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California High School Exit Exam: Math by Jerry Bobrow, Ph.D.
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California High School Exit Exam: Math by Jerry Bobrow, Ph.D.
Contributing Authors George Crowder, M.A. Adam Bobrow, B.A. Dale Johnson, M.A.
CliffsTestPrep
®
California High School Exit Exam: Math by Jerry Bobrow, Ph.D.
Contributing Authors George Crowder, M.A. Adam Bobrow, B.A. Dale Johnson, M.A.
About the Author
Publisher’s Acknowledgments
Dr. Jerry Bobrow, Ph.D., is a national authority in the field of test preparation. As executive director of Bobrow Test Preparation Services, he has been administering the test preparation programs at over 25 California institutions for the past 27 years. Dr. Bobrow has authored over 30 national best-selling test preparation books, and his books and programs have assisted over two million test-takers. Each year, Dr. Bobrow personally lectures to thousands of students on preparing for graduate, college, and teacher credentialing exams.
Editorial Project Editor: Marcia L. Johnson Senior Acquisition Editor: Greg Tubach Copy Editor: Kathleen Robinson Production Proofreader: Arielle Mennelle Wiley Publishing, Inc. Composition Services
Author’s Acknowledgments My loving thanks to my wife, Susan, and my children, Jennifer, Adam, and Jonathon, for their patience and support in this long project. My sincere thanks to Michele Spence, former chief editor of CliffsNotes, for her invaluable assistance. I would also like to thank Marcia Johnson for final editing and careful attention to the production process. CliffsTestPrep® California High School Exit Exam: Math Published by: Wiley Publishing, Inc. 111 River Street Hoboken, NJ 07030-5774 www.wiley.com
Note: If you purchased this book without a cover, you should be aware that this book is stolen property. It was reported as “unsold and destroyed” to the publisher, and neither the author nor the publisher has received any payment for this “stripped book.”
Copyright © 2005 Jerry Bobrow, Ph.D. Published by Wiley, Hoboken, NJ Published simultaneously in Canada Library of Congress Cataloging-in-Publication Data Bobrow, Jerry. California high school exit exam—math / by Jerry Bobrow ; contributing authors, George Crowder, Adam Bobrow, Dale Johnson.— 1st ed. p. cm. — (CliffsTestPrep) ISBN 0-7645-5939-7 (pbk.) 1. Mathematics—Examinations, questions, etc. 2. California High School Exit Exam—Study guides. I. Crowder, George. II. Bobrow, Adam. III. Johnson, Dale. IV. Title. V. Series. QA43.B648 2004 510'.76—dc22 2004020172 ISBN: 0-7645-5939-7 Printed in the United States of America 10 9 8 7 6 5 4 3 2 1 1B/RV/RR/QU/IN No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400, fax 978-646-8600. Requests to the Publisher for permission should be addressed to the Legal Department, Wiley Publishing, Inc., 10475 Crosspoint Blvd., Indianapolis, IN 46256, 317-572-3447, or fax 317-572-4447. THE PUBLISHER AND THE AUTHOR MAKE NO REPRESENTATIONS OR WARRANTIES WITH RESPECT TO THE ACCURACY OR COMPLETENESS OF THE CONTENTS OF THIS WORK AND SPECIFICALLY DISCLAIM ALL WARRANTIES, INCLUDING WITHOUT LIMITATION WARRANTIES OF FITNESS FOR A PARTICULAR PURPOSE. NO WARRANTY MAY BE CREATED OR EXTENDED BY SALES OR PROMOTIONAL MATERIALS. THE ADVICE AND STRATEGIES CONTAINED HEREIN MAY NOT BE SUITABLE FOR EVERY SITUATION. THIS WORK IS SOLD WITH THE UNDERSTANDING THAT THE PUBLISHER IS NOT ENGAGED IN RENDERING LEGAL, ACCOUNTING, OR OTHER PROFESSIONAL SERVICES. IF PROFESSIONAL ASSISTANCE IS REQUIRED, THE SERVICES OF A COMPETENT PROFESSIONAL PERSON SHOULD BE SOUGHT. NEITHER THE PUBLISHER NOR THE AUTHOR SHALL BE LIABLE FOR DAMAGES ARISING HEREFROM. THE FACT THAT AN ORGANIZATION OR WEBSITE IS REFERRED TO IN THIS WORK AS A CITATION AND/OR A POTENTIAL SOURCE OF FURTHER INFORMATION DOES NOT MEAN THAT THE AUTHOR OR THE PUBLISHER ENDORSES THE INFORMATION THE ORGANIZATION OR WEBSITE MAY PROVIDE OR RECOMMENDATIONS IT MAY MAKE. FURTHER, READERS SHOULD BE AWARE THAT INTERNET WEBSITES LISTED IN THIS WORK MAY HAVE CHANGED OR DISAPPEARED BETWEEN WHEN THIS WORK WAS WRITTEN AND WHEN IT IS READ. Trademarks: Wiley, the Wiley Publishing logo, CliffsNotes, the CliffsNotes logo, Cliffs, CliffsAP, CliffsComplete, CliffsQuickReview, CliffsStudySolver, CliffsTestPrep, CliffsNote-a-Day, cliffsnotes.com, and all related trademarks, logos, and trade dress are trademarks or registered trademarks of John Wiley & Sons, Inc. and/or its affiliates. All other trademarks are the property of their respective owners. Wiley Publishing, Inc. is not associated with any product or vendor mentioned in this book. For general information on our other products and services or to obtain technical support, please contact our Customer Care Department within the U.S. at 800-762-2974, outside the U.S. at 317-572-3993, or fax 317-572-4002. Wiley also published its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books.
