point to the right the necessary number of places to make the numerator a whole number. Add the same number of zeros to the right of the denominator as you moved places to the right in the numerator. Reduce where possible. Examples 20% 5 36.5% 5

20 2 1 5 5 100 10 5 36.5 365 73 5 5 100 1000 200

Changing a Fraction to a Percent Use either of two methods. First Method

Change the fraction into an equivalent fraction with a denominator of 100. Drop the denominator (equivalent to multiplying by 100) and add the % sign. Example 6 as a percent. Express 20 6 5 30 3 5 5 30% 20 5 100

Second Method

Divide the numerator by the denominator to get a decimal with two places (express the remainder as a fraction if necessary). Change the decimal to a percent. Example 6 as a percent. Express 20 .30 6 5 20q6.00 5 30% 20 60

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MATHEMATICS REVIEW

QUIZ

PERCENT PROBLEMS 1.

Change the following percents into decimals: a. b.

2.

Change the following decimals into percents: a. b.

3.

0.625 3.75

Change the following fractions into percents: a. b.

4.

37.5% 0.5%

7 8 73 200

Change the following percents into fractions: a. b.

87.5% 0.02% SOLUTIONS

1.

2.

a.

37.5% 5 0.375

b.

00.5% 5 0.005

a.

0.625 5 62.5%

b.

3.75 5 375%

| |

\ \

3.

a. b.

4.

a. b.

GRE CAT Success

0.875 7 5 8q7.000 5 87.5% 8 0.365 73 5 200q73.000 5 36.5% 200 875 35 7 5 5 1,000 40 8 2 1 0.02% 5 0.0002 5 5 10,000 5,000 87.5% 5 0.875 5

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UNIT 7

WORD PROBLEMS When doing percent problems, it is usually easier to change the percent to a decimal or a fraction before computing. When we take a percent of a certain number, that number is called the base, the percent we take is called the rate, and the result is called the percentage or part. If we let B represent the base, R the rate, and P the part, the relationship between these quantities is expressed by the following formula: P5R{B All percent problems can be done with the help of this formula. Example 1 In a class of 24 students, 25% received an A. How many students received an A? The number of students (24) is the base, and 25% is the rate. Change the rate to a fraction for ease of handling and apply the formula. 25% 5

25 1 5 100 4

P5R3B 6 1 24 5 3 4 1 1 5 6 students To choose between changing the percent (rate) to a decimal or a fraction, simply decide which would be easier to work with. In Example 1, the fraction was easier to work with because cancellation was possible. In Example 2, the situation is the same except for a different rate. This time the decimal form is easier. Example 2 In a class of 24 students, 29.17% received an A. How many students received an A? Changing the rate to a fraction yields 29.17 2917 5 100 10,000 You can quickly see that the decimal is the better choice. 29.17% 5 .2917 P5R3B 5 .2917 3 24 5 7 students

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.2917 3 24 1.1668 5.834 7.0008

196

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MATHEMATICS REVIEW

Example 3 What percent of a 40-hour week is a 16-hour schedule? 40 hours is the base and 16 hours is the part. P5R{B 16 5 R { 40 Divide each side of the equation by 40. 16 5R 40 2 5R 5 40% 5 R Example 4 A woman paid $15,000 as a down payment on a house. If this amount was 20% of the price, what did the house cost? The part (or percentage) is $15,000, the rate is 20%, and we must find the base. Change the rate to a fraction. 20% 5

1 5

P5R3B $15,000 5

1 3B 5

Multiply each side of the equation by 5. $75,000 5 B 5 cost of house

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UNIT 7

Percent of Increase or Decrease This kind of problem is not really new but follows immediately from the previous problems. First calculate the amount of increase or decrease. This amount is the P (percentage or part) from the formula P 5 R { B. The base, B, is the original amount, regardless of whether there was a loss or gain. Example By what percent does Mary’s salary increase if her present salary is $20,000 and she accepts a new job at a salary of $28,000? Amount of increase is: $28,000 2 $20,000 5 $8000 P5R{B $8000 5 R { $20,000 Divide each side of the equation by $20,000. Then: 40 8000 40 5 5 R 5 40% increase 20,000 100 100

Discount and Interest These special kinds of percent problems require no new methods of attack. Discount

The amount of discount is the difference between the original price and the sale, or discount, price. The rate of discount is usually given as a fraction or as a percent. Use the formula of the percent problems P 5 R { B, but now P stands for the part or discount, R is the rate, and B, the base, is the original price. Example 1 A table listed at $160 is marked 20% off. What is the sale price? P5R{B 5 .20 { $160 5 $32 This is the amount of discount or how much must be subtracted from the original price. Then: $160 2 $32 5 $128 sale price

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MATHEMATICS REVIEW

Example 2 A car priced at $9000 was sold for $7200. What was the rate of discount? Amount of discount 5 $9000 2 $7200 5 $1800 Discount 5 rate { original price $1800 5 R { $9000 Divide each side of the equation by $9000: 20 1800 20 5 5 R 5 20% 9000 100 100 Successive Discounting When an item is discounted more than once, it is called successive discounting. Example 1 In one store, a dress tagged at $40 was discounted 15%. When it did not sell at the lower price, it was discounted an additional 10%. What was the final selling price? Discount 5 R { original price First discount 5 .15 { $40 5 $6 $40 2 $6 5 $34 selling price after first discount Second discount 5 .10 { $34 5 $3.40 $34 2 $3.40 5 $30.60 final selling price Example 2 In another store, an identical dress was also tagged at $40. When it did not sell, it was discounted 25% all at once. Is the final selling price lower or higher than in Example 1? Discount 5 R { original price 5 .25 { $40 5 $10 $40 2 $10 5 $30 final selling price This is a lower selling price than in Example 1, where two successive discounts were taken.

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UNIT 7

Interest

Interest problems are similar to discount and percent problems. If money is left in the bank for a year and the interest is calculated at the end of the year, the usual formula P 5 R { B can be used, where P is the interest, R is the rate, and B is the principal (original amount of money borrowed or loaned). Example 1 A certain bank pays interest on savings accounts at the rate of 4% per year. If a man has $6700 on deposit, find the interest earned after 1 year. P5R{B Interest 5 rate { principal P 5 .04 { $6700 5 $268 interest Interest problems frequently involve more or less time than 1 year. Then the formula becomes: Interest 5 rate { principal { time Example 2 If the money is left in the bank for 3 years at simple interest (the kind we are discussing), the interest is 3 { $268 5 $804 Example 3 Suppose $6700 is deposited in the bank at 4% interest for 3 months. How much interest is earned? Interest 5 rate { principal { time Here the 4% rate is for 1 year. Since 3 months is Interest 5 .04 { $6700 {

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3 1 5 12 4

1 5 $67 4

200

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QUIZ

PERCENT WORD PROBLEMS 1.

Janet received a rent increase of 15%. If her rent was $785 monthly before the increase, what is her new rent?

2.

School bus fares rose from $25 per month to $30 per month. Find the percent of increase.

3.

A dress originally priced at $90 is marked down 35%, then discounted a further 10%. What is the new, reduced price?

4.

Dave delivers flowers for a salary of $45 a day, plus a 12% commission on all sales. One day his sales amounted to $220. How much money did he earn that day?

5.

A certain bank pays interest on money market accounts at a rate of 6% a year. If Brett deposits $7,200, find the interest earned after one year. SOLUTIONS

1.

Amount of increase 5 $785 3 15% 5 $785 3 .15 5 $117.75 New rent 5 $902.75

2.

Amount of increase 5 $30 2 $25 5 $5 Percent of increase 5

3.

1 5 5 5 20% 25 5

Amount of first markdown 5 $90 3 35% 5 $90 3 .35 5 $31.50 Reduced price 5 $90 2 $31.50 5 $58.50 Amount of second markdown 5 $58.50 3 10% 5 $58.50 3 .1 5 $5.85 Final price 5 $58.50 2 $5.85 5 $52.65

4.

Commission 5 $220 3 12% 5 $220 3 .12 5 $26.40 Money earned 5 $45 1 $26.40 5 $71.40

5.

GRE CAT Success

Interest 5 $7,200 3 6% 5 $7,200 3 .06 5 $432

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UNIT 7

SIGNED NUMBERS In describing subtraction of whole numbers, we said that the operation was not closed—that is, 4 2 6 will yield a number that is not a member of the set of counting numbers and zero. The set of integers was developed to give meaning to such expressions as 4 2 6. The set of integers is the set of all signed whole numbers and zero. It is the set {…, 24, 23, 22, 21, 0, 1, 2, 3, 4, …} The first three dots symbolize the fact that the negative integers go on indefinitely, just as the positive integers do. Integers preceded by a minus sign (called negative integers) appear to the left of 0 on a number line.

Decimals, fractions, and mixed numbers can also have negative signs. Together with positive fractions and decimals, they appear on the number line in this fashion:

- 23 -.5 0

212

1

All numbers to the right of 0 are called positive numbers. They have the sign 1, whether it is actually written or not. Business gains or losses, feet above or below sea level, and temperature above and below zero can all be expressed by means of signed numbers.

ADDITION If the numbers to be added have the same sign, add the numbers (integers, fractions, decimals) as usual and use their common sign in the answer: 19 1 ~18! 1 ~12! 5 119 or 19 24 1 ~211! 1 ~27! 1 ~21! 5 223 If the numbers to be added have different signs, add the positive numbers and then the negative numbers. Ignore the signs and subtract the smaller total from the larger total. If the larger total is positive, the answer will be positive; if the larger total is negative, the answer will be negative. The answer may be zero. Zero is neither positive nor negative and has no sign. Example 13 1 ~25! 1 ~28! 1 ~12! 5 ? 13 1 ~12! 5 15 25 1 ~28! 5 213 213 1 5 5 28 Since the larger total (13) has a negative sign, the answer is 28.

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MATHEMATICS REVIEW

SUBTRACTION The second number in a subtraction problem is called the subtrahend. In order to subtract, change the sign of the subtrahend and then continue as if you were adding signed numbers. If there is no sign in front of the subtrahend, it is assumed to be positive. Examples Subtract the subtrahend (bottom number) from the top number. 15 5

5 15

235 242

235 42

42 35

10

210

7

277

7

MULTIPLICATION If two and only two signed numbers are to be multiplied, multiply the numbers as you would if they were not signed. Then, if the two numbers have the same sign, the product is positive. If the two numbers have different signs, the product is negative. If more than two numbers are being multiplied, proceed two at a time in the same way as before, finding the signed product of the first two numbers, then multiplying that product by the next number, and so on. The product has a positive sign if all the factors are positive or there is an even number of negative factors. The product has a negative sign if there is an odd number of negative factors. Example 23 { (15) { (211) { (22) 5 2330 The answer is negative because there is an odd number (three) of negative factors. The product of a signed number and zero is zero. The product of a signed number and 1 is the original number. The product of a signed number and 21 is the original number with its sign changed. Examples 25 { 0 5 0 25 { 1 5 25 25 { ~21! 5 15

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UNIT 7

DIVISION If the divisor and the dividend have the same sign, the answer is positive. Divide the numbers as you normally would. If the divisor and the dividend have different signs, the answer is negative. Divide the numbers as you normally would. Examples 2 3 4 ~22! 5

3 1 51 2 2

8 4 ~2.2! 5 240 If zero is divided by a signed number, the answer is zero. If a signed number is divided by zero, the answer does not exist. If a signed number is divided by 1, the number remains the same. If a signed number is divided by 21, the quotient is the original number with its sign changed. Examples 0 4 ~22! 5 0 4 2 40 3

not defined

2 2 415 3 3 4 4 21 5 24

QUIZ

SIGNED NUMBERS PROBLEMS Perform the indicated operations:

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1.

1 6 1 (25) 1 (12) 1 (28) 5

2.

2 5 2 (24) 1 (22) 2 (16) 5

3.

23 { (15) { (27) { (22) 5

4.

9 4 (2.3) 5

204

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MATHEMATICS REVIEW SOLUTIONS

1.

16 1 ~25! 5 11 11 1 ~12! 5 13 13 1 ~28! 5 25

2.

25 2 ~24! 5 25 1 4 5 21 21 1 ~22! 5 23 23 2 ~16! 5 29

3.

23 { ~15! 5 215 215 { ~27! 5 1105 1105 { ~22! 5 2210

4.

9 4 (2.3) 5 230

POWERS, EXPONENTS,

AND

ROOTS

Exponents The product 10 { 10 { 10 can be written 103. We say 10 is raised to the third power. In general, a 3 a 3 a ... a n times is written an. The base a is raised to the nth power, and n is called the exponent. Examples 32 5 3 { 3 23 5 2 { 2 { 2 54 5 5 { 5 { 5 { 5

read “3 squared” read “2 cubed” read “5 to the fourth power”

If the exponent is 1, it is usually understood and not written; thus, a1 5 a Since a2 5 a 3 a

and

a3 5 a 3 a 3 a

then a2 3 a3 5 (a 3 a)(a 3 a 3 a) 5 a5 There are three rules for exponents. In general, if k and m are any counting numbers or zero, and a is any number,

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1 m

Rule 1:

ak 3 am 5 ak

Rule 2:

am { bm 5 (ab)m

Rule 3:

(ak)n 5 akn

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UNIT 7

Examples Rule 1:

22 { 23 5 4 3 8 5 32 and 22 3 23 5 25 5 32

Rule 2:

32 3 42 5 9 3 16 5 144 and 32 3 42 5 (3 3 4)2 5 122 5 144

Rule 3:

(32)3 5 93 5 729 and (32)3 5 36 5 729

ROOTS The definition of roots is based on exponents. If an 5 c, where a is the base and n n the exponent, a is called the nth root of c. This is written a 5 =c. The 4 symbol = is called a radical sign. Since 54 5 625, =625 5 5 and 5 is the fourth root of 625. The most frequently used roots are the second (called the square) root and the third (called the cube) root. The square root is written = 3 and the cube root is written = .

Square Roots If c is a positive number, there are two values, one negative and one positive, which when multiplied together will produce c. Example 14 { (14) 5 16

and

24 { (24) 5 16

The positive square root of a positive number c is called the principal square root of c (briefly, the square root of c) and is denoted by =c :

=144 5 12 If c 5 0, there is only one square root, 0. If c is a negative number, there is no real number that is the square root of c:

=24 is not a real number Cube Roots Both positive and negative numbers have real cube roots. The cube root of 0 is 0. The cube root of a positive number is positive; that of a negative number is negative. Examples 2{2{258 Therefore

3 852 =

23 { (23) { (23) 5 227 Therefore

3 227 5 23 =

Each number has only one real cube root.

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MATHEMATICS REVIEW

Fractional Exponents The values of k, m, and n from the three exponent rules can be expanded to include positive and negative fractions. In particular, roots can be expressed as 1 1 fractional exponents. In Rule 3, (ak)n 5 akn. Let k 5 . Then (an )n 5 a1 5 a and n 1 m m m an is the nth root of a. Rule 2, a 3 b 5 (a 3 b) , which is true when a and b are any numbers and n is an integer, can be extended to include the case in 1 which the exponent is a fraction. Suppose m 5 . Then: k 1

1

1

ak 3 bk 5 (a 3 b)k or

k k k a 3 b 5 =a 3 =b =

This last formulation justifies the simplification of square roots. If the number under the radical sign is a square number, the process will terminate in a number without the radical sign. If the number is not square, the process should terminate when the number remaining under the radical sign no longer contains a square. Example 1 Simplify =98

=98 5 =2 3 49 5 =2 3 =49

where 49 is a square number

5 =2 3 7 Therefore, =98 5 7=2 and the process terminates because there is no whole number whose square is 2. 7=2 is called a radical expression or simply a radical. Example 2 Which is larger,

~=96!

2

or

=214?

~=96!2 5 =96 3 =96 5 =96 3 96 5 96

=214 5 27 5 128 because 214 5 27 3 27 by Rule 1 or 1 because =214 5 (214)2 5 27 by Rule 3. Since 128 . 96,

=214 . ~=96!2

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UNIT 7

Example 3 Which is larger, 2=75 or 6=12? These numbers can be compared if the same number appears under the radical sign. Then the greater number is the one with the larger number in front of the radical sign.

=75 5 =25 3 3 5 =25 3 =3 5 5=3 Therefore: 2=75 5 2~5=3! 5 10=3

=12 5 =4 3 3 5 =4 3 =3 5 2=3 Therefore: 6=12 5 6~2=3! 5 12=3 Since 12=3 . 10=3, 6=12 . 2=75 Note: Numbers such as =2 and =3 are called irrational numbers to distinguish them from rational numbers, which include the integers and the fractions. Irrational numbers also have places on the number line. They may have positive or negative signs. The combination of rational and irrational numbers, all the numbers we have used so far, make up the real numbers. Arithmetic, algebra, and geometry deal with real numbers. The number p, the ratio of the circumference of a circle to its diameter, is also a real number; it is irrational, although it is approximated by 3.14159.... Instructions for taking the GRE say that the numbers used are real numbers. This means that answers may be expressed as fractions, decimals, radicals, or integers, whatever is required. Radicals can be added and subtracted only if they have the same number under the radical sign. Otherwise, they must be reduced to expressions having the same number under the radical sign. Example 4 Add 2=18 1 4=8 2 =2.

=18 5 =9 3 2 5 =9 3 =2 5 3=2 therefore 2=18 5 2~3=2! 5 6=2 and

=8 5 =4 3 2 5 =4 3 =2 5 2=2 therefore 4=8 5 4~2=2! 5 8=2

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MATHEMATICS REVIEW

giving 2=18 1 4=8 2 =2 5 6=2 1 8=2 2 =2 5 13=2 Radicals are multiplied using the rule that: k k k a 3 b 5 =a 3 =b =

Example 5

=2~=2 2 5=3!

5 =4 2 5=6 5 2 2 5=6

A quotient rule for radicals similar to the product rule is: k

k

a =a 5 k b = b

=

Example 6 9 =9 3 5 5 4 =4 2

=

QUIZ

POWERS, EXPONENTS,

ROOTS PROBLEMS =162

1.

Simplify

2.

=75 and =12 Combine =80 1 =45 2 =20 Simplify =5~2=2 2 3=5! 15=96 Divide and simplify 5=2

3. 4. 5. 6.

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AND

Find the sum of

Calculate 52 3 23

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UNIT 7 SOLUTIONS

1. 2. 3. 4.

5. 6.

=162 5 =2 { 81 5 =2 { =81 5 9=2 =75 1 =12 5 5=3 1 2=3 5 7=3 =80 1 =45 2 =20 5 4=5 1 3=5 2 2=5 5 5=5 =5~2=2 2 3=5! 5 2=10 2 3=25 5 2=10 2 3~5! 5 2=10 215 15=96 15~4=6! 60=6 5 5 5 12=3 5=2 5=2 5=2 52 3 23 5 25 3 8 5 200

ALGEBRA Algebra is a generalization of arithmetic. It provides methods for solving problems that cannot be done by arithmetic alone or that can be done by arithmetic only after long computations. Algebra provides a shorthand way of reducing long verbal statements to brief formulas, expressions, or equations. After the verbal statements have been reduced, the resulting algebraic expressions can be simplified. Suppose that a room is 12 feet wide and 20 feet long. Its perimeter (measurement around the outside) can be expressed as: 12 1 20 1 12 1 20 or 2(12 1 20) If the width of the room remains 12 feet but the letter l is used to symbolize length, the perimeter is: 12 1 l 1 12 1 l or 2(12 1 l) Further, if w is used for width, the perimeter of any rectangular room can be written as 2(w 1 l). This same room has an area of 12 feet by 20 feet or 12 { 20. If l is substituted for 20, any room of width 12 has area equal to 12l. If w is substituted for the number 12, the area of any rectangular room is given by wl or lw. Expressions such as wl and 2(w 1 l) are called algebraic expressions. An equation is a statement that two algebraic expressions are equal. A formula is a special type of equation.

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MATHEMATICS REVIEW

EVALUATING FORMULAS If we are given an expression and numerical values to be assigned to each letter, the expression can be evaluated. Example Evaluate 2x 1 3y 2 7 if x 5 2 and y 5 24. Substitute given values 2(2) 1 3(24) 2 7 5 ? Multiply numbers using rules for signed numbers 4 1 (212) 2 7 5 ? Collect numbers 4 2 19 5 215 We have already evaluated formulas in arithmetic when solving percent, discount, and interest problems. Example The formula for temperature conversion is: F5

9 C 1 32 5

where C stands for the temperature in degrees Celsius and F for degrees Fahrenheit. Find the Fahrenheit temperature that is equivalent to 20°C. F5

9 (20°C) 1 32 5 36 1 32 5 68°F 5

ALGEBRAIC EXPRESSIONS Formulation A more difficult problem than evaluating an expression or formula is to translate from a verbal expression to an algebraic one: Verbal Thirteen more than x Six less than twice x The square of the sum of x and 5 The sum of the square of x and the square of 5 The distance traveled by a car going 50 miles an hour for x hours The average of 70, 80, 85, and x

Algebraic x 1 13 2x 2 6 (x 1 5)2 x2 1 52 50x 70 1 80 1 85 1 x 4

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UNIT 7

SIMPLIFICATION After algebraic expressions have been formulated, they can usually be simplified by means of the laws of exponents and the common operations of addition, subtraction, multiplication, and division. These techniques will be described in the next section. Algebraic expressions and equations frequently contain parentheses, which are removed in the process of simplifying. If an expression contains more than one set of parentheses, remove the inner set first and then the outer set. Brackets, [ ], which are often used instead of parentheses, are treated the same way. Parentheses are used to indicate multiplication. Thus 3(x 1 y) means that 3 is to be multiplied by the sum of x and y. The distributive law is used to accomplish this: a(b 1 c) 5 ab 1 ac The expression in front of the parentheses is multiplied by each term inside. Rules for signed numbers apply. Example Simplify 3[4(2 2 8) 2 5(4 1 2)]. This can be done in two ways. Method 1

Combine the numbers inside the parentheses first: 3@4~2 2 8! 2 5~4 1 2!# 5 3@4~26! 2 5~6!# 5 3@224 2 30# 5 3@254# 5 2162

Method 2

Use the distributive law: 3@4~2 2 8! 2 5~4 1 2!# 5 3@8 232 2 20 2 10# 5 3@8 2 62# 5 3@254# 5 2162 If there is a (1) before the parentheses, the signs of the terms inside the parentheses remain the same when the parentheses are removed. If there is a (2) before the parentheses, the sign of each term inside the parentheses changes when the parentheses are removed.

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MATHEMATICS REVIEW

Once parentheses have been removed, the order of operations is multiplication and division, then addition and subtraction from left to right. Example (215 1 17) { 3 2 [(4 { 9) 4 6] 5 ? Work inside the parentheses first: (2) { 3 2 [36 4 6] 5 ? Then work inside the brackets: 2 { 3 2 [6] 5 ? Multiply first, then subtract, proceeding from left to right: 62650 The placement of parentheses and brackets is important. Using the same numbers as above with the parentheses and brackets placed in different positions can give many different answers. Example 215 1 [(17 { 3) 2 (4 { 9)] 4 6 5 ? Work inside the parentheses first: 215 1 [(51) 2 (36)] 4 6 5 ? Then work inside the brackets: 215 1 [15] 4 6 5 ? Since there are no more parentheses or brackets, proceed from left to right, dividing before adding: 1 1 215 1 2 5 212 2 2

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UNIT 7

OPERATIONS When letter symbols and numbers are combined with the operations of arithmetic (1, 2, {, 4) and with certain other mathematical operations, we have an algebraic expression. Algebraic expressions are made up of several parts connected by a plus or a minus sign; each part is called a term. Terms with the same letter part are called like terms. Since algebraic expressions represent numbers, they can be added, subtracted, multiplied, and divided. When we defined the commutative law of addition in arithmetic by writing a 1 b 5 b 1 a, we meant that a and b could represent any number. The expression a 1 b 5 b 1 a is an identity because it is true for all numbers. The expression n 1 5 5 14 is not an identity because it is not true for all numbers; it becomes true only when the number 9 is substituted for n. Letters used to represent numbers are called variables. If a number stands alone (the 5 or 14 in n 1 5 5 14), it is called a constant because its value is constant or unchanging. If a number appears in front of a variable, it is called a coefficient. Because the letter x is frequently used to represent a variable, or unknown, the times sign 3, which can be confused with it in handwriting, is rarely used to express multiplication in algebra. Other expressions used for multiplication are a dot, parentheses, or simply writing a number and letter together: 5 { 4 or 5(4) or 5a Of course, 54 still means fifty-four.

Addition and Subtraction Only like terms can be combined. Add or subtract the coefficients of like terms, using the rules for signed numbers. Example 1 Add x 1 2y 2 2x 1 3y. x 2 2x 1 2y 1 3y 5 2x 1 5y Example 2 Perform the subtraction: 230a 2 15b 1 4c 2 (2 5a 1 3b 2 c 1 d) Change the sign of each term in the subtrahend and then add, using the rules for signed numbers: 230a 2 15b 1 4c 5a 2 3b 1 c 2 d 225a 2 18b 1 5c 2 d

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MATHEMATICS REVIEW

Multiplication Multiplication is accomplished by using the distributive property. If the multiplier has only one term, then a(b 1 c) 5 ab 1 bc Example 1 9x~5m 1 9q! 5 ~9x!~5m! 1 ~9x!~9q! 5 45mx 1 81qx When the multiplier contains more than one term and you are multiplying two expressions, multiply each term of the first expression by each term of the second, and then add like terms. Follow the rules for signed numbers and exponents at all times. Example 2 ~3x 1 8!~4x2 1 2x 1 1! 5 3x~4x2 1 2x 1 1! 1 8~4x2 1 2x 1 1! 5 12x3 1 6x2 1 3x 1 32x2 1 16x 1 8 5 12x3 1 38x2 1 19x 1 8 If more than two expressions are to be multiplied, multiply the first two, then multiply the product by the third factor, and so on, until all factors have been used. Algebraic expressions can be multiplied by themselves (squared) or raised to any power. Example 3 ~a 1 b!2 5 ~a 1 b!~a 1 b! 5 a~a 1 b! 1 b~a 1 b! 5 a2 1 ab 1 ba 1 b2 5 a2 1 2ab 1 b2 since ab 5 ba by the commutative law Example 4 ~a 1 b!~a 2 b! 5 a~a 2 b! 1 b~a 2b! 5 a2 2 ab 1 ba 2 b2 5 a2 2 b2

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UNIT 7

Factoring When two or more algebraic expressions are multiplied, each is called a factor and the result is the product. The reverse process of finding the factors when given the product is called factoring. A product can often be factored in more than one way. Factoring is useful in multiplication, division, and solving equations. One way to factor an expression is to remove any single-term factor that is common to each of the terms and write it outside the parentheses. It is the distributive law that permits this. Example 1 3x3 1 6x2 1 9x 5 3x(x2 1 2x 1 3) The result can be checked by multiplication. Expressions containing squares can sometimes be factored into expressions containing letters raised to the first power only, called linear factors. We have seen that: (a 1 b)(a 2 b) 5 a2 2 b2 Therefore, if we have an expression in the form of a difference of two squares, it can be factored as: a2 2 b2 5 (a 1 b)(a 2 b) Example 2 Factor 4x2 2 9. 4x2 2 9 5 (2x)2 2 (3)2 5 (2x 1 3)(2x 2 3) Again, the result can be checked by multiplication. A third type of expression that can be factored is one containing three terms, such as x2 1 5x 1 6. Since: (x 1 a)(x 1 b) 5 x(x 1 b) 1 a(x 1 b) 5 x2 1 xb 1 ax 1 ab 5 x2 1 (a 1 b)x 1 ab an expression in the form x2 1 (a 1 b)x 1 ab can be factored into two factors of the form (x 1 a) and (x 1 b). We must find two numbers whose product is the constant in the given expression and whose sum is the coefficient of the term containing x. Example 3 Find factors of x2 1 5x 1 6. First find two numbers which, when multiplied, have 16 as a product. Possibilities are 2 and 3, 22 and 23, 1 and 6, 21 and 26. From these, select the one pair whose sum is 5. The pair 2 and 3 is the only possible selection, and so: x2 1 5x 1 6 5 (x 1 2)(x 1 3) written in either order

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MATHEMATICS REVIEW

Example 4 Factor x2 2 5x 2 6. Possible factors of 26 are 21 and 6, 1 and 26, 2 and 23, 22 and 3. We must select the pair whose sum is 25. The only pair whose sum is 25 is 1 1 and 26, and so x2 2 5x 2 6 5 (x 1 1)(x 2 6) In factoring expressions of this type, notice that if the last sign is plus, both a and b have the same sign and it is the same as the sign of the middle term. If the last sign is minus, the numbers have opposite signs. Many expressions cannot be factored. Method 1

Division 36mx2 4 9m2x or

36mx2 9m2x

5 4m21x1 5 Method 2

4x m

Cancellation 4 1 36mx 36mxx 4x 5 5 2 9m x 9mmx m 1 1 2

This is acceptable because

SD

ac a c c ac a and 5 1 so that 5 5 bc b c c bc b If the divisor contains only one term and the dividend is a sum, divide each term in the dividend by the divisor and simplify as you did in Method 2. 2 3x2 x 9x 1 3x 1 6x 9x3 3x2 6x 5 1 1 3x 3x 3x 3x 3

2

5 3x2 1 x 1 2 Example 5 This method cannot be followed if there are two or more terms in the denominator since: a a a Þ 1 b1c b c

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UNIT 7

In this case, write the example as a fraction. Factor the numerator and denominator if possible. Then use laws of exponents or cancel. Example 6 Divide x3 2 9x by x3 1 6x2 1 9x. Write as: x3 2 9x x3 1 6x2 1 9x Both numerator and denominator can be factored to give: x~x2 2 9! x~x 1 3!~x 2 3! x 2 3 5 5 2 x~x 1 6x 1 9! x~x 1 3!~x 1 3! x 1 3

QUIZ

ALGEBRA PROBLEMS

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1.

Simplify: 4[2(327) 2 4(216)]

2.

Subtract: (225x 1 4y 2 12z) 2 (4x 2 8y 2 13z)

3.

Multiply: (5x 1 2)(3x2 2 2x 1 1)

4.

Factor completely: 2x3 1 8x2 2 90x

5.

Factor completely: 32x2 2 98

6.

Divide:

x2 1 2x 2 8 x2 2 x 2 20

218

GRE CAT Success

MATHEMATICS REVIEW SOLUTIONS

1.

4[2(327) 2 4(216)] 5 4[2(24)24(8)] 5 4[28 2 32] 5 4(240) 5 2160

2.

(225x 1 4y 2 12z) 2 (4x 2 8y 2 13z) 5 225x 1 4y 2 12z 2 4x 1 8y 1 13z 5 229x 1 12y 1 z

3.

(5x 1 2)(3x2 2 2x 1 1) 5 5x (3x2 2 2x 11) 1 2(3x2 2 2x 1 1) 5 15x3 2 10x2 1 5x 1 6x2 2 4x 1 2 5 15x3 2 4x2 1 x 1 2

4.

2x3 1 8x2 2 90x 5 2x (x2 1 4x 2 45) 5 2x (x 1 9)(x 2 5)

5.

32x2 2 98 5 2(16x2 2 49) 5 2(4x 2 7)(4x 1 7)

6.

x2 1 2x 2 8 ~x 1 4!~x 2 2! 5 x2 2 x 2 20 ~x 2 5!~x 1 4! 1 ~x 1 4!~x 2 2! x 22 5 5 ~x 2 5!~x 1 4! x 25 1

EQUATIONS Solving equations is one of the major objectives in algebra. If a variable x in an equation is replaced by a value or expression that makes the equation a true statement, the value or expression is called a solution of the equation. (Remember that an equation is a mathematical statement that one algebraic expression is equal to another.) An equation may contain one or more variables. We begin with one variable. Certain rules apply to equations whether there are one or more variables. The following rules are applied to give equivalent equations that are simpler than the original: Addition: Subtraction: Multiplication: Division:

If If If If

s 5 t, then s 1 c 5 t 1 c. s 1 c 5 t 1 c, then s 5 t. s 5 t, then cs 5 ct. cs 5 ct and c Þ 0, then s 5 t.

To solve for x in an equation in the form ax 5 b with a Þ 0, divide each side of the equation by a: ax b 5 a a Then,

GRE CAT Success

yielding

x5

b a

b is the solution to the equation. a

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UNIT 7

Example 1 Solve 4x 5 8. Write

4x 8 5 x52 4 4

Example 2 Solve 2x 2 (x 2 4) 5 5(x 1 2) for x. 2x 2 (x 2 4) 2x 2 x 1 4 x14 x 24x x

5 5 5 5 5 5 5

5(x 1 2) 5x 1 10 5x 1 10 5x 1 6 6 6 24 3 2 2

Remove parentheses by distributive law. Combine like terms. Subtract 4 from each side. Subtract 5x from each side. Divide each side by 24. Reduce fraction to lowest terms.

Negative sign now applies to the entire fraction. Check the solution for accuracy by substituting in the original equation: 3 3 2(2 ) 2 (2 24) 2 2 11 23 2 2 2 11 23 1 2 6 11 2 1 2 2

S D

3 1 2) 2

0

5 (2

0

5

0

5 2 5 check 2

0

SD 1 2

QUIZ

EQUATIONS PROBLEMS Solve the following equations for x:

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1.

3x 2 5 5 3 1 2x

2.

3(2x 2 2) 5 12

3.

4(x 2 2) 5 2x 1 10

4.

7 2 4(2x 2 1) 5 3 1 4(4 2 x)

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MATHEMATICS REVIEW SOLUTIONS

1.

3x 2 5 5 3 1 2x 22x

22x

x2553 1515 x58 2.

3~2x 2 2! 5 12 6x 2 6 5 12 6x 5 18 x53

3.

4~x 2 2! 5 2x 1 10 4x 2 8 5 2x 1 10 4x 5 2x 1 18 2x 5 18 x59

4.

7 2 4~2x 2 1! 5 3 1 4~4 2 x! 7 2 8x 1 4 5 3 1 16 2 4x 11 2 8x 5 19 2 4x 11 5 19 1 4x 28 5 4x x 5 22

WORD PROBLEMS INVOLVING ONE UNKNOWN In many cases, if you read a word problem carefully, assign a letter to the quantity to be found, and understand the relationships between known and unknown quantities, you can formulate an equation in one unknown.

Number Problems and Age Problems These two kinds of problems are similar to each other. Example One number is 3 times another, and their sum is 48. Find the two numbers. Let x 5 second number. Then the first is 3x. Since their sum is 48, 3x 1 x 5 48 4x 5 48 x 5 12 Therefore, the first number is 3x 5 36. 36 1 12 5 48 check

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UNIT 7

Consecutive Number Problems This type usually involves only one unknown. Two numbers are consecutive if one is the successor of the other. Three consecutive numbers are of the form x, x 1 1, and x 1 2. Since an even number is divisible by 2, consecutive even numbers are of the form 2x, 2x 1 2, and 2x 1 4. An odd number is of the form 2x 1 1. Example Find three consecutive whole numbers whose sum is 75. Let the first number be x, the second x 1 1, and the third x 1 2. Then: x 1 ~x 1 1! 1 ~x 1 2! 5 75 3x 1 3 5 75 3x 5 72 x 5 24 The numbers whose sum is 75 are 24, 25, and 26. Many versions of this problem have no solution. For example, no three consecutive whole numbers have a sum of 74.

QUIZ

WORD PROBLEMS

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WITH

ONE UNKNOWN

1.

If 18 is subtracted from six times a certain number, the result is 96. Find the number.

2.

A 63-foot rope is cut into two pieces. If one piece is twice as long as the other, how long is each piece?

3.

Peter is now three times as old as Jillian. In six years, he will be twice as old as she will be then. How old is Peter now?

4.

The sum of two consecutive odd integers is 68. Find the integers.

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MATHEMATICS REVIEW SOLUTIONS

1.

Let x 5 the number. Then, 6x 2 18 5 96 6x 5 114 x 5 19 The number is 19.

2.

Let x 5 the length of the short piece. Then, 2x 5 the length of the longer piece. And, x 1 2x 5 63 3x 5 63 x 5 21 2x 5 42 The pieces are 21 feet and 42 feet.

3.

Let J 5 Jillian’s age now; 3J 5 Peter’s age now; J 1 6 5 Jillian’s age in 6 years; 3J 1 6 5 Peter’s age in 6 years. Then, 3J 1 6 5 2 ~J 1 6! 3J 1 6 5 2J 1 12 3J 5 2J 1 6 J56 3J 5 18 Peter is currently 18 years old.

4.

Let x 5 the first odd integer. Then, x 1 2 5 the second odd integer, and, x 1 x 1 2 5 68 2x 1 2 5 68 2x 5 66 x 5 33 x 1 2 5 35 The numbers are 33 and 35.

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UNIT 7

LITERAL EQUATIONS An equation may have other letters in it besides the variable (or variables). Such an equation is called a literal equation. An illustration is x 1 b5 a, with x the variable. The solution of such an equation will not be a specific number but will involve letter symbols. Literal equations are solved by exactly the same methods as those involving numbers, but we must know which of the letters in the equation is to be considered the variable. Then the other letters are treated as constants. Example 1 Solve ax 2 2bc 5 d for x. ax 5 d 1 2bc x5

d 1 2bc if a Þ 0 a

Example 2 Solve ay 2 by 5 a2 2 b2 for y. y~a 2 b! 5 a2 2 b2

Factor out common term.

y~a 2 b! 5 ~a 1 b!~a 2 b!

Factor expression on right side.

y5a1b

Divide each side by a 2 b if a Þ b.

Example 3 Solve for S in the equation 1 1 1 5 1 R S T Multiply every term by RST, the LCD: ST 5 RT 1 RS ST 2 RS 5 RT S~T 2 R! 5 RT S5

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RT T2R

If T Þ R

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MATHEMATICS REVIEW

QUADRATIC EQUATIONS An equation containing the square of an unknown quantity is called a quadratic equation. One way of solving such an equation is by factoring. If the product of two expressions is zero, at least one of the expressions must be zero. Example 1 Solve y2 1 2y 5 0. y~y 1 2! 5 0 y 5 0 or y 1 2 5 0

Remove common factor. Since the product is 0, at least one of the factors must be 0.

y 5 0 or y 5 22 Check by substituting both values in the original equation: ~0!2 1 2~0! 5 0 ~22!2 1 2~22! 5 4 2 4 5 0 In this case there are two solutions. Example 2 Solve x2 1 7x 1 10 5 0. x2 1 7x 1 10 5 ~x 1 5! ~x 1 2! 5 0 x1 550 x 5 25

or x 1 2 5 0 or

x 5 22

Check: (25)2 1 7(25) 1 10 5 25 2 35 1 10 5 0 (22)2 1 7(22) 1 10 5 4 2 14 1 10 5 0 So far, each quadratic we have solved has had two distinct answers, but an equation may have a single answer (repeated), as in x2 1 4x 1 4 5 0 ~x 1 2!~x 1 2! 5 0 x 1 2 5 0 and x 1 2 5 0 x 5 22 and x 5 22 The only solution is 22.

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UNIT 7

Rewriting Equations Certain equations written with a variable in the denominator can be rewritten as quadratics. Example 4 Solve 2 1 5 5 x. x 24 1 5x 5 x2 2x 1 5x 2 4 5 0 2

x2 2 5x 1 4 5 0 ~x 2 4!~x 2 1! 5 0

Multiply both sides by x Þ 0. Collect terms on one side of equals and set sum equal to 0. Multiply both sides by 21. Factor

x 2 4 5 0 or x 2 1 5 0 x 5 4 or x

51

Check the result by substitution: 4 4 2 1 5 0 4 and 2 1 5 0 1 4 1 21 1 5 5 4 24 1 5 5 1 Some equations containing a radical sign can also be converted into a quadratic equation. The solution of this type of problem depends on the principle that If A 5 B then A2 5 B2 and If A2 5 B2 then A 5 B or A 5 2B Example Solve y 5 =3y 1 4. y 5 =3y 1 4 y2 5 3y 1 4 y2 2 3y 2 4 5 0 ~y 2 4!~y 1 l! 5 0 y 5 4 or y 5 21 Check by substituting values into the original equation: y54

=3~4! 1 4 4 0 =16 40

454

y 5 21

=3~21! 1 4 21 0 =23 1 4

and 21 0

21 Þ 1

The single solution is y 5 4: the false root y 5 21 was introduced when the original equation was squared.

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MATHEMATICS REVIEW

QUIZ

EQUATION SOLVING PROBLEMS Solve the following equations for the variable indicated: 1.

Solve for W: P 5 2L 1 2W

2.

Solve for x: ax 1 b5 cx 1 d

3.

Solve for x: 8x2 2 4x 5 0

4.

Solve for x: x2 2 4x 5 21

5.

Solve for y:

=y 11 2 3 5 7 SOLUTIONS

1.

P 5 2L 1 2W 2W 5 P 2 2L W5

2.

P 2 2L 2

ax 1 b5 cx 1 d ax 5 cx 1 d 2 b ax 2 cx 5 d 2 b x~a 2 c! 5 d 2 b x5

3.

d2b a2c

~if a Þ c!

8x2 2 4x 5 0 4x~2x 21! 5 0 4x 5 0, 2x 2 1 5 0 x 5 0,

4.

1 2

x2 2 4x 5 21 x2 2 4x 2 21 5 0 ~x 2 7!~x 1 3! 5 0 x 5 7, 23

5.

=y 1 1 2 3 5 7 =y 1 1 5 10

~=y 1 1!

2

5 102

y 1 1 5 100 y 5 99

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UNIT 7

LINEAR INEQUALITIES For each of the sets of numbers we have considered, we have established an ordering of the members of the set by defining what it means to say that one number is greater than the other. Every number we have considered can be represented by a point on a number line. An algebraic inequality is a statement that one algebraic expression is greater than (or less than) another algebraic expression. If all the variables in the inequality are raised to the first power, the inequality is said to be a linear inequality. We solve the inequality by reducing it to a simpler inequality whose solution is apparent. The answer is not unique, as it is in an equation, since a great number of values may satisfy the inequality. There are three rules for producing equivalent inequalities: 1. The same quantity can be added or subtracted from each side of an inequality. 2. Each side of an inequality can be multiplied or divided by the same positive quantity. 3. If each side of an inequality is multiplied or divided by the same negative quantity, the sign of the inequality must be reversed so that the new inequality is equivalent to the first. Example 1 Solve 5x 2 5 . 29 1 3x. 5x . 24 1 3x Add 5 to each side. 2x . 24 x . 22

Subtract 3x from each side. Divide by 12.

Any number greater than 22 is a solution to this inequality. Example 2 Solve 2x 2 12 , 5x 2 3. 2x , 5x 1 9 23x , 9 x . 23

Add 12 to each side. Subtract 5x from each side. Divide each side by 23, changing sign of inequality.

1 Any number greater than 23—for example, 22 , 0, 1, or 4—is a solution to this 2 particular inequality.

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MATHEMATICS REVIEW

LINEAR EQUATIONS

IN

TWO UNKNOWNS

Graphing Equations The number line is useful in picturing the values of one variable. When two variables are involved, a coordinate system is effective. The Cartesian coordinate system is constructed by placing a vertical number line and a horizontal number line on a plane so that the lines intersect at their zero points. This meeting place is called the origin. The horizontal number line is called the x axis, and the vertical number line (with positive numbers above the x axis) is called the y axis. Points in the plane correspond to ordered pairs of real numbers. Example The points in this example are: x y 0 0 1 1 3 21 22 22 22 1

Solving Simultaneous Linear Equations Two linear equations can be solved together (simultaneously) to yield an answer (x, y) if it exists. On the coordinate system, this amounts to drawing the graphs of two lines and finding their point of intersection. If the lines are parallel and therefore never meet, no solution exists. Simultaneous linear equations can be solved in the following manner without drawing graphs. From the first equation find the value of one variable in terms of the other; substitute this value in the second equation. The second equation is now a linear equation in one variable and can be solved. After the numerical value of the one variable has been found, substitute that value into the first equation to find the value of the second variable. Check the results by putting both values into the second equation. Example 1 Solve the system 2x 1 y 5 3 4x 2 y 5 0 From the first equation, y 5 3 2 2x. Substitute this value of y into the second equation to get 4x 2 ~3 22x! 5 0 4x 2 3 1 2x 5 0 6x 5 3 1 x5 2

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UNIT 7

Substitute x 5

1 in the first of the original equations: 2

SD

2

1 1y53 2 11y53 y52

Check by substituting both x and y values into the second equation:

SD

4

1 1 ~22! 5 0 2 22250

Example 2 A change-making machine contains $30 in dimes and quarters. There are 150 coins in the machine. Find the number of each type of coin. Let x 5 number of dimes and y 5 number of quarters. Then: x 1 y 5 150 Since .25y is the product of a quarter of a dollar and the number of quarters, and .10x is the amount of money in dimes, .10x 1 .25y 5 30 Multiply the last equation by 100 to eliminate the decimal points: 10x 1 25y 5 3000 From the first equation, y 5 150 2 x. Substitute this value in the equivalent form of the second equation. 10x 1 25~150 2 x! 5 3000 215x 5 2750 x 5 50 This is the number of dimes. Substitute this value in x 1 y 5 150 to find the number of quarters, y 5 100. Check: .10~50! 1 .25~100! 5 30 $5 1 $25 5 $30

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MATHEMATICS REVIEW

QUIZ

LINEAR INEQUALITIES

AND

EQUATIONS PROBLEMS

1.

Solve for x:

12x , 5(2x 1 4)

2.

Solve for y:

6y 1 2 , 8y 1 14

3.

Find the common solution: x 2 3y 5 3 2x 1 9y 5 11

4.

A coin collection consisting of quarters and nickels has a value of $4.50. The total number of coins is 26. Find the number of quarters and the number of nickels in the collection.

5.

Mr. Linnell bought 3 cans of corn and 5 cans of tomatoes for $3.75. The next week, he bought 4 cans of corn and 2 cans of tomatoes for $2.90. Find the cost of a can of corn. SOLUTIONS

1.

12x , 5~2x 1 4! 12x , 10x 1 20 2x , 20 x , 10

2.

6y 1 2 , 8y 1 14 6y , 8y 1 12 22y , 12 y . 26

3.

x 2 3y 5 3 2x 1 9y 5 11 Multiply the first equation by 3. 3~x 2 3y! 5 3~3! 2x 1 9y 5 11 3x 2 9y 5 9 2x 1 9y 5 11 5x

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5 20 x54

231

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UNIT 7

Now substitute this answer for x in the second equation. 2~4! 1 9y 5 11 8 1 9y 5 11 9y 5 3 y5 4.

1 3

Let Q 5 the number of quarters in the collection. Let N 5 the number of nickels in the collection. Then, .25Q 1 .05N 5 4.50 Q 1 N 5 26 Multiply the top equation by 100 to clear the decimals: 25Q 1 5N 5 450 Q 1 N 5 26 Multiply the bottom equation by 25 and add: 25Q 1 5N 5 450 25Q 2 5N 5 2130 20Q

5 320 Q 5 16 N 5 10

There are 16 quarters and 10 nickels. 5.

Let c 5 the cost of a can of corn. Let t 5 the cost of a can of tomatoes. Then, 3c 1 5t 5 3.75 4c 1 2t 5 2.90 Multiply the top equation by 2, the bottom one by 25, and add: 6c 1 10t 5 7.50 220c 2 10t 5 214.50 214c 5 27.00 c 5 .50 A can of corn costs 50¢.

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MATHEMATICS REVIEW

RATIO

AND

PROPORTION Many problems in arithmetic and algebra can be solved using the concept of ratio to a c a compare numbers. The ratio of a to b is the fraction . If the two ratios and b b d represent the same comparison, we write: a c 5 b d This equation (statement of equality) is called a proportion. A proportion states the equivalence of two different expressions for the same ratio. Example 1 In a class of 39 students, 17 are men. Find the ratio of men to women. 39 students 2 17 men 5 22 women Ratio of men to women is 17/22, also written 17;22. Example 2 A fertilizer contains 3 parts nitrogen, 2 parts potash, and 2 parts phosphate by weight. How many pounds of fertilizer will contain 60 pounds of nitrogen? The ratio of pounds of nitrogen to pounds of fertilizer is 3 3 to 3 1 2 1 2 5 . 7 Let x be the number of pounds of mixture. Then: 3 60 5 7 x Multiply both sides of the equation by 7x to get: 3x 5 420 x 5 140 pounds

COMPUTING AVERAGES

AND

MEDIANS

Mean Several statistical measures are used frequently. One of them is the average or arithmetic mean. To find the average of N numbers, add the numbers and divide their sum by N. Example 1 Seven students attained test scores of 62, 80, 60, 30, 50, 90, and 20. What was the average test score for the group? 62 1 80 1 60 1 30 1 50 1 90 1 20 5 392 Since there are 7 scores, the average score is 392 5 56 7 GRE CAT Success

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UNIT 7

Example 2 Joan allotted herself a budget of $50 a week, on the average, for expenses. One week she spent $35, the next $60, and the third $40. How much can she spend in the fourth week without exceeding her budget? Let x be the amount spent in the fourth week. Then: 35 1 60 1 40 1 x 5 50 4 35 1 60 1 40 1 x 5 200 135 1 x 5 200 x 5 65 She can spend $65 in the fourth week.

Median If a set of numbers is arranged in order, the number in the middle is called the median. Example Find the median test score of 62, 80, 60, 30, 50, 90, and 20. Arrange the numbers in increasing (or decreasing) order 20, 30, 50, 60, 62, 80, 90 Since 60 is the number in the middle, it is the median. It is not the same as the arithmetic mean, which is 56. If number of scores is an even number, the median is the arithmetic mean of the middle two scores.

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MATHEMATICS REVIEW

PLANE GEOMETRY Plane geometry is the science of measurement. Certain assumptions are made about undefined quantities called points, lines, and planes, and then logical deductions about relationships between figures composed of lines, angles and portions of planes are made, based on these assumptions. The process of making the logical deduction is called a proof. In this summary we are not making any proofs but are giving the definitions frequently used in geometry and stating relationships that are the results of proofs.

ANGLES

AND

LINES

Angles A line in geometry is always a straight line. When two straight lines meet at a point, they form an angle. The lines are called sides or rays of the angle, and the point is called the vertex. The symbol for angle is ∠. When no other angle shares the same vertex, the name of the angle is the name given to the vertex, as in angle A:

An angle may be named with three letters. Following, for example, B is a point on one side and C is a point on the other. In this case the name of the vertex must be the middle letter, and we have angle BAC.

Occasionally an angle is named by a number or small letter placed in the angle.

2

Angles are usually measured in degrees. An angle of 30 degrees, written 30°, is an angle whose measurement is 30 degrees. Degrees are divided into minutes; 608 (read “minutes”) 5 1°. Minutes are further divided into seconds; 609 (read “seconds”) 5 18.

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UNIT 7

Vertical Angles When two lines intersect, four angles are formed. The angles opposite each other are called vertical angles and are equal to each other.

a and c are vertical angles. ∠a 5 ∠c b and d are vertical angles. ∠b 5 ∠d

Straight Angle A straight angle has its sides lying along a straight line. It is always equal to 180°.

∠ABC 5 ∠B 5 180° ∠B is a straight angle.

Adjacent Angles Two angles are adjacent if they share the same vertex and a common side but no angle is inside another angle. ∠ABC and ∠CBD are adjacent angles. Even though they share a common vertex B and a common side AB, ∠ABD and ∠ABC are not adjacent angles because one angle is inside the other.

Supplementary Angles If the sum of two angles is a straight angle (180°), the two angles are supplementary and each angle is the supplement of the other.

∠G is a straight angle 5 180°. ∠a 1 ∠b 5 180° ∠a and ∠b are supplementary angles.

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Right Angles If two supplementary angles are equal, they are both right angles. A right angle is one-half a straight angle. Its measure is 90°. A right angle is symbolized by N .

∠G is a straight angle. ∠b 1 ∠a 5 ∠G, and ∠a 5 ∠b. ∠a and ∠b are right angles.

Complementary Angles Complementary angles are two angles whose sum is a right angle (90°).

∠Y is a right angle. ∠a 1 ∠b 5 ∠Y 5 90°. ∠a and ∠b are complementary angles.

Acute Angles Acute angles are angles whose measurement is less than 90°. No two acute angles can be supplementary angles. Two acute angles can be complementary angles.

∠C is an acute angle.

Obtuse Angles Obtuse angles are angles that are greater than 90° and less than 180°.

∠D is an obtuse angle.

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UNIT 7

Example 1 In the figure, what is the value of x?

2 30

Since the two labeled angles are supplementary angles, their sum is 180°. ~x 1 30°! 1 2x 5 180° 3x 5 150° x 5 50° Example 2 Find the value of x in the figure.

40˚ 2

Since the two labeled angles are vertical angles, they are equal. x 1 40° 5 2x 40° 5 x Example 3 If angle Y is a right angle and angle b measures 30°158, what does angle a measure?

Since angle Y is a right angle, angles a and b are complementary angles and their sum is 90°. ∠a 1 ∠b 5 90° ∠a 1 30°158 5 90° ∠a 5 59°458

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MATHEMATICS REVIEW

LINES A line in geometry is always assumed to be a straight line. It extends infinitely far in both directions. It is determined if two of its points are known. It can be expressed in terms of the two points, which are written as capital letters. The following line is called AB.

Or, a line may be given one name with a small letter. The following line is called line k.

A line segment is a part of a line between two endpoints. It is named by its endpoints, for example, A and B.

AB is a line segment. It has a definite length. If point P is on the line and is the same distance from A as from B, then P is the midpoint of segment AB. When we say AP 5 PB, we mean that the two line segments have the same length.

A part of a line with one endpoint is called a ray. AC is a ray of which A is an endpoint. The ray extends infinitely far in the direction away from the endpoint.

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UNIT 7

Parallel Lines Two lines meet or intersect if there is one point that is on both lines. Two different lines may either intersect in one point or never meet, but they can never meet in more than one point.

Two lines in the same plane that never meet no matter how far they are extended are said to be parallel, for which the symbol is \. In the following diagram a \ b.

If two lines in the same plane are parallel to a third line, they are parallel to each other. Since a \ b and b \ c, we know that a \ c.

Two lines that meet each other at right angles are said to be perpendicular, for which the symbol is ⊥. Line a is perpendicular to line b.

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MATHEMATICS REVIEW

Two lines in the same plane that are perpendicular to the same line are parallel to each other.

Line a ⊥ line c and line b ⊥ line c. Therefore, a \ b. A line intersecting two other lines is called a transversal. Line c is a transversal intersecting lines a and b.

The transversal and the two given lines form eight angles. The four angles between the given lines are called interior angles; the four angles outside the given lines are called exterior angles. If two angles are on opposite sides of the transversal, they are called alternate angles.

∠z, ∠w, ∠q, and ∠p are interior angles. ∠y, ∠x, ∠s, and ∠r are exterior angles. ∠z and ∠p are alternate interior angles; so are ∠w and ∠q. ∠y and ∠s are alternate exterior angles; so are ∠x and ∠r. Pairs of corresponding angles are ∠y and ∠q; ∠z and ∠r; ∠x and ∠p; and ∠w and ∠s. Corresponding angles are sometimes called exterior-interior angles.

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UNIT 7

When the two given lines cut by a transversal are parallel lines: 1. the corresponding angles are equal. 2. the alternate interior angles are equal. 3. the alternate exterior angles are equal. 4. interior angles on the same side of the transversal are supplementary.

a

b

If line a is parallel to line b: 1. ∠y 5 ∠q, ∠z 5 ∠r, ∠x 5 ∠p, and ∠w 5 ∠s. 2. ∠z 5 ∠p and ∠w 5 ∠q. 3. ∠y 5 ∠s and ∠x 5 ∠r. 4. ∠z 1 ∠q 5 180° and ∠p 1 ∠w 5 180° Because vertical angles are equal, ∠p 5 ∠r, ∠q 5 ∠s, ∠y 5 ∠w, and ∠x 5 ∠z. If any one of the four conditions for equality of angles holds true, the lines are parallel; that is, if two lines are cut by a transversal and one pair of the corresponding angles is equal, the lines are parallel. If a pair of alternate interior angles or a pair of alternate exterior angles is equal, the lines are parallel. If interior angles on the same side of the transversal are supplementary, the lines are parallel. Example In the figure, two parallel lines are cut by a transversal. Find the measure of angle y.

2

3

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50˚

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The two labeled angles are supplementary. 2x 1 ~3x150°! 5 180° 5x 5 130° x 5 26° Since ∠y is vertical to the angle whose measurement is 3x 1 50°, it has the same measurement. y 5 3x 1 50° 5 3(26°) 1 50° 5 128°

Polygons A polygon is a closed plane figure composed of line segments joined together at points called vertices (singular, vertex). A polygon is usually named by giving its vertices in order.

Polygon ABCDE In the figure, points A, B, C, D, and E are the vertices, and the sides are AB, BC, CD, DE, and EA. AB and BC are adjacent sides, and A and B are adjacent vertices. A diagonal of a polygon is a line segment joining any two nonadjacent vertices. EB is a diagonal.

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UNIT 7

Polygons are named according to the number of sides or angles. A triangle is a polygon with three sides, a quadrilateral a polygon with four sides, a pentagon a polygon with five sides, and a hexagon a polygon with six sides. The number of sides is always equal to the number of angles.

The perimeter of a polygon is the sum of the lengths of its sides. If the polygon is regular (all sides equal and all angles equal), the perimeter is the product of the length of one side and the number of sides.

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MATHEMATICS REVIEW

Congruent and Similar Polygons If two polygons have equal corresponding angles and equal corresponding sides, they are said to be congruent. Congruent polygons have the same size and shape. They are the same in all respects except possibly position. The symbol for congruence is ≅.

When two sides of congruent or different polygons are equal, we indicate the fact by drawing the same number of short lines through the equal sides.

This indicates that AB 5 EF and CD 5 GH. Two polygons with equal corresponding angles and corresponding sides in proportion are said to be similar. The symbol for similar is z.

Similar figures have the same shape but not necessarily the same size. A regular polygon is a polygon whose sides are equal and whose angles are equal.

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UNIT 7

TRIANGLES A triangle is a polygon with three sides. Triangles are classified by measuring their sides and angles. The sum of the angles of a plane triangle is always 180°. The symbol for a triangle is D. The sum of any two sides of a triangle is always greater than the third side.

Equilateral Triangles Equilateral triangles have equal sides and equal angles. Each angle measures 60° because 1 (180°) 5 60°. 3

AB 5 AC 5 BC. ∠A 5 ∠B 5 ∠C 5 60°.

Isosceles Triangles Isosceles triangles have two sides equal. The angles opposite the equal sides are equal. The two equal angles are sometimes called the base angles and the third angle is called the vertex angle. Note that an equilateral triangle is isosceles.

FG 5 FH. FG Þ GH. ∠G 5 ∠H. ∠F is vertex angle. ∠G and ∠H are base angles.

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Scalene Triangles Scalene triangles have all three sides of different length and all angles of different measure. In scalene triangles, the shortest side is opposite the angle of smallest measure, and the longest side is opposite the angle of greatest measure.

AB . BC . CA; therefore, ∠C . ∠A . ∠B.

Right Triangles Right triangles contain one right angle. Since the right angle is 90°, the other two angles are complementary. They may or may not be equal to each other. The side of a right triangle opposite the right angle is called the hypotenuse. The other two sides are called legs. The Pythagorean theorem states that the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.

AC is the hypotenuse. AB and BC are legs. ∠B 5 90°. ∠A 1 ∠C 5 90°. a2 1 c2 5 b2. Examples If ABC is a right triangle with right angle at B, and if AB 5 6 and BC 5 8, what is the length of AC? AB2 1 BC2 5 AC2 62 1 82 5 36 1 64 5 100 5 AC2 AC 5 10 If the measure of angle A is 30°, what is the measure of angle C ? Since angles A and C are complementary: 30° 1 C 5 90° C 5 60°

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UNIT 7

If the lengths of the three sides of a triangle are a, b, and c and the relation a2 1 b2 5 c2 holds, the triangle is a right triangle and side c is the hypotenuse. Example Show that a triangle with sides 5, 12, and 13 is a right triangle. The triangle will be a right triangle if a2 1 b2 5 c2. 52 1 122 5 132 25 1 144 5 169 Therefore, the triangle is a right triangle and 13 is the length of the hypotenuse.

Area of a Triangle The altitude (or height) of a triangle is a line segment dropped as a perpendicular from any vertex to the opposite side. The area of a triangle is the product of one-half the altitude and the base of the triangle. (The base is the side opposite the vertex from which the perpendicular was drawn.)

Altitudes Example Find the area A of the following isosceles triangle.

In an isosceles triangle the altitude from the vertex angle bisects the base (cuts it in half).

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MATHEMATICS REVIEW

The first step is to find the altitude. By the Pythagorean theorem, 1 a2 1 b2 5 c2; c 5 13, a 5 h, and b 5 (10) 5 5. 2 h2 1 52 5 132 h2 1 25 5 169 h2 5 144 h 5 12 A5 5

1 { base { height 2 1 { 10 { 12 2

5 60

Similarity Two triangles are similar if all three pairs of corresponding angles are equal. The sum of the three angles of a triangle is 180°; therefore, if two angles of triangle I equal two corresponding angles of triangle II, the third angle of triangle I must be equal to the third angle of triangle II, and the triangles are similar. The lengths of the sides of similar triangles are in proportion to each other. A line drawn parallel to one side of a triangle divides the triangle into two portions, one of which is a triangle. The new triangle is similar to the original triangle.

DABE z DACD

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UNIT 7

Example In the following figure, if AC 5 28 feet, AB 5 35 feet, BC 5 21 feet, and EC 5 12 feet, find the length of DC if DE \ AB.

Because DE \ AB, DCDE z DCAB. Since the triangles are similar, their sides are in proportion: DC EC 5 AC BC DC 12 5 28 21 DC 5

12 { 28 5 16 feet 21

QUADRILATERALS A quadrilateral is a polygon of four sides. The sum of the angles of a quadrilateral is 360°. If the opposite sides of a quadrilateral are parallel, the quadrilateral is a parallelogram. Opposite sides of a parallelogram are equal and so are opposite angles. Any two consecutive angles of a parallelogram are supplementary. A diagonal of a parallelogram divides the parallelogram into congruent triangles. The diagonals of a parallelogram bisect each other.

∠A 1 ∠B 5 180° DABD ≅ DCDB DABC ≅ DCDA AP 5 PC BP 5 PD

AD \ BC AD 5 BC AB \ DC AB 5 DC ∠D 5 ∠B ∠A 5 ∠C

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MATHEMATICS REVIEW

Definitions A rhombus is a parallelogram with four equal sides. The diagonals of a rhombus are perpendicular to each other.

A rectangle is a parallelogram with four right angles. The diagonals of a rectangle are equal and can be found using the Pythagorean theorem if the sides of the rectangle are known.

AB2 1 BC2 5 AC2 A square is a rectangle with four equal sides.

A trapezoid is a quadrilateral with only one pair of parallel sides, called bases. The nonparallel sides are called legs.

AD \ BC. AD and BC are bases. AB and DC are legs. h 5 altitude

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UNIT 7

Finding Areas The area of any parallelogram is the product of the base and the height, where the height is the length of the altitude, a line segment drawn from a vertex perpendicular to the base.

Since rectangles and squares are also parallelograms, their areas follow the same formula. For a rectangle, the altitude is one of the sides, and the formula is length times width. Since a square is a rectangle for which length and width are the same, the area of a square is the square of its side. The area of a trapezoid is the height times the average of the two bases. The formula is: A5h

b1 1 b2 2

The bases are the parallel sides, and the height is the length of an altitude to one of the bases. Example 1 Find the area of a square whose diagonal is 12 feet. Let s 5 side of square. By the Pythagorean theorem:

s2 1 s2 5 122 2s2 5 144 s2 5 72 s 5 =72 Use only positive values because this is the side of a square. Since A 5 s2 A 5 72 square feet

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MATHEMATICS REVIEW

Example 2 Find the altitude of a rectangle if its area is 320 and its base is 5 times its altitude. Let altitude 5 h. Then base 5 5h. Since A 5 bh,

A 5 ~5h!~h! 5 320 5h2 5 320 h2 5 64 h58 If a quadrilateral is not a parallelogram or a trapezoid but it is irregularly shaped, its area can be found by dividing it into triangles, attempting to find the area of each, and adding the results.

CIRCLES Definitions Circles are closed plane curves with all points on the curve equally distant from a fixed point called the center. The symbol ( indicates a circle. A circle is usually named by its center. A line segment from the center to any point on the circle is called the radius (plural, radii). All radii of the same circle are equal.

C 5 center CP 5 radius 5 r

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UNIT 7

A chord is a line segment whose endpoints are on the circle. The diameter of a circle is a chord that passes through the center of the circle. The diameter, the longest distance between two points on the circle, is twice the length of the radius. A diameter perpendicular to a chord bisects that chord.

AB is a chord. C is the center. DCE is a diameter. FCG is a diameter. AB ⊥ DCE so AP 5 PB. A central angle is an angle whose vertex is the center of a circle and whose sides are radii of the circle. An inscribed angle is an angle whose vertex is on the circle and whose sides are chords of the circle.

∠ACB is a central angle. ∠RST is an inscribed angle.

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An arc is a portion of a circle. The symbol ∩ is used to indicate an arc. Arcs are usually measured in degrees. Since the entire circle is 360°, a semicircle (half a circle) is an arc of 180°, and a quarter of a circle is an arc of 90°.

∩ ABD is an arc. ∩ AB is an arc. ∩ BD is an arc. A central angle is equal in measure to its intercepted arc.

Perimeter and Area The perimeter of a circle is called the circumference. The length of the circumference is pd, where d is the diameter, or 2pr, where r is the radius. The number p is irrational and can be approximated by 3.14159..., but in problems dealing with circles it is best to leave p in the answer. There is no fraction exactly equal to p. Example 1 If the circumference of a circle is 8p feet, what is the radius? Since C 5 2pr 5 8p, r 5 4 feet. The length of an arc of a circle can be found if the central angle and radius are n° known. Then, the length of arc is (2pr), where the central angle of the arc is 360° n°. This is true because of the proportion: Arc central angle 5 Circumference 360°

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UNIT 7

Example 2 If a circle with a radius of 3 feet has a central angle of 60°, find the length of the arc intercepted by this central angle. 60° (2p3) 5 p feet 360°

Arc 5

The area, A, of a circle is pr2, where r is the radius. If the diameter is given instead of the radius,

SD

A5p

d 2

2

5

pd2 . 4

Example 3 Find the area of a circular ring formed by two concentric circles with radii of 6 and 8 inches, respectively. (Concentric circles are circles with the same center.) The area of the ring will equal the area of the large circle minus the area of the small circle. Area of ring 5 p82 2 p62 5 p(64 2 36) 5 28p square inches

Example 4 A square is inscribed in a circle whose diameter is 10 inches. Find the difference between the area of the circle and that of the square. If a square is inscribed in a circle, the diagonal of the square is the diameter of the circle. If the diagonal of the square is 10 inches, then, by the Pythagorean theorem, the side of the square s is =50, and the area of the square is 50 square inches.

2s2 5 100 s2 5 50

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If the diameter of the circle is 10, its radius is 5 and the area of the circle is p52 5 25p square inches. Then, the difference between the area of the circle and the area of the square is: 25p 2 50 square inches 5 25 (p22) square inches

Distance Formula In the arithmetic section, we described the Cartesian coordinate system when explaining how to draw graphs representing linear equations. If two points are plotted in the Cartesian coordinate system, it is useful to know how to find the distance between them. If the two points have coordinates (a, b) and (p, q), the distance between them is: d5

=~a 2 p!2 1 ~b 2 q!2

This formula makes use of the Pythagorean theorem. Example Find the distance between the two points (23, 2) and (1, 21).

Let (a, b) 5 (23, 2) and (p, q) 5 (1, 21). Then:

=~2321!2 1 @2 2 ~21!#2 5 =~24!2 1 ~2 1 1!2 5 =~24!2 1 32 5 =16 1 9 5 =25 5 5

d5

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UNIT 7

QUIZ

PLANE GEOMETRY PROBLEMS 1.

In triangle QRS, ∠Q 5 ∠R and ∠S 5 64°. Find the measures of ∠Q and ∠R.

2.

In parallelogram ABCD, ∠A and ∠C are opposite angles. If ∠A 5 12x° and ∠C 5 (10x 1 12)°, find the measures of ∠A and ∠C.

3.

What is the area of a trapezoid whose height is 5 feet and whose bases are 7 feet and 9 feet?

4.

In the preceding figure, CF \ BG. Find the length of CF.

5.

The hypotenuse of a right triangle is 25 feet. If one leg is 15 feet, find the length of the other leg.

6.

Find the area of a circle whose diameter is 16 inches.

7.

Find the distance between the points (21, 22) and (5, 7). SOLUTIONS

1.

∠Q 1 ∠R 1 ∠S 5 180° ∠Q 1 ∠R 1 64° 5 180° ∠Q 1 ∠R 5 116° Since ∠Q 5 ∠R, they each must have measures of 58°.

2.

The opposite angles in a parallelogram are equal. Thus, 12x 5 10x 1 12 2x 5 12 x56 Thus, 12x 5 12(6) 5 72. ∠A and ∠C both measure 72°.

3.

S D S D SD

A5h 55

b1 1 b2 2

719 16 55 5 5~8! 5 40 2 2

The area of the trapezoid is 40.

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MATHEMATICS REVIEW

4.

Since CF \ BG, DACF z DABG. 6 8 Therefore, 5 CF 12 8 CF 5 72 CF 5 9 inches.

5.

Using the Pythagorean theorem, a2 1 152 5 252 a2 1 225 5 625 a2 5 400 a 5 =400 5 20 The length of the other leg is 20.

6.

If d 5 16, r 5 8. A 5 pr2 5 p(8)2 5 64p The area of the triangle is 64p.

7.

d5

=~5 2 ~21!!2 1 ~7 2 ~22!!2 5 =62 1 92 5 =36 1 81 5 =117

The distance between the points is equal to

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=117.

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R E D A LERT QUANTITATIVE COMPARISONS STRATEGIES Quantitative Comparison questions are the only questions on the entire GRE CAT that are not arranged in the standard multiple-choice format. Instead, in each of these questions you are given two quantities, one in Column A and one in Column B. Your job, simply put, is to determine which of these two quantities is larger. The answer scheme is simple. If the quantity in Column A is larger, you should answer (A). If the quantity in Column B is larger, you should answer (B). If the two quantities are of identical size, the correct answer is (C). Finally, if it is not possible to tell which quantity is larger, the answer is (D). Occasionally there will be some additional information given to help you determine the relative size of the two quantities. This information, when given, will be centered just above the Column A and Column B entries. Following are the actual directions as they appear on the GRE. They should be memorized so that you do not waste time reading them when you take the actual test.

QUANTITATIVE COMPARISONS Numbers: All numbers used are real numbers. Figures: Position of points, angles, regions, etc., can be assumed to be in the order shown; and angle measures can be assumed to be positive. Lines shown as straight can be assumed to be straight. Figures can be assumed to lie in a plane unless otherwise indicated. Figures that accompany questions are intended to provide information useful in answering the questions. However, unless a note states that a figure is drawn to scale, you should solve these problems NOT by estimating sizes by sight or by measurement, but by using your knowledge of mathematics.

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Directions: Each of the Questions 1–15 consists of two quantities, one in Column A and one in Column B. You are to compare the two quantities and choose: (A) (B) (C) (D) Note:

if the quantity in Column A is greater; if the quantity in Column B is greater; if the two quantities are equal; if the relationship cannot be determined from the information given.

Since there are only four choices, NEVER mark (E).

COMMON INFORMATION In a question, information concerning one or both of the quantities to be compared is centered above the two columns. A symbol that appears in both columns represents the same thing in Column A as it does in Column B. Example 1 Column A

Column B

236

216

The correct answer is (A). 2 3 6 5 12, 2 1 6 5 8 Examples 2–4 refer to DPQR

PN

NQ

Example 2 Column A

Column B

PN

NQ

The correct answer is (D). Equal measure cannot be assumed, even though PN and NQ appear equal. Example 3 Column A

Column B

x

y

The correct answer is (B). We know that N is between P and Q.

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QUANTITATIVE COMPARISONS STRATEGIES

Example 4 Column A

Column B

w1z

180

The correct answer is (C). The line PQ is a straight line. To gain a better understanding of choices (A) through (D), we will now look at four examples. These examples have been selected so that the answer to the first one is (A), the answer to the second is (B), and so on. Directions: Each of the Questions 1–15 consists of two quantities, one in Column A and one in Column B. You are to compare the two quantities and choose: (A) (B) (C) (D) Note:

if the quantity in Column A is greater; if the quantity in Column B is greater; if the two quantities are equal; if the relationship cannot be determined from the information given.

Since there are only four choices, NEVER mark (E). Column A

Column B 1 54 x

1.

1

x

The correct answer is (A). The equation given in the common information can 1 1 be solved to determine that x 5 5 4. Since 1 . 5 4, the answer is (A). x x

=26 2 =10

2.

=26 2 10

The correct answer is (B). The entry in Column B is equal to 4. While we cannot exactly determine the value of Column A, if we estimate =26, and =10, we can see that its value is close to 2.

3.

162

z

The correct answer is (C). Since a triangle contains 180°, y 1 4y° 1 90° 5 180°. Thus, 5y° 5 90° and y° 5 18°. Since z 1 y 5 180, z must be 162, and the answer is (C).

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Column A

Column B 6

x 5 64 4.

2

x

The correct answer is (D). Solving the equation given as common information, we can determine that x is either 2 or 22. Thus, x is either less than or equal to 2.

HINTS

AND

STRATEGIES

FOR

QUANTITATIVE COMPARISON QUESTIONS

Before we look at some specific problem-solving strategies for Quantitative Comparison questions, let us examine some general strategies.

General Strategies 1. Remember that your goal is to do as little work as possible to answer the question. You frequently don’t need to determine the actual size of the quantities in Columns A and B to know which one is larger. As a simple example, if you have enough information to determine that the quantity in Column A is positive and the quantity in Column B is negative, then the quantity in Column A is bigger, regardless of its actual value. 2. Unlike every other type of question on the GRE CAT, Quantitative Comparison questions have only four possible choices. 3. Be sure that you understand the meaning of the answer choices. For example, choice (A) indicates that the quantity in Column A is always bigger than the quantity in Column B. If choice (A) is sometimes, but not always, bigger, the answer is choice (D). Similarly, choice (C) is the answer only if the quantities are always equal. 4. Be sure that you do only as much math as is absolutely necessary to determine which quantity is bigger. Estimate and approximate as much as possible. You can often answer a question correctly by doing very little actual mathematical computation. 5. Whenever both of the given quantities are purely numerical (contain only numbers, no letters), then both quantities have a definite size, and the answer cannot be choice (D). If you are not sure how to answer a problem with two purely numerical entries, be sure to guess either choices (A), (B), or (C).

SPECIFIC MATHEMATICAL STRATEGIES

FOR

QUANTITATIVE COMPARISON QUESTIONS

Whenever you can, eliminate common factors and terms from Column A and Column B. Then, simply compare the remaining quantities. Often, sums and products can be compared term by term, or factor by factor.

1.

Column A

Column B

(108)2 2 (13)2

(108 213)2

The correct answer is (A). The quantity in Column A, when factored, becomes (108 2 13)(108 1 13). The quantity in Column B is equal to (108 213)(108 2 13). Upon canceling the common factor of (108 2 13), we see that we are comparing (108 1 13) in Column A to (108 2 13) in Column B.

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2.

Column A

Column B

5 6 7 1 1 6 7 8

5 6 7 1 1 7 8 9

The correct answer is (A). Simply note that each term in Column A is bigger than the corresponding term in Column B. 3.

6(125)4

2(125)12

The correct answer is (C). Cancel the common factor of 125 from both sides. Then, both sides become equal to 24. Remember that you can often determine which quantity is bigger by simply estimating sizes. 221 333

4.

667 999

The correct answer is (B). Note that the quantity in Column A is less than 2 2 and that the quantity in Column B is greater than . 3 3 A Quantitative Comparison question can be treated as if it were an algebraic inequality, with your job being to position the correct inequality sign (5, ,, .) between entries. As such, you may perform any operation to both columns of the question that you can perform on both sides of an inequality. This means, whenever you wish, you can add or subtract the same number to Column A and Column B, multiply or divide both columns by the same positive number, or square both columns (if both entries are positive). This strategy can be used to change the operations of subtraction and division to the relatively less confusing operations of addition and multiplication.

5.

Column A

Column B

=89,905

300

The correct answer is (B). Square both sides. Column A becomes 89,905 and Column B becomes 90,000. 4 6. =3 =3 The correct answer is (B). Do not waste any time estimating the values of the quantities. Simply multiply both entries by =3. Column A is then equal to =3 3 4 =3 5 3, while Column B is equal to 3 =35 4. Since 4 . 3, the answer is (B). = GRE CAT Success

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RED ALERT

7.

Column A

Column B

5 1 9 1 6 7

5 6 10 2 6 7

The correct answer is (C) Eliminate the subtraction in Column B by adding

6 to 7

5 1 6 both entries. Column A then becomes 9 1 1 5 6 7 7 5 5 5 6 6 5 9 1 1 5 10 . Column B becomes 10 2 1 5 10 . 6 6 6 7 7 6 Whenever you are comparing quantities containing variables, remember to consider both positive and negative values of the variables. Similarly, remember that the variables could have fractional values. Column B

Column A 3,x,5 4,y,6 8.

x

y

The correct answer is (D). Many people might answer (B) for this, assuming that x 5 4 and y 5 5. However, remember that x and y could also be fractional. For example, x could be 4.5, while y is 4.1. If the column entries contain algebraic operations, it frequently helps to begin by performing these operations. Column A

Column B a 5 22, c 5 5

9.

3a(2b 1 5c)

2a(3b 1 5c)

The correct answer is (B). To begin, expand both expressions. The entry in Column A becomes 6ab 1 15ac. Column B becomes 6ab 1 10ac. Cancel the common factor of 6ab and you’ll see that we are actually comparing 15ac in Column A to 10ac in Column B. We know that a 5 22 and c 5 5. Thus, the entry in Column A becomes 2150, while Column B becomes 2100. See if the common information can be manipulated to a form that is similar in appearance to the entry in one of the columns. Column B

Column A 5p 1 7q 5 13 10.

15p 1 21q

40

The correct answer is (A). If you multiply both sides of the equation given as common information by 3, you will obtain 15p 1 21q 5 39. Thus, the value of the expression in Column B is 39.

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QUANTITATIVE COMPARISONS STRATEGIES

When either of the column entries contain variables, it is often very helpful to substitute numerical values for these variables and observe what happens. Any substitution you make will enable you to eliminate two of the possible answer choices. Suppose, for example, that you plug a value into the quantities, and for this particular value the quantity in Column A turns out to be bigger. This means that the answer cannot be choice (B) or choice (C). Either Column A is always larger, choice (A), or sometimes larger, choice (D). Column A

Column B sÞ1sÞ0 r21 s21

r s

11.

The correct answer is (D). Try to substitute values for r and s. If, for example, r r21 r 5 s 5 2, 5 . Thus, we know that the answer is either choice (C) or s s21 choice (D). Now let r 5 0 and s 5 2. Then, the value in Column A becomes 0, 1 and the value in Column B becomes 2 . This result indicates that the correct 2 answer is (D). Remember that powers of, roots of, and divisions by numbers between 0 and 1 behave differently than those with numbers greater than 1. For example, if you square a number larger than 1, the resulting number is larger than the original; yet, if you square a number less than 1 but greater than 0, the resulting number is smaller than the original. Also, remember that powers of even and odd numbers behave differently. The following examples illustrate some of these variations. Column A

Column B x.0

x2

12.

x3

The correct answer is (D). While intuition tells us that cubing a positive number yields a larger result than squaring the number, this result is actually true only for numbers bigger than 1. In fact, x2 5 x3 if x 5 1, and if x , 1, x2 . x3. 1 1 1 For example, if x 5 , then x2 5 and x3 5 . Thus, there is no way to tell if 2 4 8 x2 or x3 is larger. x.1 x2

13.

x3

The correct answer is (B). As long as we know that x . 1, we have x3 . x2.

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Column A

Column B 0,z,1

12 z

14.

12z

The correct answer is (A). When 12 is divided by z, 0 , z , 1, will yield a number greater than 12, while 12 multiplied by z, 0 , z , 1, will yield a number smaller than 1. Now turn to the Quantitative Comparisons practice section that follows and try your hand at what you’ve learned.

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Unit 8 QUANTITATIVE COMPARISONS REVIEW Directions: Each of the questions 1–15 consists of two quantities, one in Column A and one in Column B. You are to compare the two quantities and choose: (A) if the quantity in Column A is greater; (B) if the quantity in Column B is greater; (C) if the two quantities are equal; (D) if the relationship cannot be determined from the information given. Note:

Since there are only four choices, NEVER MARK (E).

Numbers All numbers used are real numbers. Figures Position of points, angles, regions, etc. can be assumed to be in the order shown, and angle measures can be assumed to be positive. Lines shown as straight can be assumed to be straight. Figures can be assumed to lie in a plane unless otherwise indicated. Figures that accompany questions are intended to provide information that is useful in answering the questions. However, unless a note states that a figure is drawn to scale, you should solve these problems NOT by estimating sizes by sight or by measurement, but by using your knowledge of mathematics.

Common Information In a question, information concerning one or both of the quantities to be compared is centered above the two columns. A symbol that appears in both columns represents the same thing in Column A as it does in Column B. Example 1 Column A

Column

236

216

The correct answer is (A). Since 2 3 6 5 12, 2 1 6 5 8

269

UNIT 8

Examples 2–4 refer to DPQR Example 2

Column A

Column B

PN

NQ

The correct answer is (D). Equal measure cannot be assumed, even though PN and NQ appear equal.

Example 3 x

y

The correct answer is (B). N is between P and Q.

Example 4 w1z

180

The correct answer is (C). PQ is a straight line.

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QUANTITATIVE COMPARISONS REVIEW

Directions: Each of the questions 1–30 consists of two quantities, one in Column A and one in Column B. You are to compare the two quantities and choose: (A) (B) (C) (D) Note:

if the quantity in Column A is greater; if the quantity in Column B is greater; if the two quantities are equal; if the relationship cannot be determined from the information given.

Since there are only four choices, NEVER MARK (E).

Column A

Column B

2 7 3

1 7 3

1.

S DS D 5 12

S DS D

1 1 1 2 2 10 5 3

4 15

1 3 of 3 5

.25

4 5

2.

12

3.

4.

5.

GRE CAT Success

1 ’s 16 1 3 (one-sixteenths) in of . 3 4

The number of

Average price per pound of a mixture of 3 lbs. of nuts at $1.89 per pound and 2 lbs. of pecans at $1.49 per pound

271

5 6

The number of in

6 15

1 ’s (one-sixteenths) 16

1 3 of . 4 4

Average of $1.09, $2.19, and $4.75

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UNIT 8

6.

Column A

Column B

N 3 (N 1 1)

N2 1 1

7y 5 28 7.

y

(22)2

8.

x2y

y2x

9.

6x 1 7y 5 45

xy

2 r5 z 3 3 p5 z 2 10.

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z2

p 1 r

11.

(y 1 3)2

(y 1 3)

12.

S D

6 10

3 2 5

2

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QUANTITATIVE COMPARISONS REVIEW

Column A

Column B aÞ0

13.

14.

a°

1

=7

3

3

=7

10b 2 17 5 13 9z 2 27 5 0 15.

b

z

16.

.2y 5 2.6

6y 2 5.7 5 5y 1 7.3

x

20

4 x 5

y Perimeter 5 72

17.

Area 5 320

x

y

1 x 1 3 . 26 3 18.

GRE CAT Success

218

x

273

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UNIT 8

Column A

Column B x Þ 3, x Þ 0

19.

x23 x2 2 3x

Reciprocal of x

a1b 7 5 b 2 20.

2.75

a:b

x 5 3y 21.

x:y

y:x

xÞ0 22.

2x3 2 3x2 x2

2x 2 3

xÞ2 23.

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Average of x 2 8 and 3x 1 2

274

3 2 4x2 2 9 2x 1 3

GRE CAT Success

QUANTITATIVE COMPARISONS REVIEW

Column A

Column B x5

24.

1 2

S D

x2 1 x

=3

2

2

∠BFC and ∠DFE are right angles. 25.

∠3

∠1

Questions 26 and 27 refer to the drawing below.

GRE CAT Success

26.

∠4

∠2

27.

∠4 1 ∠3

∠1 1 ∠2 1 ∠3

275

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UNIT 8

Column A

Column B

Questions 28 and 29 refer to the drawing below,

Line r is parallel to line s.

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28.

∠k

∠g

29.

∠j

∠l

30.

∠H

∠L

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QUANTITATIVE COMPARISONS REVIEW

Quick Score Answers 1. 2. 3. 4. 5.

A A B A B

6. 7. 8. 9. 10.

D C D D C

11. 12. 13. 14. 15.

D B C B C

16. 17. 18. 19. 20.

C A D C B

21. 22. 23. 24. 25.

A C C C C

26. 27. 28. 29. 30.

A C C A A

ANSWERS AND EXPLANATIONS 1. The correct answer is (A). Column A

Column B

2 7 3

1 7 3

S DS D

S DS D

5 4 3 Simplify denominator. 5 12 1 1 5 1 4 3 5 5 12 3 1 3

5 6 3 Simplify denominator. 6 15 1 1 6 1 5 3 5 6 15 3 1 3

Cancel 5’s and 4’s.

Cancel 5’s and 6’s.

4 5

5 12

5 6

6 15

1 2 7 . 7 (Consider the size of the numerators since the denominators are the 3 3 same.) 2. The correct answer is (A). Find the least-common denominator.

Column A

Column B 2 8 4 3 5 Change 15 2 30 11 8 Hence . 30 30

1 3 3 3 5 10 3 30 1 6 6 3 5 5 6 30 1 10 10 3 5 3 10 30 19 30 Subtract 1 2

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19 30 19 5 2 30 30 30 11 5 30

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UNIT 8 3. The correct answer is (B). Column A

Column B

1 3 of 3 5

Convert .25 to a fraction and reduce 25 .25 5 100 Cancel out 25’s:

Multiply

1 3 3 3 5

1 25 5 100 4

Cancel out 3’s: 1 1 3 1 3 5 3 5 5 1

Same numerator, smaller denominator—greater (larger) fraction. 4. The correct answer is (A). Column B

Column A Multiply

1 3 3 3 4

Multiply

Cancel out 3’s: 1 1 3 1 3 5 3 4 4 1

1 3 3 4 4

3 1 3 3 5 4 4 16

4 1 4 3 5 4 4 16 3 4 . 16 16 Or, simply notice that the numerator in Column A is larger than that of Column B and, therefore, must contain more sixteenths. 5. The correct answer is (B). Column B

Column A Multiply $1.89 3 3 5 $5.67 Multiply $1.49 3 2 5 $2.98

Add $1.09 1 $2.19 1 $4.75 5 $8.03 Divide $8.03 4 3 5 $2.68

Add $5.67 1 $2.98 5 $8.65 Divide $8.65 4 5 lbs. 5 $1.73 $1.73 . $2.68 6. The correct answer is (D). N 3 (N 1 1) 5 N2 1 N. Thus, N2 1 N and N2 1 1 cannot have any relationship determined since N does not have any given numerical value.