Table of Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Study Guide Checklist. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Format of CAHSEE Math . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Questions Commonly Asked About CAHSEE Math . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 How You Can Do Your Best . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 A Positive Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 The Elimination Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Avoiding the Misread . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
PART I: WORKING TOWARD SUCCESS Strategies for the Math Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Number Sense (Grade 7) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Standard Set 1.0: Working with Rational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Standard Set 2.0: Use Exponents, Powers and Roots, and Fractions with Exponents . . . . . 8 Statistics, Data Analysis, and Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Grade 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Standard Set 1.0: Analyzing Statistical Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Standard Set 2.0: Describing and Using Data Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Standard Set 3.0: Determining Probabilities and Making Predictions . . . . . . . . . . . . . . . . 16 Grade 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Standard Set 1.0: Collect, Organize, and Represent Data and Identify . . . . . . . . . . . . . . . 17 Measurement and Geometry (Grade 7) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Standard Set 1.0: Choose Appropriate Units of Measure and Conversions Between Measurement Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Standard Set 2.0: Compute Perimeter, Area, and Volume; Understand Effects of Scale Changes on These Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Standard Set 3.0: Know the Pythagorean Theorem, Understand Plane and Solid Shapes, Identify Attributes of Figures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Algebra and Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Standard Set 1.0: Using Algebraic Terminology, Expressions, Equations, Inequalities, and Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Standard Set 2.0: Interpret and Evaluate Expressions Involving Powers and Simple Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Standard Set 3.0: Graph and Interpret Linear and Nonlinear Functions . . . . . . . . . . . . . . 46 Standard Set 4.0: Solve Simple Linear Equations and Inequalities (Rational Answers) . . . 46 Mathematical Reasoning (Grade 7) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 Standard Set 1.0: Decide How to Approach Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 Standard Set 2.0: Use Strategies, Skills, and Concepts to Solve Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 Standard Set 3.0: Determine That a Solution Is Complete and Generalize to Other Situations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 Algebra I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 Standard Set 2.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 Standard Set 3.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 Standard Set 4.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 Standard Set 5.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 Standard Set 6.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
CliffsTestPrep California High School Exit Exam: Math
Standard Set 7.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 Standard Set 8.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 Standard Set 9.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 Standard Set 10.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 Standard Set 15.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
A Quick Review of Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 Symbols, Terminology, Formulas and General Mathematical Information. . . . . . . . . . . . . . . 85 Common Math Symbols and Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 Math Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 Important Equivalents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 Math Words and Phrases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 Mathematical Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 Arithmetic Diagnostic Test (Including Number Sense, Probability, Statistics and Graphs). . . 90 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Arithmetic Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Place Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Rounding Off . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Estimating Sums, Differences, Products and Quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 Decimals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 Conversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 Using Percents. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Signed Numbers (Positive Numbers and Negative Numbers). . . . . . . . . . . . . . . . . . . . . 102 Absolute Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Powers and Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Squares and Square Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 Scientific Notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 Parentheses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 Order of Operations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 Some Basic Probability and Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 Statistics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 Algebra Diagnostic Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 Algebra Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 Variables and Algebraic Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 Evaluating Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 Proportions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 Monomials and Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 Solving Quadratic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 Solving for Two Unknowns—Systems of Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 Basic Coordinate Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 Measurement and Geometry Diagnostic Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 Answers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Measurement and Geometry Review. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 Measures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 Coordinate Geometry and Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
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Table of Contents
PART II: FULL-LENGTH PRACTICE TESTS CAHSEE Practice Test #1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 Test #1—Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 Reviewing Practice Test 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 Review Chart. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 Reasons for Mistakes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 Answer Key . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 Number Sense (NS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 Statistics, Data Analysis, Probability (P) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 Algebra and Functions (AF) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 Measurement and Geometry (MG) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 Algebra I (AL) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 Mathematical Reasoning (MR). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
CAHSEE Practice Test #2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 Test #2—Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 Reviewing Practice Test 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 Review Chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 Reasons for Mistakes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 Answer Key . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 Number Sense . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 Statistics, Data, Analysis, Probability (P) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 Algebra and Functions (AF) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 Measurement and Geometry (MG). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 Algebra I (AI) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 Mathematical Reasoning (MR) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
CAHSEE Practice Test #3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 Test #3—Answers and Explanations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 Reviewing Practice Test 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 Review Chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 Reasons for Mistakes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 Answer Key . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 Number Sense . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 Statistics, Data Analysis, Probability (P) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 Algebra and Functions (AF) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 Measurement and Geometry (MG) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 Algebra I (AI). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 Mathematical Reasoning (MR) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
Arithmetic/Statistics and Probability Glossary of Terms . . . . . . . . . . . . . . . 265 Algebra Glossary of Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 Measurement and Geometry Glossary of Terms . . . . . . . . . . . . . . . . . . . . . 273 Final Preparations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 Final Preparation and Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 Finishing Touches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
v
Preface We know that passing the CAHSEE Math is important to you! And we can help. As a matter of fact, we have spent the last thirty years helping over a million test takers successfully prepare for important exams. The techniques and strategies that students and adults have found most effective in our preparation programs at 26 universities, county offices of education, and school districts make this book your key to success on the CAHSEE Mathematics. Our easy-to-use CASHEE Mathematics Preparation Guide gives you that extra edge by: ■ ■ ■ ■ ■ ■ ■
Answering commonly asked questions Introducing important test-taking strategies and techniques Reviewing the California mathematics standards Analyzing sample problems and giving suggested approaches Providing a quick review of mathematics with diagnostic tests Providing three simulated practice exams with explanations Including analysis charts to help you spot your weaknesses
We give you lots of strategies and techniques with plenty of practice problems. There is no substitute for working hard in your regular classes, doing all of your homework and assignments, and preparing properly for your classroom exams and finals. But if you want that extra edge to do your best on the CAHSEE Mathematics, follow our Study Plan and step-by-stem approach to success on the CAHSEE. Best of luck, Jerry Bobrow, Ph.D.
vii
Study Guide Checklist Check off each step after you complete it. ❑
1. Read “Mathematics Study Guide” available from your school or from the California Department of Education (CDE).
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2. Review any information or materials available online at cde.ca.gov.
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3. Look over the format of CAHSEE Math (p. 1).
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4. Read “Questions Commonly Asked About CAHSEE Math” (p. 2).
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5. Learn how you can do your best (p. 3).
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6. Carefully read “Part I: Working Toward Success” focusing on the sample problems and suggested approaches.
■
Number Sense (pp. 7–16) Statistics, Data Analysis, and Probability (pp. 16–26) Measurement and Geometry (pp. 26–45) Algebra and Functions (pp. 45–60) Mathematical Reasoning (pp. 60–68)
■
Algebra I (pp. 68–83)
■ ■ ■ ■
❑
7. Read the introductory material in “A Quick Review of Mathematics.”
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8. Take the diagnostic test in “Arithmetic” (pp. 90–91) and review any basic skills that you need to refine (pp. 93–115).
❑
9. Take the diagnostic test in “Algebra” (pp. 115–116) and review any basic skills that you need to refine (pp. 117–139).
❑ 10. Take the diagnostic test in “Measurement and Geometry: (pp. 139–142) and review any basic skills that you need to refine (pp. 145–155). ❑ 11. Take “CAHSEE Practice Test #1” (pp. 159–178). After you take the test, check your answers and analyze your results using the “Answer Key” (pp. 181), the “Review Chart” (pp. 180), and the “Answers and Explanations” (pp. 182–192). Review any basic skills that you need to refine from “A Quick Review of Mathematics” (pp. 85–155). ❑ 12. Take “CAHSEE Practice Test #2” (pp. 193–210). After you take the test, check your answers and analyze your results using the “Answer Key” (pp. 213), the “Review Chart” (pp. 212), and the “Answers and Explanations” (pp. 214–223). Review any basic skills that you need to refine from “A Quick Review of Mathematics” (pp. 85–155). ❑ 13. Take “CAHSEE Practice Test #3” (pp. 225–247). After you take the test, check your answers and analyze your results using the “Answer Key” (pp. 251), the “Review Chart” (pp. 250), and the “Answers and Explanations” (pp. 252–264). Review your weak areas and then selectively review the strategies and samples in “Part I: Working Toward Success” (pp. 5–157). ❑ 14. Read “Finishing Touches” (pp. 277).
ix
Introduction Format of CAHSEE Math The test consists of 92 questions—80 of those questions actually count toward your score. The following areas are covered (not necessarily in this order): Number Sense (NS)
14 questions
Statistics, Data Analysis, Probability (P)
12 questions
Measurement and Geometry (MG)
17 questions
Algebra and Functions (AF)
17 questions
Mathematical Reasoning (MR)
8 questions
Algebra I (AI)
12 questions
Total questions that count on score
80 questions
Plus trial questions for future tests
12 questions
Total questions for the Mathematics test
92 questions
Because this is a new test, the number of questions and the types of questions might be adjusted slightly in later tests. Also note that the trial questions can be scattered anywhere on the exam. The actual CAHSEE Math exam is given in two sessions—46 questions in each session. Although there is no time limit on the test, the approximate working times are estimated as follows: Mathematics
Approximate Working Times
Number of Questions
Session 1
1 hour 30 minutes
46
Session 2
1 hour 30 minutes
46
Both sessions will be administered on the same day with a break between the two sessions. You should be allowed to take the time you need within the school day to finish the exam.
1
CliffsTestPrep California High School Exit Exam: Math
Questions Commonly Asked About CAHSEE Math Q: What does CAHSEE Math cover? A: CAHSEE Math tests state content standards in Grades 6 and 7, and Algebra I. The exam covers number sense (including computation), statistics, data analysis (graphs and charts), probability, measurement and geometry, mathematical reasoning, and algebra. Q: How much time do I have to complete the test? A: There is no time limit for the exam. If you do not complete the test in the time period given, simply ask the proctor for additional time. You should be allowed to take the time you need within the school day to finish the exam. Q: How is the exam administered: A: The exam is administered in two sessions. Each session will have 46 questions. In each session you are only allowed to work on the questions given in that session. The approximate working time for each session should be 1 hour and 30 minutes. Q: Can I use a calculator on the exam? A: No. Calculators are not allowed on the exam unless there is a modification specified on the student’s record. Q: Can I use scratch paper on the exam? A: No. The use of scratch paper is not permitted during the exam. All scratch work must be done in the question booklet. Q: When will I first take the test? A: You will take the exam for the first time in the second part of 10th grade. Q: What is a passing score? A: Raw scores (the actual number of correctly answered questions) are converted to scaled scores ranging from 250 to 450. A passing score is 350 or higher. Because this is a new exam, you might wish to check with your school district to confirm the passing scores. Q: When do I find out if I passed? A: Score reports are mailed to the school district and to your home about two months after you take the test. Q: What if I don’t pass the exam in 10th grade? A: You have several chances to take the test as a junior and senior. Q: How should I prepare? A: Keep up with your class work and homework in your regular classes. There is no substitute for a sound education. As you get closer to your exit-level tests, using an organized test-preparation approach is very important. Carefully follow the Study Plan in this book to give you that organized approach. It shows how to apply techniques and strategies and help focus your review. Carefully reviewing the Study Guide for each exit-level exam available from your school district or the California Department of Education (CDE) also gives you an edge in doing your best. Q: Should I guess on the tests? A: Yes! Because there is no penalty for guessing, guess if you have to. If possible, try to eliminate some of the choices to increase your chances of choosing the right answer. Q: Can I write in the test booklet? A: Yes. You can do your work in the test booklet. Use the test booklet for scratch paper and to mark problems or draw diagrams. Your answer sheet must not have any stray marks, but your test booklet can be marked up.
2
Introduction
Q: Is the test given in more than one language? A: No. The test is only given in English. All students must pass CAHSEE Math in English to be eligible to get a high school diploma. Q: How can I get more information? A: More information and released exam questions can be found on the CDE’s Web site.
How You Can Do Your Best A Positive Approach Because every question is worth the same number of points, do the easy ones first. To do your best, use this positive approach: ■ ■ ■
First, look for the questions that you can answer and should get right. Next, skip the ones that give you a lot of trouble. (But take a guess.) Don’t get stuck on any one of the questions.
Here’s a closer look at this system: 1. Answer the easy questions as soon as you see them. 2. When you come to a question that gives you trouble, don’t get stuck. 3. Before you go to the next question, see if you can eliminate some of the incorrect choices to that question. Then take a guess from the choices left! 4. If you can’t eliminate some choices, take a guess anyway. Never leave a question unanswered. 5. Put a check mark in your test booklet next to the number of a problem for which you do not know the answer and simply guess. 6. After you answer all the questions, go back and work on the ones you checked (the ones that you guessed on the first time through). Don’t ever leave a question without taking a guess. There is no penalty for guessing.
The Elimination Strategy Sometimes the best way to get the right answer is to eliminate the wrong answers. As you read your answer choices, keep the following in mind: 1. Eliminate wrong answer choices right away. 2. Mark them out in your test booklet. 3. If you feel you know the right answer when you spot it, mark it. You don’t need to look at all the choices (although a good strategy for some questions is to scan the choices first). 4. Try to narrow your choices down to two so that you can take a better guess. Getting rid of the wrong choices can leave you with the right choice. Look for the right answer choice and eliminate wrong answer choices.