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QUANTITATIVE COMPARISONS REVIEW 7. The correct answer is (C). Solve 7y 5 28 y54 ~22!2 5 ~222!~22! 54 8. The correct answer is (D). No relationship can be determined since there are no values for x or y. 9. The correct answer is (D). No relationship can be determined between x and y. r p 10. The correct answer is (C). . Thus, p 3 5 p 3 r. Invert divisor and multiply 1 1 r 3 2 3 2 z 3 z. Substitute values for p and r: 3 3 z 3 z. Cancel out 2’s and 3’s: 3 2 3 2 1z2 5 z2. 11. The correct answer is (D). (y 1 3)2 5 (y 1 3)(y 1 3). Column A has two factors of (y 1 3). Column B has one factor of (y 1 3). No relationship can be determined since y has no numerical value. 12. The correct answer is (B). Column B

Column A

S D S DS D 3 2 5

2

3 5 2 5

3 15 6 5 5 10 5 25

3 9 2 5 5 25 15 9 , 25 25

13. The correct answer is (C). a° has a value of 1. 14. The correct answer is (B). Multiply both sides by 3=7. Then, Column A becomes =73 3 7 5 7 3 7 5 7. Column B becomes 3 3 3 7 5 9. = = = = 3 =7 15. The correct answer is (C). Column A

9z 2 27 5 0 1 27 1 27 9z 5 27

10b 2 17 5 13 1 17 1 17 10b 5 30 (additive inverse) b53

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Column B

279

(additive inverse) z53

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UNIT 8 16. The correct answer is (C). Column A .2y 5 2.6 Divide by .2 y 5 13

Column B 6y 2 5.7 5 5y 1 7.3 25y 1 5.7 5 5y 1 5.7 (additive inverse) y 5 13.0

17. The correct answer is (A). Column A

Column B y~20! 5 area 20y 5 320

2l 1 2w 5 p 4 2 x 1 2~x! 5 72 5 8 x 1 2x 5 72 5 10x 8 x1 5 72 5 5 18 x 5 72 5

SD

Divide by 20 y 5 16

18 . Invert the divisor and 5 5 5 x 5 72 3 multiply by 18 18

Divide by

Cancel out 18’s: 4 5 72 3 x5 1 18 1 x 5 20 18. The correct answer is (D). Solve 1 x 1 3 . 26 3 23 23 1 x . 29 3 Multiply by 3 3

S

D

1 x . 29 3 x . 227

Thus x . 227, but we have no way of telling if x . 218.

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QUANTITATIVE COMPARISONS REVIEW 19. The correct answer is (C). Column A

Column B

Reciprocal of x Factor

x23 1 x23 5 5 2 x 2 3x x~x 2 3! x 1 x

1 x 20. The correct answer is (B). Solve

a1b 7 5 product extremes equals product of b 2

means 2~a 1 b! 5 7b 2a 1 2b 5 7b 2 2b 5 2b ~additive inverse! Divide by 2b 2a 5b 5 2b 2b a 5 a:b 5 5 b 2 a a:b 5 5 2.50 b 2.50 , 2.75 21. The correct answer is (A). Solve x 5 3y. Divide by y:

x 3 5 y 1

Column A

Column B

x:y 5 3:1 3 1 . 1 3

y:x 5 1:3

22. The correct answer is (C). Factor

2x3 2 3x2 x2~2x 2 3! 5 5 2x 2 3. x2 x2

23. The correct answer is (C). Column B

Column A x 2 81 3x 1 2 2 4x 2 6 2

Factor

4x2 2 9 ~2x 2 3!~2x 1 3! 5 2x 1 3 2x 1 3

Reduce and cancel out 2x 2 3.

Reduce 2x 2 3

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UNIT 8 24. The correct answer is (C). Column A Substitute x 5 x2 1 x 1 2 1 1 5 2 2 1 1 5 1 4 2 3 5 4

Column B

S D S= DS= D

1 inches 2

=3 2

SD

2

3

5

5 5

2

3

2

=9 4 3 4

25. The correct answer is (C). ∠3 1 ∠2 5 90° complementary angles and ∠2 1 ∠1 5 908 complementary ∠3 5 ∠1 angles complementary to the same angle are equal. 26. The correct answer is (A). ∠4 5 ∠1 1 ∠2 exterior angle is equal to the sum of two nonadjacent interior angles ∠4 . ∠2. 27. The correct answer is (C). ∠4 1 ∠3 5 180° supplementary angles 5 180°. ∠1 1 ∠2 1 ∠3 5 180° sum of angles in a triangle 5 180°. 28. The correct answer is (C). ∠k 5 ∠g alternate interior angles are equal. 29. The correct answer is (A). ∠l 5 55° alternate interior angles are equal. ∠j 1 55° supplementary angles 5 180° ∠j 5 125° ∠j . ∠l 30. The correct answer is (A). Column A

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Column B

∠H 1 50°1 50° 5 180°. Sum of the angles in a triangle 5 180°.

∠L 1 70° 1 50° 5 180°. Sum of the angles in a triangle 5 180°.

∠H 1 100 5 180 ∠H 5 80

∠L 1 120 5 180 ∠L 5 60 ∠L 5 60

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R E D A LERT DATA ANALYSIS STRATEGIES Data Analysis questions are based on data contained in tables and graphs. Typically, all five questions in this portion of the test are based on one or two tables or graphs. In each question you will need to make use of a portion of the data contained in the table or graph. On the following pages you will find a thorough review of graph and table reading and analysis. Before you go over this information, read and remember the following suggestions and reminders: 1. If a problem involves more than one graph or table, make sure to use the appropriate one to answer the question you are working on. 2. Many of the graphs on the GRE CAT contain a tremendous amount of information. Do not try to understand such a graph all at once. Instead, analyze it carefully, focusing on important information such as: • the title of the graph and any subtitles; • the quantities represented by the axes of a line or bar graph, or by the sectors of a circle graph; • any numerical scaling factor use (for example, “all numbers in millions”); • any legends indicating what particular symbols or shadings represent; and • the range of numerical values that the graph represents. 3. Some questions that appear to involve a lot of computation can actually be answered quickly by estimating and approximating values. Always remember that one of the multiplechoice answers must be correct. 4. Make sure that you do not confuse decimals and percents. Some graphs may indicate percents along their axes; others may contain actual numerical values. 5. As always, use only information given in the graphs to help you answer the questions. Do not ever use any outside knowledge that you may have. 6. If possible, see if answers can be visualized. For example, if a problem asks for the average of two of the numbers on a graph, try to estimate the midpoint of the two numbers instead of doing the actual computation.

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Unit 9 DATA ANALYSIS REVIEW Tables and graphs give visual comparisons of amount. They show relationships between two or more sets of information. It is essential to be able to read tables and graphs correctly.

TABLES Tables present data corresponding to classifications by row and column. Tables always state the units (thousands of people, years, millions of dollars, for example) in which the numbers are expressed. Sometimes the units are percents. Both specific and general questions can be answered by using the information in the table. (Notice that in this table the numbers are given in thousands, so that the number speaking German at home, for example, is not 1,261 but 1,261,000.)

Persons 5 Years Old and Over Speaking Various Languages at Home, by Age (Numbers in thousands: civilian noninstitutional population)

Language spoken at home Total Percent Speaking English only Speaking other language Chinese French German Greek Italian Japanese Korean Filipino Polish Portuguese Spanish Yiddish Other Not reported

Persons 5 years old and over

5 to 13 years

14 to 17 years

18 to 24 years

25 to 44 years

45 to 64 years

65 to 74 years

75 years and over

176.319

30,414 15.1 15.4

15,955 7.9 8.0

27,988 13.9 14.1

59,385 29.6 29.5

43,498 21.7 21.5

15,053 7.5 7.4

8,519 4.2 4.0

17,985

14.4

6.9

12.6

30.8

21.8

7.5

6.0

514 987 1,261 365 1,354 265 191 419 731 245 8,768 234 2,651 6,508

12.5 8.1 5.4 16.7 7.5 7.9 16.2 10.7 2.7 15.9 20.2 8.5 10.0 11.1

5.8 5.5 7.1 4.9 4.9 6.8 5.8 5.3 1.4 8.6 8.8 0.4 4.9 8.4

15.8 10.2 10.8 10.4 8.1 7.9 17.8 8.6 3.7 12.2 15.4 3.0 10.8 13.5

34.8 29.9 24.3 38.1 19.3 27.2 35.6 40.8 13.8 33.9 34.6 15.8 30.3 26.9

21.2 30.4 27.4 21.9 31.5 36.6 19.9 20.3 45.7 22.0 15.8 20.9 23.3 25.1

6.8 9.9 12.8 4.4 15.1 9.4 3.7 7.2 21.6 3.7 3.1 29.1 10.1 9.5

3.1 6.0 12.2 3.6 13.7 3.8 1.0 6.9 10.9 3.3 2.2 21.8 10.6 5.6

200.812

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Example 1 What language is spoken at home by almost one-half of those not speaking English at home? Spanish; 8,768/17,985 is about 48%. Example 2 What language has the highest percent of its speakers in the 45- to 64-year-old age bracket? Polish, with 45.7% Example 3 How many persons between the ages of 18 and 24 speak Korean at home? There are 191,000 of all ages speaking Korean, of which 17.8% are between 18 and 24: .178 3 191,000 5 33,998 persons

GRAPHS Bar Graphs Bar graphs may be horizontal or vertical, but both axes are designed to give information. The height (or width) of the bar is proportional to the number or percent represented. Bar graphs are less accurate than tables but give a quick comparison of information. There may be only two variables, as in the following graph. One is the year and the other is the percentage of the labor force made up of women. Women as a Percentage of the Labor Force

Example Between which 10 years does the chart show the greatest percent increase of women in the labor force? For each of the 10-year periods there is some increase. Subtract each percent from the one to the right of it; four subtractions. The greatest increase, 4.6%, occurs between 1970 and 1980. In this bar graph, percents are written at the top of each bar. This is not always the case. If the numbers are not given, you must read across, using a ruler or card, to the relevant axis and estimate the height.

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UNIT 9

Bar graphs such as the following one can compare two sets of data for varying years. This graph shows, for example, that 86.8% of the male population 16 years old and over was in the labor force in 1950. In that same year, 33.9% of the female population was in the labor force. It gives different information than the previous graph. Percentage of Population 16 Years Old and Over in the Labor Force

Example Explain the apparent discrepancy for the year 1990 between the percentage for women in this graph (47.8%) and that in the previous graph (40.3%). This graph shows that 47.8% of all women were in the labor force in 1990—that is, 47.8% of 100 women were working. The previous graph showed that 40.3% of 100 workers, or 40.3%, were women. There is no discrepancy. The populations are different. These graphs are similar to bar graphs, but each bar contains more than one kind of information and the total height is the sum of the various components. The following graph gives percentages for college graduates on the bottom and high school graduates on the top. There might well be other gradations, such as “some college” above the college section and “some high school” above the high school section as well.

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DATA ANALYSIS REVIEW

Percent of Persons 25 Years Old and Over Who Were High School and College Graduates, by Region: 1980, 1990, and 2000

Example 1 For each of the 3 years, which region consistently has the lowest percentage of college graduates? North Central Example 2 Which region has the lowest total educational attainment for each of the 3 years? South Example 3 In 2000, which region had the highest percentage of high school graduates, and what was it? For 2000, subtract the percent for college graduates from the total percent; four subtractions. The highest is the West, with 75.8% 2 20.1% 5 55.7%. Example 4 Which region had the greatest percentage increase of college graduates between 1990 and 2000? The West, with 20.1% 2 13.8% 5 6.3%.

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UNIT 9

Circle Graphs—Pie Charts Percent Distribution of Voters in the Last Election by Years of School Completed and Family Income

Years of School Completed

Family Income (Restricted to Persons Living in Primary Families)

Circle graphs, also known as pie charts, show the breakdown of an entire quantity, such as a college budget, into its component parts. The circle representing 100% of the quantity is cut into pieces, each piece having a certain percentage value. The sum of the pieces is 100%. The size of the piece is proportional to the size of the percent. To make a circle graph, you must have an instrument called a protractor, which measures degrees. Suppose the measured quantity is 10% of the whole. Because 10% of 360° is 36°, a central angle of 36° must be measured and radii drawn. This piece now has an area of 10% of the circle. When answering questions on circle graphs, compare percentages. Example 1 Of those who voted in the last election, what percentage attended college at some time? This information is in the first graph. Add 20.6% to 19.3% to get 39.9%. Example 2 Of those who voted in the last election and who reported their income levels, what percentage had a family income below $10,000? This information is in the second graph. Add 4.2% to 11.6% to get 15.8%.

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DATA ANALYSIS REVIEW

Line Graphs Like bar graphs, line graphs follow vertical and horizontal information axes, but the line graph is continuous. There may be a single broken line or there may be several, comparing three or four stocks or incomes or, as in the case of this graph, numbers of workers in selected occupations. The line graph shows trends: increasing, decreasing, or not changing. In the graph below, the actual number of people in an occupation in a given year must be estimated. For example, the number of social workers in 1985 seems to be 14 million, and the number of white-collar workers for the same year is about 40 million. Number of Workers in Selected Occupations (In Millions)

W

Example 1 In 1995, what was the total number of workers in all four occupations? Estimate each number by comparison with the values at the left. Then add the four. Estimates: farm workers, 3 million; social workers, 18 million; blue-collar workers, 35 million; and white-collar workers, 52 million. Total, 108 million. Since the scale on graphs is usually marked in large increments, since the lines used are often thick, and since the estimates must often be made on the side of the graph far from the scale, use whole numbers as much as possible when estimating. Use only the fraction one-half (1/2) if your judgment tells you something less than a whole number should be used. Because all the information must be estimated, units less than one-half will not significantly affect your answer. Do not spend time trying to figure out the precise number on the scale. A reasonable estimate should let your answer be within 1 or 2 percent on either side of the correct answer choice. As part of your strategy for dealing with graphs, look at the answer choices to get an idea of the magnitude of your estimate before doing the estimating. Choose the answer choice closest to your estimate.

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UNIT 9

Example 2 If your estimate is 97 million and the answer choices are 3 million, 0.5 million, 90 million, 103 million, and 98 million, choose 98 million as your answer. Use this bar graph to answer the following questions. Student Enrollments: State U. vs. Thomas U.

QUIZ

BAR GRAPH QUESTIONS 1.

What was the enrollment at State U. in 1980?

2.

In 1990, how many more students were enrolled at State U. than at Thomas U.?

3.

If the average tuition at State U. in 2000 was $6,500, what was the total revenue received in tuition at State U. that year?

4.

In 1990, 74% of the students enrolled at State U. were males. How many males attended State U. in 1990?

5.

Find the percent of increase in enrollment at Thomas U. from 1980 to 1990. SOLUTIONS

1.

8,000 students

2.

14,000 2 12,000 5 2,000 students

3.

12,000 3 $6,500 5 $78,000,000

4.

14,000 3 74% 5 14,000 3 .74 5 10,360 students

5.

Increase in enrollment 5 12,000 - 10,000 5 2,000 Percent of increase in enrollment 5 2,000 4 10,000 5

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Practice Test 1 ANALYTICAL WRITING MEASURE Directions: Present your perspective on the issue below, using relevant reasons and/or examples to support your views. Time—45 minutes. (Note: In the actual GRE, you will be given a choice of two issues.) “Schools should be responsible only for teaching academic skills and not for teaching sex education.” Directions: Discuss how well-reasoned you find this argument. Time—30 minutes. The following appeared in an announcement by the manufacturer of Roland Rainwear, a clothing manufacturer: “Since a competing lower-priced product line of outdoor clothing, TrekOut Gear, was started five years ago, Roland Rainwear’s sales have declined by 10,000 units per month. The best way to get more people to purchase Roland Rainwear is to reduce its price below that of TrekOut Gear, at least until sales increase to former levels. The increased sales of Roland Rainwear will attract more distributors to carry our product.”

291

Practice Test 2 ANALYTICAL WRITING MEASURE Directions: Present your perspective on the issue below, using relevant reasons and/or examples to support your views. Time—45 minutes. (Note: In the actual GRE, you will be given a choice of two issues.) “The reading of books is not important today. People can learn as much by watching television or movies as they can by reading books.” Directions: Discuss how well reasoned you find this argument. Time—30 minutes. The following appeared in the editorial section of a newspaper in the country of Montour: “The practice of officially changing seat-belt laws on the highways—whether requiring them or eliminating them—is a dangerous one. Consider what happened over the past decade whenever neighboring Serenia changed its seat-belt laws: an average of 9 percent more automobile accidents occurred during the week following the change than had occurred during the week preceding it—even when the seat-belt law was required. This statistic shows that the change in seat-belt law adversely affected the alertness of drivers.”

292

Practice Test 1 VERBAL ABILITY SENTENCE COMPLETION Directions: Each sentence below has one or two blanks, each blank indicating that something has been omitted. Beneath the sentence are five lettered words or sets of words. Choose a set of words for each pair of blanks that best fits the meaning of the sentence as a whole. 1.

While the instructor claimed his comments were ______ to a mild warning, his students recognized his tone as ______ and responded with haste. (A) tantamount. .scathing (B) equivalent. .embrocation (C) maladroit. .inimical (D) reticulate. .circumspect (E) countermand. .loquacious

2.

Although typically unpleasant and ______ in business decisions out of his purview, the junior vice president was often rewarded for his remarkable ______. (A) bungling. .demagoguery (B) onerous. .gerrymandering (C) finical. .dissipation (D) exemplar. .ignominy (E) officious. .perspicacity

3.

Even though he was only a neophyte in the business, he effectively used ______ methods to ______ himself with the manager. (A) audacious. .reprove (B) obsequious. .ingratiate (C) obdurate. .extradite (D) stentorian. .vacillate (E) desultory. .embrangle

4.

In ancient Greece, ______ warfare between city-states decimated the population with ______ loss of lives. (A) equitable. .needless (B) quixotic. .cynical (C) recondite. .immoderate (D) splendiferous. .superfluous (E) internecine. .prodigious

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5.

Only someone with incredible ______ is likely to find ______ in a potentially disastrous contretemps. (A) faux pas. .topiary (B) dichotomy. .guerdon (C) erudition. .morass (D) sanguinity. .serendipity (E) savoir-faire. .acumen

6.

The irate customer became ______, and then he ______ against the mechanic’s unfair practices and outrageous fees for car repairs. (A) mendacious. .seethed (B) lachrymose. .jeered (C) apoplectic. .fulminated (D) stultified. .scoffed (E) desolate. .denigrated

ANTONYMS Directions: Each item below consists of a word printed in capital letters, followed by five lettered words or phrases. Choose the lettered word or phrase that is more nearly opposite in meaning from the word in capital letters. Since some of the questions require you to distinguish fine shades of meaning, be sure to consider all the choices before deciding which one is best.

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1.

SVELTE (A) limber (B) lissome (C) adipose (D) vile (E) disingenuous

2.

ENCOMIUM (A) panegyric (B) censure (C) declamation (D) oratory (E) pastiche

3.

CONNOISSEUR (A) sycophant (B) zealot (C) hedonist (D) pariah (E) tyro

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4.

CELERITY (A) splendor (B) alacrity (C) luster (D) indolence (E) declivity

5.

DISSENSION (A) desolation (B) strife (C) mandate (D) concord (E) dearth

6.

REPREHENSIBLE (A) docile (B) inculpable (C) dubious (D) usurious (E) diffident

7.

DELETERIOUS (A) volatile (B) enigmatic (C) fulsome (D) esoteric (E) salutary

8.

OBFUSCATE (A) simulate (B) palliate (C) compliment (D) elucidate (E) portend

9.

INVEIGLE (A) inveigh (B) repulse (C) malinger (D) herald (E) impugn

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READING COMPREHENSION Directions: Each passage in this group is followed by questions based on its content. After reading a passage, choose the best answer to each question. Answer all questions following a passage on the basis of what is stated or implied in that passage.

PASSAGE 1 Line

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There are 160 million people in the United States who wear contact lenses or glasses, many of whom are tired of the inconvenience, ongoing expense, and ineffectiveness of prescription lenses. More and more Americans, almost 500,000, will have undergone corrective surgery to improve their vision. The surgical process is called LASIK (laser-assisted in situ keratomileusis). The procedure can take less than 15 minutes and is performed by an ophthalmologist. The results are impressive, with seven out of ten people having their vision corrected to 20/20. The majority of the remainder of people could drive without the use of corrective lenses. In the future, experts are expecting that the majority of people will see better than 20/20. The procedure begins by marking the cornea with ink, then dropping a liquid anesthetic into the patient’s eye. The ink is water soluble and harmless and is used to help reposition the flaps of the cornea. Then a suction ring stabilizes the eye and pressure is applied to allow for a clean cut by the microkeratome (the cutting instrument), which glides across the moistened surface of the cornea, cutting the outer layers and stopping automatically. An uncut section acts as a hinge. The attached cornea is then lifted, and the layers below are excised by the laser. A computer guides the laser as it reshapes the cornea. To correct farsightedness, a piece of tissue shaped like a doughnut ring is removed. If the cornea’s center is trimmed, thereby making it flatter, nearsightedness is corrected. The hinged flap is then put back in position. While the procedure is effective for the majority of patients, even those with an astigmatism, other LASIK patients have had problems. A diminishing of contrast has made driving a car more difficult, especially at night. Everyone experiences glare and halo effect in the beginning, but, for some, it becomes a permanent disability. Also, as people age, most need bifocals for closer work. This is due to a condition called prebyopia (lenses in the eyes lose their ability to curve enough; therefore it is difficult to focus on close objects). LASIK can’t help or prevent this condition. There are other alternatives being explored to improve vision. INTACS are crescent shaped rings that are removable. They are placed within the cornea, leaving it intact. It works well for minimal nearsightedness, but is not effective for extreme nearsightedness, farsightedness, or astigmatism. CUSTOM LASIK is adjusted for specific differences in an individual’s cornea, lens, and retina. It corrects both nearsightedness (sometimes to 20/10) and farsightedness with or without astigmatism. The main disadvantage is that it permanently alters the curve of the cornea. INTRAOCULAR LENSES are implanted behind the iris or the cornea. These are also removable and leave the cornea in

296

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40

45

GRE CAT Success

one piece; however, they may cause injury to the cornea, intraocular infection, and cataracts. They are effective for correcting both high degrees of nearsightedness and farsightedness. Although the risks are small for LASIK surgery, they are real and serious; therefore, one must consider the reasons for opting for the surgery. Americans seem to have a penchant for the “cure,” the quick fix, and, as a result, people may rush too quickly for what they see to be a panacea. When it comes to eye surgery that could possibly cause worse problems than it “cures,” no matter how small the risk, it is wise to explore all of the options. 1.

The name of the eye surgery, LASIK, as it appears in the regular surgery and the CUSTOM LASIK surgery, is (A) an acronym for the names of the doctors that developed the procedure. (B) the brand name of the surgical instrument that is used for the procedure. (C) an acronym for the surgical procedure that tells how the surgery is done. (D) the generic name of the anesthetic used. (E) the Latin prefix of a longer medical term.

2.

The (A) (B) (C) (D) (E)

3.

The difference between correcting nearsightedness and farsightedness depends on the (A) marking of the cornea with ink. (B) pressure exerted by the suction ring. (C) size of the microkeratome. (D) repositioning of the cornea flaps. (E) way that the cornea is either cut or trimmed.

4.

Of all the procedures described in the article, which of the following are effective for correcting vision with or without astigmatism? (A) LASIK and INTACS (B) INTEROCULAR LENSES and INTACS (C) CUSTOM LASIK and INTACS (D) LASIK and CUSTOM LASIK (E) INTEROCULER LENSES and LASIK

cornea is marked with ink (line 11) in order to align the suction ring to attach it to the eyeball. determine where the layers below the cornea are to be excised. measure the movement of the eye during surgery. aid in replacing the hinged flaps of the cornea. measure swelling of the eye after the operation has been completed.

297

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PASSAGE 2 Line

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Researchers have been trying to find new methods to combat one of the peskiest of natural enemies in nature: the fire ant. They are known for their aggressiveness and the painful stings and bites they deliver. Fire ants have displaced ants that are native to the United States. The huge mounds that they inhabit are now being seen around the South. The pests came from South America; it is believed that they arrived in the 1930s aboard ships that docked at Gulf ports, such as Mobile, Alabama. Since they have no natural enemies outside of their original habitat, they have been able to multiply more readily and now are in 12 Southern states and Puerto Rico, occupying more than an astonishing 300 million acres. Fire ants have also been discovered in New Mexico and California. Because of prolific breeding, their numbers alone, not to mention their tenacity, preclude eradicating them entirely, but researchers are trying to control them. One way to achieve this, possibly, is to use their natural enemies imported from their original habitat. For example, there is a fly that preys on fire ants by beheading them. Also, there is a disease that is a natural enemy of the fire ant because it substantially reduces the egg production of the queen. However, any insect introduced into the United States for the purpose of controlling other insects has to go through a strict quarantine process developed by the U.S. Department of Agriculture under the supervision of experts in the field of entomology. Ironically, farmers do not mind the insects and find them useful because they actually kill other crop pests. However, when fire ant colonies and the larvae, which are about the size of a pin point, are disturbed, the results can be disastrous with the likelihood of multiple stings. The dangerous truth is that one percent of the population can be fatally allergic to the venom of the insects. The insects pose other problems like shorting electrical connections and burrowing that can damage the foundation of roads resulting in pot holes on the surface. Although the pest cannot be eradicated, the fight against the fire ant will continue with USDA entomologists studying what disease will effectively limit their numbers. 1.

An antonym for “prolific,” as it is used in line 11, is (A) fecund. (B) laborious. (C) luxuriant. (D) barren. (E) lugubrious.

2.

The (A) (B) (C) (D) (E)

word “entomology” (line 20) means the study of agriculture. infectious diseases. insects. reproduction. import/export protocol.

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3.

It is (A) (B) (C) (D) (E)

ironic that farmers do not mind the insects because the foundation of roads can be damaged. people can be injured. electrical connections can be shorted. people can be fatally allergic to the venom of fire ants. insects usually destroy crops, not kill other crop pests.

4.

The (A) (B) (C) (D) (E)

tone of the article is one of desperation over an enigmatic problem. defeat because fire ants cannot be completely destroyed. victory over the success of new methods of control. reconciliation in limiting their numbers. confusion because of the failure to keep fire ants out of the country.

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The hippocampus is located inside the brains of both humans and mice; hence, research is being done on the latter to see if its function can be improved. Both the hippocampus and the medial temporal lobe are critical in the process of learning by helping transform short-term memory into permanent memory—one of the cornerstones in the foundation of the learning process itself. Scientists are currently experimenting with increasing more of a single gene, NR2B, that aids in building a protein called NMDA (N-methyl D-aspartate). This very consequential protein, located at the end of dendrites (vein-like projections that extend from nerve and brain cells), functions as a receptor for detailed chemical signals. The function of these signals is to condition brain cells to fire in patterns that repeat. These repeating patterns are what the brain acknowledges as memories. The NR2B gene is copious in the hippocampus portion of the brains of mice that are young; however, after they reach sexual maturity, the amount is greatly reduced. Researchers are experimenting with increasing NR2B in adult mice thereby giving the mature animals the same learning skills of the young mice. NMDA receptors need two independent signals: one is in the form of a glutamate molecule that is transmitted by the axon of a cell that is close by; the second one is an electric signal that is released within the cell itself. The signals unblock the NMDA receptors allowing calcium to come in, and this assists in the formation of memories. Mice, genetically altered by splicing the gene that creates NR2B into their DNA, consistently performed better in learning and memory tests than the control mice used in experiments. In addition, the sensitivity of their brain cells increased in response to new and different stimuli. No researchers have experimented with the NR2B gene in humans in a similar manner; however, drugs may be developed that increase its function. By doing this, the ability to learn and remember would be increased even with the onset of aging. Also, this research may produce new therapies for the treatment of memory problems that are the result of illness or injury, learning disabilities, and, even, Alzheimer’s disease.

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PRACTICE TEST 1

1.

It is (A) (B) (C) (D) (E)

2.

In line 12, the word “copious” most closely means (A) replete. (B) meager. (C) mutated. (D) interspersed. (E) vicious.

3.

When NR2B is increased in adult mice, it improves long term memory by (A) aiding in the building of a protein called NMDA that acts as a receptor. (B) unblocking receptors and permitting calcium to enter. (C) producing an electric signal within the cell structure. (D) forming glutamate molecules that signal brain cells. (E) increasing the size of the hippocampus and medial temporal lobe.

4.

Humans may benefit from these studies because (A) splicing of NR2B genes is being done on human DNA. (B) mice and humans both have a hippocampus located in their brains. (C) NMDA protein based receptors are being created in the human brain. (D) therapeutic drugs are being developed that augment the function of the NR2B gene. (E) the NR2B gene is abundant in the brain of young humans.

desirable to have brain cells fire in repeating patterns because the process uses fewer receptors to record memory. they are acknowledged as memories. the time needed to memorize is decreased. NMDA receptors are unblocked. calcium is released.

ANALOGIES Directions: In each of the following questions, five lettered pairs of words or phrases follow a related pair of words or phrases. Select the lettered pair that best expresses a relationship similar to that expressed in the original pair. 1.

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EPISTLE : LETTER :: (A) ode : poem (B) novel : nonfiction (C) puzzle : cryptogram (D) diary : repartee (E) lyric : refrain

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2.

PLIANT : YIELDING :: (A) relinquish : pertinacious (B) steep : strong (C) schism : abyss (D) scull : yacht (E) pilfer : filch

3.

NUCLEUS : EXTRANEOUS :: (A) verbose : prolix (B) clemency : pardon (C) callow : cosmopolitan (D) bungle : boast (E) dilatory : widen

4.

RECORDER : TUNE :: (A) radio : medley (B) clarinet : melody (C) microphone : sing (D) harp : complain (E) snare : drum

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SLOUGH : SWAMP :: (A) fen : bog (B) snake : skin (C) shed : rest (D) mud : medicine (E) slumber : insomnia

6.

RAPIER : EPEÉ :: (A) tomahawk : wickiup (B) grenade : artillery (C) javelin : competition (D) carbine : rifle (E) catapult : siege

7.

DERACINATE : ERADICATE :: (A) ruminate : spurn (B) impugn : endorse (C) grovel : glorify (D) mollify : mitigate (E) quibble : masticate

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Practice Test 2 VERBAL ABILITY SENTENCE COMPLETION Directions: Each sentence below has one or two blanks, each blank indicating that something has been omitted. Beneath the sentence are five lettered words or sets of words. Choose a set of words for each pair of blanks that best fits the meaning of the sentence as a whole. 1.

Making a(n) ______ visit to his ______, the unknown thief sought reassurance that no one had discovered his crime. (A) prurient. .domicile (B) nefarious. .morass (C) irrefutable. .acumen (D) surreptitious. .cache (E) supercilious. .tirade

2.

Some regard ______ as a ______ for the year 2000. (A) millennium. .misnomer (B) imprecation. .moniker (C) bruit. .bellwether (D) euphoria. .malaise (E) volition. .malapropism

3.

Despite her eagerness to see DaVinci’s Mona Lisa in the Louvre, she ______ indifference to appear ______. (A) goaded. .ineffable (B) feigned. .sophisticated (C) deigned. .provincial (D) imbued. .bibulous (E) obfuscated. .munificent

4.

On election day, the presidential candidate’s ______ supporters braved the ______ weather in Maine to vote in the primary. (A) officious. .nebulous (B) motley. .ribald (C) moribund. .placid (D) partisan. .fungible (E) staunch. .inclement

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5.

Almost lulled to sleep by the candidate’s ______ speech, the audience seemed to fall into a state of ______. (A) convoluted. .ecstasy (B) vociferous. .apostasy (C) turgid. .torpor (D) acerbic. .sloth (E) discursive. .effervescence

6.

The old farmer’s ______ labor and wisdom about planting invariably produced ______ fields of soybeans. (A) erudite. .evanescent (B) assiduous. .verdant (C) obdurate. .palpable (D) incipient. .labyrinthine (E) infinitesimal. .scanty

ANTONYMS Directions: Each item below consists of a word printed in capital letters, followed by five lettered words or phrases. Choose the lettered word or phrase that is more nearly opposite in meaning from the word in capital letters. Since some of the questions require you to distinguish fine shades of meaning, be sure to consider all the choices before deciding which one is best.

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SATURNALIAN (A) sardonic (B) ascetic (C) nascent (D) envisaged (E) insolvent

2.

MELLIFLUOUS (A) sugary (B) resonant (C) risible (D) supercilious (E) cacophonous

3.

PECCADILLO (A) sin (B) pedantry (C) virtue (D) riata (E) condolence

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4.

VACUOUS (A) multifarious (B) amorous (C) clandestine (D) perspicacious (E) inane

5.

GAUCHE (A) adroit (B) devious (C) prosaic (D) volatile (E) dormant

6.

SPURIOUS (A) bogus (B) hackneyed (C) affable (D) authentic (E) peremptory

7.

RATIOCINATION (A) schism (B) fallacy (C) arithmetic (D) paradigm (E) redress

8.

INTRANSIGENT (A) meretricious (B) muddled (C) cooperative (D) nonchalant (E) willful

9.

MAVERICK (A) reagin (B) nomenclature (C) conformist (D) heroic (E) tattoo

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READING COMPREHENSION

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“Body Art” (using the body as a human canvas) has been documented throughout recorded history, and some anthropologists believe that it even precedes it. The hackneyed phrase “since time immemorial” might actually apply in this case, as so many diverse peoples throughout time have tattooed, pierced, painted, scarred, or even reshaped their bodies. The relationship to modern day society is that there has been a resurgence, especially in tattooing and body piercing, among teenagers. But on a less “exotic” level, people style their hair, put on make-up, wear facial hair, and pluck eyebrows regularly. Enhancing one’s appearance, especially to positively affect one’s self-perception and to attract members of the opposite sex, is a common practice that is repeated around the world. However, body art has traditionally been ritualistic in nature, conveying marital status, designating aspects of gender and age group, recognizing achievements, or to mark rites of passage like puberty. Most important, body art has been used in religious rites and rituals, even to summon forth spirits. In New Zealand, the Maori, as part of “pagan” ritual, tattoo their faces in elaborate, ornate patterns. In Ethiopia, the Hamar warrior scars his body by cutting himself with a razor, thereby creating raised scar tissue. The scar tissue heals with a somewhat lighter color, and they are arranged in symmetrical patterns. This is done to acknowledge the killing of an enemy or a very dangerous animal. The young women stretch their earlobes by inserting “plugs” made of rolled leaves. The size of the holes is gradually increased by putting in ever increasingly larger clay plates. When Kenyan teenage boys are circumcised, they draw ritual designs on each other’s faces with white chalk and wear brass pendants on the sides of their faces. For centuries in China, young girls had their feet bound. This painful and crippling procedure stunted the growth of them to a mere five inches long for an adult woman. Small feet were considered aesthetically pleasing. Today, however, most body art is done for the sake of fashions and trends, with very little ritualistic sensibility unless it is an expression of rebellion against authority figures like parents, teachers, or society as a whole. An exception would be the ritual of tattooing and scarring to identify fellow gang members. This body art is a rite of passage that allies friends and identifies foes. 1.

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In line 3, the word “hackneyed” most closely means (A) choppy. (B) metaphorical. (C) indecorous. (D) trite. (E) unique.

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2.

In the first paragraph, it can be inferred that “time immemorial” means (A) a celebration of time. (B) since the beginning of recorded history. (C) longer than anyone can remember. (D) to memorialize time. (E) an endless future.

3.

In today’s society, body art is practiced primarily (A) to enhance one’s appearance. (B) as a ritual and rite. (C) to designate age and gender status. (D) as a recognition of achievement. (E) to convey marital status.

4.

The (A) (B) (C) (D) (E)

tone of the article does which of the following? Condemns the practice of body art Minimizes its ubiquity Promotes the practice of body art Acknowledges its universality Warns of its harmful effects

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Speculation about the cause of global warming continues, and the controversy centers on whether increases in temperature are “natural” or the result of ever-increasing pollution. Either way, calculations show that by the end of the next century, the earth’s mean temperature will likely rise between 1.3 degrees Centigrade and 4.0 degrees Centigrade (2.3 degrees Fahrenheit and 7.2 degrees Fahrenheit). One major consequence of this warming will be rising sea levels. The Greenhouse Effect is when increasing carbon emissions, as a result of burning billions of tons of fossil fuels around the world, traps more heat and makes the global temperature rise. As the sun’s rays go through the atmosphere, it heats the earth’s surface. This heat is radiated as infrared rays, and while some of it escapes, the remainder becomes trapped by an increasing amount of carbon dioxide, along with other greenhouse gases. According to some scientists, one of the indicators of this phenomenon is the shrinking Quelccaya ice cap located in Peru, South America. It has lost one fifth of its mass over the last two decades. It is incontrovertible that the earth’s temperature has increased about one degree Fahrenheit over the last 12 decades, and the increasing temperature has resulted in the lessening of the ice cap. What is debatable is whether this lessening is part of some larger shift in climate or whether it has resulted from greenhouse gases. To combat the problem of greenhouse gases, several fronts are being attacked simultaneously. Research and development needs to be intensified to find renewable sources of energy, like solar and wind power. Fuel efficiency for popular vehicles, like vans, SUVs, and light trucks, needs to be improved in addition to more rigorous emission standards. Similarly, emission standards for industries that use fossil fuels for power must also be stricter. The federal government should reduce its need for and use of energy significantly in an effort to cut

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emissions. Lastly, both state and federal government could give tax credits and incentives for individuals and companies that use alternative sources of energy. Unfortunately, even if all further emissions are ended immediately, greenhouse gases will stay in the atmosphere for a hundred years, and global warming will increase. According to environmentalists, however, that does not preclude working on the problem as diligently as possible because it will be a serious dilemma for the twenty-first century. 1.

The (A) (B) (C) (D) (E)

2.

Greenhouse gasses heat up the earth because they (A) magnify the rays of the sun, making them hotter. (B) trap infrared heat in the atmosphere. (C) reduce the size of the Quelccaya ice cap. (D) produce infrared rays that heat up the earth. (E) are produced by burning fossil fuels.

3.

In line 15, “incontrovertible” most closely means (A) indecisive. (B) improbable. (C) indubitable. (D) imprudent. (E) incredible.

4.

The word “fronts” (line 19) completes a metaphor that compares finding potential solutions to (A) curing a disease. (B) waging a war. (C) winning a race. (D) predicating climatic changes. (E) solving a puzzle.

essence of the debate regarding global warming is whether it is a reality or not. the temperature will increase over the next century. the Quelccaya ice cap is shrinking. conversion from the Centigrade scale to Fahrenheit is accurate. it is the result of nature or the Greenhouse Effect.

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Hurricanes are a means to transfer heat from the equator to the poles. After waters off the coast of Africa are warmed to more than 81 degrees Fahrenheit, with the right combination of winds and accumulated thunderstorms, a hurricane is born. The water temperature is essential, but it is the winds that play the significant role. If they are weak easterlies and strong westerlies, as produced by El Niño, then clouds will have their tops cut off, but if there is still air between two wind belts and a stronger easterly flow, as found in La Niña, hurricanes form more readily. This is one reason why the 1999 hurricane season was so active. The presence of La Niña increased the chance of the United States being hit by 75%, as compared to the 25% chance with El Niño.

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When water evaporates, the moist vapor rises, then expands, then cools. It condenses into cloud droplets and then forms rain. Heat that is locked within the vapor is released, and this lowers air pressure, causing more rain and thunderstorms. The winds in these accumulating thunderstorms begin to move in the direction of the earth’s rotation and circulate in a counterclockwise motion. If this takes place in the Southern Hemisphere, it is a clockwise direction. This mass of whirling storms is called a tropical cyclone. As the winds accelerate in this spiral, the storm is designated as a hurricane when they reach a speed of 74 m.p.h. There is a column at the center of these multiple thunderstorms. It consists of calm air and is called the “eye” of the hurricane. It is surrounded by the fiercest storms of the system. There are other factors that affect the formation of hurricanes. In addition to the presence of El Niño and La Niña conditions in the equatorial Pacific, rainfall amounts in West Africa, the intensity of stratospheric winds, and significant changes in ocean circulation also affect how storms develop and to what degree. Therefore, predicting where hurricanes will appear is difficult. Other factors that may be a bit more esoteric are the overall effects of a natural shift in climate and whether global warming, induced by the pollutants of technology, transportation, and industrialization, will also contribute significantly. For example, global warming may make storms more severe by loading them with more rain. This would cause even more extensive flooding. If global sea levels rise, the surge effect of waves would increase, thereby causing more damage and destruction. Even though the number of typhoons in the Pacific and hurricanes in the Atlantic may vary in proportion to one another, there will probably be about 80 such events in any one year. 1.

Respectively, La Niña and El Niño affect the creation of hurricanes by (A) increasing the amount of rain in storm systems and raising the temperature of still air. (B) decreasing the amount of evaporation and lowering the temperature of still air. (C) producing weak easterlies and strong westerlies and diminishing clouds and increasing the formation of storms. (D) producing strong westerlies and weak easterlies and increasing the formation of storms and diminishing clouds. (E) producing strong easterlies and weak easterlies and increasing the formation of storms and diminishing clouds.

2.

A tropical cyclone is defined by the (A) amount of rain that it contains. (B) speed of the winds. (C) direction of the movement of the winds. (D) eye at the center of the thunderstorms. (E) amount of heat locked within the vapor.

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3.

All of the following factors affect the formation of hurricanes EXCEPT (A) global warming. (B) rainfall amounts. (C) stratospheric winds. (D) phases of the moon. (E) natural shift in climate.

4.

It can be inferred that the difference between a typhoon and a hurricane is the (A) ferocity of the winds. (B) geographical location. (C) amount of rain. (D) rate of evaporation. (E) temperature of water.

ANALOGIES Directions: In each of the following questions, five lettered pairs of words or phrases follow a related pair of words or phrases. Select the lettered pair that best expresses a relationship similar to that expressed in the original pair.

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QUISLING : TRAITOR :: (A) redress : lawyer (B) vaunt : braggart (C) scuttle : coal (D) poltroon : coward (E) bushel : peck

2.

PURLOIN : BEQUEATH :: (A) adumbrate : apotheosis (B) amalgamate : commingle (C) nullify : satiate (D) prevaricate : cajole (E) polemic : consensus

3.

LINTEL : POSTERN :: (A) pandemic : universal (B) esprit de corps : le mot juste (C) joist : threshold (D) maelstrom : phenomenal (E) sill : window

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4.

POD: WHALES :: (A) pride : lions (B) covey : geese (C) quail : flock (D) bees : swarm (E) pack : leopards

5.

MATTOCK : FARM :: (A) digging : spade (B) appetizer : dessert (C) painter : easel (D) mason : trowel (E) adze : carpentry

6.

KIT (A) (B) (C) (D) (E)

7.

BADMINTON : SHUTTLECOCK :: (A) wicket : croquet (B) wedge : golf (C) bridge : billiards (D) hockey : puck (E) trump : bridge

: FOX :: deer : fawn gosling : goose lamb : ewe salamander : newt eagle : fledgling

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Quick Score Answers Practice Test 1: Verbal Ability Sentence Completion 1. 2. 3. 4. 5. 6.