Here’s a closer look at the elimination strategy.
3
CliffsTestPrep California High School Exit Exam: Math
Take advantage of being allowed to mark in your test booklet. As you eliminate an answer choice from consideration, make sure to mark it out in your test booklet as follows: A ? B C ? D Notice that some choices are marked with question marks, signifying that they are possible answers. This technique helps you avoid reconsidering those marked-out choices you have already eliminated and helps you narrow down your possible answers. These marks in your test booklet do not need to be erased!
Avoiding the Misread One of the most common errors is the misread, that is, when you simply misread the question. A question could ask, if 3x + x = 20, what is the value of x + 2? This question doesn’t ask for the value of x, but rather the value of x + 2. A question could ask, which of the following is the best estimate of 511 × 212? Here, you are looking for the best estimate. A question could be phrased as follows: What is the probability that a spinner will not stop on green if you spin it one time? The word not changes the preceding question significantly. To avoid misreading a question (and therefore answering it incorrectly), simply circle or underline what you must answer in the question. For example, do you have to find x or x + 2? Are you looking for what can happen or what cannot happen? To help you avoid misreads, circle or underline the questions in your test booklet in this way: If 3x + x = 20, then what is the value of x + 2? Which of the following is the best estimate of 511 × 212? What is the probability that the spinner will not stop on green if you spin it one time? (Sometimes the test has key words underlined for you.) And, once again, these circles or underlines in your test booklet do not have to be erased.
A Quick Review of Basic Strategies 1. 2. 3. 4. 5. 6.
4
Do the easy problems first. Don’t get stuck on one problem—they’re all of equal value. Eliminate answers—mark out wrong answer choices in your test booklet. Avoid misreading a question—circle or underline important words. Take advantage of being allowed to write in the test booklet. No penalty for guessing means “never leave a question without at least taking a guess.”
PART I
W O R K I N G TOWAR D SUCCESS This section emphasizes how to approach question types that you will be seeing on the CAHSEE Math. Sample problems are followed by complete explanations and important test-taking techniques and strategies. Read this section carefully. Underline or circle key techniques. Make notes in the margins to help you understand the strategies, suggested approaches, and question types. Part I includes a variety of samples and suggested approaches from each math category: Number Sense Statistics, Data Analysis, Probability Measurement and Geometry Algebra and Functions Mathematical Reasoning Algebra I
Strategies for the Math Test Before you take a careful look at the standards with samples and suggested approaches, let’s review some specific testtaking strategies for the Math Test. These strategies can be very helpful in solving a problem. Circle or Underline. Take advantage of being allowed to mark on the test booklet by always circling or underlining what you’re looking for. This ensures that you are answering the right question. Pull Out Information. Pulling information out of word problems often gives you a better look at what you’re working with; therefore, you gain additional insight into the problem. Work Forward. If you quickly see the method to solve the problem, then do the work. Work Backward. In some instances, it’s easier to work from the answers. Don’t disregard this method because it at least eliminates some of the choices and can give you the correct answer. Eliminate. From initial information, or using common sense, you might be able to eliminate some of the answers. If you can eliminate an answer, mark it out immediately in the question booklet. Substitute Simple Numbers. Substituting numbers for variables can often be an aid to understanding a problem. Substitute simple numbers because you have to do the work. Use 10 or 100. Some problems deal with percent or percent change. If you don’t see a simple method for working the problem, try using the values of 10 or 100, and see what you get. Be Reasonable. Sometimes you immediately recognize a simple method to solve a problem. If this is not the case, try a reasonable approach and then check the answers to see which one is most reasonable. Sketch a Diagram. Sketching diagrams or simple pictures can also be very helpful in problem solving because the diagram might tip off either a simple solution or a method for solving the problem. Mark in Diagrams. Marking in or labeling diagrams as you read the questions can save you valuable time. Marking can also give you insight into how to solve a problem because you have the complete picture clearly in front of you. Approximate. If it appears that extensive calculations are going to be necessary to solve a problem, check to see how far apart the choices are and then approximate. The reason for checking the answers first is to give you a guide to how freely you can approximate. Glance at the Choices. Some problems might not ask you to solve for a numerical answer or even an answer including variables. Rather, you might be asked to set up the equation or expression without doing any solving. A quick glance at the answer choices helps you know what is expected.
Number Sense (Grade 7) There are 14 problems involving number sense on the CAHSEE. These problems are grouped together. The areas covered include:
Standard Set 1.0: Working with Rational Numbers ■ ■ ■ ■ ■ ■ ■ ■ ■
Use scientific notation—read, write, and compare. Perform operations with whole numbers, fractions, and decimals (terminating)—add, subtract, multiply, and divide. Apply the rules for powers and exponents. Convert fractions to decimals and percents. Compute with fractions, decimals, and percents. Estimate. Calculate percentage increase and decrease. Solve problems involving discounts, markups, commissions, and profits. Compute simple and compound interest. 7
Part I: Working Toward Success
Standard Set 2.0: Use Exponents, Powers and Roots, and Fractions with Exponents ■ ■ ■ ■ ■ ■ ■
Understand negative whole-number exponents. Multiply and divide exponents with a common base. Add and subtract fractions finding common denominators by using factors. Multiply, divide, and simplify rational numbers using exponent rules. Understand perfect square numbers. Estimate square roots. Define and determine absolute value.
Samples with Suggested Approaches 1. 2.9 × 103 = A. B. C. D.
.290 29 290 2,900
This is a straightforward conversion question. If you know how to do a problem, work forward carefully. The simplest method is probably to move the decimal point three places to the right and add zeros. Since this is the process when multiplying by a power of 10, simply move the decimal point the same number of places as the power of 10, as follows:
2.9 × 103 = 2.900
[three places to the right]
Another method is to first, change 103 to 10 × 10 × 10, which equals 1,000. Next, multiply 2.9 by 1,000, giving 2,900. The correct answer is D. Also, 2.9 × 103 is written in scientific notation. To change from scientific notation, simply move the decimal point according to the exponent of 10. 2. The speed of light is approximately 300,000,000 miles per second. How is this speed represented in scientific notation? A. B. C. D.