A E B E D C

Reading Comprehension

Antonyms 1. 2. 3. 4. 5. 6. 7. 8. 9.

Passage 1 1. C 2. D 3. E 4. D

C B E D D B E D B

Passage 2 1. D 2. C 3. E 4. D

Analogies 1. 2. 3. 4. 5. 6. 7.

A E C B A D D

Passage 3 1. B 2. A 3. A 4. D

Practice Test 2: Verbal Ability Sentence Completion 1. 2. 3. 4. 5. 6.

D A B E C B

Reading Comprehension

Antonyms 1. 2. 3. 4. 5. 6. 7. 8. 9.

Passage 1 1. D 2. C 3. A 4. D

B E C D A D B C C

Passage 2 1. E 2. B 3. C 4. B

Analogies 1. 2. 3. 4. 5. 6. 7.

D E E A E B D

Passage 3 1. E 2. C 3. D 4. B

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PRACTICE TEST 1

ANSWERS AND EXPLANATIONS

SAMPLE RESPONSES—ANALYTICAL WRITING MEASURE PRACTICE TEST 1 Score 5–6 Issue Response Many would argue that sex education should be taught only in the home or at religious institutions—that schools, especially public schools, do not have the right, the resources, or the appropriate set of moral guidelines to be teaching impressionable youth the difference between “right” and “wrong” regarding sex or birth-control methods. However, others feel that if schools do not prioritize the teaching of appropriate personal sexual behaviors to students, no one else will, and society will suffer as a consequence. In the following essay, I will argue that both homes and schools are suitable places for instruction in sex education. Behaviors learned in youth are generally carried through to adulthood. If school-aged children are not taught basic grammar, for example, they will often struggle far into adulthood with poor writing and speaking skills. Likewise, if children are not drilled on what constitutes appropriate sexual behavior, they will struggle with making choices grounded in safety and common sense. Sex education must come both from home and from school. If one does not reinforce the other, the child can become frustrated and confused as to what is expected regarding appropriate sexual behavior, safety, and protection. While every moral compass points in a slightly different direction, it is only by a matter of degrees—generally accepted social and ethical values can be consistently reinforced in school and at home even if the parties at each institution have slightly variable codes of sexual conduct. For example, values such as respect for sexual partners, abstinence, and safe sex can be taught, albeit in different ways, both at home and in school. There are those who will insist that schools cannot possibly instill the sexual values of an individual family, religion, or religious community. I agree that school is not the appropriate place for religion-based sex education. Religion-based sex education, at the discretion of the parents, should be conducted outside of the public school system. However, just as many individuals have morality codes that are similar, many religions share values and appropriate behaviors that can be taught in school, free from religious overtones. Finally, parents trust schoolteachers to teach their children life skills such as reading, writing, and critical thinking. These are skills that the children will continue to develop and use throughout the rest of their lives, and to some extent they will determine how “successful” a person becomes. Why would sex education—information that could prevent pregnancy, disease, or even death—be less important to the teachers, the schools, or the children? This information may determine the sexual behavior of each student as a mature person, and it may have more influence than academico.

Score 5–6 Argument Response While the manufacturer of Roland Rainwear may be looking for reasons for the decline in sales, the argument that TrekOut Gear has caused a decrease in sales of Roland Rainwear is not convincing. It contains several fundamental logical flaws. Assumptions are made about the current manufacturing and sales environment for outdoor clothing that may or may not be accurate. The following essay will discuss these flaws, flesh out the specious assumptions, and reveal how the argument could have been made in a more effective manner.

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ANSWERS AND EXPLANATIONS First, it cannot possibly be determined that the sales of Roland Rainwear have declined solely because of the appearance of a lower-priced competing product. Many questions need to be asked. Does TrekOut Gear truly compete with Roland Rainwear? Does the TrekOut Gear product line contain the same specific articles of clothing? Does it use the same or similar textiles? Does it have the same level of quality? Do the companies use the same marketing strategies? Because this information is not given, it is not truly known whether or not TrekOut Gear actually competes with and seizes sales from Roland Rainwear. Competing manufacturers in a local market can certainly affect sales of products, but multiple manufacturers with different products can actually enhance overall product consumption. More information regarding the products of both manufacturers is needed in order to make the assumption that TrekOut Gear is responsible for a decline in sales for Roland Rainwear. Also, in the five-year time period mentioned in the argument, many other factors besides the introduction of TrekOut Gear may have been present to cause a decrease in the sales of Roland Rainwear. For example, marketing changes may have occurred, staff turnover could have affected the quality of the product, population decline could have taken place in the area, population demographics may have changed, and quality and quantity of advertisements may have been altered by general economic conditions or changes in Roland Rainwear’s sales force. The third assumption made in the argument is that if the price of Roland Rainwear is lowered below that of TrekOut Gear, more people will purchase Roland Rainwear. But consumers may or may not purchase Roland Rainwear if the price is dropped. The factors listed above will play a role in sales. And if Roland Rainwear is an inferior product, consumers will probably not spend discretionary income on it. However, if Roland Rainwear is a superior product and the sole reason that people have stopped purchasing it is the high price, then the assumption that a price decrease will boost sales would be accurate. The final two assumptions the argument makes are that sales will return to preTrekOut Gear levels if there is a price decrease for Roland Rainwear and that distributors will increase their purchasing of Roland Rainwear if general sales increase. Both of these events could take place, but they will not necessarily take place. Sales may never reach previous levels for a number of reasons, some of which are discussed above, and distributors may or may not increase their purchasing of Roland Rainwear because of increased sales. The argument makes the leap from sales to distribution without evidence to support a cause–effect relationship. Overall, the argument as stated is not credible. The argument could have been strengthened if the manufacturer had referenced circumstances in addition to the introduction of TrekOut Gear in the past five years. Without this information to correct the core assumptions, this argument cannot be considered convincing.

PRACTICE TEST 2 Score 5–6 Issue Response The statement above claims that television and movies, as a resource for information, are as effective as books and have lessened the importance of books. In the following essay, I will argue that television and movies, for several reasons, should not be considered the primary sources of historical information, interpretations of current events, or even entertainment. Rather, books should play a principal, but not solitary, role in gaining perspective on history, current events, and entertainment, and other media outlets should be important, but secondary, resources for information gathering and entertainment. First, the statement “People can learn as much by watching television or movies as they can by reading books” ignores the other alternatives to books for the presentation of

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PRACTICE TEST 2 information. This is, in fact, quite untrue. We are actually in an age of information overload—we can choose from books, television, movies, satellite services, newspapers, magazines, Internet sources, and multiple other outlets. A reasonable combination of some or all of these media should be used in the accrual of current or historical information, as well as for entertainment, but reliance solely on television and movies would limit a person’s ability to have a well-rounded perspective on any issue or event. Second, reading books has several advantages over television or movies for the stimulation of a developing or adult brain. Television and movies present interpretations for a viewer via images and sounds. Information is fed to the viewer—watching television or a movie is a passive activity that requires little, if any, interaction from the viewer. Books, on the other hand, require the reader to interpret the information for himor herself. The reader must use visual and auditory imagination, logic, and previous knowledge to complete the picture and tie the words, sentences, paragraphs, and chapters together into meaningful or useful information. Third, television or movies, especially those geared toward historical events or issues, are a re-creation of what has occurred in the past. They represent one person’s (or group’s) adaptation of written history. Historical and fictional books written during a certain time period, on the other hand, are one step closer to the actual events. While these books may also represent a personal or group interpretation of the events, they are closer to being “eye-witness” accounts. Television and movies do have many benefits. News coverage of current events is more immediate on television than in newspapers or books. For entertainment purposes, fantastic technological imagery and sound is available in movies. However, as passive behavior, television and movie viewing cannot stimulate the imagination, challenge the knowledge, or encourage growth the way that books can. Unlike television and movies, books, both fiction and nonfiction, excite the mind, test the psyche, and stimulate the reader to use interactive cognitive processes that allow the mind to grow by generating new ideas, thoughts, and perspectives. In conclusion, I have discussed why television and movies should not be considered the primary sources of historical information, interpretation of current events, or entertainment. Books should play a principal, but not solitary, role in gaining perspective on history, current events, and entertainment. Other media outlets, such as television and movies, are important, but secondary, resources for information gathering and entertainment.

Score 5–6 Argument Response The writer of this argument states that changing the seat-belt law on highways is dangerous. This conclusion is based on several misleading notions, including irrelevant comparisons, extraneous statistics, and unfounded assumptions. The essay below will explain why the argument is tenuous, and what elements could have been added or deleted to clarify and validate it. The author of the argument asks the reader to, “Consider what happened over the past decade whenever neighboring Serenia changed its seat-belt laws: an average of 9 percent more automobile accidents occurred during the week following the change than had occurred during the week preceding it—even when the seat-belt law was required.” This claim is fallacious for several reasons. First, to address the statistics cited, the author does not indicate how many times the seat-belt law was implemented or eliminated. If the rate of accidents in the week after a change in the seat-belt law occurred consistently, dozens of times, there could be significance to the claim. However, if this happened only once or twice, the significance of the claim is greatly reduced. Also, statistics for only two weeks were noted. What about the importance of the other 50 weeks of each year? The comparison timeframe and lack of numerical data is insufficient to make such a bold analysis.

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ANSWERS AND EXPLANATIONS Additionally, the author of the argument does not take into consideration the other phenomena that could have added to or even completely accounted for the 9-percent increase in traffic accidents in the week following the implementation or elimination of the seat-belt law. Factors such as weather, physical road conditions, road construction, time of year, holiday traffic, and the specific highway(s) the statistics are taken from are just a few of the other aspects of traffic that could have caused an increase or decrease in accidents. Another speculative assumption the author makes is that accidents were caused by the “alertness of drivers.” There is no evidence presented to link increase or decrease of speed with driver alertness, and there is also no evidence presented that associates alertness with accidents or lack of accidents. These illogical jumps in rhetoric are not supported by the claims in the argument. The argument could be strengthened by an expansion of statistics to include a wider timeframe for looking at the average number of accidents per week and also by including specific numerical data. The argument would also be enhanced by reviewing traffic patterns and accidents in locations other than in a single country. Neighboring Serenia may or may not have characteristically “normal” numbers of traffic accidents, it may or may not have comparable highway systems, and it may or may not have similar climate and geographical circumstances, all of which would be important factors in studying the effects of seat-belt law changes on traffic accidents. Overall, the line of reasoning in this argument is much too narrow to be convincing.

PRACTICE TEST 1—VERBAL ABILITY Sentence Completion 1. The correct answer is (A). Tantamount means “equivalent in value or significance,” and scathing means “bitterly severe.” Notice the use of “while” in the introductory clause; this subordinate conjunction is typically used to set up a contrast. Your choice of answer for this item should, therefore, include a pair that demonstrates a contrast. In choice (B), equivalent will work for the first blank, but embrocation, which means “lotion” or “liniment,” is not logical for the second blank. Choice (C), maladroit, means “clumsy,” and inimical means “hostile” or “harmful”; neither is logical for this sentence. Choice (D), reticulate, means “intricate,” while circumspect is “cautious” or “prudent.” Again the terms are not logical in this sentence. Choice (E), countermand, means “to revoke a former command by a contrary order; loquacious means “talkative.” These terms do not logically fit in the sentence either. 2. The correct answer is (E). Officious means “meddlesome” or “impertinent,” and perspicacity is “shrewdness” or “acuity of mental vision.” Again, notice the use of “although” at the beginning of the introductory adverb clause. You should recognize an implicit contrast in the two blanks in the item. Choice (A), bungling, is “botching” or “mishandling,” while demagoguery pertains to a leader who makes use of popular prejudices, false claims, and false promises to get elected. Choice (B), onerous means “burdensome,” and gerrymandering means “to divide an area into political units to give special advantages to one group.” This choice does not logically complete the blanks either.

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PRACTICE TEST 1 3. The correct answer is (B). Obsequious means “unctuous, flattering, submissive, subservient,” and ingratiate means “to cajole, wheedle, and court.” The word “neophyte” in the sentence means “novice” or “beginner.” Again, there is an implicit contrast set up in the sentence: even though he is a newcomer, he is apparently using flattery to gain the manager’s attention. Choice (A), audacious, means “daring” or “reckless,” while reprove means “to scold” or “to criticize.” Choice (C), obdurate, means “inflexible” or “stubborn,” and extradite means “to banish” or “to expel.” Choice (D), stentorian, means “extremely loud,” while vacillate means “to waver” or “to shift.” Choice (E), desultory, means “haphazard” or “superficial” while embrangle means “to confuse” or “to entangle.” Again, only choice (B) offers terms that can logically complete the sentence. 4. The correct answer is (E). Internecine means “deadly” while prodigious means “vast” or “inordinate.” Choice (A), equitable, means “fair” or “just,” and needless can easily be recognized as meaning “without cause.” Choice (B), quixotic, which is derived from the literary character Don Quixote, means “idealistic to an impractical degree,” while cynical means “pessimistic” and “distrustful.” Choice (C), recondite can be compared to something “incomprehensible or obscure,” and immoderate means “excessive.” Choice (D), splendiferous, is another form of “splendid,” while superfluous means “unnecessary.” 5. The correct answer is (D). Sanguinity means “confident” or “optimistic,” while serendipity means “the gift of finding valuable or agreeable things not sought for.” In the sentence itself, contretemps means “an embarrassing occurrence or mishap”; understanding the meaning of this word is helpful, but the word disastrous offers a clear clue that contretemps is something negative. Choice (A), faux pas, a French expression that translates as “false step,” means “a blunder,” while topiary is “the art of trimming trees or shrubs into ornamental shapes.” Choice (B), dichotomy, is “a division into two exclusive groups,” and guerdon is “a reward” or “recompense.” Choice (C), erudition, is “learning,” especially knowledge gained through reading books; morass is “a marsh or swamp.” Choice (E), savoir-faire, another French term, means “tact” or “social adroitness,” and acumen is “keenness of perception or discrimination, especially in practical matters.” 6. The correct answer is (C). Apoplectic means “highly excited” or “frenetic,” while fulminated means “denounced” or “exploded.” Context clues within the sentence include these words: irate (“angry”), unfair, and outrageous. Also notice the use of the preposition “against” in the sentence. Sometimes a word in the sentence, such as this preposition, can help you narrow your choices because certain verbs are used with some prepositions. Choice (A), mendacious, means “dishonest” or “lying.” Choice (B), lachrymose, means “tearful,” while jeered means “taunted.” Choice (D), stultified, means “to cause to appear stupid” or “to invalidate”; scoffed means “to show contempt” or “to sneer.” Choice (E), desolate, means “lonely,” “gloomy,” or “lifeless,” while denigrated means “defamed.”

Antonyms 1. The correct answer is (C). Adipose means “fat.” Svelte means “slender” or “slim.” Choice (A), limber, means “flexible.” Choice (B), lissome, also means “nimble” or “flexible.” Choice (D), vile, means “disgusting,” and choice (E), disingenuous, means “calculating” or “giving a false appearance.”

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ANSWERS AND EXPLANATIONS 2. The correct answer is (B). Encomium is “glowing praise.” Choice (B), censure, means to “criticize.” Choice (A), panegyric, is an elaborate, often poetic compliment. Choice (C), declamation, is a speech or recitation; choice (D), oratory, also pertains to speech. Choice (E), pastiche, is an artistic work that imitates the style of another piece of art, music, or literature. 3. The correct answer is (E). A connoisseur is an expert, while choice (E), tyro, is a beginner. Choice (A), sycophant, is someone who tries flattery to obtain his objectives. Choice (B), zealot, is a fanatic. Choice (C), hedonist, is one devoted to pleasure or happiness. Choice (D), pariah, is an outcast. 4. The correct answer is (D). Celerity means “speed,” while indolence is laziness or slowness to develop. Choice (A), splendor, means “magnificence.” Choice (B), alacrity, is speed, a synonym for “celerity.” Choice (C), luster, means “radiance” or “glow.” Choice (E), declivity, is a downward inclination. 5. The correct answer is (D). Dissension is “disagreement,” while concord means “harmony.” Choice (A), desolation, is “loneliness.” Choice (B), strife, is a synonym for “dissension.” Choice (C), mandate, is a formal order. Choice (E), dearth, is a lack of something. 6. The correct answer is (B). Reprehensible means “guilty,” while inculpable means “blameless.” Choice (A), docile, means “obedient.” Choice (C), dubious, is “doubtful.” Choice (D), usurious, pertains to money lending at exorbitant interest rates, and choice (E), diffident, means “timid.” 7. The correct answer is (E). Salutary, means “healthy” or “curative.” Deleterious is “harmful.” Choice (A), volatile, means “explosive” or “fickle.” Choice (B), enigmatic, means “puzzling.” Choice (C), fulsome, means “disgusting.” Choice (D), esoteric, means “confidential” or “understood by only a few.” 8. The correct answer is (D). Elucidate means “to explain.” Obfuscate means “to confuse.” Choice (A), simulate is “to imitate.” Choice (B), palliate, is “to excuse” or “to abate.” Choice (C), compliment, means “an expression of esteem or respect” or, as a verb, “to offer a flattering remark.” Choice (E), portend, is “to indicate” or “to bode, serve as an omen.” 9. The correct answer is (B). Inveigle means “to win over by flattery.” The opposite logically is “to repulse,” “to drive back,” or “to repel.” Choice (A), inveigh, is “to protest or complain bitterly.” Choice (C), malinger, means “to pretend to be ill so as to avoid work.” Choice (D), herald, is “to announce.” Choice (E), impugn, means “to assail” or “to attack.”

Reading Comprehension Passage 1 1. The correct answer is (C). In the first paragraph, the acronym is explained in the parentheses with the first two letters standing for the process of using a laser, rather than a scalpel. Choice (A) can be eliminated because there are no names of doctors in the article, nor is it stated that the acronym represents names. Choice (B) is incorrect because, although the technical name of the surgical instrument is mentioned in the second paragraph (microkeratome), its brand name is never given. Choice (D) is incorrect because the anesthetic is not mentioned by name in the article, nor is it stated that the acronym stands for the name of the anesthetic. Choice (E) is incorrect because the term is not a prefix but an acronym with each letter representing a term. Also, the medical term for the procedure is mentioned in paragraph one.

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PRACTICE TEST 1 2. The correct answer is (D). It is described in the second paragraph. Choice (A) is incorrect, even though the suction ring is positioned after the cornea and is marked with ink, the sequence of the steps does not make them interrelated. Choice (B) is incorrect because it is the exterior layers of the cornea that are marked with the ink. Once those layers are rolled back, the underlying layers would not bear the marks; therefore, they could not aid in the excising of that tissue. Choice (C) is incorrect as it has already been stated in paragraph two that the eye has been stabilized (rendered immobile) by the suction ring, so there is no need to measure movement of the eye. Choice (E) is incorrect because nowhere in the article does it mention swelling of the eye as a side effect of this surgical procedure. 3. The correct answer is (E). Paragraph two reveals how the cornea must be cut or trimmed, retrospectively, to relieve farsightedness and nearsightedness. Choice (A) is incorrect because a previous choice reveals the purpose of marking the cornea with ink. In test design, this is called scaffolding, when one question builds on another. Choice (B) is incorrect as the purpose of the suction ring is to stabilize the eye, not correct the vision. Choice (C) is incorrect because the size of the microkeratome is discussed in the article, only its motion is addressed. Choice (D) is also incorrect because in the sequence of the procedure, the repositioning of the cornea flaps comes after the adjustment to the layers of tissue below the cornea that make the correction in the vision. 4. The correct answer is (D). It is the only combination of procedures that categorically states that the procedures are effective in correcting vision even when there is an astigmatism. Choice (A) is incorrect because, although LASIK is effective even with an astigmatism, INTACS is not effective for “extreme nearsightedness, farsightedness, or astigmatism.” Choice (B) is incorrect because the description of INTEROCULER does not say whether it is effective in cases of astigmatism, and INTACS definitely are not effective, as stated in the description. Choice (C) is incorrect because it again uses INTACS as a choice and while CUSTOM LASIK is correct, INTACS is not. Similarly, Choice (E) is not correct because while LASIK can be used on patients with astigmatisms, the description of INTEROCULAR LENSES does not specifically address this issue.

Passage 2 1. The correct answer is (D). “Antonym” means opposite, and in the first sentence of paragraph two there is a context clue. The fire ants can’t be eradicated because of their number; therefore, prolific means their fertility and ability to breed. Choices (A) and (C) are all synonyms for “prolific.” Choices (B) and (E) are not related to the word. “Laborious” means burdensome, and “lugubrious” means somber. 2. The correct answer is (C). Since the article is about an insect, the fire ant, it would be logical to assume that experts in the study of insects would be the correct choice. Also, the Greek word “entomon” means insect and provides the root of the word. 3. The correct answer is (E). Irony is when the opposite of what one expects occurs. In this case, one would expect that the insects would be hazardous to the crops, but they aren’t. They are a help by destroying other pests. Choices (A), (B), (C), and (D) are incorrect because the farmers, along with the general population, would “mind” the destruction and danger of the fire ants as expressed in those choices. This is not ironic but expected. The way in which they do not mind the fire ants, as stated above, is ironic.

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ANSWERS AND EXPLANATIONS 4. The correct answer is (D). It is stated in paragraph two and four that while the insects cannot be eradicated, they can, at least, be controlled. Therefore, researchers are reconciled to limiting the number of fire ants, rather than destroying them completely. Choice (A) is incorrect because researchers are not desperate. Also, the problem is not “enigmatic” (mysterious); it is clear, and remedies are being developed, as stated in the article. Choice (B) is incorrect because a compromise has been reached in not attempting to destroy all fire ants but to control their numbers, so the researchers have not been defeated. Choice (C) is incorrect because a victory has not been won with the new methods of control, only a partial success has been achieved. Choice (E) is incorrect because of the current spread of fire ants in terms of numbers and geography, the fact that they have been here since the 1930s, and that they have no natural enemies makes preventing their arrival in the United States irrelevant.

Passage 3 1. The correct answer is (B). Choice (A) is incorrect because the number or receptors relative to memory is not discussed in the article. Choice (C) is incorrect because the correlation between time and memory is not referenced in the article. Choices (D) and (E), although mentioned in the article, are not the products of repeated firing of brain cells. 2. The correct answer is (A). “Replete” means abundant; therefore, it is synonymous with “copious.” Also, there is a context clue in the second part of the sentence established by the trigger word “however.” What follows is the opposite of what “copious” means: “however, after they reach sexual maturity, the amount is greatly reduced.” Choice (B) is incorrect because it is the opposite in meaning. The word “meager” means a small amount. Choices (C) and (E) are incorrect because they have no relationship to the meaning of “copious,” although they are somewhat synonymous with each other. The word “mutated” means altered, and “vicious” means “corrupt.” Choice (D) is incorrect, although related, because “interspersed” means scattered, and small amounts of a substance can be “scattered” as well as large amounts. 3. The correct answer is (A). This is stated in the first sentence of paragraph two. Choices (B), (C), and (D) are mentioned in the article but are not effects of the gene NR2B. The reference in choice (E) to “increasing the size of the hippocampus and medial temporal lobe” are not mentioned in the article. 4. The correct answer is (D). This is stated in the concluding paragraph. Choice (A) is incorrect because it is stated in the concluding paragraph that no such experiments have been done on humans. Choice (B) is incorrect because, although it is a true statement, it is simplistic. The complex relationship between the experiments on mice and the application to humans has not been clearly established. Choice (C) is incorrect because while it is a given that the NR2B gene is producing the protein NMDA in the human brain, the benefit will come from finding ways to increase the production of that protein as was done by gene splicing in the mice. Choice (E) is incorrect. While the article states that this gene is abundant in the brains of young mice, it does not state the same about young humans. Even if one was to infer this to be true, the benefit will come from sustaining the high levels of this gene in adult humans.

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PRACTICE TEST 1

Analogies 1. The correct answer is (A). Epistle is a “formal, elegant” letter, so the relationship here is “epistle” is a kind of letter. Choice (A), ode, is a formal kind of poem. Choice (B), novel is fiction, not nonfiction. In choice (C), the order of the terms is reversed. A cryptogram is a kind of puzzle, but the original pair of terms presents the relationship in the opposite order. Choice (D), repartee, means “wit” or “clever language.” A diary is a personal journal or record, not a kind of repartee. Choice (E), lyric, means the words to a song or a kind of poem; it is not a kind of refrain. Again, the proper order is reversed. 2. The correct answer is (E). Pilfer and filch mean “to steal.” Pliant and yielding are synonyms. Choice (A), relinquish, means “to give up,” while pertinacious means “obstinate” or “unyielding.” In this pair, then, the second term happens to be an antonym for yielding in the original pair. Be careful not to be misled by such traps. Choice (B), steep and strong, are not necessarily synonyms. Choice (C), schism is a “separation” or “discord,” while abyss is “a bottomless pit” or “great gulf.” Chasm, not schism, is a synonym for abyss. In choice (D), both choices refer to kinds of watercraft; however, a scull is a small boat propelled by oars, while a yacht is significantly larger. 3. The correct answer is (C). Nucleus and extraneous are antonyms. Callow means “youthful, immature,” while cosmopolitan means “worldly, sophisticated.” Choice (A), verbose and prolix, which are synonyms, mean “wordy.” Choice (B), clemency and pardon, are also similar in meaning but different in degree. Clemency means “mildness,” while pardon is a broader term including “forgiveness” or “to free from penalty.” Choice (D), bungle, means “to mishandle or botch”; boast, meaning “to brag,” is not the opposite. Choice (E), dilatory, means “tardy” or “causing delay”; widen is not the opposite of this term. 4. The correct answer is (B). A recorder is a musical instrument on which one can play a tune. A clarinet is a woodwind musical instrument on which one can play a melody. Both recorders and clarinets are instruments played by mouth. Choice (A), radio, is not a musical instrument, offers a number of programs, not just music; a medley is a musical composition made of a series of songs. One does not, however, play a medley on a radio in the same way that one can play a recorder or clarinet. Of course, your first interpretation of recorder may have associated this term with tape recorder, so you may have been misled. Be sure to check all of the possible choices carefully. Choice (C), microphone, is a device used to amplify sound, and one can use it to sing. Again, however, a microphone is not played as a musical instrument. Choice (D), harp, is a musical instrument, but it is a percussion instrument, one played by plucking strings, not by mouth. Harp can also be a verb, meaning “to complain,” but the relationship is still not the same as that between the original pair. Choice (E), snare, is a kind of drum. 5. The correct answer is (A). Slough and swamp are synonyms. Fen is “low marshy land,” and bog is “wet spongy ground.” Choices (B), (C), and (D) offer no logical relationships. Choice (E) presents antonyms in slumber and insomnia, which means “an inability to sleep.”

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ANSWERS AND EXPLANATIONS 6. The correct answer is (D). Rapier and epée are both kinds of swords used in fencing. Carbine and rifle are both kinds of guns. Choice (A), tomahawk is a kind of weapon, but a wickiup is a kind of hut used by nomadic Native Americans. Choice (B), grenade is a type of explosive missile while artillery includes weapons for discharging missiles. Choice (C), javelin, is a light spear once used in war or hunting but now used in athletic competition; competition is the way in which the javelin is used. Choice (E), catapult, is an ancient military device used for hurling missiles; siege is a kind of battle strategy, not a weapon. 7. The correct answer is (D). Deracinate and eradicate are synonyms, meaning “to uproot.” mollify and mitigate are synonyms meaning “to alleviate.” Choice (A), ruminate, means “to contemplate,” and spurn means “to reject.” Choice (B), impugn, means “to deny” or “to attack as false,” but endorse means “to approve.” In choice (C), these two choices are not true antonyms, but they are opposite in general definition. Grovel means “to crawl” or “to abase oneself” while glorify means “to shed radiance on.” Choice (E), quibble means “to bicker”; masticate means “to chew.”

PRACTICE TEST 2—VERBAL ABILITY Sentence Completion 1. The correct answer is (D). The general context of the sentence, with reference to a “criminal” whose crime had not been discovered, helps us recognize that the missing words can also be negative. Surreptitious means “sneaky, secret, hidden,” while cache is “a secure place of storage.” In choice (A), prurient means “lascivious,” and domicile is “a place of residence.” The second term fits the sense of the sentence, but the first is illogical. In choice (B), nefarious means “wicked,” while morass is “a marsh or swamp.” In this pair, the first seems appropriate, but the second is illogical. In choice (C), irrefutable means “incontrovertible,” while acumen means “keenness of perception in practical matters.” Neither of these choices are relevant. In choice (E), supercilious means “proud” or “haughtily contemptuous,” while tirade is “a lengthy speech, often marked by intemperate language.” The first choice can be relevant if the criminal is arrogant, but the second term is illogical. 2. The correct answer is (A). Millennium is the popular term for a thousandth anniversary or a period of one thousand years; misnomer is a wrong name or designation. In choice (B), imprecation is a curse, and moniker is a nickname. In choice (C), bruit means “rumor” or “noise,” while bellwether is “a leader” or “one who takes the lead or initiative.” In choice (D), euphoria is “an often unaccountable feeling of well-being or elation,” and malaise is “an indefinite feeling indicative of or accompanying the onset of an illness.” Neither of these words is a logical choice. In choice (E), volition means “the act of making a choice” or “will,” and malapropism is “the use of a word sounding somewhat like the word intended but ludicrously wrong in the context.” Again, neither term is appropriate.

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PRACTICE TEST 2 3. The correct answer is (B). The opening phrase sets up an important contrast. Feigned means “to pretend,” while sophisticated is appropriate to complete the sentence’s meaning that she tried to look worldly or experienced in art appreciation. Only this pair logically completes the sentence. In choice (A), goaded means “urged or spurred,” while ineffable means “indescribable.” In choice (C), deigned means “condescended or stooped,” while provincial means “narrow-minded or unsophisticated.” In choice (D), imbued means “to permeate or infuse”; bibulous means “highly absorbent” or “inclined to drink.” In choice (E), obfuscated means “made obscure or darkened,” and munificent means “liberal” or “characterized by generosity.” 4. The correct answer is (E). Staunch means “steadfast in loyalty” or “faithful,” while inclement means “stormy.” Only this choice includes a term that can describe the weather. In choice (A), officious means “meddlesome or impertinent”; nebulous means “cloudy, hazy, or vague.” In choice (B), motley means “a jester or fool” or “clothing worn by a harlequin or jester”; ribald means “coarse” or “using broad, indecent humor.” In choice (C), moribund means “being in a dying state,” while placid means “serene or calm.” In choice (D), partisan means “supporter” or “follower,” and fungible means “interchangeable.” 5. The correct answer is (C). Turgid means “bombastic or pompous” and torpor is “lethargy or dullness.” The phrase “almost lulled asleep” is a big clue that the correct choice will somehow pertain to sleep. Only two choices offer something similar: torpor, choice (C), and sloth, choice (D). In choice (A), convoluted is “twisted”; ecstasy can mean “transport, rapture, even joy.” In choice (B), vociferous means “boisterous, blatant, or clamorous,” while apostasy is “renunciation of religious faith.” In choice (D), acerbic means “bitter or sharp,” and sloth is “laziness or indolence.” In choice (E), discursive means “rambling or verbose,” but effervescence is “liveliness or exhilaration.” 6. The correct answer is (B). The sentence refers to a farmer and soybeans so the correct answer will somehow logically pertain to farming. Assiduous means “diligent” or “steadily attentive,” and verdant means “green with growing plants.” In choice (A), erudite means “scholarly or learned,” and evanescent means “short-lived” or “vanishing or transient.” In choice (C), obdurate means “inflexible or unyielding,” while palpable means “noticeable” or “perceptible.” In choice (D), incipient means “beginning or commencing,” while labyrinthine means “intricate” or “involved.” In choice (E), infinitesimal means “immeasurably small;” scanty means “insufficient or meager.”

Antonyms 1. The correct answer is (B). Ascetic, means “austere, strict self-denial.” Saturnalian means “an unrestrained, often licentious celebration.” In choice (A), sardonic means “bitter, mocking,” while choice (C), nascent means “beginning to develop.” In choice (D), envisaged means “having a mental picture of something” and choice (E), insolvent, means “impoverished.” 2. The correct answer is (E). Cacophonous means “harsh sounding.” Euphonious means “pleasant sounding.” In choice (A), sugary pertains to a sweet flavor, while choice (B), resonant, means “echoing or continuing to sound.” In choice (C), risible means “laughable” and in choice (D), supercilious means “haughtily contemptuous.” 3. The correct answer is (C). Virtue means “merit or valor.” Peccadillo is “a slight offense.” In choice (A), sin is a synonym for peccadillo, while in choice (B), pedantry refers to “one who parades his learning.” In choice (D), riata is a “lariat or lasso,” and in choice (E), condolence means “pity” or “an expression of sympthy.”

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ANSWERS AND EXPLANATIONS 4. The correct answer is (D). Perspicacious means “shrewd” or “mentally keen.”Vacuous means “empty” or “inane, stupid.” In choice (A), multifarious means “diverse,” and in choice (B), amorous means “being in love.” In choice (C), clandestine means “secret” or “hidden,” while in choice (E), inane is a synonym for vacuous. 5. The correct answer is (A). Adroit means “dextrous” or “marked by shrewdness in coping with difficulty.” Gauche means “awkward” or “lacking grace.” In choice (B), devious means “tricky” or “crooked,” while in choice (C), prosaic means “unimaginative” or “commonplace.” In choice (D), volatile means “changeable, fickle” and in choice (E), dormant means “sleeping or latent.” 6. The correct answer is (D). Authentic means “genuine” or “veritable.” Spurious means “false or fraudulent.” In choice (A), bogus is a synonym for spurious, meaning “false.” In choice (B), hackneyed means “commonplace,” while in choice (C), affable means “pleasant or gracious.” In choice (E), peremptory means “imperative” or “haughty.” 7. The correct answer is (B). Fallacy is “a false idea” or “trickery.” Ratiocination means “reasoning” or “the process of exact thinking.” In choice (A), schism is “a division or separation.” In choice (C), arithmetic is “a branch of mathematics,” while in choice (D), paradigm is “an example or a pattern.” In choice (E), redress means “retribution” or “reparation.” 8. The correct answer is (C). Intransigent means “refusing to compromise” or “irreconcilable.” In choice (A), meretricious means “gaudy” or “tawdrily attractive.” In choice (B), muddled means “confused.” In choice (D), nonchalant means “indifferent” or “apparently unconcerned,” while in choice (E), willful means “obstinate” or “unruly.” 9. The correct answer is (C). Conformist describes one who is ‘obedient or compliant.” Maverick is “a motherless calf” or “an independent individual.” In choice (A), reagin is “a kind of antibody in the blood,” and in choice (B), nomenclature means “a system of names or symbols.” In choice (D), heroic pertains to bravery and honor. In choice (E), tattoo is a type of “indelible body decoration” or “a rapid rhythmic tapping.”

Reading Comprehension Passage 1 1. The correct answer is (D). In the first paragraph, line 3 has the context clue “might actually apply.” It can be inferred that a “hackneyed phrase” is one that is over-used; therefore, “trite” would be correct as it can mean stale or commonplace. Choice (A) is incorrect, although it is misleading because “choppy” seems synonymous with the root “hack.” Choices (B) and (C) are incorrect as they are not synonyms for “hackneyed.” Choice (E) is incorrect because it is an antonym for the word. 2. The correct answer is (C). When looking for context clues to help define words, it is helpful to look at the sentences that precede and follow the sentence in which the word appears. In this case, the preceding sentence contains the context clue “some anthropologists believe that it (body art) precedes it” (recorded history). Therefore, “time immemorial” means preceding recorded history, which is the memory of humankind (“longer than anyone can remember”). Choice (A) is incorrect even though it mentions time; the word “celebration” is irrelevant. Choice (B) is incorrect because it is the opposite of being “longer than anyone can remember.” Choice (D) is incorrect, although it is misleading because “memorialize” looks close to “immemorial.” However, the prefix “im” means “not” as in “impossible,” “not” possible, so “immemorial” means “not” memorialized. Choice (E) is incorrect because it refers to the future, not the past.

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PRACTICE TEST 2 3. The correct answer is (A). It is stated in the first paragraph and reinforced in the concluding paragraph. Choice (B) is incorrect. Although mentioned briefly in the concluding paragraph, it is not the primary reason for practicing body art today. Choices (C), (D), and (E) are more practices of the past than contemporary society. 4. The correct answer is (D). The main point of the article is that the practice of body art is universal with many peoples in different places throughout time. Choice (A) is incorrect. Although the second paragraph states that the practice of binding feet in China was “painful and crippling,” the general tone of the passage is not one of condemnation. Choice (B) is the opposite of the correct answer as the article emphasizes the ubiquity (commonness) of body art. Choice (E) is incorrect; although in the case of binding feet harmful effects are mentioned again, the general tone is not one of warning.

Passage 2 1. The correct answer is (E). It is stated in the first paragraph and is reiterated in the last sentence in the second paragraph. Choice (A) is incorrect because the article states that global warming is a reality. Choice (B) is incorrect because there are calculations, as stated in the second paragraph, that indicate a “likely rise.” Choice (C) is incorrect because the ice cap is shrinking as stated in the second paragraph. Choice (D) is incorrect. Although conversions are provided in the article, there is not a debate over this conversion. 2. The correct answer is (B). The Greenhouse Effect is described in the second paragraph. Choices (A) and (D) are incorrect because the greenhouse gasses do not produce either effect; they only trap infrared gasses. Choice (C) is incorrect because although the greenhouse gasses trap infrared heat that increases global temperature, it is the increased temperature that melts the ice cap, not the gasses themselves. Choice (E) is incorrect because it is not the actual burning of the gases that produces the heat of global warming. 3. The correct answer is (C). “Incontrovertible” means “not doubted” just as “indubitable” does; therefore, they are synonymous. Choices (A), (B), and (E) are incorrect because although they begin with the prefix “in” or “im,” which means “not,” they mean not “decisive,” “probable,” or “credible.” That makes the answer choices antonyms (opposites) of “incontrovertible.” 4. The correct answer is (B). “Fronts” means the front lines of a battle or war. The context clues of “combat” (used as a verb) and “attacked” complete the metaphor or comparison. Choices (A), (C), and (E) are incorrect because the metaphors do not pertain to the word “fronts” when combined with the context clues. Choice (D) is incorrect because it misrepresents “fronts” as weather fronts, which is not supported by the context clues.

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ANSWERS AND EXPLANATIONS

Passage 3 1. The correct answer is (E). La Niña has strong easterly winds that increases the formation of hurricanes, and, conversely, El Niño has weak easterly winds that diminish clouds. Choice (A) is incorrect because neither system, as described in this article, has an effect on the amount of rain or the raising of the temperature of still air. Choice (B) is incorrect because neither system, as described in this article, has an effect on the amount of evaporation and lowering the temperature of still air. Choice (C) is incorrect because La Niña does not produce weak easterlies and diminishing clouds and, while El Niño does produce strong westerlies, it does not increase the formation of hurricanes. Choice (D) is incorrect because La Niña does not produce strong westerlies. 2. The correct answer is (C). The second paragraph describes how the movement of wind in the direction of the earth’s rotation causes a whirling mass of storms that is called a tropical cyclone. Choices (A), (B), (D), and (E) are incorrect. Although the information appears in the article, it is not relevant to the definition of a tropical cyclone. 3. The correct answer is (D). It is the exception because it is not mentioned in the article, and all other choices are referred to in the third paragraph as factors affecting the formation of hurricanes. 4. The correct answer is (B). The fourth paragraph states that typhoons are in the Pacific, and hurricanes are in the Atlantic. Therefore, it is geographical location that determines the terminology. Choices (A), (C), (D), and (E) are incorrect. Although the information appears in the article, it is not relevant to the difference between a typhoon and a hurricane.

Analogies 1. The correct answer is (D). Poltroon is synonymous with coward. A quisling is a traitor, so the terms in this pair are synonyms. In choice (A), redress can mean “to remedy or correct” as well as “reparation or restitution.” A lawyer may seek redress , but the terms are not synonyms. In choice (B), vaunt means “to boast or brag,” which is what a “braggart” might do, but again, the relationship between these terms is not synonymous. In choice (C), scuttle can be “a metal pail for carrying coal” so it has a logical relationship with coal, but these terms are not synonyms. In choice (E), bushel and peck are both measurement terms; however, a bushel contains four pecks so the relationship is not synonymous. 2. The correct answer is (E). Purloin means “to steal or filch,” while bequeath means “to give or leave by will”; therefore, the terms are antonyms. Polemic is “an aggressive attack on the opinions of another,” while consensus is “unanimity or general agreement.” In choice (A), adumbrate means “to foreshadow or suggest,” while apotheosis means “deification or elevation to divine status. In choice (B), amalgamate means “to mix or to merge into a single body,” and commingle means “to blend”; therefore, these terms are synonyms. In choice (C), nullify means “to invalidate or negate,” while satiate means “to glut or gorge.” In choice (D), prevaricate means “to deviate from the truth,” while cajole means “to persuade with flattery.”

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PRACTICE TEST 2 3. The correct answer is (E). Lintel is a horizontal architectural structure above an opening, while postern is a back door or gate. The terms, roughly speaking, present a relationship of “support :: door.” A sill is a horizontal piece at the base of a window. Although lintel refers to a structure above the door and a sill is found at the base of a window, both are architectural terms related to support. In choice (A), pandemic typically refers to something occurring at a wide geographic area; universal is virtually synonymous. In choice (B), both phrases are French terms incorporated into English, but the relationship between these phrases is not relevant to the relationship between the original pair. Esprit de corps is “common spirit existing among members of a group, inspiring enthusiasm, devotion, and strong regard for the honor of the group.” Le mot juste refers to “exactly the right word.” In choice (C), a joist is a small timber used to support a floor or ceiling, while a threshold is “the plank, stone, or timber that lies under a door.” In choice (D), a maelstrom is “a violent whirlpool”; phenomenal may describe the maelstrom, but the relationship between this pair is not the same as the one between the original pair. 4. The correct answer is (A). Pod is the term for a group of whales. The relationship between these two terms, then, is collective noun : single. Notice that all of the choices offer animal terms, and some have the correct collective form for the noun. However, only choice (A) presents the terms in the correct order as the original pair. In choice (B), the collective noun for geese is “gaggle,” not “covey.” In choice (C), this pair presents the terms in the incorrect order as well as offers the improper collective noun. The correct collective noun for quail is “covey.” In choice (D), this pair offers a correct pair of terms, but the order is backward. In choice (E), pack is the collective noun for wolves, not leopards. 5. The correct answer is (E). Adze, is a cutting tool and is used in carpentry. Mattock is “a digging and grubbing tool with the features of an adze, ax, and pick,” and it is used on a farm. The relationship, therefore, is tool :: place. In choice (A), a spade is used for digging, but this relationship is tool :: activity. In choice (B), appetizer begins a meal while dessert comes at the end. This pair is not relevant at all to the original pair. In choice (C), a painter uses an easel, but the relationship here is user :: tool. In choice (D), a mason uses a trowel, a tool used to spread masonry material, such as concrete. The relationship of the words in choice (D) is user :: tool. 6. The correct answer is (B). A gosling is a young goose, and the order in which the terms is presented is correct. A kit is a young fox. In choice (A), even though a fawn is a young deer, the relationship is reversed in this pair. In choice (C), while both of these are terms relating to animals, a lamb is a young sheep while a ewe is a female sheep. In choice (D), another pair is reversed here: a newt is a young salamander. In choice (E), another reversed pair is presented: a fledgling is a newly hatched eagle. 7. The correct answer is (D). Badminton is a kind of game played with rackets and a shuttlecock, which is hit back and forth over a net. The relationship between this pair of words is game :: moving part. In choice (A), a wicket is “an arch or hoop” through which a ball is hit in the game of croquet. In choice (B), a wedge is type of club in the game of golf. In choice (C), a bridge is a device used to steady a cue in the game of billiards.

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Practice Test 1 QUANTITATIVE ABILITY Directions: Each of the Questions 1–21 consists of two quantities, one in Column A and one in Column B. You are to compare the two quantities and choose: (A) (B) (C) (D)

if the quantity in Column A is greater; if the quantity in Column B is greater; if the two quantities are equal; if the relationship cannot be determined from the information given.

Note: Since there are only four choices, NEVER MARK (E). Numbers: All numbers used are real numbers. Figures: Position of points, angles, regions, etc. can be assumed to be in the order shown, and angle measures can be assumed to be positive. Lines shown as straight can be assumed to be straight. Figures can be assumed to lie in a plane unless otherwise indicated. Figures that accompany questions are intended to provide information useful in answering the questions. However, unless a note states that a figure is drawn to scale, you should solve these problems NOT by estimating sizes by sight or by measurement, but by using your knowledge of mathematics.

1.

Column A

Column B

0.02 3 0.004

0.00008

S DS DS DS D p q

2.

p2v

q r

r t

t 51 v 0

327

PRACTICE TEST 1

3.

Column A

Column B

x

75

The sum of a group of numbers is 4,550. The average of the group of numbers is 325. 4.

The number of numbers in the group

15

840 5 5.

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880 8n 2

n

6.

The number of different positive divisors of 18

7.

The number of integers that satisfy both 5y 2 15 , 0 and the inequality 5 2 3y , 11

8.

The percent of 105 that 103 is

328

The number of different positive divisors of 28

2

0.01

GRE CAT Success

QUANTITATIVE ABILITY

Column A

Column B x . 0, y . 0

9.

y4x3

(xy)4

ax2 2 y3 5 12 ax2 1 y3 5 3 10.

30

a2x4 2 y6

11.

B1C

2A

2x 1 2y 51 x 1 2y 12.

13.

GRE CAT Success

1

xy

The average (arithmetic mean) of y 1 4, 2y 1 4 and 1 2 3y

329

The average (arithmetic mean) of 0, 3, and 6

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PRACTICE TEST 1

Column A

Column B

14.

x

50

15.

A bonus of $750 plus a 12% increase in annual salary

A bonus of $800 plus an 11.5% increase in annual salary

16.

The greatest even factor of 180 that is less than 90

The greatest odd factor of 180

a , 0, b . 0, c . 0 17.

(2a)(2b)(2c)

2(abc)

p.0 18.

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29 p

p 29

330

GRE CAT Success

QUANTITATIVE ABILITY

Column A

Column B

19.

x1y

z1w

20.

S D S D 1 20 1 4 1 3 3 7 23 7 23

1 7

The time it takes Jimmy to drive 300 miles at a rate of 52 miles per hour

The time it takes Bobby to drive 240 miles at a rate of 40 miles per hour

21.

GRE CAT Success

331

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PRACTICE TEST 1

Directions: Each of the questions 22–39 has five answer choices. For each of these questions, select the best of the answer choices given. 22.

(2.3 (A) (B) (C) (D) (E)

3 103) 1 (2.3 3 102) 1 (2.3 3 10) 5 255,300 25,530 2,553 255.3 25.53

23.

If a (A) (B) (C) (D) (E)

5 3, then what is the value of (a2)3 2 a? 12 15 240 243 726

24.

If x5 is odd and (x 1 y)5 is even, then which of the following must be odd? I. x 1 y II. xy III. x2y2 (A) I only (B) II only (C) I and II only (D) II and III only (E) I, II, and III

25.

Which of the following ratios is the same as the ratio of 1 1 2 to 3 ? 4 4 (A) 3 to 4 (B) 3 to 7 (C) 4 to 3 (D) 9 to 13 (E) 13 to 17

26.

If

a a2b 5 2, what is the value of ? b b 3 2 21 1 2 1 3 2

(A) 2 (B) (C) (D) (E)

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332

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QUANTITATIVE ABILITY

27.

If the parallelogram pictured above has an area of 30, then what is the value of N? (A)

=5 =10

(B) (C) 5 (D) 6 (E) 10 28.

During a sale at an office supply store, for every box of paper clips purchased for 15 cents, a second box can be purchased for 4 cents. How many boxes of paper clips did Paul buy if he spent 91 cents on boxes of paper clips? (A) 6 (B) 7 (C) 8 (D) 9 (E) 10

29.

Which of the following is equal to (A) (B) (C) (D)

=75x10y14? =3x6y8

25x4y6 5x4y6 25x2y3 5x2y3

(E) 2xy=xy 30.

The length of a side of an equilateral triangle is equal to the diameter of a area of the circle circle. What is the value of the ratio ? area of the triangle (A) p : =3 (B) 4p : =3 (C) (D) (E)

GRE CAT Success

=3 : p =3 : 4p 2p : =3

333

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PRACTICE TEST 1

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31.

What is the remainder when 63 is divided by 8? (A) 0 (B) 1 (C) 2 (D) 3 (E) 5

32.

If b boxes of paper clips cost c cents, how much will 5 boxes cost, at the same rate? (A) 5bc 5c (B) b c (C) 5b 5b (D) c b (E) 5c

33.

If one side of a square decreases by 10% and the adjacent side increases by 30%, by what percent does its area increase? (A) 15% (B) 17% (C) 20% (D) 22% (E) 25%

34.

If y 5 x(x (A) 9 (B) 27 (C) 81 (D) 243 (E) 729

2 1)2

, and x 5 3, then y 5

334

GRE CAT Success

QUANTITATIVE ABILITY

Data Analysis Questions Questions 35–39 pertain to the following chart. Average Undergraduate Budgets, 1998–99 Tuition & Fees

Books & Supplies

Room & Board

Transportation

Other Expenses

Total Expenses

1,600

620

*

*

*

N/A

1,600

620

2,000

1,000

1,200

6,420

7,300

660

4,600

550

1,000

14,110

7,300c

660

2,200

880

1,200

12,240

3,250

660

4,500

600

1,400

10,410

Commuter

3,250

660

2,100

1,000

1500

8,510

Out-of-State

8,400

660

4,500

600

1,400

15,560

14,500

670

5,800

550

1,000

22,520

14,500

670

2,100

860

1,200

19,330

Sector Two-Year Public Resident Commuter Two-Year Private Resident Commuter Four-Year Public Resident

Four-Year Private Resident Commuter

GRE CAT Success

35.

If a commuter decides to attend a two-year private college instead of a two-year public college, she can expect her total expenses to be (A) approximately the same. (B) almost twice as much. (C) almost three times as much. (D) slightly less. (E) There is not enough information to determine the amount.

36.

How much more does a resident attending a two-year private college pay for room and board than a commuter does? (A) $330 (B) $2,200 (C) $2,400 (D) $3,700 (E) $6,800

37.

Approximately what percent of a four-year public college commuter’s total expenses are for transportation? (A) 4% (B) 6% (C) 8% (D) 12% (E) 15%

335

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PRACTICE TEST 1

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38.

Approximately what percent of a four-year private resident’s college budget is used to pay non-tuition and fee expenses? (A) 16% (B) 20% (C) 24% (D) 28% (E) 36%

39.

Tuition and fees and room and board account for approximately what percent of a two-year public college commuter’s expenses? (A) 44% (B) 52% (C) 56% (D) 62% (E) 68%

336

GRE CAT Success

Practice Test 2 QUANTITATIVE ABILITY Directions: Each of the Questions 1–21 consists of two quantities, one in Column A and one in Column B. You are to compare the two quantities and choose: (A) if the quantity in Column A is greater; (B) if the quantity in Column B is greater; (C) if the two quantities are equal; (D) if the relationship cannot be determined from the information given. Note: Since there are only four choices, NEVER MARK (E). Numbers: All numbers used are real numbers. Figures: Position of points, angles, regions, etc. can be assumed to be in the order shown, and angle measures can be assumed to be positive. Lines shown as straight can be assumed to be straight. Figures can be assumed to lie in a plane unless otherwise indicated. Figures that accompany questions are intended to provide information useful in answering the questions. However, unless a note states that a figure is drawn to scale, you should solve these problems NOT by estimating sizes by sight or by measurement, but by using your knowledge of mathematics.

Column A

Column B x1y5y

1.

0

x

Mr. Norwalk borrowed $1,000 for one year. Mr. Palisano borrowed $500 for two years. 2.

The amount of interest Mr. Norwalk owes on his loan

337

The amount of interest Mr. Palisano owes on his loan

PRACTICE TEST 2

3.

Column A

Column B

40 percent of 19

39 percent of 20

A rhombus has a base of 10 and a height of 4 4.

5.

6.

The area of the rhombus

The perimeter of the rhombus

(58 2 35)2

(58)2 2 (35)2

The greatest prime factor of 243

The greatest prime factor of 180

4x2 1 y2 5 24xy 7.

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x

2y 2

338

GRE CAT Success

QUANTITATIVE ABILITY

Column A

Column B

8.

q1r

p1s

9.

=20

3 220 =

10.

c1b

a1d

x11 51 y 11.

x

y

5a 2 b 5 1 12.

GRE CAT Success

a

b

339

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PRACTICE TEST 2

Column A

Column B

13.

140 1 =3

~=3 1 =3!4

14.

AB 1 1

AC

U, V, and W are consecutive even integers, and U , V , W 15.

V1W21

U1V11

WX 5 1 16.

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6

The perimeter of WXYZ

340

GRE CAT Success

QUANTITATIVE ABILITY

17.

Column A

Column B

.667 0.166

2 3 1 6

y2 5 x2 2 1 xÞ0 18.

x4 1 1

y4

n.0 19.

10n 10n 1 1

10n 1 1 10n 1 2

20.

~2=7 1 3!~2=7 2 3!

19

Two students spent a total of 35 hours on a school project. One of the students spent 25% fewer hours on the project than the other. 21.

GRE CAT Success

The difference in the numbers of hours spent by each

341

6

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PRACTICE TEST 2

Directions: Each of the questions 22–39 has five answer choices. For each of these questions, select the best of the answer choices given. 22.

The I. II. III. (A) (B) (C) (D) (E)

23.

The perimeter of a square is equal to six times the circumference of a circle. What is the ratio of the area of the square to the area of the circle? (A) 1 : 9p (B) 3p : 1 (C) 9p : 1 (D) 1 : 3p (E) 1 : 1 p 7 The equation 5 is satisfied by how many different pairs of integers p and q? q 8 (A) One (B) Two (C) Four (D) Eight (E) Infinitely many

24.

25.

26.

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product of three consecutive integers must be Divisible by 3 Divisible by 5 Divisible by 6 I only II only I and II only I and III only I, II, and III

What is the largest of 11 consecutive integers whose sum is 0? (A) 211 (B) 25 (C) 5 (D) 9 (E) 11 B 5 A, then, in terms of A, B, and C, D 5 C1D (A) B 2 AC 2 A 1 (B) AC 2 B B 2C (C) A (D) AC 2 B A 2C (E) B If

342

GRE CAT Success

QUANTITATIVE ABILITY

27.

In the circle above, XY is a diameter. If XZ 5 5, what is the area of the circle? (A) 50p (B) 25p 25p (C) 2 (D) 5p=2 (E) 25

28.

29.

30.

GRE CAT Success

If a, b, x, and y represent different positive integers between 1 and 10, a2b ? what is the largest possible value of x1y 4 (A) 1 5 1 (B) 2 5 1 (C) 2 3 1 (D) 2 2 (E) 4 1 What is the maximum number of cubes with sides of a length s that could 2 fit inside a cube with sides of length 2s? (A) 8 (B) 16 (C) 32 (D) 64 (E) 128 If x (A) (B) (C) (D) (E)

1 y 5 0, what is the value of (x 2 2y)2 2 (2x 2 y)2? 9 3 0 23 210

343

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PRACTICE TEST 2

31.

The ratio (A) (B) (C) (D) (E)

32.

3 9 1 6 7

to to to to to

SD SD

1 of 3 1 1 3 7 6

6

1 to 3

7

is

The average (arithmetic mean) of five numbers is 26. After one of the numbers is removed, the average (arithmetic mean) of the remaining numbers is 25. What number has been removed? (A) 20 (B) 25 (C) 26 (D) 30 (E) 32

W

33.

What is the perimeter of pentagon VWXYZ shown above? (A) (B) (C) (D) (E)

34.

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53 58 60 66 70

?22? 1 ?7? 1 ?22 1 7? 5 (A) 0 (B) 5 (C) 10 (D) 14 (E) 18

344

GRE CAT Success

QUANTITATIVE ABILITY

Questions 35–39 refer to the following graph.

Unsubsidized Stafford Loans Subsidized Stafford Loans

2000 2001

GRE CAT Success

35.

What was the total amount of federal aid awarded through Pell Grants in 2000–2001? (A) $10 million (B) $8 million (C) $6 million (D) $1 million (E) $100,000

36.

What is the ratio of the amount of federal aid awarded through Family Education Loans to the amount of federal aid awarded through Ford Direct Loans and Specially Directed Aid? (A) 10 to 1 (B) 5 to 3 (C) 5 to 2 (D) 2 to 1 (E) 5 to 1

37.

What was the total amount of federal aid awarded through Unsubsidized Stafford Loans in 2000–2001? (A) $16 million (B) $14 million (C) $10 million (D) $6 million (E) $4 million

38.

What percent of the federal aid awarded to postsecondary students in 2000–2001 was awarded through Subsidized Stafford Loans? (A) 50% (B) 15% (C) 12.5% (D) 10% (E) 6.5%

345

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PRACTICE TEST 2

39.

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If the amount of federal aid awarded through PLUS Loans in 2000–2001 was projected to increase by 10% in 2001–2002 and by an additional 20% in 2002–2003, what is the projected amount of federal aid to be awarded through PLUS Loans in 2002–2003? (A) $13,300,000 (B) $13,200,000 (C) $1,330,000 (D) $1,320,000 (E) $1,230,000

346

GRE CAT Success

ANSWERS AND EXPLANATIONS

Quick Score Answers Practice Test 1 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

C C B B A C A A D B C A C

14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26.

C D A B D C A B C E D D D

Practice Test 2 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39.

A D D A A B B C B C D E C

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

C D B C B B C D B D B C B

14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26.

C A A A A C C B D C E C C

27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39.

C C D C A D D D C B E C D

ANSWERS AND EXPLANATIONS

PRACTICE TEST 1—QUANTITATIVE ABILITY 1. The correct answer is (C). When multiplying two decimal numbers, the number of digits to the right of the decimal point in the product must equal the sum of the number of digits to the right of the decimal point in the two numbers being multiplied. Since 0.02 has two digits to the right of the decimal point, and 0.004 has three digits to the right of the decimal point, 0.02 3 0.004 5 0.00008. 2. The correct answer is (C). Canceling the q’s, r’s, and t’s from the product on the p left hand side of the given information leaves the equation 5 1. This equation can v only be true if p 5 v. Therefore, p 2 v 5 0. 3. The correct answer is (B). Since there are 180° in a straight angle, we have 40 1 x 1 x 5 180 or 2x 5 140. This tells us that x 5 70. 4. The correct answer is (B). The average of a group of numbers can be computed by the formula: Average 5 (Sum of the Numbers) 4 (The Number of Numbers). Substituting the information that is given, we obtain: 325 5 4,550 4 (The Number of Numbers). Therefore, the number of numbers in the group must be 4,550 4 325 5 14. 880 5 880 2 n, we can see that 840 must be equal 8n n . Therefore, 40 5 80 2 n, or n 5 40.

5. The correct answer is (A). Since to 880

2

6. The correct answer is (C). The divisors of 18 are 1, 2, 3, 6, 9, and 18. The divisors of 28 are 1, 2, 4, 7, 14, and 28. Thus, each number has 6 divisors.

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PRACTICE TEST 1 7. The correct answer is (A). The inequality 5y 2 15 , 0 is satisfied by 5y , 15 or y , 3. The inequality 5 2 3y , 11 is satisfied by 23y , 6 or y . 22. Thus, both inequalities are satisfied by the integers 21, 0, 1, and 2. 8. The correct answer is (A). 103 is equal to 1,000, and 105 5 100,000. To find the 1,000 3 100% 5 1%. Then, 1 . 0.01. percent of 100,000 that 1,000 is, compute 100,000 9. The correct answer is (D). There is no way to tell which expression is larger. For example, if x 5 y 51, then A and B are equal. But, if x 5 y 5 2, then B is larger. 1 Also, if x 5 y 5 , then A is bigger. 2 10. The correct answer is (B). In order to solve this problem, simply note that (ax2 2y3)(ax2 1 y3) 5 a2x4 2 y6. Thus, the value of the expression in Column B is equal to 12 3 3 5 36 . 30. 11. The correct answer is (C). Using the fact that when two parallel lines are cut by a transversal, corresponding angles are congruent, we see that A 1 C. By the vertical angle theorem, B 5 C. Thus, A 5 B 5 C. Clearly, then, B 1 C 5 2A. 12. The correct answer is (A). By multiplying both sides of the equation in the given information by x 1 2y, we obtain 2x 1 2y 5 x 1 2y. From this, we see that 2x 5 x or x 5 0. If x 5 0, then xy 5 0 also. 13. The correct answer is (C). To find the average of y 1 4, 2y 1 4, and 1 2 3y, begin by adding these three expressions together: (y 1 4) 1 (2y 1 4) 1 (1 2 3y) 5 9. Next, divide by 3, giving an average of 3. Since the average of 0, 3, and 6 is also 3, the values in the two columns are the same. 14. The correct answer is (C). Since angle BDE is supplementary to a 120° angle, its measure must be 60°. Similarly, the measure of angle BED must be 70° since it is supplementary to two angles which total 110°. Finally, since there are 180° in a triangle, x must equal 50. 15. The correct answer is (D). Without any specific knowledge of what the annual salaries were, it is impossible to answer the question. 16. The correct answer is (A). Note that 180 5 2 3 2 3 3 3 3 3 5. The greatest even factor is 90, since 180 5 2 3 90. The greatest even factor less than 90 would be 60, since 180 can also be written as 3 3 60. The greatest odd factor of 180, on the other hand, would be 3 3 3 3 5 5 45. 17. The correct answer is (B). Note that (2a)(2b)(2c) 5 8abc. Thus, we need to compare the size of 8abc to the size of 2abc. We are given that a is negative, but b and c are positive. Thus, abc is negative, and 8 times a negative number is always less than 2 times a negative number. Thus, 2abc is larger than 8abc. 18. The correct answer is (D). If p 5 1, clearly, the entry in Column B would be bigger. However, if p 5 29, the two entries would be equal.

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ANSWERS AND EXPLANATIONS 19. The correct answer is (C). Consider the diagram below, with angles and points as labeled.

Since ABC forms a triangle, a 1 b 1 c 5 180. Similarly, since ABE forms a triangle, a 1 d 1 e 5 180. Therefore, a 1 b 1 c 5 a 1 d 1 e, or b 1 c 5 d 1 e. Now, since angles ABC and CBD are supplementary, we have b 5 180 2 x. Similarly, c 5 180 2 y, d 5 180 2 z, and e 5 180 2 w. Substituting these expressions into b 1 c 5 d 1 e, we get (180 2 x) 1 (180 2 y) 5 (180 2 z) 1 (180 2 w). Manipulating this equation, yields: ~180 2 x! 1 ~180 2 y! 5 ~180 2 z! 1 ~180 2 w! 360 2 x 2 y 5 360 2 z 2 w 2x 2 y 5 2z 2 w or (multiplying by 21 ) x 1 y 5 z 1 w. 20. The correct answer is (A). There is no need to perform a lot of arithmetic to solve 1 this problem. Simply take the expression in Column A and factor out the from 7 4 24 1 20 1 24 1 is 5 . Since both terms. The expression then becomes 7 23 23 7 23 23 1 greater than 1, the expression in Column A is greater than . 7

S DS

D S DS D

21. The correct answer is (B). The formula to use in motion problems is d 5 rt, that d is, distance 5 rate 3 time. This equation can be rewritten as t 5 . Jimmy’s time, r 240 300 ' 5.8 hours, whereas Bobby’s time is 5 6 hours. then, is 52 40 22. The correct answer is (C). (2.3 3 103) 1 (2.3 3 102) 1 (2.3 3 10) 5 (2.3 3 1,000) 1 (2.3 3 100) 1 (2.3 3 10) 5 2,300 1 230 1 23 5 2,553. 23. The correct answer is (E). Since (a2 )3 5 a6, we need to evaluate a6 2 a, with a 5 3. 36 2 3 5 729 2 3 5 726.

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PRACTICE TEST 1 24. The correct answer is (D). The only way x5 can be odd is if x is odd. The only way (x 1 y)5 can be even is if (x 1 y) is even. Since x is odd, y must be odd as well. The sum of two odd numbers is even, so I is even. The product of two odd numbers is odd, so II is odd. Similarly, x2y2 is odd. 1 1 25. The correct answer is (D). Writing 2 as 2.25 and 3 as 3.25, we can see that we 4 4 are looking for a ratio that is equal to that of 2.25 to 3.25. Writing this ratio as a fraction: 9 2.25 2.25 3 100 225 225 4 25 5 5 5 5 . 3.25 3.25 3 100 325 325 4 25 13 26. The correct answer is (D).

a2b a b a 5 2 5 21522151 b b b b

27. The correct answer is (A). The height of the parallelogram is 3N, and the length of its base is 2N. Its area, therefore, is (3N)(2N) 5 6N2. Since the area is 30, we have 6N2 5 30, or N2 5 5. Therefore, N 5 =5. 28. The correct answer is (D). Two boxes of paper clips cost 19 cents. Note that 19 3 4 5 76. Therefore, Paul could have paid for 8 boxes of paper clips for 76 cents, and still have had 15 cents left to purchase one more box. Therefore, the total number of boxes purchased would be 9.

=75x10y14 5 =3x6y8

29. The correct answer is (D).

75x10y14 5 3x6y8

=

=25x4y6 5 5x2y3

30. The correct answer is (A). Let D 5 the diameter of the circle. Then,

SD

D would be 2

D 2 pD2 . As far as the triangle 5 2 4 is concerned, its side is also equal to D. As the diagram below shows, the height of D the triangle is =3. 2

the radius of the circle, and the area would be p

Its area, then, is

SD

1 D2 =3. The ratio, then, is given by 2 2

pD2 p pD2 4 5 . 5 2 2 D =3 D =3 =3 4

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ANSWERS AND EXPLANATIONS 31. The correct answer is (A). Simply note that 63 5 6 3 6 3 6 5 2 3 3 3 2 3 3 3 2 3 3 5 (2 3 2 3 2) 3 (3 3 3 3 3) 5 8 3 27. Thus, when 63 is divided by 8, the remainder is 0. 32. The correct answer is (B). To answer this question, we need to write and solve a proportion: 5c b boxes 5 boxes 5 Now, cross multiply: b 3 x 5 5c, or x 5 c cents x cents b 33. The correct answer is (B). Let the side of the original square equal x. Then, the original area of the square is x2. The new sides of the square can be written as .9x and 1.3x, so the new area of the square is (.9x)(1.3x) 5 1.17x2. Clearly, the area of the square has increased by .17 5 17%. 2

2

2

34. The correct answer is (C). If x 5 3, we have y 5 x(x 2 1) 5 3(3 2 1) 5 32 5 34 5 81. 35. The correct answer is (B). The total expenses for a commuter at a two-year public college are $6,420, while the total expenses for a commuter at a two-year private college are $12,240. The best answer is, thus, “almost twice as much.” 36. The correct answer is (C). A resident attending a two-year private college pays $4,600, while a commuter pays $2,200. The difference is $4,600 2 $2,200 5 $2,400. 37. The correct answer is (D). The transportation expense is $1,000 out of a total of 1000 3 100% $8,510. Rounding $8,510 to $8,500 gives us an approximate answer of 8500 10 3 100% ' 11.7% ' 12%. 5 85 38. The correct answer is (E). The easiest way to do this problem is to find what 14,500 percent tuition and fees represent, and subtract this percentage from 100%. 22,520 3 100% ' 64%. Therefore, the percent spent on non-tuition and fee expenses is 100% 2 64% 5 36%. 39. The correct answer is (C). The total tuition and fees and room and board expenses are $1,600 1 $2,000 5 $3,600. The total expenses are $6,420 ' $6,400. Thus, the 9 3,600 3 100% 5 3 100% ' 56%. percent is 6,400 16

PRACTICE TEST 2—QUANTITATIVE ABILITY 1. The correct answer is (C). Since y 5 y, the only way that x 1 y can possibly equal y is if x 5 0. 2. The correct answer is (D). We cannot tell who owes more since we know nothing about the interest rates of the loans. 3. The correct answer is (B). Forty percent of 19 5 0.4 3 19 5 7.6, while 39 percent of 20 5 0.39 3 20 5 7.8. 4. The correct answer is (C). The four sides of a rhombus are equal, so, if the base is 10, then the perimeter is 40. The area is equal to base 3 height, which is also 40.

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PRACTICE TEST 2 5. The correct answer is (B). If you approach this problem correctly, it is not necessary to do very much computation. First, note that (58 2 35)2 5 232. On the other hand, (58)2 2 (35)2 can be easily simplified if we first factor it as the difference of two squares: (58)2 2 (35)2 5 (58 2 35)(58 1 35) 5 (23)(93). Clearly, (23)(93) is larger than 232. 6. The correct answer is (B). Since 243 5 35, the greatest prime factor of 243 is 3. On the other hand, it is easy to see that 180 is divisible by 5 since it ends in 0. 7. The correct answer is (C). Rewrite the equation in the given information as 4x2 1 4xy 1 y2 5 0, and factor the left hand side to get: (2x 1 y)2 5 0. The only 2y . way this equation can be true is if 2x 5 2y, that is, if x 5 2 8. The correct answer is (D). Since there are 180° in a triangle, we know that p 1 s 1 q 1 r 5 180. We also know (from looking at the two smaller triangles) that p 1 q 5 90, and r 1 s 5 90. However, we have no information about the sizes of these angles, nor do we have any information about the lengths of the sides. As such, we can reach no conclusions about the relative size of q 1 r and p 1 s. 9. The correct answer is (B). We can solve this problem by estimating. Clearly, =20 3 is somewhere between 4 and 5, since 42 5 16, and 52 5 25. As for =220, it must 3 3 be somewhere between 6 and 7, since 6 5 216, and 7 5 343. 10. The correct answer is (D). All we know from the diagram is that a 5 d, and c 5 b. We have no way to compare a 1 d to c 1 b. 11. The correct answer is (B). The fact that

x11 5 1 tells us that x 1 1 5 y. This y

means that y is 1 bigger than x. 12. The correct answer is (C). The only way that 5a a 2 b 5 0, that is, if a 5 b.

2 b

can be equal to 1 is if

13. The correct answer is (B). Since =3 is about 1.7, the entry in Column A is about 141.7. As for Column B, we can see that =3 1 =3 is 2=3. To quickly raise this expression to the fourth power, we can square it twice. First, when we square 2=3, we get 4 3 3 5 12. Squaring again gives us 144. The entry in Column B is larger. 14. The correct answer is (C). AB is the hypotenuse of right triangle OAB. Since the legs of the triangle are of length 3 and 4, the hypotenuse AB is 5, and AB 1 1 is 6. Since AC extends from (23, 0) to (3, 0), it also has a length of 6. 15. The correct answer is (A). If U, V, and W are consecutive even integers, V can be written as U 1 2, and W 5 U 1 4. Thus, V 1 W 2 1 5 (U 1 2) 1 (U 1 4) 2 1 5 2U 1 5, whereas U 1 V 1 1 5 U 1 (U 1 2) 1 1 5 2U 1 3. 16. The correct answer is (A). The perimeter that we are looking for is given by WX 1 XY 1 YX 1 WZ. We are given that WX 5 1, and since triangle WXZ is isosceles, WZ 5 1 as well. Further, since triangle WXZ is right, the hypotenuse XZ is equal to =2. Next, since triangle XYZ is isosceles, ZY is also equal to =2. Finally, since triangle XYZ is also right, the hypotenuse XY is equal to ~=2!~=2! 5 2. The perimeter of the figure, then, would be 1 1 1 1 2 1 =2 5 4 1 =2. Since =2 ' 1.4, the perimeter is about 4 1 1.4 or 5.4. This is less than 6.

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ANSWERS AND EXPLANATIONS 17. The correct answer is (A). The quickest way to solve this problem is to note that 1 2 .667 is slightly larger than , while 0.166 is slightly less than . Therefore, the 3 6 fraction in Column A has a numerator larger than that of Column B, and a denominator smaller than that of Column B. Overall, this makes the fraction in Column A larger. 18. The correct answer is (A). Take the equation y2 5 x2 2 1 , and square both sides: (y2)2 5 (x2 2 1)2. From this , we see that y4 5 x4 2 2x2 1 1, which can be rewritten as y4 5 (x4 1 1) 2 2x2. Now, since we know that x Þ 0, 2x2 must be positive. Since x4 1 1 minus a positive number equals y4, x4 1 1 must be larger than y4. 19. The correct answer is (C). Using the rule for the division of numbers with 1 exponents, it is easy to see that both of the given fractions are equal to . 10 20. The correct answer is (C). ~2=7 1 3!~2=7 2 3! 5 ~2=7!2 2 6=7 1 6=7 2 9 5 19 21. The correct answer is (B). Let x 5 the number of hours spent by the student who worked the larger number of hours. Then, the other student worked .75x hours, and we have the equation: x 1 .75x 5 35, or 1.75x 5 35. Solving for x, we get x 5 20. Thus, the other student worked 15 hours, and the difference is 5 hours. 22. The correct answer is (D). Since every third integer is divisible by 3, any group of three consecutive integers must contain one that is divisible by 3. The product of such a group of integers, then, will contain a factor of 3 and, thus, be divisible by 3. In the same fashion, any group of three consecutive integers must also contain (at least) one that is divisible by 2. Therefore, when a group of three consecutive integers is multiplied together, the product must contain a factor of 2 as well as a factor of 3. Since a number that is divisible by 2 and 3 must also be divisible by 6, the product of three consecutive integers must also be divisible by 6. However, only every fifth number is divisible by 5, so it is possible that a group of three consecutive integers will not contain one that is divisible by 5. In this case, the product will not be divisible by 5 either. 23. The correct answer is (C). The formula for the perimeter of a square is 4s, where s is the length of the side. The formula for the circumference of a circle is 2pr, where r is the radius of the circle. Thus, we are given that 4s 5 6(2pr) or 4s 5 12pr. Dividing both sides of this equation by 4 gives us s 5 3pr. Now, the area of the square is given by s2, which, by the work above, is equal to (3pr)2 5 9p2r2. The area of the circle is, of course, pr2. Thus, the ratio of the area of the square to the area of the circle is: 9p2r2 9p 5 pr2 1 24. The correct answer is (E). There are infinitely many pairs of numbers in the ratio of 7 to 8. If p 5 7n, and q 5 8n, with n equal any number, p and q will have the desired ratio. 25. The correct answer is (C). The only 11 consecutive integers that add up to 0 are 25, 24, 23, 22, 21, 0, 1, 2, 3, 4, 5. The largest of these numbers is 5.

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PRACTICE TEST 2

26. The correct answer is (C). Begin by taking

B 5 A and multiplying both sides C1D

by C 1 D to obtain the equation: B 5 A(C 1 D). Now, distribute on the right hand side: B 5 AC 1 AD. Subtract AC from both sides: B 2 AC 5 AD. To finish, divide both sides by A: D5

B 2 AC B 5 2C A A

27. The correct answer is (C). Since any triangle inscribed in a semicircle is a right triangle, triangle XYZ is right, with the right angle at Z. Further, since angle X is the same size as angle Y, the triangle is also isosceles. We are given that XZ is 5; thus, using the properties of isosceles right triangles, XY 5 5=2. Since XY is the diameter of the 5 circle, the radius of the circle is =2. The area of the circle is: 2

S D SD

A 5 pr2 5 p

5 =2 2

2

5p

25p 25 5 2 2

28. The correct answer is (C). This problem can be solved only with a bit of trial and error. Clearly, to maximize the size of the fraction, we’d like to make the numerator as big as possible and the denominator as small as possible. The smallest that the denominator can be is 1 1 2 5 3. However, if we make the denominator 3, the largest that the denominator can be is 10 2 3 5 7. The value of the fraction would 1 7 be 5 2 . Another fraction to try would be the one with the largest possible 3 3 numerator, which would be 10 2 1 5 9. However, in this case, the smallest the 1 9 denominator could be would be 2 1 3 5 5, making the fraction or 2 , which is 5 5 1 less than 2 . Similar experimentation with other possibilities clearly shows that the 3 1 largest value is 2 . 3

SD

1 1 3. 1 3 5 s . The 29. The correct answer is (D). The volume of a cube with side s is 2 2s 8 volume of a cube with side 2s is 8s3.The question, thus, is how many cubes with a 1 volume of s3 will fit in a cube with a volume of 8s3. We can determine this by 8 dividing: 8s3 8 5 5 8 3 8 5 64 13 1 s 8 8 30. The correct answer is (C). If x 1 y 5 0, x 5 2y. Substituting this into the given expression yields ~x 2 2y!2 2 ~2x 2 y!2 5 ~2y 2 2y!2 2 ~22y 2 y!2 5 ~23y!2 2 ~23y!2 5 9y2 2 9y2 50

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ANSWERS AND EXPLANATIONS

SD SD

1 3 31. The correct answer is (A). Note that 1 3

6

7

5

1 3 5 . Thus, the ratio is 3 to 1. 1 1 3

32. The correct answer is (D). Call the five numbers a, b, c, d, and e. Thus, we know that a1b1c1d1e 5 26, or a 1 b 1 c 1 d 1 e 5 130. 5 a1b1c1d 5 25, or Now, let’s say that e is the number which was removed. Then, 4 a 1 b 1 c 1 d 5 100. Looking at these two equations, we can see that e 5 30. 33. The correct answer is (D). In the given pentagon, we know the lengths of all of the sides except WX. However, if we “complete” the pentagon, making it into a rectangle, as shown below, we can see that WX is the hypotenuse of the right triangle with legs 5 and 12. Since the hypotenuse of such a triangle is 13, the perimeter is 13 1 22 1 10 1 8 1 13 5 66.

34. The correct answer is (D). Note that ?22? 5 2, ?7? 5 7, and ?22 1 7? 5 ?5? 5 5. Thus, the expression is equal to 2 1 7 1 5 5 14. 35. The correct answer is (C). Pell Grants make up 15% of the total aid money of $40 million. 15% of $40 million 5 0.15 3 $40 million 5 $6 million 36. The correct answer is (B). The easiest way to do this is by working directly with the percents instead of the actual values. Family Education Loans account for 50% of the federal aid money. Ford Direct Loans and Specially Directed Aid account for a total of 30% of the loan money. The ratio is, thus, 50 to 30, which can be simplified to 5 : 3. 37. The correct answer is (E). Unsubsidized Stafford Loans account for 40% of the Ford Loan money. The amount of Ford Loan money is 25% of $40 million 5 $10 million. 40 percent of this amount is $4 million. 38. The correct answer is (C). Subsidized Stafford Loans account for 50% of the Ford Direct Loan money, which, in turn, is 25% of the federal loan money. Now, simply note that 50% of 25% is 12.5%. 39. The correct answer is (D). PLUS Loans account for 10% of the $10 million Ford Direct Loan money, or $1 million. A 10% increase will raise this amount to $1.1 million. Raising this amount by 20% gives $1.32 million 5 $1,320,000.

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Appendices

Appendix A THE GRE CAT SUCCESS MATH REVIEW Perhaps more than any other subject, math creates a gulf between classes of students. Generally speaking, there are students who think of themselves as “good at math,” who do well in all the usual math subjects and often take advanced classes in their junior and senior years of high school. They may go on to major in math-intensive college subjects like chemistry, biology, economics, or engineering. Then there are the others, more numerous, who are a little afraid of math. In high school they take only those math classes they are required to take and breathe a sigh of relief when they pass. They’re more likely to major in the humanities, history, or any other field where the only numbers involved are the prices of textbooks. Here’s the good news. The test-makers know that the GRE will be taken by hundreds of thousands of students in both categories. They’ve deliberately designed the exam to be fair to both. As a result, many of the math topics that some students find most intimidating—such as trigonometry and calculus—do not appear on the test. Since many students are never exposed to these subjects, the test-makers wouldn’t consider it fair to test them. Instead, they restrict their questions to topics that virtually all high school students study in the ninth and tenth grades. This doesn’t mean that all of the GRE math questions are easy. But it does mean that it’s highly unlikely that you’ll be tested on any topic you never learned in high school. Of course, if you haven’t studied math in the years since then, you may have some serious catching up to do. But at least you have this comfort: You knew all this stuff once; now all you have to do is remember it. This appendix is designed to help. In the GRE CAT Success Math Review, we’ve selected the fifty math topics most frequently tested on the exam. For each, we’ve created a mini-lesson reviewing the basic facts, formulas, and concepts you need to know. We’ve also provided an example or two of how these concepts might be turned into test questions. You’ll probably find that you are comfortable with many of the topics included in the “Nifty Fifty” that follow. If so, great. Make a note of the other topics—the ones you find confusing, tricky, or difficult. Perhaps you never quite mastered those concepts when they were presented in class, or you’ve forgotten the details in the intervening years. In your study between now and the day of the GRE, concentrate on reviewing and practicing these topics. You can boost your GRE math score significantly by mastering as many of your personal “math demons” as possible.

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ARITHMETIC

TOPIC 1. NUMBERS

AND THE NUMBER LINE We can think of the real numbers as points on a line. Usually, we draw a horizontal line, with one point chosen to represent zero. All the positive numbers are to the right of zero, and all the negative numbers are to the left. The numbers get larger as you go from left to right. 24

23

22

21

0

1

2

3

4

The farther you get from zero, the larger the absolute value of a number. The absolute value of a number is the number without its sign; it can be thought of as the number’s distance from zero. So numbers far to the left on the number line are negative numbers with large absolute values. Remember, in comparing negative numbers, the one with the larger absolute value is the smaller number!

Example 1

On the number line shown below, where is the number that is less than D and half as far from D as D is from G? A

B

C

D

E

F

G

H

I

Solution

First, any number less than D must lie to the left of D. (Get it? Left 5 Less!) The 1 distance from D to G is 3 units. Thus, the point we want must be 1 units to the 2 left of D; that is, halfway between B and C.

Example 2

On the number line shown below, which point corresponds to the number 2.27? 2.2

A

B

C

D

E

F

G

H

I

2.3

Solution

Since the labeled end points are 2.2 and 2.3, the ten intervals between must each represent one tenth of the difference. Hence the “tick marks” must represent hundredths. That is, A 5 2.21, B 5 2.22, and so on. Thus, we know that G 5 2.27.

TOPIC 2. LAWS

OF ARITHMETIC AND ORDER OF OPERATIONS In carrying out arithmetic or algebraic operations, you should use the famous mnemonic device Please Excuse My Dear Aunt Sally. The operations of Powers, Exponents Multiplication, Division, Addition, and Subtraction should be carried out in that order, reading from left to right. If we want to indicate a change in the order of operations, we place the operation in parentheses, creating one number. In other words, calculate the number in parentheses first. Thus, 16 2 3 3 4 5 16 2 12 5 4, because we multiply before adding. If we want the number 16 2 3 to be multiplied by 4, we must write (16 2 3) 3 4 5 13 3 4 5 52.

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APPENDIX A

The basic laws of arithmetic were defined for whole numbers but carry over to all numbers. You should know all of them from past experience. They are: • The commutative law. This says that it doesn’t matter in which order you add or multiply two numbers. That is, a 1 b 5 b 1 a and ab 5 ba. • The associative law (also called the Regrouping law). This law says that it doesn’t matter how you group the numbers when you add or multiply more than two. That is, a 1 (b 1 c) 5 (a 1 b) 1 c, and a(bc) 5 (ab)c. Remember, enclosing the numbers in parentheses indicates that the operation within the parentheses should be done first. • The distributive law for multiplication over addition. This law can be stated as a(b 1 c) 5 ab 1 ac, which means that you can add first and then multiply, or multiply each term in the sum by a and then add the results. It doesn’t matter which you choose—the value of the answer will be the same. • The properties of zero and one. Zero times any number is zero. Zero added to any number leaves the number unchanged. One times any number leaves the number unchanged. Finally, it is very important to know that if the product of several numbers is zero, at least one of the numbers must be zero. • The additive inverse (or opposite). For every number n, there is a number 2n such that n 1 (2n) 5 0. • The multiplicative inverse. For every number n except 0, there exists a 1 1 number such that ~n! 5 1. Division by n is the same as multiplican n 1 tion by , and division by zero is never allowed. n

SD

31B if B 5 3? What value may B NOT have? 4{ 3 2 3B

Example

What is the value of

Solution

The fraction bar in a fraction acts as a “grouping symbol” like parentheses, meaning that we should calculate the numerator and denominator separately. That is, we should read this as (3 1 B) 4 (4 3 3 2 3 3 B). When B 5 3, the numerator is 3 1 3 5 6 and the denominator is 12 2 3 3 3 5 12 2 9 5 3. Therefore, the fraction 6 is 5 2. Since we cannot divide by zero, we cannot let 4 3 3 2 3 3 B 5 0. But in 3 order for this to be zero, 4 3 3 5 3 3 B. By the commutative law, B 5 4. Thus, the only value that B cannot have is 4.

TOPIC 3. DIVISIBILITY RULES A factor or divisor of a whole number is a number that divides evenly into the given number, leaving no remainder. For example, the divisors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24 itself. A proper divisor is any divisor except the number itself. Thus, the proper divisors of 24 are 1, 2, 3, 4, 6, 8, and 12.

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If you want to know whether k is a divisor of n, just try to divide n by k and see whether there is any remainder. If the remainder is zero, then n is divisible by k. However, there are several useful rules for testing for divisibility for certain small numbers. These are summarized in Table A.1. Table A.1 Rules for Testing Divisibility Number 2 3 4 5 6 8 9 10

Is Divisible by a Number N if . . . N is even; that is, its last digit is 2, 4, 6, 8, or 0. The sum of the digits of N is divisible by 3. The last two digits form a number divisible by 4. The number’s last digit is 5 or 0. The number is divisible by 2 and 3. The last three digits form a number divisible by 8. The sum of the digits of N is divisible by 9. The number’s last digit is 0.

For example, consider the number 7,380. It is divisible by all the numbers in the table except 8. Do you see why? To start, 7,380 is divisible by 10 and 5 because the last digit is 0. It is divisible by 2 because it is even, and by 4 because 80 is divisible by 4. However, it is not divisible by 8 because 380 isn’t. In addition, the sum of its digits is 18, which is divisible by 3 and by 9. Since it is divisible by both 2 and 3, it is also divisible by 6.

Example

Which numbers in the following list are divisible by 3, 4, and 5 but not by 9? 15,840 20,085 23,096 53,700 79,130

Solution

The easiest thing to look for is divisibility by 5. Is the last digit 5 or 0? By inspection, we eliminate 23,096, whose last digit is 6. We want the number to be divisible by 4, which means it must be even and its last two digits must form a number divisible by 4. That knocks out the one ending in 5 (which is odd), and 79,130, because 30 is not divisible by 4. This leaves 15,840 and 53,700. The digits of 15,840 add up to 18, while those of 53,700 total 15. Both are divisible by 3, but 15,840 is also divisible by 9. Therefore, only 53,700 meets all the conditions.