3.0 × 108 miles per second 3.0 × 109 miles per second 30 × 107 miles per second 300 × 106 miles per second
Underline or circle the key words in the question. The key words here are scientific notation. A number written in scientific notation is a number equal to or greater than 1, but less than 10 multiplied by a power of 10. So you can quickly eliminate Choices C and D because they do not start with a number between 1 and 10. Next, be aware of the number you are working with or converting, in this case 300,000,000. To change this number to scientific notation, simply place the decimal point to get a number between 1 and 10, and then count the zeros to the right of the decimal to get the power of 10.
8
Strategies for the Math Test
300,000,000 = 3 00,000,000 × 108 Since there are eight zeros, the power of 10 is 8. Therefore, this speed is correctly represented in scientific notation as 3.0 × 108 miles per second. The correct answer is A. 3. A family of four goes out to dinner at a restaurant. Three family members order appetizers at $5 each. Each family member orders an entrée at a cost of $14 each and a drink at $2 each. Two family members order dessert at a cost $6 each. If nothing else is ordered by the family, how much money do they spend at the restaurant in total? A. B. C. D.
$79 $83 $91 $96
You are looking for the total spent. In this case, the key word is actually pointed out for you. Next, pull out important information as follows: 3 appetizers at $5 each [3 × 5 = $15] 4 entrees at $14 each [4 × 14 = $56] 4 drinks at $2 each [4 × 2 = $8] 2 desserts at $6 each [2 × 6 = $12] Now simply add the amounts. 15 56 8 + 12 91 The total amount spent by the family is $91. The correct answer is C. 4. 7 - c 1 + 1 m = 8 2 4 1 A. 8 B. 1 4 1 C. 2 D. 5 8 This problem involves simply adding and subtracting fractions, but be careful. Since you have three fractions with different denominators, you must find the LCD (lowest common denominator). In this case each denominator divides into 8 evenly, so the LCD is 8. Next, you must convert each fraction to an equivalent fraction with a denominator of 8.
9
Part I: Working Toward Success
1 = 4 and 1 = 2 2 8 4 8 Now you have
7- 4+2 8 c8 8m
Work inside the parentheses first. 7- 6 =1 8 c8m 8 The correct answer is A. 5. To add the fractions 3 and 9 , you must find a common denominator. Which of the following is the prime5 20 factored form of the lowest common denominator (LCD)? A. B. C. D.
2×5 2 × 10 2×2×5 5 × 20
The key words are prime-factored form. You can quickly eliminate Choices B and D because they are not in prime-factored form—composed only of prime numbers. Working from the remaining answers, A and C, you can see that Choice A, 2 × 5 (which is 10), cannot be the common denominator for 3 and 9 . So eliminate 20 5 Choice A, leaving Choice C. The correct answer is C. If you want to work the problem forward, find the LCD of 5 and 20. Since 20 is a multiple of 5, the LCD is 20. The prime factors of 20 can be found using a factor tree as follows:
20 2 × 10 2 × 2 × 5 So the prime factorization of 20 is 2 × 2 × 5. 6. Which of the following results in a positive number? A. B. C. D.
(–2) × (–3) (–2) × (–3) × (–4) (–2) + (–3) + (–4) (–2) + (–3) + (4)
First, circle or underline what you are looking for, in this case, a positive number. If you get an answer that you are absolutely sure is right, you don’t need to look at the rest of the choices. In this case, Choice A is a negative times a negative, which is always a positive, so the correct answer must be A. There is no need to look at the other choices.
10
Strategies for the Math Test
For your information, the other choices work out as follows: Choice B: A negative times a negative times a negative is always negative. (An odd number of negative signs in multiplication or division is negative.) Choice C: Because you are adding only negative numbers, the answer must be negative. Choice D: (–2) + (–3) + (4) is the same as (–5) + (4), which is –1. 7. If John gets 3 out of 5 questions correct on his math test, what percentage does he get correct? A. B. C. D.
35% 40% 60% 66%
First, note what you are looking for, percentage correct. Next, pull out information. 3 out of 5 or 3 5 Since 3 = 60% , the correct answer is C. You should memorize some fraction-to-decimal-to-percent equivalents, 5 such as: 1 = .20 = 20% , 2 = .40 = 40% , and so on. 5 5 Another method is to simply work out the percentage. To find what percentage 3 out of 5 is, divide 3 (questions answered correctly) by 5 (questions attempted). This leaves you with a success rate of 0.6, or 60%. By the way, you can eliminate Choices A and B if you realize that 3 out of 5 is greater than 50%. 8. Little Billy weighed 60 pounds last year. This year he weighs 75 pounds. What is the percentage increase of Billy’s weight from last year to this year? A. B. C. D.
15% 20% 25% 75%
First, underline what you are looking for, percentage increase. To find percentage increase or decrease, use the following formula: Difference Starting point Pulling information out of the problem gives you: 75 - 60 = 15 = 1 = .25 = 25% 60 60 4 The correct answer is C.
11
Part I: Working Toward Success
Memorize some fraction-to-decimal-to-percent equivalents. A recommended list follows: 1 = .10 = 10% 10 2 = 1 = .20 = 20% 10 5 3 = .30 = 30% 10 4 = 2 = .40 = 40% 10 5 5 = 1 = .50 = 50% 10 2 6 = 3 = .60 = 60% 10 5 7 = .70 = 70% 10 8 = 4 = .80 = 80% 10 5 9 = .90 = 90% 10 1 = 1.00 = 100% 1 = .25 = 25% 4 3 = .75 = 75% 4 9. Carol’s factory produced 40,000 disks in 1987. In 1986, her factory produced 50,000 disks. What was the percent decrease in production from 1986 to 1987? A. B. C. D.
10% 20% 25% 30%
First, you are looking for percent decrease. To find percent decrease, set up a ratio by dividing the change in production between the two years by the total production in the starting year: 50, 000 - 40, 000 10, 000 1 = = 50, 000 50, 000 5 Now change 1 to a percent. 5 Divide 5 into 1, which gives .20 or 20%. The correct answer is B. 10. Patrice puts $600.00 in a bank. If her money earns 12% simple interest every year, how much interest does she make at the end of 5 years? A. B. C. D.
$60 $72 $300 $360
Underline what you are looking for, simple interest. The basic formula for simple interest is: Interest = principal × rate × time (The time is in years.)
12
Strategies for the Math Test
Next, pull out important information: Principal = $600 Rate = 12% or .12 Time = 5 years Now plug these numbers into the formula: Interest = (600)(.12)(5) = 360 The correct answer is D. 11. Two thousand dollars is deposited in a savings account that pays 6% annual interest compounded semiannually. To the nearest dollar, how much is in the account at the end of the year? A. B. C. D.
$2,060 $2,120 $2,122 $2,247
Although compound interest problems can take a little extra work, they are straightforward. You should work this problem step by step. A 6% annual interest rate, compounded semi-annually (every half year) is the same as a 3% semi-annual interest rate. At the end of the first half of the year, the interest on $2,000 at 3% is: $2,000 × .03 = $60 So the new balance at the end of the first half of the year is: $2,000 + $60 = $2,060 At the end of the first full year, the interest on $2,060 at 3% is: $2,060 × .03 = $61.80 ≈$62 So the new balance at the end of the first full year is: $2,060 + $62 = $2,122 The correct answer is C. Choice B is correct only if the problem involves simple interest, not interest compounded semi-annually. 12. The regular price of a baseball glove at Big Three Sporting Goods is $50. If the glove is on sale for 30% off, what is the sale price of the baseball glove? A. B. C. D.
$20 $25 $30 $35
13
Part I: Working Toward Success
First, you are looking for the sale price. Next, pull out information. Regular price: $50 Sale: 30% off Now multiply the regular price by the percent off to get the amount saved. 50 × 30% = 50 × .30 = 15 Since the regular price is $50 and the amount saved is $15, the sale price must be $35. The correct answer is D. -3 13. 10 - 6 10 A. B. C. D.
= 10–3 10–2 102 103
This is a straightforward mechanical problem. You must know the rules for dividing numbers of the same base with exponents. When you divide numbers with exponents and the bases of the numbers are the same, then you keep the same base and subtract the exponents. For example, xa divided by xb is xa–b. -3 In this case, 10 - 6 = 10 - 3 ' 10 - 3 ] - 6 g = 10 - 3 + 6 = 10 3 . 10
The correct answer is D. 14. (63)9 = A. B. C. D.
66 612 627 6729
This is another straightforward mechanical problem. You must know the rule for taking the power of a number that already has an exponent. The rule is to simply multiply the exponents. In algebraic terms, (xa)b = xab. Using the numbers in the problem you have: (63)9 = 6(3 x 9) = 627 The correct answer is C. 15. (3)–2 = A. B. C. D.
14
–6 -1 9 1 6 1 9
Strategies for the Math Test
Similar to the previous problem, this is a straightforward mechanical problem. You must know the rule for negative exponents. To remove the negative sign in front of the exponent, drop the number and the exponent under the number 1 in a fraction. In algebraic terms, x - a = x1a . Using the numbers in the problem you have: 3 - 2 = 12 = 1 9 3 The correct answer is D. 16. 55 × 53 = A. B. C. D.
58 515 258 2515
To answer this straightforward mechanical problem, you must know the rule for multiplying numbers of the same base with exponents. To multiply numbers of the same base with exponents, simply keep the same base number and add the exponents. In algebraic terms, ax × ay = a(x + y). Using the numbers in the problem you have: 55 × 53 = 5(5+3) = 58 The correct answer is A. 17. The square root of 90 is between A. B. C. D.
9 and 10 10 and 11 11 and 12 12 and 13
This problem is most easily answered by working backward, from the answers. Start with Choice A, 9 and 10. If you square 9, that is, 9 × 9, you get 81, which is below 90. Next, try squaring the second number, 10, and you get 100. Since 90 is between 81 and 100, the square root of 90 is between 9 and 10. The correct answer is A. You can work this problem forward by approximating the square root of 90. First, find the closest perfect square number below 90. That is 81. Next, find the closest perfect square number above 90, which is 100. Since 90 is between 81 and 100, it falls somewhere between 9 and 10. 9
10
81 < 90 < 100 18. If x = 6, what is the value of x? A. B. C. D.
–6 or 0 –6 or 6 0 or 6 0 or 12
15
Part I: Working Toward Success
You can work this problem forward, using the definition of absolute value. If you know that absolute value refers to actual distance on a number line, and not direction, then it is evident that x can be –6 or 6. The correct answer is B. You can also work this problem by plugging in the answers, but you still need to know how to work with absolute values. 19. What is the absolute value of –9? A. B. C. D.
–9 –3 1 9 9
This question is just testing your knowledge of the definition of absolute value. Since absolute value refers to actual distance on a number line, and not direction (positive or negative), the absolute value of –9 is 9. The correct answer is D.
Statistics, Data Analysis, and Probability There are 12 problems involving statistics, data analysis, and probability on the CAHSEE. These problems are grouped together. The areas covered include:
Grade 6 Standard Set 1.0: Analyzing Statistical Measurements ■ ■ ■
Mean Mode Median
Standard Set 2.0: Describing and Using Data Samples ■ ■
Identify claims. Evaluate validity of claims.
Standard Set 3.0: Determining Probabilities and Making Predictions ■ ■ ■ ■ ■ ■
16
Represent outcomes for compound events; Express probability of each outcome. Use ratios, proportions, and decimals (between 0 and 1) and percentages (between 0 and 100) to represent outcomes. Verify that probabilities are reasonable. Know that if P is the probability of an event occurring, then 1 – P is the probability of an event not occurring. Understand independent and dependent events and how they are different.
Strategies for the Math Test
Grade 7 Standard Set 1.0: Collect, Organize, and Represent Data and Identify Relationships ■ ■ ■
Know various forms of data displays. Display data and compare data. Describe and display data on a scatterplot.
Samples with Suggested Approaches 1. The Chicago Bulls scored 98, 110, 112, and 120 points in four consecutive games. What is the mean score for the Chicago Bulls in these four games? A. B. C. D.
110 112 120 440
Underline the key words mean score. Now that you know what you are looking for, pull out important information. 98, 110, 112, 120 To find the mean score, you must find the total of the scores and then divide that total by the number of games played. 98 + 110 + 112 + 120 = 440 total points Since four games were played, divide 440 by 4, giving a mean score of 110. The correct answer is A. 2. The following table shows Teresa’s scores on her first-semester economics exams.