TOPIC 4. DIVISIBILITY

IN ADDITION, SUBTRACTION, AND MULTIPLICATION If you add or subtract two numbers that are both divisible by some number k, then the new number formed will also be divisible by k. Thus, 28 and 16 are both divisible by 4. If you take either their sum, 44, or their difference, 12, they too are divisible by 4.

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APPENDIX A

If you multiply two numbers together, any number that divides either one divides the product. If j divides M and k divides N, then jk divides MN. If both have a common divisor, then the product is divisible by the square of that number. Thus, 21 3 15 5 315 is divisible by 7, because 7 divides 21, and by 5, because 5 divides 15. It is also divisible by 35 5 5 3 7, and by 9, because 9 5 32, and 3 divides both 21 and 15!

Example 1

If m is divisible by 5, what is the largest number that divides 5m 1 25?

Solution

Since m is divisible by 5, 5m can be divided by 25. Therefore, the sum 5m 1 25 can be divided by 25.

Example 2

If a and b are whole numbers, and 3a 5 2b, which of the following must be true? I. II. III. (A) (B) (C) (D) (E)

Solution

a is divisible by 2 b is divisible by 2 b is divisible by 3 I only II only III only I and II only I and III only

If 3a equals 2b, then 3a must be divisible by 2, which means a must be divisible by 2, since 3 is not. Similarly, 2b must be divisible by 3, which means b must be divisible by 3, since 2 is not. Thus, the correct answer is (E)—both I and III must be true. Notice that II need not be true; since b 5 3, a 5 2 is a perfectly satisfactory solution.

TOPIC 5. EVEN NUMBERS

AND ODD NUMBERS Even numbers are those that are divisible by 2: 0, 2, 4, 6, . . . . Odd numbers are those that are not divisible by 2: 1, 3, 5, . . . . Certain simple results follow from these definitions, and they can be very useful. They are as follows: • If you add or subtract two even numbers, the result is even. • If you add or subtract two odd numbers, the result is even. • Only when you add or subtract an odd number and an even number is the result odd. Thus, 4 1 6 is even, as is 7 2 3. But 4 1 3 is odd. • When you multiply any whole number by an even number, the result is even. • Only when you multiply two odd numbers will the result be odd. Again, (4)(6) and (4)(7) are both even, but (3)(7) is odd.

Example 1

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If 3x 1 4y is an odd number, is x odd or even—or can’t you tell?

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Solution

4y must be even, so for the sum of 3x and 4y to be odd, 3x must be odd. Since 3 is odd, 3x will be odd only if x is odd. Hence, x is odd.

Example 2

If 121 2 5k is divisible by 3, can k be odd?

Solution

The fact that a number is divisible by 3 does not make it odd. (Think of 6 or 12.) Therefore, 121 2 5k could be either odd or even. It will be odd when k is even and even when k is odd. (Do you see why?) Thus, k could be odd or even. For example, if k 5 2, 121 2 5k 5 111, which is divisible by 3; if k 5 5, 121 2 5k 5 96, which is divisible by 3.

TOPIC 6. COMPARING FRACTIONS a c 3 9 and are defined to be equal if ad 5 bc. For example, 5 b d 4 12 because (3)(12) 5 (4)(9). This definition, using cross-multiplication, is very useful in solving algebraic equations involving fractions. However, for working with numbers, the important thing to remember is that multiplying the numerator (top) and the denominator (bottom) of a fraction by the same number (other than zero) results in a fraction equal in value to the original fraction. Thus, by 3 ~3!~3! 9 multiplying the numerator and the denominator by 3, we have 5 5 . 4 ~3!~4! 12 Similarly, dividing the numerator and denominator of a fraction by the same number (other than zero) results in a fraction equal in value to the original fraction. It is usual to divide through the numerator and the denominator by the greatest common factor of both the numerator and the denominator to “simplify the fraction to simplist form.” Thus, by dividing numerator and denominator by 5, 15 15 4 5 3 we have 5 5 . 25 25 4 5 5 For positive numbers, if two fractions have the same denominator, the one with the greater numerator is greater. If two fractions have the same numerator, 5 is less the one with the least denominator is the greater one. For example, 19 8 8 8 than , but is greater than . 19 17 19

Two fractions

Example 1

If b and c are both positive whole numbers greater than 1, and

5 b 5 , what are c 3

b and c?

Solution

Using cross-multiplication, bc 5 15. The only ways 15 can be the product of two positive integers is as (1)(15) or (3)(5). Since both b and c must be greater than 1, one must be 3 and the other 5. Trying both cases, it is easy to see that the only possibility is that b 5 3 and c 5 5, and both fractions are equal to 1.

Example 2

Which is greater,

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APPENDIX A

Solution

The first fraction has a greater numerator, but it also has a greater denominator, so we rewrite both with the common denominator 35 by multiplying the numera4 3 tor and denominator of by 5 and the numerator and denominator of by 7, to 7 5 20 21 3 yield and , respectively. Now it is easy to see that is the greater fraction. 35 35 5

Example 3

Which is larger,

Solution

First of all, it does not matter where you put the negative sign—numerator, denominator, or opposite the fraction bar; if there is one negative sign, the fraction is negative. Remember, in comparing negative numbers, the one with the greater absolute value is the least number. So start by ignoring the signs, and compare the absolute values of the fractions. If the two fractions had a common denominator (or numerator), it would be easy. So multiply the numerator and 6 12 13 denominator of by 2 to yield and it is easy to see that is the greater 11 22 22 26 fraction. Hence, is the greater number. 11

26 13 or ? 11 222

TOPIC 7. MULTIPLICATION

AND DIVISION OF FRACTIONS When multiplying two fractions, the result is the product of the numerators divided by 3 2 6 a c ac the product of the denominators. In symbols, z 5 . Thus, z 5 . Don’t forget b d bd 7 5 35 that the resulting fraction can be simplified to simplest form by dividing common 3 10 2 5 . factors in both the numerator and denominator, such as z 5 9 3 3 2 of what Sidney earns, and Sidney earns of what Paul earns. 4 3 What fraction of Paul’s salary does Jasmine earn?

Example 1

Jasmine earns

Solution

Using J, S, and P to stand for the people’s earnings respectively, we have: 3 2 S 5 P; J 5 S 3 4

S DS D

SD

3 3 4 12 4 Thus, P 5 S, and S 5 J. So P 5 J; P 5 J, which means that P is 2 3 2 3 6 twice J. Hence, Jasmine’s earnings are one-half Paul’s. When dividing fractions, simply multiply by the reciprocal of the divisor. Remember, the divisor is the number you are dividing by (usually the second one named), or the denominator in a “built-up” fraction. In symbols: a c a d ad 4 5 z 5 b d b c bc

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or: a b c d

5

a d ad z 5 b c bc

For example: 3 4 3 11 33 4 5 z 5 5 11 5 4 20 3 as many pairs 5 of jeans as Alfred. What fraction of Marco’s number of pairs does Patty have?

Example 2

Patty has half as many pairs of jeans as Alfred has, and Marco has

Solution

Using P, A, and M to stand for the number of jeans each person owns respectively, we have 1 3 P 5 A; M 5 A 2 5 thus, 1 A P 2 1 5 5 5 5 z 5 M 3 2 3 6 A 5 Thus, Patty has

5 as many pairs as Marco. 6

TOPIC 8. ADDITION

AND SUBTRACTION OF FRACTIONS To add (or subtract) fractions with the same denominator, simply add (or sub3 8 5 3 2 5 1 5 and 2 5 . However, if tract) the numerators. For example, 17 17 17 17 17 17 the denominators are different, you must first rewrite the fractions so they will have the same denominator. That is, you must find a common denominator. Most books stress that you should use the least common denominator (LCD), which is the least common multiple (LCM) of the original denominators. This will keep the numbers smaller. However, any common denominator will do! To find the least common denominator, you must first understand what a least common multiple is. Given two numbers M and N, any number that is divisible by both is called a common multiple of M and N. The least common multiple (LCM) of the two numbers is the least number that is divisible by both. For example, 108 is divisible by both 9 and 12, so 108 is a common multiple—but the LCM is 36.

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APPENDIX A

For small numbers, the easiest way to find the LCM is to mentally, or in writing, list the multiples of each until you find the first common (or shared) multiple. For example, for 9 and 12 we have: 9 18 27 36 45 . . . 12 24 36 48 60 . . . We see that the first number that appears in both lists is 36. However, the traditional method, which is really the method that translates most readily into algebra, requires that you find the prime factorization of the numbers. Every whole number is either prime or composite. A prime is a whole number greater than 1 which has exactly two factors (divisors), namely 1 and the number itself. All numbers that are not prime are composite. Every composite number can be factored into primes in a unique way. To find an LCM by the method of prime factorization, you must find the least number that contains all the factors of both numbers. Thus, 9 factors as (3)(3), and 12 factors as (2)(2)(3). The least number to use all the prime factors of both has to have factors (3)(3)(2)(2) 5 36. This process can also extend to sets of more than two numbers. Thus, the LCM of 12, 15, and 20 must contain all the prime factors of all three numbers: (2)(2)(3), (3)(5), (2)(2)(5). That is, (2)(2)(3)(5) 5 60.

Example 1

Find the LCM for 18 and 30.

Solution

Using prime factorization, 18 5 (2)(3)(3); 30 5 (2)(3)(5). Since the factors of 2 and 3 are common to both numbers, we need only multiply in one extra 3 to get the factors of 18, and a 5 to get the factors of 30. Thus, the LCM 5 (2)(3)(3)(5) 5 90.

Example 2

Mario figures that, working every day, he could finish a certain task in 20 days. Angelo figures that, working every day, he could finish the same task in 25 days. What fraction of the task could they get done by working on it together for seven days?

Solution

In 7 days, Mario would do

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7 of the entire task. In the same week, Angelo would 20 7 7 7 do of the entire task. Therefore, together they do 1 of the entire task. 25 20 25 To add these two fractions, we must find a common denominator. Find the 28 63 35 1 5 5 0.63 or 63%. LCD, which is 100. Thus, 100 100 100 Do you understand those last two equalities? If not, you should read the next section carefully.

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TOPIC 9. FRACTIONS, DECIMALS,

AND PERCENTS Every fraction can be expressed as a decimal, which can be found by division. Those fractions for which the prime factorization of the denominator involves only 2s and 5s will have terminating decimal expansions. All others will have 3 3 repeating decimal expansions. For example, 5 0.15, and 5 0.27272727 . . . 20 11 To rename a number given as a decimal as a fraction, you must know what the decimal means. In general, the decimal represents a fraction with denominator 10, or 100, or 1000, . . . , where the number of zeros is equal to the number of digits to the right of the decimal point. Thus, for example: 0.4 means

4 2 5 ; 10 5

0.52 means

13 52 5 ; and 100 25

0.103 means

103 . 1000

Decimals of the form 3.25 are equivalent to mixed numbers, thus, 3.25 5 3 1

25 100

1 1 5 3 . For purposes of addition and subtraction, mixed numbers can be use4 4 ful, but for purposes of multiplication or division, it is usually better to rename a mixed number as an improper fraction (one whose numerator is larger than its de43 3 nominator). That is, 2 5 . 20 20 2 How did we do that? Formally, we realize that 2 5 , and we add the two 1 2 3 fractions and using the common denominator 20. More simply, we multiply 1 20 the whole number part (2) by the common denominator (20), and add the numerator of the fraction (3) to get the numerator of the resulting improper fraction. That is, (2)(20) 1 3 5 43. 531

0.56 a simplified to simplest form is , where a and b are positive whole num1.26 b bers, what is b?

Example 1

If

Solution

Rewriting both numerator and denominator as their fractional equivalents, 0.56 5 56 14 26 13 63 5 , and 1.26 5 1 1 5 1 5 . We now accomplish the division by 100 25 100 50 50 14 50 4 multiplying by the reciprocal of the divisor. Thus, 5 and b 5 9. 25 63 9 Of course, you could also do this example by renaming the numerator and denominator of the original fraction as whole numbers. That is, multiply numerator and denominator by 100 to move both decimal points two places to the right,

S DS D

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APPENDIX A

0.56 56 5 . Now you can divide out the common factor of 14 in the 1.26 126 4 numerator and the denominator to simplify the fraction to . 9 Notice that by using long division or dividing on a calculator you will find 4 0.56 5 0.4444444 . . . , which you might recognize as . that 1.26 9 Remember, per cent means per hundred (from Latin centum 5 hundred). So, 30 , or, as a decimal, 0.30. for example, 30% means 30 per hundred, or, as a fraction, 100

thus:

Example 2

In a group of 20 English majors and 30 history majors, 50% of the English majors and 20% of the history majors have NOT taken a college math course. What percent of the entire group have taken a college math course?

Solution

Start with English. Since 50% 5 0.50, 50% of 20 5 (0.50)(20) 5 10. For history, 20% 5 0.20, 20% of 30 5 (0.20)(30) 5 6. Hence, a total of 16 out of 50 people in the group have not taken math, which means that 34 have. As a fraction, 34 out 34 of 50 is 5 0.68 5 68%. 50

ALGEBRA

TOPIC 10. ADDITION

AND SUBTRACTION OF SIGNED NUMBERS To add two numbers of the same sign, just add them and attach their common sign. So 7 1 9 5 16, and (27) 1 (29) 5 216. You could drop the parentheses and instead of (27) 1 (29) you could write 27 2 9, which means the same thing. In other words, adding a negative number is the same thing as subtracting a positive number! When adding numbers of opposite sign, temporarily ignore the signs, subtract the lesser from the greater, and attach to the result the sign of the number with the greater absolute value. Thus, 9 1 (23) 5 6, but (29) 1 3 5 26. Again, we could have written 9 1 (23) 5 9 2 3 5 6, and (29) 1 3 5 29 1 3 5 26. When subtracting, change the sign of the “second” number (the subtrahend), and then use the rules for addition. Thus, 7 2 (23) 5 7 1 3 5 10, and 27 2 3 5 27 1 (23) 5 210.

Example

Evaluate 2A 2 (2B) when A 5 25 and B 5 26.

Solution

All the negative signs can be confusing. However, if you remember that “minus a negative is a plus”, you can do this in two ways. The first is to realize that, if B 5 26, then 2B 5 16, and if A 5 25, then 2A 5 5 Thus, 2A 2 (2B) 5 5 2 6 5 21. Alternatively, you can work with the variables first: 2A 2 (2B) 5 2A 1 B 5 2(25) 1 (26) 5 5 2 6 5 21.

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TOPIC 11. MULTIPLICATION

AND DIVISION OF SIGNED NUMBERS If you multiply two numbers with the same sign, the result is positive. If you multiply two numbers with opposite signs, the result is negative. Furthermore, the exact same rule holds for division. Thus, (24)(23) 5 112, and (24)(3) 5 212. For division, it doesn’t matter which is negative and which positive, thus (26) 4 (2) 5 23 and (6) 4 (22) 5 23, but (26) 4 (22) 5 13. This also means that if you have a string of multiplications and divisions to do, if the number of negative factors is even, the result will be positive; if the number of negative factors is odd, the result will be negative. Of course, if even one factor is zero, the result is zero, and if even one factor in the denominator (divisor) is zero, the result is undefined.

Example 1

If A 5 (234,906 2 457,219)(35)(2618) and B 5 (22356) (289,021)(23125), which is greater, A or B?

Solution

Don’t actually do the arithmetic! 457,219 is greater than 234,906, so the difference is a negative number. Now, you see that A is the product of two negative numbers and a positive number, which makes the result positive. B is the product of three negative numbers and must be negative. Any positive number is greater than any negative number, and so A is greater than B.

Example 2

AB is a negative number, and N is negative, which of the following are MN possible? If

I. II. III. (A) (B) (C) (D) (E)

Solution

A is positive, but B and M are negative. A, B, and M are all negative. A, B, and M are positive. I only II only I and II only I and III only II and III only

To determine the sign of the fraction, just think of A, B, M, N as four factors. Knowing that N is negative, the product of the other three must be positive in order that the result be negative. The only possibilities are that all three are positive, or one is positive and the other two are negative. This corresponds to cases I and III. Thus, the correct answer is (D).

TOPIC 12. LAWS

OF EXPONENTS In an expression of the form bn, b is called the base and n is called the exponent or power. We say, “b is raised to the power n.” Notice, b1 5 b, and hence the power 1 is usually omitted.

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APPENDIX A

If n is any positive integer, then bn is the product of n number of b’s. For example, 43 is the product of three 4s, that is, 43 5 4 3 4 3 4 5 64. Certain rules for operations with exponents are forced upon us by this definition. They are: • bm 3 bn 5 bm 1 n. That is, when multiplying powers of the same base, keep the base and add the exponents. Thus, 32 3 33 5 32 1 3 5 35 5 243. • (ab)(n) 5 anbn and

SD a b

n

5

an . bn

That is, to raise a product or a quotient to a power, raise each factor to that power, whether that factor is in the numerator or denominator. Thus, (2x)3 5 23x3 5 8x3 and

SD 2 x

3

5

23 8 35 3 x x

• (bm)n 5 bnm. That is, to raise a power to a power, retain the base and multiply exponents. Thus, (23)2 5 26 5 64. •

bn bn 1 n2m 5 b if n . m, and m m 5 m2n if n , m. b b b That is, to divide powers with the same base, retain the base and subtract exponents. For example, 45 5 43 5 64 42 and 42 1 1 55 35 4 4 64

TOPIC 13. ZERO

AND NEGATIVE EXPONENTS For various technical reasons, x0 5 1 for all x except x 5 0, in which case it is undefined. With this definition, one can define b2n in such a way that all the laws of exponents given above still work even for negative powers! The definition is 1 one that you should not only know but know how to use. It is: b2n 5 n. Now b x3 1 you have the choice of writing 5 as 2 or as x22. x x

Example Solution

Which is greater, 1.10 or x0 1 x24, if x 5 2? If x 5 2, x0 5 20 5 1; and x24 5 224 5

1 1 5 0.0625. Hence, x0 1 x24 5 45 2 16

1.0625, which is less than 1.10.

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TOPIC 14. EVEN POWERS

AND ODD POWERS Even powers of real numbers cannot be negative. Thus, x2 is positive, except for x 5 0, when it is zero. Note that 232 means 2(32) 5 29. If you want the square of 23, which equals 19, you must write it (23)2. Odd powers are positive or negative depending upon whether the base is positive or negative. Thus, 23 5 8, but (22)3 5 28. Zero to any power is zero, except that zero to the zero is undefined.

Example 1

If x , 0 and y . 0, what is the sign of 24x4y3?

Solution

x4 is positive, because it has an even power. y3 is positive because y is, and 24 is obviously negative. The product of two positives and a negative is negative. Thus, 24x4y3 is negative.

Example 2

If x4 1 3y2 5 0, what is the sign of 2x 2 6y 1 1?

Solution

Since neither x4 nor 3y2 can be negative, the only way their sum can be zero is if both x and y are zero. Therefore, 2x 2 6y 1 1 5 11, which is positive.

TOPIC 15. AVERAGES There are three common measurements used to define the predominant value— loosely thought of as the average—of a collection of numbers. However, when you see the word average with no other explanation, it is assumed that what is meant is the arithmetic mean. The average in this sense is the sum of the T numbers divided by the number of numbers in the collection. In symbols, A 5 . n So, for example, if on four math exams you scored 82, 76, 87, and 89, your average math score at this point is (82 1 76 1 87 1 89) 4 4 5 334 4 4 5 83.5.

Example 1

At an art show, Eleanor sold six of her paintings at an average price of $70. At the next show, she sold four paintings at an average price of $100. What was the overall average price of the ten paintings?

Solution

You can’t just say the answer is 85, the average of 70 and 100, because we do not have the same number of paintings in each group. We need to know the overall total. Since the first six paintings sold for an average price of $70, the total received T for the 6 was $420. Do you see why? 70 5 ; this means T 5 (6)(70) 5 420. In the 6 same way, the next 4 paintings must have brought in $400 in order to average $100 apiece. Therefore, we have a total of 10 paintings selling for $420 1 $400 5 $820, $820 5 $82. and the average is 10

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APPENDIX A

Example 2

Erica averaged 76 on her first four French exams. To get a grade of B for the course, she must have an 80 average. What grade must she get on the next exam to bring her average to 80?

Solution

If Erica’s average is 76 on four exams, she must have a total of (4)(76) 5 304. In order to average 80 on five exams, her total must be (5)(80) 5 400. Therefore, she must score 400 2 304 5 96 on her last exam. The other two quantities that are sometimes used in ways similar to the average are the median, or the middle number when the numbers are arranged in increasing order, and the mode, the most common number.

Example 3

Which is greater, the mean minus the median or the median minus the mode, for the set of nine integers {1, 2, 2, 2, 3, 5, 6, 7, 8}?

Solution

The median (middle number) is 3, the mode is 2, and the mean is (1 1 2 1 2 1 2 1 3 1 5 1 6 1 7 1 8) 4 9 5 4. Thus, the mean minus the median is 4 2 3 5 1. And the median minus the mode is 3 2 2 5 1. Thus, the two quantities are equal.

TOPIC 16. MEASURES

OF DISPERSION In analyzing data, in addition to trying to measure average value, it is also important to measure the dispersion or “spread” of the numbers. The two sets A 5 {11, 12, 12, 12, 13} and B 5 {1, 2, 12, 17, 28}, both have the same average (12), but they have considerably different spreads. The amount of dispersion clearly has a major impact on how one interprets the significance of the data. The simplest way to measure the dispersion is to look at the range. The range is the difference between the greatest and least values in the set. Thus, for set A, the range is 13 2 11 5 2, while for B it is 28 2 1 5 27. A more complicated and more widely used measure is the standard deviation, which is the square root of the average squared deviation from the mean. To compute the standard deviation, follow these steps: 1.

Find the mean.

2.

For each data point, subtract the mean from the data value and square the result.

3.

Average the squares you have just found.

4.

Take the square root of the result.

Example

Find the standard deviations for sets A and B above.

Solution

For set A, the average is 12. For the five values, the differences when you subtract 12 from each are 21, 0, 0, 0, and 1. Their squares are 1, 0, 0, 0, and 1. The average of these numbers is 2 4 5 5 0.4, and the standard deviation is about 0.63 (since the square root of 0.4 is close to 0.63).

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For set B, the average is also 12, but the differences when you subtract 12 from each data point are 211, 210, 0, 5, and 16. Their squares are 121, 100, 0, 25, and 256. The average of these numbers is 502 4 5 5 100.4, and the standard deviation is around 10.02. Quite a difference!

TOPIC 17. RATIO

AND PROPORTION A fractional relationship between two quantities is frequently expressed as a ratio. It can be written as a fraction or in the form b : a (read “b is to a”). A proportion is a statement that two ratios are equal. To say, for example, that the ratio of P passing to failing students in a class is 5 : 2 means that if we set up the fraction F 5 it should simplify to . If we write this statement as P : F :: 5 : 2, we read it “P is 2 P 5 to F as 5 is to 2,” and it means 5 . F 2 Often, a good way to work with information given in ratio form is to represent the actual numbers as multiples of the same number.

Example

The ratio of Democrats to Republicans in a state legislature is 5 : 7. If the Legislature has 156 members, all of whom are either Democrats or Republicans (but not both), what is the difference between the number of Republicans and the number of Democrats? (A) (B) (C) (D) (E)

Solution

14 26 35 37 46

Let the number of Democrats be 5m and the number of Republicans be 7m, so that D : R :: 5m : 7m 5 5 : 7. The total number of members is 5m 1 7m 5 12m, which must be 156. Therefore, 12m 5 156, and m 5 13. Of course, the difference is 7m 2 5m 5 2m 5 2(13) 5 26. Hence, the answer is choice (B).

TOPIC 18. SOLVING LINEAR EQUATIONS To solve a linear equation, remember these rules: • You can add or subtract the same quantity from both sides of an equation and the equation will still be true and have the same roots (i.e., possible values). • You can multiply or divide both sides of an equation by any number except zero and the equation will still be true and have the same roots. Use these two properties to isolate the unknown quantity on one side of the equation, leaving only known quantities on the other side. This is known as solving for the unknown.

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APPENDIX A

Example 1

If 14 5 3x 2 1 and B 5 6x 1 4, what is the numerical value of B?

Solution

From the first equation, 3x 2 1 5 14. Adding 1 to both sides: 3x 2 1 5 14 151 5 15

3x

Dividing both sides by 3: x 5 5. Of course, the question asked for B, not x. So we substitute x 5 5 into B 5 6x 1 4 and get B 5 6(5) 1 4 5 34. 2x 1 2 5 a, and y 5 2x 1 6, find an expression for y in terms of a. 3

Example 2

If

Solution

How do we solve this? We realize that, if we knew what x was in terms of a, then we could substitute that expression for x into y 5 2x 1 6 and have y in terms of a. 2x In other words, we want to solve 1 2 5 a for x. 3 Multiply both sides by 3 to clear the fractions. Be careful: when you multiply by 3, be sure to use the distributive law and multiply every term on both sides by 3. You should now have 2x 1 6 5 3a. Now add 26 to both sides of the equation: 2x 1 6 5 3a 26 5 26 5 3a 2 6

2x

Now divide by 2: x5

3a 23 2

Substituting: y 5 3a 2 6 1 6 y 5 3a

TOPIC 19. SOLVING LINEAR INEQUALITIES If a number M is less than another number N, this means that N 2 M is positive. That is, when you subtract a lesser number from a greater number, the result is positive. In symbols, M , N or N . M. Notice that the “sense arrow” always points towards the lesser number. Another way of saying this is to note that, on the number line, M lies to the left of N. This means, in particular, that any negative number is less than any positive number. It also implies that, for negative numbers, the one with the greater absolute value is the lesser number.

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To solve linear inequalities (also called inequations), remember these rules: • You can add or subtract the same quantity from both sides of an inequation, and the inequation will still be true in the same sense. Thus, 14 . 7, and 14 2 5 . 7 2 5. • You can multiply or divide both sides of an inequation by any positive number and the inequation will still be true in the same sense. Thus, 3 , 8 and (6)(3) , (6)(8). • You can multiply or divide both sides of an inequation by any negative number and the inequation will still be true, but with the sense reversed. Thus, 4 , 9; but if you multiply by (–2), you get 28 . 218. Remember, for negative numbers, the one with the greater absolute value is the lesser number. Notice that these rules hold whether you are working with , (is less than) and . (is greater than) or ≤ (is less than or equal to) and ≥ (is greater than or equal to). You can use these rules to isolate the unknown quantity on one side of the inequality, leaving only known quantities on the other side. This is known as solving for the unknown.

Example 1

For what values of x is 12 2 x ≥ 3x 1 8?

Solution

We solve this just like an equation. Start by adding the like quantity (x 2 8) to both sides in order to group the x terms on one side and the constants on the other; thus: 12 2 x ≥ 3x 1 8 x285x28 4 ≥ 4x Now divide by 4, which does not change the sense of the inequality, yielding: 1≥x Hence, the inequality will be true for any value of x less than or equal to 1 and false for any number greater than 1. For example, for x 5 3, 12 2 x 5 9, and 3x 1 8 5 17, and the inequality is not satisfied.

Example 2

If A , 2 2 4B, can you tell how large B is in terms of A? Can you tell how small B is?

Solution

We are really being asked to solve the inequality for B. To start, we add 22 to both sides, thus: A

, 2 2 4B 22 5 22

A 2 2 , 24B

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APPENDIX A

Next, divide by 24, remembering to reverse the inequality, thus: ~A 2 2! .B 24 ~2 2 A! .B 4 Notice two things here. When we changed the denominator on the left side from 24 to 14, we also changed the sign of the numerator by changing (A 2 2) to (2 2 A). Of course, what we really did, in effect, was to multiply numerator and denominator by 21. Also, this tells us what B is less than but tells us nothing about what B might be greater than. For example, if A were 6, then B , 21, but B could be 2100 or 21000 or anything more negative.

TOPIC 20. SOLVING TWO LINEAR EQUATIONS

IN TWO UNKNOWNS Many word problems lead to equations in two unknowns. Usually, one needs as many equations as there are unknowns to solve for all or some of the unknowns, but there are exceptions. You should know two methods for solving two equations in two unknowns. They are the method of substitution and the method of elimination by addition and subtraction. We shall illustrate both methods by example. The first example uses the method of substitution.

Example 1

Mrs. Green and her three children went to the local movie. The total cost of their admission tickets was $14. Mr. and Mrs. Arkwright and their five children went to the same movie and had to pay $25. What was the cost of an adult ticket and what was the cost of a child’s ticket?

Solution

Expressing all amounts in dollars, let x 5 cost of an adult ticket, and let y 5 cost of a child’s ticket. For the Greens: x 1 3y 5 14 For the Arkwrights: 2x 1 5y 5 25 The idea of the method of substitution is to solve one equation for one variable in terms of the other, and then substitute this solution into the second equation. So we solve the first equation for x, because that is the simpler unknown to isolate: x 5 14 2 3y And substitute this solution into the second equation: 2(14 2 3y) 1 5y 5 25 This gives us one equation in one unknown that we can solve:

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28 2 6y 15y 5 25 2y 5 23 y53 Now that we know that y 5 3, we put this into x 5 14 2 3y to get: x 5 14 2 3(3) 5 5 Thus, the adult tickets were $5 each, and the children’s tickets were $3 each. Following is an example using the method of elimination.

Example 2

Paula and Dennis both went to the bakery. Paula bought 3 rolls and 5 muffins for a total cost of $3.55. Dennis bought 6 rolls and 2 muffins for a total cost of $3.10. What is the price of one roll?

Solution

Let us express all amounts in cents. Let r 5 cost of a roll; let m 5 cost of a muffin. Paula paid 3r 1 5m 5 355; Dennis paid 6r 1 2m 5 310. The idea of the method of elimination is that adding equal quantities to equal quantities gives a true result. So we want to add some multiple of one equation to the other one so that, when we sum the two equations, one variable will be eliminated. In this case, it is not hard to see that if we multiply the first equation by 22, the coefficient of r will become 26. Now, if we add the two equations, r will drop out. 22 times the first equation is: 26r 2 10m 5 2710 The second equation is: 6r 1 2m 5 310 Adding the two equations: 28m 5 2400 Dividing by 28: m 5 50. We now substitute this value into either of the two equations. Let’s use the second: 6r 1 ~2!~50! 5 310 6r 5 210 r 5 35 Thus, muffins are 50¢ each and rolls are 35¢.

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APPENDIX A

TOPIC 21. WORD PROBLEMS

IN ONE OR TWO UNKNOWNS There are word problems of many different types. Some require special knowledge. Others, like age or coin problems, involve only common sense. For example, for consecutive integer problems, you need to remember simply that consecutive integers differ by 1, so a string of such numbers can be represented as n, n 1 1, n 1 2, . . . . Consecutive even integers differ by 2, so a string of such numbers can be represented as n, n 1 2, n 1 4, . . . . Consecutive odd integers also differ by 2! So a string of such numbers can also be represented as n, n 1 2, n 1 4, . . . . Rate-time-distance problems require you to know and use the formula d 5 rt; that is, distance equals rate times time. Here are some examples of various kinds of word problems.

Example 1

Sally is 6 years older than Manuel. Three years ago, Sally was twice as old as Manuel. How old is Sally today?

Solution

If you have trouble setting up the equations, try plugging in numbers. Suppose that Sally is 20. If Sally is 6 years older than Manuel, how old is Manuel? He is 14. You get from 14 to 20 by adding 6. So if S is Sally’s age and M is Manuel’s, S 5 M 1 6. Three years ago, Sally was S 2 3, and Manuel was M 2 3. So, from the second sentence, we see that S 2 3 5 2(M 2 3), or S 2 3 5 2M 2 6, or S 5 2M 2 3. Now, substituting S 5 M 1 6, M 1 6 5 2M 2 3, and M 5 9; which means Sally is 9 1 6 5 15.

Example 2

Three consecutive odd integers are written in increasing order. If the sum of the first and second integers and twice the third integer is 46, what is the second integer?

Solution

Calling the smallest number x, the second is x 1 2, and the third is x 1 4. Therefore: x 1 ~x 1 2! 1 2~x 1 4! 5 46 x 1 x 1 2 1 2x 1 8 5 46 4x 1 10 5 46 4x 5 36 x59 Hence, the middle number is 9 1 2 5 11.

Example 3

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1 It took Andrew 1 hours to drive from Aurora to Zalesville at an average speed of 2 50 miles per hour. How fast did he have to drive back in order to be home in 80 minutes?

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Solution

The distance from Aurora to Zalesville is given by d 5 rt 5 (50)(1.5) 5 75 miles. 4 1 Since 80 minutes is 1 hour and 20 minutes, or 1 hours, we must solve 75 5 r. 3 3 Multiplying by 3, we have 225 5 4r, and dividing by 4, we get r 5 56.25 miles per hour.

TOPIC 22. MONOMIALS

AND POLYNOMIALS When we add a collection of algebraic and arithmetic expressions, each expression is called a term. Monomial means one term. For example, we might say that 2x 1 3y2 1 7 is the sum of three terms or three monomials. Technically, if we enclose an algebraic expression in parentheses, it becomes one term, so that we could say that (x 1 2y) 1 (3x 2 5y2) is the sum of two monomials. But usually when we talk about a monomial, we mean a term that is just the product of constants and variables, possibly raised to various powers. Examples might be 7, 2x, 23y2, and 4x2z5. The constant factor is called the coefficient of the variable factor. Thus, in 23y2, 23 is the coefficient of y2. If we restrict our attention to monomials of the form Axn, the sums of such terms are called polynomials (in one variable). Polynomials with two terms are called binomials, and those with three terms are called trinomials. Expressions like 3x 1 5, 2x2 2 5x 1 8, and x4 2 7x5 2 11 are all examples of polynomials. The highest power of the variable that appears is called the degree of the polynomial. The three examples just given are of degrees 1, 2, and 5, respectively. In evaluating monomials and polynomials for negative values of the variable, the greatest pitfall is keeping track of the negative signs. Always remember that, in an expression like 2x2, the power 2 is applied to the x and the negative sign in front should be thought of as (21) times the expression. If you want to have the power apply to 2x, you must write (2x)2.

Example

Find the value of 3x 2 x3 2 x2, when x 5 22.

Solution

Substitute 22 every place you see an x, thus: 3(22) 2 (22)3 2 (22)2 5 26 2 (28) 2 (14) 5 26 1 8 2 4 5 22

TOPIC 23. COMBINING MONOMIALS Monomials with identical variable factors can be added or subtracted by adding or subtracting their coefficients. So 3x2 1 4x2 5 7x2, and 3x4 2 9x4 5 26x4. When you multiply monomials, take the product of their coefficients and take the product of the variable parts by adding exponents of factors with like bases. So we have (24xy2)(3x2y3) 5 212x3y5. Monomial fractions can be simplified to simplest form by dividing out common factors of the coefficients and then using the usual rules for subtraction of exponents in division.

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APPENDIX A

Example Solution

Combine into a single monomial 9y 2

~6y3! . ~2y2!

The fraction simplifies to 3y, and 9y 2 3y 5 6y.

TOPIC 24. COMBINING POLYNOMIALS

AND MONOMIALS Polynomials are added or subtracted by just combining like monomial terms in the appropriate manner. Thus, (2x2 1 5x 2 3) 1 (3x2 1 5x 2 12) is summed by removing the parentheses and combining like terms to yield 5x2 1 10x 2 15. If there is a negative sign in front of a polynomial within parentheses, be careful to change the signs of all the terms within the parentheses when you remove the parentheses. Consider: (2x2 1 5x 2 3) 2 (3x2 1 5x 2 12) 5 2x2 1 5x 2 3 2 3x2 2 5x 1 12 5 2x2 1 9 Did you notice that 2x2 2 3x2 5 21x2, but the “1” is not shown? To multiply a polynomial by a monomial, use the distributive law to multiply each term in the polynomial by the monomial factor. For example, 2x(2x2 1 5x 2 11) 5 4x3 1 10x2 2 22x. When multiplying a polynomial by a polynomial, you are actually repeatedly applying the distributive law to form all possible products of the terms in the first polynomial with the terms in the second. The most common use of this is in multiplying two binomials, such as (x 1 3)(x 2 5). In this case, there are four terms in the result: x 3 x 5 x2; x(25) 5 25x; 3 3 x 5 3x; and 3 3 (25) 5 215; but the two middle terms are added together to give 22x. Thus, the product is x2 2 2x 2 15. This process is usually remembered as the FOIL method. That is, form the products of First, Outer, Inner, Last, as shown in the figure below. O F

(x + 3)(x - 5) = x2 + (-5x + 3x) - 15 I L

Be sure to remember the special cases: (x 1 a)2 5 (x 1 a)(x 1 a) 5 x2 1 2ax 1 a2 And: (x 2 a)2 5 (x 2 a)(x 2 a) 5 x2 2 2ax 1 a2

Example

If m is an integer, and (x 2 6)(x 2 m) 5 x2 1 rx 1 18, what is the value of m 1 r?

Solution

The product of the last terms, 6m, must be 18. Therefore, m 5 3. If m 5 3, then the sum of the outer and inner products becomes 26x 2 3x 5 29x, which equals rx. Hence, r 5 29, and m 1 r 5 3 1 (29) 5 26.

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TOPIC 25. FACTORING MONOMIALS Factoring a monomial from a polynomial is simply reversing the distributive law. For example, if you are looking at 3x2 2 6xy, you should see that 3x is a factor of both terms. Hence, you could just as well write this as 3x(x 2 2y). Multiplication using the distributive law will restore the original formulation.

Example

If x 2 5y 5 12, which is greater, 15y 2 3x or 235?

Solution

We can see that 15y 2 3x 5 23(x 2 5y); hence, it must equal 23(12) 5 236, which is less than 235.

TOPIC 26. TRINOMIAL FACTORING

AND QUADRATIC EQUATIONS When you multiply two binomials (x 1 r)(x 1 s) using the FOIL method, the result is a trinomial of the form x2 1 bx 1 c, where b, the coefficient of x, is the sum of the constants r and s, and the constant term c is their product. Trinomial factoring is the process of reversing this multiplication. For example, to find the binomial factors of x2 2 2x 2 8, we need to find two numbers whose product is 28 and whose sum is 22. Since the product is negative, one of the numbers must be negative and the other positive. The possible factors of 8 are 1 and 8, and 2 and 4. In order for the sum to be 22, we must choose 24 and 12. Thus, x2 2 2x 2 8 5 (x 2 4)(x 1 2). This technique can sometimes be used to solve quadratic equations. If you have an equation like x2 2 7x 1 6 5 0 you can factor the trinomial. You need two numbers whose product is 16 and whose sum is 27. Since the product is positive, both numbers must have the same sign, and since the sum is negative, they must both be negative. It is not hard to see that 26 and 21 are the correct options. Thus, the equation becomes (x 2 1)(x 2 6) 5 0 The only way a product of numbers can be zero is if one of the numbers is zero. Thus, either: x2150 x51

or x 2 6 5 0 or

x56

Example

The area of a rectangle is 60 and its perimeter is 32. What are its dimensions?

Solution

As you probably know, the area of a rectangle is calculated by multiplying its length by its width (see Topic 36). Calling the dimensions L and W, we have LW 5 60, and 2L 1 2W 5 32. Dividing by 2: L 1 W 5 16. Therefore, L 5 16 2 W, which we substitute in LW 5 60, giving us:

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APPENDIX A

~16 2 W!W 5 60 16W 2 W2 5 60 Grouping everything on the right side, we have: 0 5 W2 2 16W 1 60 Now, factoring: 0 5 (W 2 10)(W 2 6) yields W 5 10 or W 5 6. Of course, if W 5 6, L 5 10, and if W 5 10, L 5 6. So, the dimensions are 6 3 10.

TOPIC 27. THE QUADRATIC FORMULA Some quadratic equations are not solvable by factoring using rational numbers. For example, x2 1 x 1 1 has no factors using whole numbers, so x2 1 x 1 1 5 0 has no rational roots (solutions.) In other cases, rational roots exist but they are difficult to find. For example, 12x2 1 x 2 6 5 0 can be solved by factoring, but it is not easy to see that: 12x2 1 x 2 6 5 (3x 2 2)(4x 1 3) Setting each factor equal to zero: 3x 2 2 5 0

or

4x 1 3 5 0

2 3 or x 5 2 . What can you do when faced with such a situation? You 3 4 use the quadratic formula: For any equation of the form ax2 1 bx 1 c 5 0, the 2b 6 =b2 2 4ac roots are x 5 2a yields x 5

TOPIC 28. THE DIFFERENCE

AND THE SUM OF TWO SQUARES When you multiply (a 2 b) by (a 1 b) by the FOIL method, the middle term exactly cancels out, leaving just a2 2 b2. Thus, the difference of two squares, a2 2 b2 5 (a 2 b)(a 1 b). For example, x2 2 16 can be thought of as x2 2 42 5 (x 2 4)(x 1 4). This makes it easy to find 1012 2 992 as (101 2 99)(101 1 99) 5 2(200) 5 400. However, binomials such as x2 1 16, which are the sum of two squares, cannot be factored.

Example 1

If x and y are positive integers, and x 2 2y 5 5, which of the following is the value of x2 2 4y2? (A) 0 (B) 14 (C) 45

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Solution

Since x2 2 4y2 5 (x 2 2y)(x 1 2y) 5 5(x 1 2y), x2 2 4y2 must be divisible by 5. Therefore, 14 is not possible. If the result is to be zero, x 1 2y 5 0, which means y 5 22x, so that both numbers cannot be positive. Hence, the expression must equal 45, which you get for x 5 7 and y 5 1.

Example 2

If x and y are positive integers, and y2 5 x2 1 7, find y.

Solution

If we rewrite the equation as y2 2 x2 5 7 and factor, we have (y 2 x)(y 1 x) 5 7. Thus, 7 must be the product of the two whole numbers, (y 2 x) and (y 1 x). But 7 is a prime number that can only be factored as 7 times 1. Of course, (y 1 x) must be the larger of the two, hence, y 1 x 5 7, and y 2 x 5 1. Adding the two equations gives us 2y 5 8; y 5 4. (Of course, x 5 3, but we weren’t asked that.)

TOPIC 29. OPERATIONS

WITH RADICALS The square root of a number N, written =N, is a number that when squared produces N. Thus, =4 5 2, =9 5 3, =16 5 4, and so on. You should be aware that =0 5 0 and =1 5 1. Square roots of negative numbers are not real numbers. The symbol = is called a radical, and many people refer to =N as “radical N.” When we write =N, it is understood to be a positive number. So when you are faced with an algebraic equation like x2 5 4, where you must allow for both positive and negative solutions, you must write x 5 6=4 5 62, where, as you know, 6 is read “positive or negative.” All positive numbers have square roots, but most are irrational numbers. Only numbers that are perfect squares like 4, 9, 16, 25, 36, . . . have integer square roots. If you assume that you are working with non-negative numbers, you can use certain properties of the square root to simplify radical expressions. The most important of these rules is =AB 5 =A z =B. This rule can be used to advantage in either direction. Reading it from right to left, we may write =3 z =12 5 =36. But you should also know how to use this rule to simplify radicals by extracting perfect squares from “under” the radical. Thus,

=18 5 =9 z 2 5 3=2 The key to using this technique is to recognize the perfect squares in order to factor them out in a sensible manner. Thus, it would do you little good to factor 18 as 3 3 6 in the preceding example, since neither 3 nor 6 is a perfect square.

Example

If

Solution

Since 10 5 =100 and =5 z =x 5 =5x, we know that 5x 5 100 and x 5 20. But 20 5 4 3 5, so =20 5 2=5. Hence, the two quantities are equal.

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=5 z =x 5 10, which is larger, =x or 2=5?

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APPENDIX A

GEOMETRY

TOPIC 30. ANGLES, COMPLEMENTS

AND SUPPLEMENTS An angle is formed when two rays originate from the same point. Angles are usually measured in degrees or radians. We shall use only degree measure, which is customary on the GRE.

C

x˚

B

A ABC = x˚

m

A straight angle has a degree measure of 180°. Any two angles that sum to a straight angle are called supplementary. Thus, 80° and 100° are supplementary. Two equal supplementary angles are 90° each, and a 90° angle is called a right angle. Two angles that sum to a right angle are called complementary. Thus, 25° is the complement of 65°. Angles less than 90° are called acute, and angles between 90° and 180° are called obtuse. The sum of all the angles around a given point must total to 360°. B

A

C

Straight Angle, m

ABC = 180˚

A

B

C

Right Angle, m

ABC = 90˚

z˚ a˚

a+b+c+...+z = 360

b˚ c˚ d˚

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Example 1

Find x in the figure below. D

x°+40 x°

A

Solution

B

C

Since ∠ABD is a right angle, so is ∠DBC. Thus, x 1 (x 1 40) 5 90. Removing parentheses: x 1 x 1 40 5 90; 2x 5 50; x 5 25

Example 2

Find x in the diagram below.

2xO O

3x

xO

2xO+ 48O

Solution

8x 1 48 5 360 8x 5 312 x 5 39

TOPIC 31. THE ANGLES

IN A TRIANGLE The sum of the measures of the three angles in any triangle is 180°, which is the measure of a straight angle. This fact is usually combined with other properties in the solution of geometric problems.

Example

In triangle ABC, m∠B is 30° more than twice m∠A, and the degree measure of ∠C is equal to the sum of the other two angles. How many degrees are there in the smallest angle of triangle ABC?

Solution

Calling the degree measure of ∠A x, we have the following: x 5 number of degrees in ∠A 2x 1 30 5 number of degrees in ∠B x 1 ~2x 1 30! 5 3x 1 30 5 number of degrees in ∠C

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APPENDIX A

Summing, we have x 1 2x 1 30 1 3x 1 30 5 180. Combining like terms: 6x 1 60 5 180 6x 5 120 x 5 20 Clearly, 2x 1 30 and 3x 1 30 are larger than x, so the smallest angle is 20°.

TOPIC 32. ISOSCELES

AND EQUILATERAL TRIANGLES A triangle with two sides of equal length is called an isosceles triangle. The angles opposite the equal sides (as shown in the following figure) are equal in measure, and if two angles are equal, then the triangle is isosceles. If all three sides are equal, it is called an equilateral triangle. In an equilateral triangle, each angle is 60°. B

Isosceles Triangle AB = BC

Equilateral Triangle AB = AC = BC

B 60°

x°

x°

A

60° C

A

60°

C

Here is a good example of how this fact can be used in a problem.

Example

If in triangle ABC as shown below, AC 5 BC and x ≤ 50, what is the smallest possible value of y? C x°

y° A

Solution

B

Since the sides AC and BC are of equal length, the two base angles, ∠A and ∠B, must also be equal. The three angles must total 180°. Hence, x 1 2y 5 180, 1 ~180 2 x! which means that y 5 x. The smallest possible value for y is 5 90 2 2 2 achieved when x is as large as possible; that is, when x 5 50, for which y 5 65.

SD

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TOPIC 33. OTHER TRIANGLE PROPERTIES In a triangle, the sum of the lengths of any two sides must exceed the third. Thus, you cannot draw a triangle with sides of lengths 3, 6, and 10, because 3 1 6 , 10. In addition, in comparing any two sides of a triangle, the longer side will be opposite the larger angle.

Example 1

A triangle has sides of 5, 12, and x. If x is an integer, what is the minimum possible perimeter of the triangle?

Solution

In order to form a triangle, the sum of any two sides must exceed the third. Therefore, x 1 5 . 12, which means that x . 7. The smallest integer greater than 7 is 8. Hence, the minimum possible perimeter is 5 1 12 1 8 5 25. (Can you see why the maximum length of the perimeter is 33?)

Example 2

In the triangle shown below, AB 5 BC. Which is longer, AC or AB? B

A

70O C

Note: Diagram not drawn to scale

Solution

Since the triangle is isosceles, the base angles are equal. Thus, m∠A 5 m∠C 5 70°. This implies that m∠B 5 40°, in order to reach the full 180° in the triangle. And that means that AB . AC, because it is the side opposite the larger angle.

TOPIC 34. VERTICAL ANGLES When two lines intersect, two pairs of vertical angles are formed (see the following figure). The “facing” pairs are equal and, of course, each pair on one side of the line add up to 180°.

y° x°

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x°

y°

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APPENDIX A

Example

In the diagram below, which is larger, x 1 y or w 1 z? B

yO O

A

x

wO

Solution

C

zO

We know that the sum of the angles in any triangle is 180°. Letting the measure of ∠ABC be m, we have in the upper triangle x 1 y 5 180 2 m. Similarly, looking at the larger triangle, we know that w 1 z 5 180 2 m. Therefore, x 1 y 5 w 1 z.

TOPIC 35. PARALLEL LINES

AND TRANSVERSALS If you start with two lines parallel to one another (that is, running in the same direction), and draw a line that crosses them, the crossing line is called a transversal. The intersections of the transversal with the parallel lines create several sets of related angles. In particular, the corresponding angles, labeled B in the diagram below, and the alternate interior angles, labeled A in the diagram, are equal. B

B

A A

Combining these properties with your knowledge about vertical angles and the angles in a triangle can lead to interesting examples.

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Example 1

In the diagram below, l1 is parallel to l2. Find x. 26°

x°

Solution

l

1

l

2

l

1

l

2

Label the diagram as shown below: D

A x°

C

E

26°

B

We see that m∠DCE 5 26°, which makes m∠ACB 5 26°. Since triangle ABC is a right triangle, x is the complement of 26°, or 64°.

Example 2

In the diagram shown below, l1 is parallel to l2. Find x. 32O

xO 66O

Solution

Extend the line AB, as shown in the following diagram. H

32O

D

C

E

G xO

B

66O F

A

Look at the angles in triangle BCD. As alternate interior angles, m∠BCE 5 m∠BAF 5 66°, so the supplement in the triangle, ∠C 5 114°. As vertical angles, m∠CDB 5 m∠HDG 5 32°. Therefore, in the triangle, m∠D 5 32°. Since the three angles in the triangle must sum to 180°, m∠B5 34°. x is the supplement to 34°, that is, 146°.

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APPENDIX A

TOPIC 36. RECTANGLES

AND PARALLELOGRAMS A parallelogram is a quadrilateral (a four-sided figure) in which the pairs of opposite sides are parallel. The opposite angles will be equal, and the opposite sides will be of equal length (see the figure below). x°

y°

H x°

y° L

The area of a parallelogram is calculated by multiplying the length times the height. That is, A 5 LH, as labeled in the diagram. If the angles in the parallelogram are right angles, we have a rectangle. For a rectangle of length L and width W, the area is A 5 LW, and the perimeter (the distance around) is P 5 2L 1 2W.

W

L

For example, the area of a rectangular garden that is 20 yards long and 10 yards deep is (20)(10) 5 200 square yards. However, to put a fence around the same garden requires 2(20) 1 2(10) 5 60 running yards of fencing (the perimeter of the rectangle). These relatively easy formulas can lead to some tricky questions.

Example 1

If sod comes in 4 3 4 foot squares costing $3.50 per square, how much will it cost to sod the lawn shown in the figure below? A

12

B 12 C

16

D

40

F

E

Distances are in feet. You may assume that all angles that appear to be right angles are right angles.

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Solution

Complete the rectangle as shown in the figure below. A

12

B

G

12 C

16

D

40

F

28

E

We see that the large rectangle AGEF is 40 3 28 5 1120 square feet. The smaller rectangle BGDC is 12 3 16 5 192 square feet. Hence, the area that must be sodded is 1120 2 192 5 928 square feet. Each 4 3 4 piece of sod is 16 square feet. Therefore, we need 928 4 16 5 58 squares of sod at $3.50 each. (58)(3.50) 5 $203.

Example 2

Rectangle 1 has an area of 64, and Rectangle 2 has an area of 16. Which rectangle has a larger perimeter?

Solution

You can’t tell! For Rectangle 1, LW 5 64. If it were a square, each side would be 8 and the perimeter would be 32, but the length could be any number greater 64 than zero if W is . Thus, you could make its perimeter (virtually) as large as you L wish! For example, the rectangle could be 64 3 1, with a perimeter of 130. For Rectangle 2, LW 5 16. If it were a square, each side would be 4 and the perimeter would be 16, but, again, the length could be any number greater than 16 zero if W is . For example, the rectangle could be 16 3 1, with a perimeter of L 34, which is less than 130 but greater than 32.

TOPIC 37. THE PYTHAGOREAN THEOREM The Pythagorean Theorem is probably the most famous geometric relationship. It tells us that the square on the hypotenuse (the longest side) of a right triangle is equal to the sum of the squares on the other two sides (or legs). In symbols, we usually remember this as:

c

a

c2 = a2 + b2

b

There are other important cases that yield non-integer solutions for the Pythagorean Theorem. For example, the hypotenuse of a triangle with one leg of length 1 and the other of length 2 can be found by computing c2 5 12 1 22, that is, c2 5 5 and c 5 =5.

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APPENDIX A

Example 1

Find x in the diagram below. B

x C

8 6

A

6

D

Solution

Using the Pythagorean Theorem, in triangle ACD, the theorem tells us that 62 1 62 5 c2. Hence, c2 5 72. In triangle ABC, if we let x represent the length of BC, 72 5 c2 5 x2 1 82. That is, x2 5 72 2 64 5 8. Thus, x 5 =8 5 2=2.

Example 2

A rectangle has one side with a length of 6 and a diagonal with a length of 10. What is its perimeter?

Solution

Notice that the diagonal of a rectangle divides the rectangle into two identical right triangles. Hence, the other side can be found by the Pythagorean Theorem. We recognize that a side of 6 and a diagonal of 10 imply that we have a 6–8–10 right triangle, so the unknown side is 8. The perimeter of the rectangle is, therefore, 2(6) 1 2(8) 5 28.

TOPIC 38. THE AREA

OF A TRIANGLE In any triangle, you can construct a line from any vertex perpendicular to the opposite side, although sometimes that side may have to be extended outside the triangle, as shown in the second illustration below. This line is called the altitude or height of the triangle; the side to which the altitude is drawn is called the 1 base. The area of a triangle is given by the formula A 5 bh, where b 5 length of 2 the base and h 5 the length of the altitude. A

A h=5

h=5 B

b = 12

C

B

b = 12

C

1 Both triangles shown above have the same area: A 5 ~12!~5! 5 30. 2

Example

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In triangle ABC, AB 5 6, BC 5 8, and AC 5 10. Find the altitude from vertex B to AC.

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Solution

Since the sides are 6–8–10, the triangle is a disguised 3–4–5 right triangle, with AC being the hypotenuse. Drawing the triangle (see the diagram below), we see that, by using the two legs as base and height, the area of the triangle must be 1 A 5 ~6!~8! 5 24. By using the hypotenuse and the unknown altitude, the area 2 1 must be A 5 ~10!~h! 5 5h. Therefore, 5h 5 24 and h 5 4.8. 2 B

8

6

h

A

10

C

TOPIC 39. SPECIAL RIGHT TRIANGLES There are two special right triangles whose properties you should be familiar with. The first is the isosceles right triangle, also referred to as the 45°–45°–90° triangle. By definition, its legs are of equal length, and its hypotenuse is =2 times as long as the leg. 45O

2

30O 2L

L

L

45O

L 3

60O

L

The other important right triangle is the 30°–60°–90° triangle. You can see by dropping an altitude that this is half of an equilateral triangle. Hence, the shorter leg is half the hypotenuse, and the longer leg (the one opposite the 60° angle) is =3 times the shorter leg.

Example

Find the area of the region shown in the diagram below. D C

3 45 o

A

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3 3

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APPENDIX A

Solution

Since BC 5 3 and AB 5 3 3 =3, we know that triangle ABC is a 30°–60°–90° right triangle. Hence, we know that AC 5 6, and taking half the product of the 9 1 legs, the triangle has an area of 3 3 3 3=3 5 =3. Since triangle ADC is 2 2 6 . Again, an isosceles right triangle with a hypotenuse of 6, each leg must be =2 1 6 taking half the product of the legs, the triangle has an area of 2 =2 6 9 18 5 5 9. Adding the two areas, we have 9 1 =3. 2 2 =2

SD

S D

SD

S DS D

TOPIC 40. OTHER POLYGONS Any geometric figure with straight line segments as sides is called a polygon. To find the perimeter of a polygon, simply add together the lengths of all the sides. Of course, it may require some thinking to figure out each length. To find the area of a polygon, connect the vertices by line segments to divide the polygon into triangles and sum the areas of these triangles.

Example

Find the area of figure ABCDE shown in the diagram below. 2

B

2

A

C

2

2 E

Solution

D

2 2

Drawing BE and BD divides the region into three triangles as shown in the diagram below. 2

B

2

A

C

2

2 E

2 2

D

Triangles ABE and BCD are both 45°–45°–90° right triangles, making BE 5 BD 5 2=2. This makes the central triangle an equilateral triangle. The area 1 of each of the two outer triangles is ~2!~2! 5 2; so the two together have area 4. 2 The center triangle has a base of 2=2. If you draw the altitude, you get a 30°–60°–90° right triangle with a shorter leg of =2, which makes the height 1 =3 times that, or =6. This gives an area of 2~2=2!~=6! 5 =12 5 2=3. Hence, the total area of the polygon is 4 1 2=3.

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TOPIC 41. BASIC PROPERTIES

OF CIRCLES A line segment from the center of a circle to any point on its circumference is called a radius. All radii of the same circle are equal in length. A line segment that passes through the center of the circle and cuts completely across the circle is called a diameter, and it is, of course, twice as long as any radius. Thus, d 5 2r. Any line cutting across a circle is called a chord, and no chord can be longer than the diameter. Any portion of the circle is called an arc. The degree measure of an arc is the measure of the central angle subtended by it, as shown in the following diagram.

O

Example

x°

x°

If the arc PS in the diagram below has a degree measure of 62°, is the chord PS longer or shorter than the radius of the circle? P

O S

Solution

Since all radii are equal, triangle OPS is isosceles, and the angles at P and S must be equal. Suppose each is x. Now, 2x 1 62 5 180. Hence, x 5 59. Therefore, PS is opposite the largest angle in the triangle and must be the longest side. Therefore, PS is longer than a radius.

TOPIC 42. THE AREA

AND CIRCUMFERENCE OF A CIRCLE For a circle of radius r, the circumference (the distance around the circle) is given by the formula C 5 2pr. The area of the circle is given by the formula A 5 pr2. In both formulas, p (the Greek letter pi) is a constant, a number whose value is 22 approximately 3.1416 (or about ). 7

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APPENDIX A

Example 1

Find the area of the shaded region shown in the figure below. The curved side is a semicircle.

15

15

12

Solution

The dotted line completes the rectangle, which is 12 3 15 5 180 square units. The radius of the circular arc must be 6, since its diameter is 12. The area of the whole circle would be pr2 5 p(62) 5 36p. Hence, the area of the semicircle is half of that, or 18p. Subtracting, the area of the shaded region is 180 2 18p.

Example 2

The larger circle shown in the figure below has an area of 36p. Find the circumference of the smaller circle.

Solution

The larger circle has an area of AL 5 p(r)2 5 36p. This means that r2 5 36 and r 5 6. The diameter of the smaller circle equals the radius of the larger one, so its 1 radius is ~6! 5 3. Its circumference must be CS 5 2p(3) 5 6p. 2

TOPIC 43. VOLUMES

OF SOLID FIGURES A solid figure with straight-line edges and flat surfaces is called a polyhedron. The surfaces bounding the solid are called faces. Thus, edges have lengths and surfaces have areas, and the solid has a surface area, which is the sum of the areas of all its faces. A solid figure also has a volume, expressed in cubic units. You should be familiar with the following formulas for volumes of regular figures: • A rectangular solid is a polyhedron with rectangular faces at right angles to one another.

H L W

V 5 LWH 5 Length 3 Width 3 Height

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• A cube is a rectangular solid with all edges of equal length s; that is, L 5 W 5 H 5 s. Therefore, V 5 s3. • A right circular cylinder is a solid with a circular base and a side perpendicular to the base (like a soda can). The volume is the area of the base times the height, or V 5 pr2h. r

h

Example 1

Find the length of a rectangular solid of height 6 that is twice as long as it is wide, if its volume is the same as that of a cube with a total surface area of 864 square inches.

Solution

Let x 5 the width. Now 2x 5 length. The volume of the rectangular solid is V 5 6(x)(2x) 5 12x2. Since the cube has six square faces, its total surface area is 6 times the area of one face. In symbols, 6s2 5 864. Dividing by 6, s2 5 144; s 5 12. Hence, the volume of the cube is 123 5 1728. Since the two solids have the same volume, 12x2 5 1728; x2 5 144; x 5 12. The length, which is twice the width, is thus 24.

Example 2

Which has a greater volume, a rectangular solid that is 6 feet long and has a square base of side 4, or a cylinder with a length of 7 feet and a diameter of 4?

Solution

The volume of the rectangular solid is V 5 (4)(4)(6) 5 96. The radius of the 22 cylinder is 2, so its volume is VC 5 p(2)2(7) 5 28p. Since p 5 about , 28p 5 7 about 88. Therefore, the rectangular solid is larger.

COORDINATE GEOMETRY

TOPIC 44. THE MIDPOINT FORMULA Given two points P(x1,y1) and Q(x2,y2), the midpoint M of the PQ has coordix 1 x2 y 1 y2 ; yM 5 1 nates: xM 5 1 2 2 Expressing the same idea in words: To find the coordinates of the midpoint, simply average the coordinates of the end points. For example, the mid312 5 4 1 22 2 point between (3,4) and (2,–2) is xM 5 5 ; yM 5 5 5 1. 2 2 2 2 5 Hence, the midpoint is ,1 5 (2.5,1). 2

S D

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APPENDIX A

Example 1 Solution

If (2,6) is the midpoint of the line segment connecting (21,3) to P(x,y), which is larger, 2x or y? ~x 2 1! , or 2 4 5 x 2 1; x 5 5. Similarly, we know that the average of y and 3 must be 6. ~y 1 3! , or 12 5 y 1 3; y 5 9. Since 2x 5 10, 2x . y. Thus, 6 5 2

We know that the average of x and 21 must be 2. That is, 2 5

Example 2

If b , 6, is (3,b) closer to P(0,2) or to Q(6,10)?

Solution

We see that (3,6) is the midpoint of PQ. Therefore, in the x-direction, (3,b) will be equidistant from both P and Q. However, if b , 6, then b must be closer to 2 than to 10. Therefore, (3,b) is closer to (0,2) than to (6,10).

TOPIC 45. THE DISTANCE FORMULA Given two points P(x1,y1) and Q(x2,y2), the distance from P to Q is given by the formula: d 5 =~x1 2 x2!2 1 ~y1 2 y2!2 In words, the distance is the square root of the sum of the change in x squared plus the change in y squared . For example, the distance from (6,2) to (3,21) is: d 5 =~6 2 3!2 1 ~2 2 ~21!!2 Thus: d 5 =32 1 32 5 =9 1 9 5 =18 5 3=2

Example 1

The point (t,21) lies on a circle with a radius of 5 and its center at (4,2). What are the possible values of t?

Solution

Since every point on the circle must be 5 units from the center, we know that (t,21) must be 5 units from (4,2). Using the distance formula,

=~t 2 4!2 1 ~21 2 2!2 5 =t2 2 8t 1 16 1 9 5=t2 2 8t 1 25 5 5 Squaring both sides, we have t2 2 8t 1 25 5 25. We subtract 25 from both sides to yield t2 2 8t 5 0, which factors as t(t 2 8) 5 0. This gives us two possibilities: t 5 0 or t 5 8.

Example 2

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The point (4,t) is equidistant from (1,1) and (5,3). What is the value of t?

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Solution

Since the distances from the two given points are the same, we use the distance formula twice and equate the results, thus:

=~4 2 1!2 1 ~t 2 1!2 5 =~5 2 4!2 1 ~3 2 t!2 =9 1 ~t2 2 2t 1 1! 5 =1 1 ~9 2 6t 1 t2! =10 2 2t 1 t2 5 =10 2 6t 1 t2 Squaring both sides: 10 2 2t 1 t2 5 10 2 6t 1 t2 Subtracting t2 1 10 leaves 22t 5 26t; 4t 5 0; t 5 0.

TOPIC 46. THE SLOPE

OF A LINE Given two points P(x1,y1) and Q(x2,y2), the slope of the line passing through P and Q is given by the formula: m5

y1 2 y2 x1 2 x2

In other words, this says that the slope is the change in y divided by the change in x. For example, the slope of the line passing through (6,4) to (3,21) is: m5

~4 2 F21G! 5 5 ~6 2 3! 3

Example The points (21,21), (3,11), and (1,t) lie on the same line. What is the value of t?

Solution

Since the slope of a line is the same for any two points on the line, using (21,21) and (3,11), we must have a slope of: m5

11 2F21G 12 5 53 3 2F21G 4

Now, using the pair (21,21) and (1,t): 35

t 2F21G ~t 1 1! 5 1 2F21G 2

Multiplying by 2, 6 5 t 1 1; t 5 5.

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APPENDIX A

COUNTING AND PROBABILITY

TOPIC 47. THE ADDITION PRINCIPLE

FOR COUNTING If a set A has m elements, and a set B has n elements, and the two sets have no elements in common, then the total number of elements in the two sets combined is m 1 n. But if there are k elements common to the two sets, then the total in the combined set is m 1 n – k. That is, you must take into account the double counting of elements common to both groups. This kind of situation is usually handled most easily by displaying the given information in a Venn diagram.

Example 1

Helena applied to 12 colleges for admission. Sergei applied to 10 colleges. Between them, they applied to 16 different colleges. How many colleges received applications from both students?

Solution

We let H be the set of colleges to which Helena applied, and S be those to which Sergei applied. Letting x be the number that are common to both sets, the following Venn diagram displays the data. H=

S=

12

12 – x

x

10

10 – x

The central region includes the colleges common to both groups, and we can see that the total is (12 2 x) 1 x 1 (10 2 x) 5 16. Removing parentheses and combining like terms, we have 22 2 x 5 16; x 5 6. Sometimes problems of this type can involve more than two sets.

Example 2

A survey of voters shows that 43% listen to radio news reports, 45% listen to TV news reports, and 36% read a daily newspaper. What is the maximum percentage of voters surveyed that do all three?

Solution

If the three sets were totally disjoint, that is, had no overlap, the sum of the percentages would be 100%. The extent of various kinds of overlap will show up as an excess over 100%. Everyone in two of the three categories will be counted twice, and everyone common to all three will be counted three times. If we total 43 1 45 1 36, we find that we have accounted for 124% of the voters, a 24% overcount. Therefore, the number common to all three cannot be greater than one third of that, or 8%. This maximum of 8% is reached only if no one falls into two out of three categories, so that the entire overcount is the result of people in all three.

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TOPIC 48. THE MULTIPLICATION PRINCIPLE

FOR COUNTING If a process can be broken down into two steps, and the first step can be performed in m ways; and if, for each of those ways, the second step can be performed in n ways; then the total number of ways of performing the complete process is T 5 mn. This is known as the multiplication principle for counting. For example, suppose that a jar contains five blocks of five different colors. We pick a block, record the color, and then pick a second block without replacing the first. The number of ordered color combinations is (5)(4) 5 20. This process extends to more than two steps in the natural way.

Example 1

The following diagram is a road map from Abbottsville to Cartersburg. How many different routes can you follow to drive from Abbottsville to Cartersburg if you go through Batestown only once? Batestown

Abbottsville

Cartersburg

Solution

You have 3 choices of roads from Abbottsville to Batestown and 4 choices of roads from Batestown to Cartersburg. Hence, by the multiplication principle, the total number of possible routes is 3 3 4 5 12.

Example 2

How many different 3-digit license plate numbers can be produced if the first digit on any license may not be 0?

Solution

By the natural extension of the multiplication principle to a three-step process, we see that we have 9 choices for the first digit, 10 choices for the second, and 10 choices for the third. Thus, the total is 9 3 10 3 10 5 900.

TOPIC 49. PERMUTATIONS (ARRANGEMENTS) As a natural extension of the multiplication principle, it is not hard to show that the number of distinct permutations, or arrangements of n distinguishable objects in a row, is n factorial; that is: n! 5 n(n21)(n22) 3 . . . 3 2 3 1 For example, there are 4! 5 4 3 3 3 2 3 1 524 ways of arranging the four symbols #, *, @, and % in a straight line.

Example 1

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If the five starting members of a basketball team are lined up randomly for a photograph, what is the chance that they will appear in order of height from shortest to tallest, left to right?

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APPENDIX A

Solution

There are 5 distinguishable people in the group, who can be arranged in 5! 5 5 3 4 3 3 3 2 3 1 5 120 ways. In only one of these ways will they be in the 1 correct size order. Therefore, the chance is . 120

Example 2

In how many ways can 3 men and 3 women be seated in theater seats if the seating must alternate men and women, starting with a woman?

Solution

The 3 women can be arranged in the first, third, and fifth seats in 3! 5 6 ways. However, for each of these six possibilities, the 3 men can be seated in the remaining seats in 3! 5 6 ways. Hence, there are really 6 3 6 5 36 ways altogether.

TOPIC 50. PROBABILITY To find the probability of any random event, divide the number of outcomes favorable to the event by the total number of possible outcomes. For example, if a bag contains 12 blue marbles and 9 red marbles, the probability that a marble selected at random is blue is the number of blue marbles divided by the total number of marbles: 12 4 5 21 7

Example

A box contains five blocks numbered 1, 2, 3, 4, and 5. Johnnie picks a block and replaces it. Lisa then picks a block. What is the probability that the sum of the numbers they picked is even?

Solution

Since each person had 5 choices, there are 25 possible pairs of numbers. The only way the sum could be odd is if one person picked an odd number and the other picked an even number. Suppose that Johnnie chose the odd number and Lisa the even one. Johnnie had 3 possible even numbers to select from, and for each of these, Lisa had 2 possible choices, for a total of (3)(2) 5 6 possibilities. However, you could have had Johnnie pick an even number and Lisa pick an odd one, and there are also 6 ways to do that. Hence, out of 25 possibilities, 12 have an odd 13 total, and 13 have an even total. The probability, then, is . 25

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Appendix B THE GRE CAT SUCCESS STRESS-BUSTING GUIDE by Mary-Jo D. Weber, M.S. Psychiatric Nurse Practitioner

THE ROLE OF STRESS IN PEAK TEST PERFORMANCE Let’s face it—if you’re like most people, you’re not really looking forward to taking the GRE. In fact, the very thought of the test might make your stomach queasy and your neck and shoulders tight. You might become aware of your heart beating, and your hands might get clammy. You might even feel restless and be tempted to close this book right now and get a snack! All these physical responses to the stress of test-taking can work for you or against you. Conditioned by millions of years of evolution, your body has developed a complex natural reaction, sometimes called the fight or flight response, that comes into play whenever you feel physically or psychologically threatened. This reaction has a very real value in getting you ready to meet whatever challenge you face, whether it’s a menacing stranger on a dark street, an auditorium full of people waiting to hear you deliver a speech, or a standardized exam. The adrenaline and other hormones that are released when you are under stress can get you ready for peak performance. They arouse your senses to increased sensitivity, alert your brain cells to pay attention, sharpen your mental focus, increase the amounts of energizing oxygen delivered to all parts of your body, and raise the levels of glucose available to fuel your brain. These chemical processes account for the sense of excitement you feel when you’re under stress. Some people—artists who thrill to public applause, for example, or world-class athletes—actually relish this state of physical and mental arousal, and even the average person finds it exhilarating, though perhaps scary, too. The problem comes when these responses get out of control, freezing your thoughts and leaving you feeling uncomfortably tense or anxious. When that happens, you may develop “tunnel vision,” a narrowing of perception that hampers your awareness of what’s around you; you may even feel that your mind is “going blank,” as if your brain is on overload and is starting to shut down. Fortunately, you can manage your stress so that the natural stress response will sharpen your focus without limiting your perspective or closing off your

403

APPENDIX B

options, making you more creative and imaginative and helping you to retrieve more of the useful information stored in your memory. This appendix will give you specific, scientifically tested techniques to use in the weeks before the exam, while you are studying, and on the very day you take the GRE. If you practice these methods, you may find yourself almost looking forward to the opportunity to tackle the test—and beat it!

PREPARING FOR PEAK PERFORMANCE Top athletes find that mental preparation is as important to their success in competition as practicing their specific athletic skills. The field of sports psychology has taught us a lot about how you can best prepare for your test. Practicing your academic skills is something like a basketball player working on his foul shot or a swimmer perfecting her stroke: it’s essential, but it’s not enough. The best performers don’t stop there. They also use relaxation and visualization techniques to ensure that they’ll be able to apply their skills and to respond effectively and creatively to the challenges that game day will bring their way. Similar relaxation and visualization techniques can help keep you from freezing up and allow you to efficiently handle whatever comes your way in the test-taking situation. They can also help you experience the test as a positive challenge, not a looming source of terror. You will maintain a degree of comfort and be able to manage the unpleasant symptoms of anxiety without letting them overwhelm you.

LEARNING

TO

RELAX First, you’ll need to learn to relax whenever you decide you want to. Yes—for most people, it’s a skill that must be learned. If you’re like most people, you probably tend to keep going—with work, play, or just hanging around—until you’re physically exhausted, and then you crash. It’s not the most efficient way to harness the energy in your body and mind. Instead, if you learn to relax whenever you want to, you’ll be able to modulate your stress responses so that you’ll feel only the amount of anxiety you need to wake up your brain cells and perform your best. The following exercise is a good way to start. If you’re feeling any sense of tiredness or anxiety—after a couple of hours of studying, for example—this is a better way to refresh yourself than napping or taking a TV break. It takes only a few minutes and will leave you feeling energized and alert. You can either read through this suggested exercise and then try it, or, even better, get a friend with a pleasant voice to read it aloud to you while you try it. (Later, you can return the favor.) As you go through the exercise, feel free to alter it in any way that seems pertinent to your individual situation.

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Relaxation Exercise Start by sitting comfortably with both feet flat on the floor. Take some time to notice how the floor is supporting your feet. Allow the surface on which you are resting to support you completely. Take all the time you need to notice the comfort and security of this. Next, turn your mind toward your breathing. Don’t try to change it; just observe it. Observe how effortless your breathing is, realizing that, with every exhalation, the tension of the day is flowing out, and, with every inhalation, revitalizing oxygen is flowing in to nourish all parts of your body. Turn your attention again to your feet. Notice that they are comfortably resting on the floor. Notice that feeling of comfort spreading up to your ankles, calves, and knees. Feel how securely the chair is supporting your thighs, your buttocks, your lower back, and your upper back. Your hands may be in your lap or at your sides. Allow them to open, and as you continue to breathe comfortably and naturally, experience any tension flowing down from your shoulders to your arms and out your finger tips. You will notice that the more relaxed you can keep your hands, the more relaxed and alert you will be. Notice whether your eyes are open or closed. Either way is fine. Take some time now to notice the muscles around your eyes, in your cheeks, and around your mouth. Notice whatever expression you naturally have on your face, whether it’s frowning, neutral, or smiling. Don’t feel you need to change it. Allow these muscles to soften. Close your eyes if you wish. Feel the heaviness of your jaw, and don’t try to hold it up. As you continue to notice the comfort in your body, pay attention to your neck and scalp. If you perceive any tension there, allow it to flow out with your next breath. Scan your body now, and if you find any areas of discomfort or tension, notice that, as you breathe, any tension or discomfort flows out with each exhalation, while energy flows in with each inhalation. Now, as you continue to enjoy the comfort of your body securely supported by the chair and energized by your breathing, imagine that you are in a special, favorite place. It may be the beach, or the mountains, or your room, or just inside yourself. Notice how comfortable and alert you are to all the things that make that place pleasurable for you. Notice the sights, sounds, smells, and feelings that make the place so nice. Take your time enjoying your special place. When you are ready to return to the room in which you are sitting, gently reorient yourself, experiencing your calm alertness and renewed enthusiasm for all your endeavors. Now that you’re back from your special place, notice how revitalized you feel. With practice, this process of relaxation will become easier and quicker, but you already have noticed that from the very first attempt. You can recharge your batteries in a way that is even better than a nap because your alertness will be increased and your focus will be sharpened. Try using this method of relaxation whenever the pressure of studying is making you feel exhausted or tense. You’ll find yourself learning more—and enjoying it more, too.

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APPENDIX B

VISUALIZING SUCCESS The next step is to add visualization of the test-taking situation and your desired successful outcome. It’s a favorite preparation strategy of many successful athletes and entertainers; they find that visualizing themselves hitting the perfect tennis stroke or playing a difficult piano concerto with fluency and ease makes it much easier to actually perform that way. It works for test-takers, too. In preparing for the computerized GRE, it’ll help if you visit the testing center beforehand, so you’ll know what to expect; but you can use our visualization exercise even if you don’t have a chance to make such a visit. Here’s how it works.

Visualization Exercise Repeat the relaxation exercise. This time, however, after you’ve imagined your special place and while you’re alertly and attentively noting the sights, sounds, and feelings that you experience there, imagine that you are entering the test room. Notice the rows of cubicles and the computer monitor and keyboard at each location. Take your seat, noting how comfortably your feet are supported by the floor, how your body is supported by the chair, and how your breathing is energizing your body and mind. You see that the computer monitor contains the directions for the GRE exam. You review them and you begin your work, knowing that you have prepared for this test and that you will correctly answer all the questions you need to in order to achieve your desired goal. You experience just the right amount of anxiety you need to feel in order to achieve your peak performance. As you work, the questions appear familiar and interesting to you. You look forward to reading each question because you know you will find it to be an interesting challenge to the skills you’ve been learning and practicing. You work efficiently, and, when you come to the end of the test, you experience the sense of a job well done. At this time, reorient to the room. You may also use this exercise before going to sleep, perhaps after a strenuous study session. If you do the exercise in bed, when it’s completed, simply allow yourself to drift into a refreshing sleep. During the weeks while you are studying for your exam, you can ensure peak performance by practicing relaxation and visualization every day. The best time for many people is at night, just before sleep. This will help your learning because your mind is working even while you sleep. Thus, if you practice visualizing test-taking success before you go to sleep, your brain will probably continue to process that information while you sleep, reinforcing the positive message. Some people find it effective to make an audiotape of the relaxation and visualization exercises, to be played while they fall asleep.

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TECHNIQUES OF POWER STUDYING If you’re an athlete, a musician, or an actor, you know how important your physical condition is to your training. What you eat or drink before practice will influence how effective your training will be. The same is true of test preparation.

PHYSICAL CONDITIONING

FOR EFFECTIVE STUDY Everyone knows his or her own best time of the day for studying. For some people, it’s early in the morning, before class or work; for others, it’s late at night. Whatever time you favor, make sure you’re in peak condition before you hit the books. Any alcohol intake within the past 12 to 24 hours (depending upon the amount) will decrease your mental functioning. Other mood-altering chemicals can interfere with your ability to learn and think critically for a much longer time, sometimes for as long as a month after use. A word to the wise, or to those who want to be: lay off drinking and drugs when preparing for a crucial exam. Nutrition is just as important for studying as it is for athletic training, because thinking and learning are physical functions of your body, carried out by the cells of your brain. It so happens that your brain cells work on glucose (sugar) only. That said, it would be incorrect to deduce that your diet while studying should consist of candy bars and sodas. But you do learn and think best when your brain has a steady stream of glucose. The best way to ensure this is to eat high-protein and so-called “complex carbohydrate” foods before studying and about every 3 hours during studying. Fruit, lean meat and fish, vegetables, pasta with low-fat sauce, cereal, crackers, bread, and legumes (such as peas and beans) are the foods associated with high mental performance. These foods will keep your blood sugar steady at an optimal level. Avoid greasy or fatty foods; the work of digesting them actually pulls oxygen-rich blood away from your brain and toward your digestive system. (So pizza, although you may love it, is really not the best study food. Wait till after the exam, and then treat yourself to a pie with your favorite toppings—as a reward for the high score you’ve earned.) Rest is often neglected when people are studying very hard, but this is a mistake. Research has shown that people who are in a state of chronic sleep deprivation just don’t think or perform very well. When hospitals shorten the working shifts of their medical residents, the doctors suffer fewer mental lapses and make better treatment decisions. The same applies to anyone working with his or her brain. The optimal amount of sleep varies from one person to another, but few people do well on less than 6 hours a night over any extended period of time, and most people thrive on 7 or 8 hours a night. Don’t pull an all-nighter; get the sleep you need, and you’ll find yourself learning more, and more easily, the next morning.

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REDUCING STRESS WHEN YOU STUDY When you sit down to study, you can improve your memory and creativity by practicing the relaxation and visualization exercises earlier in this appendix. If you don’t want to spend that much time, there is a brief technique that can be used just before studying or as needed during breaks in your study sessions. It takes only a few minutes. It works best if you have already experienced the longer exercise, and the more familiar you become with the long exercise, the better this short one will work.

The 3-Minute Relaxation Technique Sit comfortably wherever you like, with your feet flat on the floor. Rest your hands on your lap or desk. Place the thumbs and index fingers of each hand together. Close your eyes. Take a moment to notice your breathing. After a few seconds, turn your attention to the pressure of your thumbs on your index fingers. The pressure can be light or firm. Notice that pressure as you inhale and exhale several times; really notice how your fingers feel. Then, as you exhale, release the pressure, relaxing your hands, feeling your tension and fatigue flowing out with your exhalation and melting down your arms and out your fingertips. Continue to focus on your breathing for a few minutes. When you are ready, open your eyes and reorient yourself to your surroundings. This quick exercise will enable you to capture a sense of calm alertness whenever you need it.

MUSIC

TO

LEARN BY Many people enjoy listening to music while they are studying. There is nothing wrong with music as long as you like it and don’t find it distracting, but there are some things to consider when you choose music for your study sessions. Music can help you to concentrate by filtering out extraneous sounds or thoughts. For this purpose, it is helpful if the music is familiar, so listening to a selection of favorite CDs would help you to study more efficiently than listening to the radio (which will probably play both familiar and unfamiliar selections, as well as interjecting a stream of chatter and commercials that may well be distracting). Music can also set a helpful mood while you’re studying. It’s a very personal choice, but in certain recent studies, when research subjects listened to classical music, particularly the works of Mozart, just before and during cognitive (mental) tasks, their performance on those tasks improved. You might want to experiment with different kinds of music to see which help you to concentrate best.

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THE GRE CAT SUCCESS STRESS-BUSTING GUIDE

STRESS-BUSTING STRATEGIES FOR TEST-DAY AND TEST-DAY MINUS ONE Hopefully you’re not reading these suggestions for peak performance for the first time the day before your test. Ideally you’ll have used the ideas we’ve presented for developing your personal test-preparation plan and you will be following it, more or less closely, in the weeks leading up to the exam. In particular, repeatedly visualizing success in the weeks before the test will motivate you to study and to view the test as a challenge you can meet, not a disaster in the making.

THE NIGHT BEFORE On the day before the test, make sure you have your admission form, your identification, a pen and pencil, and your directions to the test site ready and available for the morning. Make sure you know how to get to the test center and how long it will take you to get there given the expected weather conditions and traffic patterns. If you control these “petty” details, they won’t inject an unnecessary note of anxiety or uncertainty on the morning of the exam. It’s really helpful if you can do something pleasant and relaxing the night before the test. Avoid alcohol and other mood-altering chemicals, and get into bed early. If you’ve made a tape of your relaxation and visualization exercises, have it playing as you drift off into a refreshing sleep. However, if you’re going to be worrying about that one last math rule or analogy technique while you’re watching that movie—or if you’re feeling that you haven’t suffered enough yet to appease the testing gods so they’ll allow you to get your highest score—here are some tips for last-minute studying. Don’t try to reread this book or do any kind of comprehensive review. At this point that will only increase your anxiety and convince you, incorrectly, that all the studying you have already done was inadequate and futile. Instead, decide how long you can study while still getting a good night’s sleep. If you need 9 hours to feel rested, your score will be boosted more by getting the full 9 hours than by another few hours of studying. Then pick a few topics you can comfortably cover in the amount of time you have left. Choose topics you are good at but feel can use a little more polishing, keeping in mind that you don’t need a “perfect” score or to get every question correct. Always remember, it is really all the previous study you’ve done that will determine your score; you’re only reviewing the night before to appease the testing gods. When it’s time to go to sleep, set your alarm clock to give you plenty of time to get ready, eat breakfast, and get to the testing site without rushing. Then practice your relaxation and visualization exercises, and have pleasant dreams of test-taking victory.

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THE MORNING

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OF THE EXAM In the morning, eat something. Yes, we know that you never eat breakfast—but just take our word on this one. Study after study has demonstrated that people perform better on tests when they have eaten about 30 to 60 minutes beforehand. The foods you should eat are the same as those recommended during studying— high protein, low fat, and complex carbohydrate. Bring a piece of fruit or a power bar to the test center to eat during the brief mid-exam break. During the exam itself—especially during the short breaks provided between test sections—you might try using the 3-minute relaxation technique if you feel fatigued or stressed. All these tips, if practiced along with the study outlined in this book, will ensure your optimal performance on the GRE. They’ll also help you attain an even higher goal: to feel balanced and sane before and after the test.

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Appendix C APPLYING TO GRADUATE SCHOOL Applying to graduate school is a time-consuming and costly process that can become overwhelming if you do not narrow your selection of graduate programs to a manageable number. Once you have narrowed your selection of graduate programs, it’s time to prepare and assemble your applications. If you have not already done so, request an application and information packet from each program to which you plan to apply. When you look over these materials, you will see that there is a lot of work involved in applying to graduate school. It may take you a year or more to assemble and submit all the necessary information, especially if you’re an international student or you’ve been out of school for a few years. Because the process is complicated and time consuming, you should start well in advance.

TIMETABLE In general, it’s advisable to start the application process at least a year and a half before you plan to enroll. Allow yourself even more time if you are applying for national fellowships or if you are applying to a health-care program through your college’s evaluation committee. In these cases, you may need to start two years before matriculation in order to meet all of the deadlines for test scores, letters of recommendation, and so on. Application deadlines for fall admission may range from August—one full year prior to your planned enrollment—to late spring or summer for programs with rolling admissions. However, most programs require that you submit your application between January and March of the year in which you wish to start. Be careful when you check the deadlines in the application packet. Different programs at the same university may have different deadlines. In addition, if you are applying for financial aid, you should leave yourself extra time to assemble all of the financial information you’ll need to support your request for assistance. Applicants for aid usually have to send in the entire application by an earlier date than the deadline for those not seeking aid. Be certain that you understand which deadline applies to you. After all, what’s the point of being admitted if you cannot afford to attend? It’s usually advantageous to apply early if you can. For one thing, an early application demonstrates strong interest and motivation on your part, especially when a program uses rolling admissions. Even more important, however, applying early means that the department or program will evaluate your application when it still has a full budget of funding to award. When you apply late, you may not be awarded full or even partial funding because the department has already used up

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its resources. “Deadlines are really important if you want to get funding,” says Suzette Vandeburg, Assistant Vice Provost for Graduate Studies at the State University of New York at Binghamton. “You may be highly qualified but lose out if you miss a deadline.” This does not mean that you will necessarily miss out on funding if you just squeeze in by the program’s deadline, but given the competition for financial aid, why gamble? While you could get lucky, you may be in for some weeks of nail biting until a program makes all of its awards. “When I applied to graduate school, I noticed that many positions were offered early, especially in departments that were very small and very competitive,” says Cindy Liutkus, a Ph.D. candidate in geology at Rutgers University in New Jersey. “Because I hadn’t applied as early as most people, I had to wait until two of the departments obtained rejections from their early offers before I knew whether I would receive a teaching assistantship or a research assistantship.” Still, you shouldn’t rely on luck for something so important—apply early!