Date September 27 October 6 October 12 November 2 November 14 November 21 December 4 December 13
Score 90% 80% 90% 87% 83% 91% 88% 96%
What is her median score on the economics exams? A. 87 1 2 B. 88 C. 89 D. 90
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Part I: Working Toward Success
Focus on what you are looking for—in this case, the median score. To find the median, first put the scores in numerical order. 80, 83, 87, 88, 90, 90, 91, 96 Next, count in halfway. Since there are an even number of scores, you must take the average of the two middle scores. The average of 88 and 90 is 89. The correct answer is C. 3. The following chart shows the amount of money collected during a school’s magazine sale.
Magazine Sale Income Cost of magazine
$10
$12
$15
$19
Number sold
18
9
5
5
Which of the following is the mode of the cost of the magazines sold during the magazine sale? A. B. C. D.
$5 $10 $12 $15
You are looking for the mode. The mode is the cost that appears the most. Eighteen $10 magazines were sold. The correct answer is B. 4. The following graph shows the acceleration test results of the Roadster II. Acceleration Test Results of the Roadster II 100 90 80
miles per hour
70 60 50 40 30 20 10 0 1
2
3
4
5
seconds
6
7
8
According to the preceding line graph, the Roadster II accelerated the most between A. B. C. D.
18
1 and 2 seconds 2 and 3 seconds 3 and 4 seconds 4 and 5 seconds
Strategies for the Math Test
The key words here are accelerated the most. To answer this question, you must understand how the information is presented. The numbers on the left side of the graph show the speed in miles per hour. The information at the bottom of the graph shows the number of seconds. The movement of the line can give important information and show trends. The more the line slopes upward, the greater the acceleration. The greatest slope upward is between 3 and 4 seconds. The Roadster II accelerates from about 40 to about 80 miles per hour in that time. The correct answer is C. The following graph shows how John spends his monthly paycheck.
20% in the bank
25% entertainment
20% car and bike repair
15% his hobby 10% misc. items
10% school supplies
How John Spends His Monthly Paycheck
5. John spends one quarter of his monthly paycheck on A. B. C. D.
his hobby car and bike repair entertainment school supplies
Focus on what you are looking for, one quarter of his monthly paycheck. To answer this question, you must be able to read the graph and apply some simple math. The information is given in the graph. Each item is given along with the percent of money spent on that item. Because one quarter is the same as 25%, entertainment is the answer you are looking for. The correct answer is C. Use the preceding circle graph to answer the following question: 6. If John receives $100 on this month’s paycheck, how much does he put in the bank? A. B. C. D.
$2 $20 $35 $60
To answer this question, you must again read the graph carefully and apply some simple math. John puts 20% of his income in the bank. Twenty percent of $100 is $20. So, he puts $20 in the bank. The correct answer is B.
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Part I: Working Toward Success
Number of Delegates Committed to Each Candidate
Candidate 1
Candidate 2
Candidate 3
Candidate 4
0
200
400
600
800
Delegates
7. Based on the preceding graph, Candidate 1 has approximately how many more delegates committed than Candidate 2? A. B. C. D.
150 200 250 400
To understand this question, you must be able to read the bar graph and make comparisons. The graph shows the “Number of Delegates Committed to Each Candidate,” with the numbers given along the bottom of the graph in increases of 200. The names are listed along the left side. Candidate 1 has approximately 800 delegates (possibly a few more). The bar graph for Candidate 2 stops about three quarters of the way between 400 and 600. Now, consider that halfway between 400 and 600 is 500. So, Candidate 2 is at about 550. 800 – 550 = 250 The correct answer is C.
20
Strategies for the Math Test
8. The following graph shows the average decibel levels associated with various common sources of noise. Listening to high levels of noise for a long time can damage the eardrum and cause loss of hearing.
Decibel Levels of Common Noise Sources 130 120 110 100
decibels
90 80 70 60 50 40 30 20 10 gunfire
nearby thunder
passing train
nightclub
orchestra
factory
noisy office
shout
0
According to the preceding graph, which of the following sources of noise is most likely to damage the eardrum? A. B. C. D.
an orchestra a passing train nearby thunder gunfire
To answer this question, you must be able to read the graph and understand the information included. The decibels of noise are listed along the left-hand side in increases of 10. The common sources of noise are listed along the bottom of the graph. Since gunfire has the highest decibel rating, it is the loudest of the choices, and is therefore the most likely to damage the eardrum. The correct answer is D.
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Part I: Working Toward Success
9. The following diagram shows a spinner that is equally divided into four sections.
red
yellow
green
blue
In spinning the spinner only once, what is the probability of spinning red, yellow, or blue? A. B. C. D.
1 4 1 3 1 2 3 4
Underline what you are looking for, probability of red, yellow, or blue. To set up this probability problem, remember the formula: number of desired outcomes = probability total possible outcomes There are 3 chances out of 4 total possibilities of spinning either red, yellow, or blue. Thus, the probability is 3 . 4 The correct answer is D.
22
Strategies for the Math Test
10. Use the equally spaced spinner in the following figure to answer this question.
8
1
7
2
6
3 5
4
In one spin, what is the probability of getting a prime number? A. B. C. D.
1 8 1 2 5 8 3 4
There are four prime numbers on the spinner—2, 3, 5, 7. (1 is not a prime number.) Since there are 8 possibilities, the probability is 4 out of 8 or 1 . 2 The correct answer is B. 11. Three green, four red, and seven yellow marbles are placed in a bag. If a yellow marble is selected first and not replaced in the bag, what is the probability of selecting a yellow marble on the second draw? A. B. C. D.
3 7 4 7 1 2 6 13
First, focus on what you are looking for, probability of a yellow marble on the second draw. Since there are a total of 14 marbles and 7 are yellow, then the probability of selecting a yellow marble on the first draw is 7 . If a 14 yellow marble is selected on the first draw, then 6 yellow marbles are left out of 13 total marbles. The correct answer is D, 6 . 13
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Part I: Working Toward Success
12. What is the probability of tossing a penny twice so that both times it lands heads up? A. B. C. D.
1 8 1 4 1 3 1 2
The probability of throwing a head in one throw is: chances of a head =1 total chances ^1 head + 1 tailh 2 Since you are trying to throw a head twice, multiply the probability for the first toss, 1 , by the probability for 2 the second toss (again 1 ). Thus, 1 # 1 = 1 , and 1 is the probability of throwing heads twice in two tosses. 2 2 2 4 4 Another way of approaching this problem is to look at the total number of possible outcomes:
1. 2. 3. 4.