WHAT HAPPENS TO YOUR APPLICATION? Before we discuss the elements of an application, it will be helpful for you to understand what happens to your application once you submit it. In general, your application is handled first by an admissions office and then by the admissions committee that makes the decisions.

THE ROLE

OF THE ADMISSIONS OFFICE If you are applying to an academic program that is part of a university, you will be sending your application to a centralized graduate studies admissions office. This office is responsible for establishing your file and coordinating all of the activity related to your application. As parts of your application arrive, an admissions staff member places each item in your file and records its receipt. In a well-run office, the staff will also notify you if anything is missing. Once your application is complete, the admissions office forwards a copy of it to the department or program’s admissions committee for evaluation and recommendation. University graduate admissions offices usually act as clearinghouses for applications, but in some cases they have the authority to reject an applicant or to waive a university requirement for an exceptional candidate. For example, they can turn down an applicant whose qualifications are clearly below university standard (e.g., someone with an extremely low GPA for all four college years). They can also bar an application from further consideration if it is incomplete.

THE ROLE

OF THE ADMISSIONS COMMITTEE The members of the admissions committee are the people on whom your future depends. They are the small group of department or program faculty members and administrators who review and evaluate each applicant and decide not only who gets in, but who gets funding. Admissions committees usually have at least

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four sources of information on which to base their decision about your application: your transcripts, your test scores, your personal essay, and your letters of recommendation. The importance of each of these sources will vary from admissions committee to admissions committee and also will vary among the members of a committee. Their decision-making processes will vary as well. Let’s take a look at how two actual admissions committees work to give you an idea of what goes on behind closed doors.

A Program in the Social Sciences The admissions committee for this program receives about 120 applications per year. A staff member extracts certain data, including undergraduate school attended, degree earned, and area of concentration; grade point average; GRE scores; and field of interest. This information is placed on a cover sheet and attached to the application. This cover sheet is the first thing the admissions committee members will see when they pick up an application. A few weeks after the application deadline, the committee meets for a day-long marathon of reading and assessing applications. The applications are divided into two groups—master’s-degree candidates and doctoral-degree candidates—and are handled separately. Each application is passed around for each committee member to read. While the applicant’s essay and letters of recommendation are fresh in each person’s mind, the committee makes a decision on the candidate. Usually, there is agreement on the acceptance or rejection of an applicant, but occasionally members of the committee have a difference of opinion on a particular candidate. In that case, if no one is willing to back down, a decision on the candidate may be deferred until the candidate can be interviewed. Or the candidate may be accepted on a conditional basis. At the same time that the accept/reject decision is being made, a tentative decision on department funding is also made. After all of the applications have been evaluated, the committee goes through the applications in the acceptance pile again, adjusting the funding decisions that they made on the first round. In this committee, a great deal of weight is placed on the personal essay. Members of the committee are looking for evidence that a candidate is focused and committed. “We look to see whether the applicant knows why he or she is applying to our graduate program in a specific way, not just as a next step in life while they’re figuring out what to do,” comments one member of the committee. “When a student knows what our strengths are, and how their interests fit into our program, we are impressed.”

A Program in the Hard Sciences In this department, the admissions committee consists of five faculty members, one from each major division of the department. Each faculty member reviews the applications of the students interested in his or her area of specialization. In addition, the chair of the committee reviews all the applications. Periodically, the committee meets to discuss and make decisions on the applicants. Since the

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department has the resources to fund all first-year students, an acceptance automatically means the student will have financial support. In this committee, grade point average and letters of recommendation are weighted heavily. A minimum GPA of 3.0 is required, although extenuating circumstances are considered if the GPA is uneven; a typical example is a low freshman year GPA, which the committee may decide to overlook. The letters of recommendation are important because the committee learns about the student’s undergraduate research experience and relevant summer internships from them. Of the four main elements of the application, the essay is the least important to this committee. As long as it is coherent and gives an indication of the student’s interest in research, the members pay little further attention to it. This particular committee does not try to make a match between every single applicant and a faculty member with similar research interests. “Students often change their minds once they get here,” comments a member of the committee. This is unlike other programs, where the admissions committees may turn down an excellent applicant solely because there is no faculty member to work with the student. In other words, if there is no faculty member who shares the student’s area of interest and wants to work with him or her, the applicant will be rejected.

ELEMENTS OF AN APPLICATION From our description of the various admissions committees, you can see that you cannot always tell which parts of your application will be considered most important by a particular admissions committee. For that reason, you should work hard to make each element of your application the best it can possibly be. For each program to which you apply, you will have to submit a number of items to make your application complete. For most programs, these include: • An application form • Undergraduate and other transcripts • Graduate admissions test scores • Letters of recommendation • Personal essay(s) • Tape, portfolio, writing sample, or audition for fine arts and design applicants • The application fee In addition, a personal interview may be required for some programs, although for most an interview is optional. Be sure you read the information packet thoroughly so you understand what each program expects of you. They may require additional items, such as a résumé. The basic elements of an application, except the personal essay, are discussed below. Since the personal essay is such an important—and difficult—item to compose, we’ve devoted a whole chapter to tips on writing it (see Appendix D).

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THE APPLICATION FORM On the application form, you provide basic information, such as the program or department to which you are applying; your name, social security number, address, and contact information; your citizenship status; your demographic background (usually optional); your current employer and position; your educational background; names of people who are providing references (ask them first!); and admissions test dates. Sometimes, the application form also includes a section for applying for financial aid. A separate application form for financial aid may be necessary. Be sure you understand what forms you need to submit and to whom if you are applying for aid. If possible, you should type the information on the application form or fill out the form on line at the program’s Web site. If neither option is available, print your entries neatly. Be sure you do not accidentally omit information and doublecheck to make sure there are no spelling errors. Remember that you will be competing against people whose forms are complete, legible, and error free. “Take your time filling out all the necessary information, no matter how tedious it may be,” advises Tammy Hammershoy, who is earning a master’s degree in English at Western Connecticut State University. “Read everything very carefully, and follow all instructions. If you really want to get into the program of your choice, be patient and careful when filling out application forms and other materials.” Some of the tedium may be relieved if you are applying to professional schools and can use one of the national application services, which are described later in the appendix.

TRANSCRIPTS As proof of your academic background, you will need to submit official transcripts from each college and university you have attended, even if you have taken just one course from that institution. To request official transcripts, contact the registrars of your undergraduate college and other institutions you have attended. Be sure to allow two or three months for your request to be processed. It will save time if you call ahead to find out what the fee for each transcript is and what information they need in order to pull your file and send the transcript to the proper recipient. Then you can enclose a check for that amount with your written request. When you review your transcript, look for weaknesses that may need explanations. For example, a low GPA one semester, a very poor grade in a course, or even a below-average overall GPA may hurt your chances of acceptance unless you have good reasons for them. You can explain any shortfalls in your transcripts in your personal essay, cover letter, or addendum to the application (see Appendix D). If you have been out of school for years and have been successful in your professional and postgraduate endeavors, do not assume that a poor undergraduate GPA will not count against you because it’s ancient history. For example, one 58-year-old prospective graduate student who had an A2 average in his previous

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master’s program but a C average as an undergraduate found that the A2 did not cancel out the C. He had to take a semester of master’s-level courses and achieve a minimum B average before he was admitted to the new master’s program as a matriculating student.

STANDARDIZED TEST SCORES Like your GPA, your admissions test scores are numbers that pop right out of your application and tell the admissions committee something about you before they have even begun reading your file. Your scores give the admissions committee a way to compare your performance to that of every other applicant to the program, even though you all may have attended very different colleges with very different instructional and grading standards. Although your GRE scores may not be directly relevant to the field in which you are planning to work, such as one of the sciences, the scores do predict how well you can cope with the types of tasks graduate students face all the time— reading, analytical thinking, and writing. “Over the years, we have found that students with poor verbal scores do not have the ability to read and write at the graduate level,” says Gail Ashley, Professor of Geological Sciences at Rutgers University in New Jersey. Still, it is rare for an admissions committee to reject an applicant solely on the basis of poor test scores. “We have no statistical cutoff,” says Thomas Rochon, dean of the School of Politics and Economics at Claremont Graduate University in California, referring to both GRE scores and GPA. “But low scores mean that the admissions committee may just look through the application very quickly.” Or, the admissions committee may scrutinize an application with low scores even more thoroughly to see if other qualifications compensate for poor test performance. You should plan on taking the GRE about a year before you plan to enroll. This will give you plenty of time for score reports to be submitted and plenty of time to retake the test if your first set of scores is lower than you had hoped. When you register for a graduate admissions test, you can request that the testing service send your official scores to the institutions you designate on the registration form. If you later decide to apply to additional programs and need more score reports, you can then request these in writing. Needless to say, there is a fee for each score report.

LETTERS

OF

RECOMMENDATION You will have to provide letters of recommendation for each program to which you apply. These letters are important because, like the personal essay, they give the members of the admissions committee a more personal view of you than is possible from your grades and test scores. Good letters of recommendation can tremendously increase your chances of admission and funding and lukewarm letters can harm your application. So it’s important to approach the task of choosing and preparing your letter writers in a thoughtful and timely fashion. In fact, it’s a good idea to start asking for references at least six months before your application deadline. “Contact the people who will be writing letters

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of recommendation well in advance of application deadlines,” suggests Felecia Bartow, an M.S.W. candidate at Washington University in St. Louis. “Many professionals and academics are extremely busy, and the more time that you can give them to work on your recommendation, the more it will reflect who you are.” Starting early will also give you an opportunity to follow up with those who recommend you before the application deadlines.

Choosing People to Write Recommendations Most of your recommendations should be from faculty members, because (1) they are in the best position to judge you as a potential graduate student and (2) members of the admissions committee will consider them peers and will be more inclined to trust their judgment of you. Having professors write your letters is absolutely essential if you are applying to academic programs. If you cannot make up the full complement of letters from faculty members or if you are applying to professional programs, you can ask employers or people who know you in a professional capacity to write references for you. In fact, if you are applying to professional programs, having letters of recommendation from those already practicing in the field is a plus. But try to find people who have a relationship with the field you are entering. It won’t do you much good to have a glowing letter of recommendation from your manager at the insurance company if you are applying to a program in history or social work. “The most important thing to remember is that you want the writers of these letters to be very familiar with you and your work,” advises Cindy Liutkus. “As I was choosing professors to ask for letters, many people gave me advice as to who would write the best letter. Some suggested that the chair of the department carries the most weight, even if he or she doesn’t know you very well. Others said to ask the dean of the school, but once again, since he didn’t know me very

When you are trying to decide whom to ask for recommendations, keep these criteria in mind. The people you ask should: • have a high opinion of you. • know you well, preferably in more than one context. • be familiar with your field. • be familiar with the programs to which you are applying. • have taught a large number of students (or have managed a large number of employees) so they have a good basis upon which to compare you (favorably!) to your peers. • be recognized by the admissions committee as someone whose opinion can be trusted. • have good writing skills. • be reliable enough to write and mail the letter on time. A tall order? Yes. It’s likely that no one person you choose will meet all of these criteria, but try to find people who come close to this ideal.

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well, I was skeptical as to the quality of the letter. Instead, I chose a professor from each of my major disciplines, namely my thesis adviser and my favorite undergraduate geology professor. I needed a third, and had a lot of trouble deciding whom to ask. I eventually chose the woman in the geology department whom I respected the most. . . . Although I had only one class with her, I felt she would give the most honest and straightforward account of my undergraduate accomplishments, my personality and work habits, and my goals for the future.”

Approaching Your Letter Writers Once you’ve decided whom you plan to ask for references, be diplomatic. Don’t simply show up in their offices, ask them to write a letter, and give them the recommendation forms. Plan your approach so that you leave the potential recommendation writer, as well as yourself, a graceful “out” in case one reacts less than enthusiastically. On your first approach, you should remind the potential recommendation writers about who you are (if necessary) and then ask whether they think they can write you a good letter of recommendation. This gives them a chance to say no. If they agree but hesitate or seem to be less than enthusiastic, you can thank them for agreeing to help you. Later, you can write them a note saying that you won’t need a letter of recommendation after all. On the other hand, if any of them seem genuinely pleased to help you, you can then make an appointment to give them the recommendation forms and the other information they will need. A confidential letter usually has more validity in the eyes of the admissions committee.

Waiving Your Right to See a Letter The recommendation forms in your application packets contain a waiver. If you sign the waiver, you give up your right to see the letter of recommendation. Before you decide to sign it, discuss the waiver with each person who is writing you a reference. Some people will write you a reference only if you agree to sign the waiver and they can be sure the letter is confidential. This does not necessarily mean they intend to write a negative letter; instead, it means that they think a confidential letter will carry more weight with the admissions committee. In fact, they are right. A confidential letter usually has more validity in the eyes of the admissions committee. From the committee’s point of view, an open letter may be less than candid because the letter writer knows you will read it. In general, it’s better to waive your right to see a letter. If this makes you anxious in regard to a particular recommendation writer, do not choose that person to write a letter.

Helping Your Letter Writers Once a faculty member or employer has agreed to write a letter of recommendation for you, they will want to write something positive on your behalf. No matter how great you are, this won’t be possible if the letter writer cannot remember you and your accomplishments very well. “Help faculty members write a more

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What should you do if the letter writer asks you to draft the letter? Accept gracefully. Then pretend you are the writer and craft a letter extolling your virtues and accomplishments in detail. Remember, if the letter writer does not like what you’ve written, he or she is free to change it in the final draft.

effective letter by reminding them of what you’ve done,” advises Teresa Shaw, Associate Dean for Arts and Humanities at Claremont Graduate University in California. “Letters that are not specific are ineffective letters.” When you meet with your letter writers to give them the forms, use the opportunity to provide them with information about yourself. Bring a short résumé that highlights your academic, professional, and personal accomplishments. List the course(s) you took with them, the grades you received, and any significant work you did, such as a big research paper or lab project. The résumé can be the basis of a conversation with the letter writer, amplifying your notable accomplishments. “Many of the people I asked to write me recommendation letters found it helpful if I wrote down a list of my accomplishments and my plans,” recalls Jenn Wagaman, a master’s candidate in public communications at the University of Alaska at Fairbanks. “Even though these people knew me, they wrote better letters because they had the exact information right in front of them.” You can help your letter writers by filling in as much of the information as you can on the letter of recommendation forms. It’s also a nice gesture to provide stamped, addressed envelopes for the letters if they are to be mailed directly to the programs or to you for inclusion in your application. Be sure your letter writers understand what their deadlines are. In other words, do everything you can to expedite the process, especially since you may be approaching your professors at the beginning of the fall semester, when they are the busiest. Finally, send thank-you notes to professors and employers who have come through for you with letters of recommendation. Remember that you are hoping someday to be their colleague in academia or a profession. Cementing good relationships now can help you in the future.

Using the Placement Office Most college placement offices will handle letters of recommendation on behalf of students. The office establishes a file for each student using this service, and they place a copy of the letter of recommendation from each professor in the student’s file. When you are ready, request that the placement office send a copy of each letter to each program to which you are applying. This service is convenient for professors because it relieves them of the responsibility of sending out multiple copies. On the down side, letters written for a placement office file often have a generic, “one size fits all” approach. You may be better off begging your professors to write individual letters that are specific to the programs to which you are applying.

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If you are an undergraduate and unsure of your plans for graduate school, you can ask your professors to write you letters of recommendation now, when you are still fresh in their minds. Have the letters placed in your file in the placement office and ask the office to keep your file active. Although there may be a fee for this service, it’s worth it. When you do apply to graduate school a few years down the road, you will already have several letters of recommendation that you can use.

If You’ve Been Out of School for Years What should you do if you have been out of school for years and have lost touch with your professors? If you established a file of letters of recommendation at the placement office when you were an undergraduate, you will now reap the benefit of your foresight. But if you did not, there are several things you can do to overcome the problems associated with the passage of time. First, if a professor is still teaching at your alma mater, you can get in touch by mail or e-mail, reminding the person who you are, what you’ve done since they taught you, your plans for graduate school, and a résumé. Tell the professor what you remember most about their courses. Most professors keep their course records for at least a few years and can look up your grades. If you are still near your undergraduate institution, you can make your request in person. “I arranged to meet one of my college professors for coffee to talk about what I had been doing in the five years since she had me as a student,” says Felecia Bartow. “It gave me a chance to bring her up to date on my experience, and it gave her a lot more information with which to write her recommendation.” Once you’ve made this initial approach, you can then call and ask whether the professor thinks he or she can write a strong recommendation for you. Another strategy if you’ve been out of school for a while is to obtain letters of recommendation from faculty members teaching in the programs to which you plan to apply. In order to obtain such a letter, you may have to take a course in the program before you enroll so that the faculty member gets to know you. Members of an admissions committee will hesitate to reject a candidate who has been strongly recommended by one of their colleagues. Finally, if you are having trouble recruiting professors to recommend you, call the programs to which you are applying and ask what their policy is for applicants in your situation. They may waive the letters of recommendation, allow you to substitute letters from employers, or ask you to take relevant courses at a nearby institution in order to obtain letters from faculty members. Remember, if you are applying to an academic program, letters from employers will not carry as much weight with the admissions committee as letters from faculty members. In fact, many academics are not at all impressed by work experience because they feel it does not predict how successful you will be as a graduate student.

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PORTFOLIOS, WRITING SAMPLES, TAPES,

AND AUDITIONS If you are applying to programs in the fine arts, design, or architecture, you will be required to demonstrate your artistic or design abilities by means of a portfolio, writing sample, videotape, audiotape, or audition. Students in the visual arts must present a portfolio of their work, usually in slide format, so that admissions committee members can judge their talent, conceptual ability, level of competence, and technical skill. Film students may have to submit a film on which they have worked. Students applying to an M.F.A. program in writing will generally have to submit a sample of their written work. If you are an undergraduate or recent college graduate, take the opportunity to work with a professor in compiling your submission. Your professor will have a good idea of what will impress an admissions committee. Most music and performing arts students must audition as part of the application process. In fact, the more selective the institution, the more important and brief the audition generally is. At the most competitive schools, an audition may last only 5 to 10 minutes. Although some schools hold regional auditions in different cities, in most cases you are expected to travel to the school. If travel would be a hardship, some schools may permit you to send a tape of your performance in lieu of an audition. Still, if you can audition in person, you should do so. In addition to giving the faculty members an opportunity to judge your abilities, an audition gives you the opportunity to evaluate a school first-hand. Since the requirements and instructions for auditions and tapes vary considerably, be sure you understand what each program expects of you in order to prepare properly for your auditions.

INTERVIEWS In most cases, an interview is not necessary. However, if you think you do well in interviews, you can call each program and request an interview. A good interview may be an opportunity to sway the admissions committee in your favor. Human nature being what it is, an excellent half-hour interview may loom larger in the minds of admissions staff and faculty members than four years of average grades. Most interviewers are interested in the way you approach problems, think, and articulate your ideas, and so they will concentrate on questions that will reveal these aspects of your character, not on questions that test your technical knowledge. They may ask you controversial questions or give you hypothetical problems to solve. Or they may ask about your professional goals, motivation for graduate study, and areas of interest—much the same material that is in your personal essay. Remember that interviewers are interested more in how you think than in what you think. When you prepare for an interview, it would be helpful if you have already written your personal essay, because the thought processes involved in preparing the essay will help you articulate many of the issues that are likely to come up in an interview. It is also helpful to do your homework on the program, so if the opportunity arises for you to ask questions, you can do so intelligently. Finally, be sure you are dressed properly. That means dressing as if you are going to a professional job interview.

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FEES Each application must be accompanied by a fee, or your papers are likely to sit in the admissions office without action. If you cannot afford the fee, you can ask the admissions office and your undergraduate financial aid office for a fee waiver. If you are applying to half a dozen schools, you can see that the costs will mount quickly. In addition to the program application fees, you must pay transcript fees, test fees, score report fees, photocopying fees, mailing costs, and travel costs if you are interviewing or auditioning. “Put aside some money for the process—it will cost more than you expect, especially if you are interviewing,” suggests Jennifer Cheavens, a Ph.D. candidate in clinical psychology at the University of Kansas at Lawrence. The application process may cost hundreds of dollars, even more if you are applying to many schools.

SUBMITTING YOUR APPLICATION As we mentioned at the beginning of the appendix, you should submit your completed applications well before they are due. Be sure to keep a copy of everything. You can either mail the application to the admissions offices, or you can file portions of it on line through the programs’ Web sites. Remember, however, that some elements of the application, such as the fee and official transcripts, will still need to be mailed in paper form. Also note that most schools that accept online applications simply print them and process them as if they had come in by regular mail. Try to submit all of your materials at once; this simplifies the task of compiling and tracking your application at the admissions office. If that’s impossible, as it is for many students, keep track of missing items and forward them as soon as possible. Remember that if items are missing, your application is likely to sit in the admissions office. According to Suzette Vandeburg at the State University of New York at Binghamton, incomplete applications are held for a year and then they are tossed.

FOLLOWING UP It’s important that you check on the status of your applications, especially if you don’t receive acknowledgment that an application is complete. Give the admissions office a couple of weeks to process your application and then call to find out if it’s complete. Usually the missing items are transcripts or letters of recommendation. “Don’t assume,” warns Rose Ann Trantham, Assistant Director of Graduate Admissions and Records for the University of Tennessee at Knoxville. “Follow up—not every week, but call periodically.” Suzette Vandeburg agrees; she advises applicants to be proactive about their applications. “Check in periodically,” says Vandeburg. “E-mail is a great way to check on your application.” Cindy Liutkus remembers how anxious she was about her applications. “The application process is definitely nerve-wracking,” Liutkus says. “I was always worried that something wouldn’t make it on time. I eventually sent stamped

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postcards along with every application and asked that the department secretary check the package and send the card along if everything was okay.” Not content with that, Liutkus made sure by following up with e-mail, too.

IN SUMMARY Preparing a thorough, focused, and well-written application is one of the most important tasks you will ever undertake. In addition to gaining you admission to a graduate program that can help you achieve your goals, a good application may win you enough monetary support to finance your degree. With these future benefits in mind, work on your applications as if they are the most important things you can possibly be doing—because they are.

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Appendix D WRITING A GOOD PERSONAL ESSAY The application to graduate school is not all numbers and outside evaluations. Admissions committees are also interested in finding out about you as an individual and in more intangible qualities, like your ability to write a good essay. Thus, the personal essay is the part of the application in which you can take control and demonstrate who you are and why you deserve to be admitted. Other parts of your application—test scores, GPA, and undergraduate transcripts—may reflect your academic ability but not much else. The letters of recommendation are beyond your control once you’ve chosen the writers. But a good personal essay can make you stand out from other applicants. It can show the committee the qualities that will make you an excellent graduate student and professional. In other words, the essay is your showcase and you should make the most of it. Even if you can write superb prose in your sleep, you still need to know what to write. In this appendix, you’ll get a step-bystep guide to preparing the personal essay.

WHAT THE ADMISSIONS COMMITTEE LOOKS FOR When they read an essay, the members of an admissions committee look for evidence that you are prepared for graduate school, have demonstrated intellectual or professional growth, and are focused and interested in a particular field. They want to know what you hope to get out of your graduate education. They are not particularly interested in your personal, psychological development. Autobiographical details and feelings, unless they help explain your intellectual and professional interests, are not relevant to an admissions committee. Instead, they want to know whether you will make a good student and colleague. The admissions committee gleans most of this information from what you write. But they can also tell a lot from how you write. If your writing is clear and conveys your ideas effectively, you are demonstrating your ability to communicate. If your writing is free of grammatical and spelling errors, you are demonstrating your attention to detail. Good writing skills are essential for a graduate student in any field, so a poorly written essay can hurt an application. A well-written statement, on the other hand, will help your case.

REQUIREMENTS VARY

FROM FIELD TO FIELD The essays required of graduate applicants vary widely. For some programs, you may just have to explain in one or two paragraphs why you want to go to graduate school. In the sciences and engineering, for example, the essay may simply be a means of conveying information about your research experience and

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interests. “I don’t think my essay was all that remarkable,” recalls a Ph.D. candidate in physics at Harvard University. “Grad schools in science look for a clearminded view on the student’s part as to their research plans.” J. W. Viers, Director of Graduate Studies for the chemistry department at Virginia Tech, explains, “The personal essay doesn’t carry much weight. We are simply looking for an indication that the applicant has done research and enjoys doing research.” For applicants in literature or the arts, the personal essay may be judged on its creativity or style as well as the information it conveys. Melany Kahn, who has an M.F.A. in film from New York University, recalls that she tried to be imaginative when writing her essay. “At NYU, it’s imperative to show imagination and the ability to tell a story, not all your technical experience.” Megan McAfee, an M.F.A. candidate in costume technology at Virginia Tech, says, “I think my essay was really unusual since I wasn’t going to go to a place that didn’t like me and I wasn’t going to puff myself up to be something I’m not. But I think that’s probably an attitude that someone in theater can take.”

A WORD

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ABOUT CREATIVITY A word of warning about creativity. It’s fine to be creative within the confines of the genre. But admissions committees usually look askance at essays that are too creative. The English literature applicant whose essay was written in light verse undermined her application. Resist the temptation to be clever or cute; it’s likely to backfire.

WRITING PROCESS

TO CLARIFY YOUR THOUGHTS Needless to say, the type of introspection required to write an effective essay is difficult and articulating its results can be even harder. Brenda Bennett, who earned a master’s degree in special education from Cambridge College in Massachusetts, recalls, “[The personal essay] forced me to evaluate my own belief system and philosophy of education. How different I am today than only seven years ago when I started in this field. It was a real eye-opening experience.” Leslie Nelman, a Master of Arts candidate in translation and interpretation at the Monterey Institute of International Studies in California, struggled to clarify her background and goals. “The ‘Statement of Purpose’ was supposed to include ‘educational and career objectives’ and describe how I acquired my language proficiency. All of this in 600 words or less. Once you get started, 600 words isn’t really that long! It is difficult articulating exactly what it is you want to do with the rest of your life, whether you’re 52 (as I was) or 22. It took several false starts and about three days to put it together.”

BE YOURSELF . . . The most common words of advice from most admissions directors about writing the personal essay are to be yourself. Remember, you are seeking to be accepted by a program that is a good match for you. If you disguise who you really are in an effort to impress an admissions committee, you are doing yourself—and

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them—a disservice. “Be who you are. Don’t try to second-guess what the committee is looking for,” Donna Lau Smith advises. “We can tell.” Cindy Liutkus, a Ph.D. candidate in geology at Rutgers, struggled with her essay for several weeks. “Eventually, though, I just wrote my thoughts and feelings. . . . I think grad schools read an essay that’s written from the heart and they realize that the student is passionate about their work, dedicated to their future in that field, and comfortable with expressing themselves.” In this introduction, Liutkus is candid about the late flowering of her interest in geology. She does not sidestep the problems that arose because of the late declaration of her major. In other words, she is straightforward and honest about her interest in geology and the shortcomings in her preparation. This essay helped Liutkus gain admission to several programs in geology. Here is the beginning of Liutkus’s essay: The decision to further my education in the area of geological sciences came later in my college career than for many others. I was originally a philosophy major with only a minor in geology, even though it is an area of study which has fascinated me since I was a child. . . . However, as I continued to take more advanced classes in the field, I became more convinced that a major in geology would better represent my true interests. It was during my junior year abroad, while studying in Melbourne, Australia, that the decision to double major in both philosophy and geology became final. However, with such a late decision, I was caught between a rock and a hard place. I had made the decision to follow my true interest but now had to play catch-up with the other students in the field in order to fulfill the requirements of the major. Despite doubling up on lab sciences during my senior year, I still have some gaps in the areas of calculus, chemistry, and physics. However, I plan to complete these classes upon enrollment in a graduate program. So, be honest. If you demonstrate self-knowledge by presenting your strengths as well as your limitations, your essay will be a true reflection of who you are.

. . . BUT BE DIPLOMATIC Honesty is important, but so is diplomacy. Try not to reveal weaknesses in your personality, such as laziness, dishonesty, or selfishness. Don’t say you want to attend a program because it’s cheap, within commuting distance, or you know you can get in. Even though these things may be true, they are not reasons with which the admissions committee will be sympathetic. Instead, frame your points in a positive light: you can fulfill its admission requirements because you have the proper prerequisites; you live nearby and know of its reputation; and so on.

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WRITE A STRONG OPENING When you write your essay, put yourself in the position of an admissions committee member who may read fifty or a hundred essays in a day or two. By the end of all this reading, the poor individual may be bored to tears and would be pleased by any essay that simply engages his or her interest. How are you going to accomplish this? By writing an opening that grabs the reader’s attention.

DESCRIBE

AN IMPORTANT EXPERIENCE “Many essays begin, ‘I would like to go to graduate school because . . . ,’ ” complains Gladys Fleckles, Director of Graduate Studies at California State University in Fullerton. “This is boring! Instead, describe a pivotal experience that made you decide that graduate school is what you want to do.” Nestor Montilla, who is pursuing a master’s degree in public administration at John Jay College of Criminal Justice in New York, described an experience that strengthened his desire to go to graduate school. My foremost reason [for pursuing graduate education] is my passion for public service. . . . My aspiration was partially fulfilled after Ruth Messinger, Manhattan Borough President, appointed me as a member of Planning Board 12. Thanks to this opportunity, my commitment to public service has reinforced my quest for civic responsibility. Indeed, the spirit of helping our community to be a better place to live is what inspires me to pursue graduate studies. By mentioning his planning board appointment, Montilla shows that he has a real interest in and experience with public service, both of which are solid reasons for pursuing a degree in public administration. The opening is also the place to set forth any unusual experience you have had that has contributed significantly to the person you are today. The experience may be growing up poor, being an Olympic athlete, or moving to the United States at the age of 14. Whatever the experience is, show how it has formed your character and life and how it relates to the graduate education you now want to pursue.

BE SPECIFIC What if you have not had a defining moment or experience that sparked your interest in graduate studies? Then write an opening that is specific enough to have some real interest. Here is the first paragraph from Leslie Nelman’s statement of purpose. In applying for admission to the translation and interpretation program at Monterey Institute of International Studies, I’m following through on a goal I set for myself over thirty years ago. I have always been fascinated by language, first by my mother tongue, then by other languages, once I became aware of their existence. The ‘cold war’ that was raging when I was young was a war of words. It occurred to me then that misunderstandings were likely when everything had to be translated for the international policy makers. Surely translators and interpreters played a key role in the fate of nations! And so my fascination with language grew into a vague career goal.

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In this paragraph, Nelman describes the development of her interest in language and her realization of the importance that language plays in international affairs. The paragraph is specific enough to engage the reader’s interest and make him or her want to continue. The key is to remember that specific details are usually more interesting than general statements. Be specific and you’ll have a better chance of connecting with your readers.

TELL HOW YOUR STORY INTERSECTS WITH THEIRS If you are applying to several programs, you will be tempted to write a boilerplate essay. Resist the temptation. Admissions committees grow adept at picking out the generic personal statements. “It always shows if it’s a boilerplate statement,” says Teresa Shaw, Associate Dean for the Arts and Humanities at Claremont Graduate University in California. “The personal statement should be written to the particular school. It’s impressive if a student shows he knows the program, its research areas, and its professors.”

DESCRIBE WHY YOU ARE

A MATCH FOR THEM Remember that when you were evaluating programs, you were looking for a good match for you. The personal essay is the place where you can explain to the admissions committee why you are a good match for them. According to Gladys Fleckles, “The key question is: Why should you be selected over anyone else? Tell about your skills and interest in the program. Be specific. Why would you be an asset to their program?” The story of your intellectual and professional development and your goals should culminate in your reasons for choosing this particular program. Your reasons should reflect a knowledge of the program’s faculty members, key research areas, and other characteristics. Jennifer Cheavens, a Ph.D. candidate in clinical psychology at the University of Kansas at Lawrence, wrote a different version of her essay for each program to which she applied. “I . . . found that tailoring the autobiographical statement to the different schools that I was applying to was helpful in helping me really decide that I matched with a program,” says Cheavens. “I tried to make the personal statement specific to each program for two reasons. One, so they would know I was really interested and had done my homework. Two, so I was sure that was somewhere I could be for five years.” In the following paragraphs of her essay, Cathy Chappell, who earned a master’s degree in educational foundations at the University of Cincinnati, explains why she wants to continue in its doctoral program. I continue to be attracted to the interdisciplinary aspect of the Educational Foundations Graduate Program. I appreciate the broad range of perspectives in the study of education and feel particularly drawn to the sociology and anthropology disciplines. The Education Foundations Department is concerned with the problems of urban education and I find the emphasis on group dynamics and interaction to have great functional value. An examination of the patterns of subordination and domination of groups in society,

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although occurring in every institutional setting, reveals disparities in equality that are profound and salient in the urban educational setting. Dominant groups structure these institutions to maintain their dominance and subordinate groups must struggle for equality . . . I find the prospect of continued study in the Educational Foundations program very exciting and am impressed with the department’s professors and their progressive, interactive scholarship. If accepted into the Doctoral Program, I plan to expand my knowledge base, to increase and hone my research and writing abilities, and to contribute significant analysis to the study, understanding, and amelioration of structural and individual causes of educational disparities. I appreciate the opportunity to extend my study to include other departments within the College and University and I look forward to expanding my limited teaching experience. My ultimate goals, as of yet undecided, include research within school systems and communities and a possible position in higher education. I feel that my background and aspirations fit well within the philosophies and goals of the Department of Educational Foundations. I am eager and prepared to continue as a dedicated and enthusiastic member of the graduate student body of the Department.

USE

THE

CATALOG

AS A RESOURCE Although Chappell had the benefit of being familiar with the doctoral program because she was a master’s student, you can use the knowledge you’ve gained from researching the program if you don’t know it first-hand. In particular, the program catalog can be a good resource when you are writing this section of the essay. “I read the Simmons College School of Library and Information Science’s catalog thoroughly several times before deciding to apply. The time was well spent,” recalls a master’s candidate in library and information science. “It is important to know what specific programs, services, faculty member expertise, and resources a grad school has to offer before writing the essay. The admissions committee will be looking for a good fit for their program.” In his personal statement, Bob Connelly, who earned an Ed.D. in educational administration from Seton Hall University in New Jersey, explains why he would be an asset to a cohort-based executive degree program in which students, who are all educational administrators, enter as a group and stay together during the time it takes to earn the degree. Now that I have made the decision to commit to the program, I need to balance optimism with the realities of meeting success in the program. I will outline personal qualities that are predictors of success. I believe that I possess the intellect, pragmatism, discipline, drive, and determination as well as the interpersonal skills to be a success in this doctoral program. I define success as completing the program within the prescribed time and contributing to the professional growth and development not only of myself but of the entire cohort that will be venturing through this exciting new program planned by Seton Hall.

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In this paragraph, Connelly explains why he is a good match for the Seton Hall program’s instructional design. In this case, being a good match involves more than academic considerations. It involves the self-discipline to keep up with a paced program and the interpersonal skills to contribute to the educational experience of the cohort.

DESCRIBE YOUR GOALS In most essays, you will have to explain how a graduate degree will help you achieve your goals. Even if you are not sure exactly what you want to do professionally, describe what you might be interested in doing once you receive the degree. Indicating that you have a purpose in obtaining a graduate degree shows that you are focused and motivated and have a real sense of the possibilities. Nestor Montilla’s objective in pursuing a master’s degree in public administration was to enable him to do meaningful work in the public sector. Here is how he explained his aspirations in his personal essay. Having a clear sense of what I want to be, I planned ahead and enrolled at John Jay College [as an undergraduate] to prepare myself academically with a public service career in mind. My . . . goal is to acquire the political skills and the academic credentials to explore career opportunities at the managerial and teaching levels in our government agencies. I aim to become a more productive, competent professional to help in both the rebuilding of our decaying communities and in the development of more feasible and effective means to better manage our public institutions. Undoubtedly, an advanced degree will help me to accomplish my plans, as the above considerations indicate. Then, after exploring career opportunities in the public sector, I will be professionally ready to pursue a doctoral or law degree as part of my ultimate career objectives. Notice that Montilla indicates both a short-term goal and possible long-term goals that his degree would help him achieve. Cindy Liutkus’s goal is long-term—to earn a Ph.D. in order to teach at the university level and continue to do field work. She explains what she hopes to accomplish with graduate study. Through my graduate study, I intend to expand my knowledge of geology while focusing on and conducting research in the areas of sedimentology and stratigraphy. In addition to my interest in these areas, my teaching assistantships and tutor position have been extremely gratifying, especially when I am able to motivate others and help them understand. Because of this, and due in large part to the influence of my undergraduate professor, Edward Cotter, Ph.D., I have decided that my ultimate goal is to teach at the college level. Teaching, for me, provides a vehicle for stimulation and exchange of ideas, an opportunity to remain current with related literature, and an appropriate atmosphere for original research design. Furthermore, continued field work would enhance my contribution to the classroom. In short, teaching will fulfill all my personal and professional goals.

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A final example is from Leslie Nelman’s personal statement. In her concluding paragraph, Nelman ties together her background, her proposed studies, and her career objectives. What I hope to do at this point in my life is integrate what I have learned in the business world with my true passion, (the German) language. I’ve spent the better part of this last year working intensively to refresh my language fluency and my understanding of contemporary German issues, politics, attitudes, etc. More specifically, I attended an eight-week advanced language course in Freiburg last summer, then spent two months each in Munich and Berlin, where I took individual instruction at the Goethe Institute while sharing an apartment with a single working woman. If my language skills are deemed adequate to the task, I would like to expand and fine-tune this background with the rigorous translation/interpretation training that the Monterey Institute offers. Given my personal circumstances, I do not have expectations of becoming an interpreter at the UN or the European Union; I anticipate, rather, working on a freelance basis on legal/finance/insurancerelated assignments, be it as interpreter or translator. I do look forward to finally playing a role in bridging the communication gap.

EXPLAIN SHORTCOMINGS IN YOUR BACKGROUND There is a difference of opinion on whether or not the personal essay is the place to explain any weaknesses in your academic or professional preparation when you are not directly asked to do so. Some people think that the essay should concentrate on a positive presentation of your qualifications. They feel that an explanation of poor GRE scores, for example, belongs in an addendum or cover letter. Others think that the essay is the place to address your application’s weaknesses. Perhaps a good rule of thumb is to address any weaknesses or shortcomings that are directly relevant to your proposed work in your field in the essay. For example, in her essay Cindy Liutkus explained why she lacked some necessary courses to pursue a graduate degree in geology and how she planned to make them up. On the other hand, if the weak spot in your application is not directly related to your field of study, you may prefer to address it in an addendum or cover letter. For example, if like many college freshmen you had a poor GPA, you can explain this separately. Try to put a positive spin on it, too. Explain, for example, how your GPA in your major was much higher, or how your GPA improved as you matured. Essentially, your decision as to where to address your weaknesses will depend on their importance and relevance to your pursuit of a graduate degree.

DRAFTING AND EDITING

FOLLOW

THE INSTRUCTIONS When you sit down to draft your essay, the first thing you should make sure of is that you are answering the question posed on the application. Be sure you read the instructions for each program’s personal statement carefully. Small differences

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in wording can affect how you approach writing the essay. Read these two sets of instructions for the personal statement. 1.

Please state your reasons for deciding to pursue a graduate degree in the field you have chosen. Include references to your past study and research in your chosen field, your plans for study at the university, including problems and issues you want to address, and your personal goals.

2.

The personal statement is an important part of your application. It is your opportunity to provide information about your background, interests and aspirations, and how they relate to your proposed academic program. In your statement, describe your reasons for pursuing graduate study, the program you hope to follow at the university, and the strengths and weaknesses of your preparation for graduate study. All personal statements should be double-spaced and typed, two to three pages.

These instructions cover more or less the same ground, but the second school asks you to describe the strengths and weaknesses of your preparation for graduate study, whereas the first school merely asks you to describe your past study. When writing the essay for the second school, therefore, you must be sure to address your preparation in greater detail. You will have to both describe and evaluate your readiness—or lack thereof.

DON’T WRITE TOO MUCH

OR TOO LITTLE The second thing you should keep in mind as you begin your draft is the length of the essay. Often, the length is specified; for example, the second school’s instructions above indicate that the statement should be two to three double-spaced pages. What should you do if length is not specified, as it is not in the first set of instructions above? Then write one to two typed pages. An essay that is shorter than one page does not allow room for you to develop your ideas. And an essay that is longer than two pages becomes a chore for the admissions committee to read. Finally, when you write your first draft, do not waste space by repeating information that the admissions committee can get from other parts of your application, like your transcript or résumé. Use the essay to provide them with new information or to highlight particular accomplishments.

REVIEW

THE

FIRST DRAFT Once you have drafted your essay, read the question again. Has your draft answered the question fully? If the essay is incomplete, go back and fill in the missing material. Then ask people for feedback. Although your spouse and friends may be helpful, you may get more valuable suggestions from faculty members who know you and who also know what a personal essay should be like. Ask whether you’ve included things you should leave out or should add things you’ve forgotten. Is the tone right? Have you achieved the right balance between boasting and being too modest? Are there any problems with organization, clarity, grammar, or spelling?

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Felecia Bartow, an M.S.W. candidate at Washington University in St. Louis, gave her drafts to several people. “It helped to have a couple of people (from different disciplines) read various drafts of my essays in order to give me feedback on the clarity and conciseness of my writing.” Jim Lipuma, a Ph.D. candidate in environmental science at the New Jersey Institute of Technology, recommends, “Proofreading goes without saying, but always read it to yourself, have someone else read it, and then read it aloud to someone. This will show all the problems, highlight the areas that need work, and allow for any weaknesses to be exposed.”

PREPARE THE FINAL DRAFT Once you’ve revised the essay and are satisfied with your final draft, ask someone with a sharp eye to proofread it for you. The final draft should be absolutely free of grammar and spelling errors, so do not rely on grammar- or spell-checks to find all of the errors. Once you are done, be sure to keep backup files as well as a hard copy. Although you won’t be able to use the whole essay for all your applications, you may be able to use parts of it. If you do work this way, be absolutely sure when you submit the final essays that you have not made any careless editing mistakes. “If you’re applying to multiple schools, make sure that you don’t have any ‘cut and paste’ errors in your application,” warns Neill Kipp, a Ph.D. candidate in computer science at Virginia Tech. “If you apply to Florida State in one letter and the University of Florida in another but forget to change every occurrence of the university name, then count on being the semester-long laughingstock of the admissions office.” Finally, if you are submitting the statement on separate sheets of paper rather than on the application form itself, put your name, social security number, and the question on the essay, and type “see attached essay” on the application form.

MAKE IT YOURS If after reading this appendix you are still daunted by the prospect of writing your personal statement, just put the whole task aside for a few days. You will find that the ideas, suggestions, and excerpts you’ve just read will trigger some mental activity and that soon you will have some ideas of your own to jot down. Also remember that it’s not necessary to have an exotic background or a dramatic event to recount in order to write a good essay and gain admission to a program. Admissions committees are looking for diversity—in gender, race, ethnicity, nationality, and socioeconomic status, to name some obvious characteristics. But they are also looking for people with diverse life experiences to add richness to their student body. Your background, which may seem perfectly ordinary to you, nevertheless has unique and relevant elements that can be assets to the program you choose. Your task is to identify and build upon these elements to persuade the admissions committee that you should be selected.

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