First Toss H H T T
Second Toss H T H T
Thus, there are four different possible outcomes. There is only one way to throw two heads in two tosses. Thus, the probability of tossing two heads in two tosses is 1 out of 4 total outcomes, or 1 . The correct answer is B. 4 13. Brenda is at a baseball hat store. In this store, there are only five different choices for hat color: red, orange, yellow, green, and blue. There are only four different choices for teams: Dodgers, Indians, Yankees, and Cubs. Each team hat is made in each color. If Brenda is handed a hat from the store at random, what is the probability that it is an orange Cubs hat? A. B. C. D.
1 25 1 20 1 5 1 4
You are looking for an orange Cubs hat. The probability that the hat is orange is 1 . The probability that the hat 5 is a Cubs hat is 1 . To find the probability that the hat is both an orange hat and a Cubs hat, you multiply the 4 probability of each, 1 # 1 = 1 because these are independent probabilities. Therefore, you know that the 4 5 20 probability that the hat is an orange Cubs hat is 1 . The correct answer is B. 20
24
Strategies for the Math Test
14. Bob Newhart is the host of a game show that has four contestants. Bob has four cards behind his back. One is blue, one is green, one is yellow, and one is orange. Each contestant can make only one selection and must keep that selection. The contestant who selects the green card wins a new sports car. The first contestant, Koji, selects a card. If the card he selects is yellow, what is the probability that the second contestant, Keiko, wins the new sports car? A. B. C. D.
1 8 1 4 1 3 1 2
To answer this question, you must follow the information given carefully. When Koji picks a card, the probability of his getting the green card is 1 . Since he eliminates one card that is not 4 the green card, there are only three cards remaining, and one of them is green. Therefore, the probability that Keiko picks the green card and wins a new sports car is 1 . The correct answer is C. 3 15. A small container is filled with buttons. Each button is the same size and shape. The container has six yellow buttons, three green buttons, five blue buttons, and four brown buttons. Ricardo takes out three green buttons and does not put them back into the container. If Ricardo selects another button at random, what is the probability that the button is yellow? A. B. C. D.
1 18 1 6 2 5 3 5
You should first focus on what you are looking for, the probability of selecting a yellow button (after the three buttons are removed). Next, pull out important information: 6 yellow 3 green 5 blue 4 brown Note the action in the problem—three green buttons are removed. Now you should carefully attack the problem. A total of 18 buttons are in the container at the start. If the 3 green buttons are removed, only 15 buttons remain in the container. You know that 6 are yellow, 5 are blue and, 4 are brown. Therefore, the probability of picking a yellow button out of those 15 in the container is 6 , which reduces to 2 . The correct answer is C. 15 5
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Part I: Working Toward Success
3 2 1
50 40 (per person)
C.
$ Spent weekly at market
A.
No. of cars per household
16. Which of the following scatterplots represents a positive correlation?
30 20 10 10
40
D.
30 20 10
10
20
30
40
No. of cigarettes smoked
B.
No. of people inside the club
1 2 3 4 No. of drivers per household
20
30 40 No. of people
50
60
50 40 30 20 10 10
20
30
40
As the number of cars increases, the number of drivers increases steadily. When the dots show a pattern going up to the right, the scatterplot shows a positive correlation. The correct answer is A.
Measurement and Geometry (Grade 7) There are 17 problems involving measurement and geometry on the CAHSEE. These problems are grouped together. The areas covered include:
Standard Set 1.0: Choose Appropriate Units of Measure and Conversions Between Measurement Systems ■ ■ ■ ■ ■
Compare weights, capacities, geometric measures, times, and temperatures in different measurement systems. Understand scale models and drawings. Solve problems using measures expressed as rates and products. Check units of solutions. Check reasonableness of answers.
Standard Set 2.0: Compute Perimeter, Area, and Volume; Understand Effects of Scale Changes on These Measures ■
■ ■
26
Find the perimeter and area of basic two-dimensional figures—triangles, parallelograms, rectangles, squares, and circles. Find the surface area and volume of basic three-dimensional figures—prisms and cylinders. Estimate and compute the area of complex and irregular two- and three-dimensional figures using basic geometric shapes.
Strategies for the Math Test
■
■ ■ ■
Compute the perimeter, total surface area, and volume of basic three-dimensional objects made from rectangular solids. Understand how scale factor affects the surface area and volume of a solid. Relate measurement changes involving scale drawings. Convert between units.
Standard Set 3.0: Know the Pythagorean Theorem, Understand Plane and Solid Shapes, Identify Attributes of Figures ■ ■ ■ ■ ■
Plot simple figures on coordinate graphs. Determine dimensions and areas of simple figures on coordinate graphs. Translate and reflect images on coordinate graphs. Use the Pythagorean theorem and its converse. Understand the congruence of figures and the relationships between sides and angles. The following formulas are not provided on the exam, you should know them.
Area of a parallelogram: A = bh (where b is the base and h is the height) Area of a triangle: c A = 1 bh m 2 Volume of a rectangular solid: V = lwh (where l is the length, w is the width, and h is the height) Circumference of a circle: C = πd (where d is the diameter) Other formulas are provided.
Samples with Suggested Approaches 1. One meter is— A. B. C. D.
10 centimeters 100 decimeters 1,000 millimeters 10,000 kilometers
You should know some of the basic conversions: 1,000 millimeters = 1 meter. The correct answer is C. Other basic conversions are as follows: 10 decimeters = 1 meter 100 centimeters = 1 meter 1,000 millimeters = 1 meter 1 kilometers = 1 meter 1000 or 1 kilometer = 1,000 meters
27
Part I: Working Toward Success
2. Jillian weighs 60 kilograms (kg). Approximately how many pounds (lbs) does she weigh? (1 kg ≈ 2.2 lbs) A. B. C. D.
25 lbs 30 lbs 120 lbs 132 lbs
First, you are looking for how many pounds. Next, pull out important information: Jillian: 60 kg 1 kg ≈ 2.2 lbs Now, since 1 kg is approximately 2.2 lbs, you should multiply 60 by 2.2 lbs. 60 × 2.2 = 132 lbs So Jillian weighs approximately 132 lbs. The correct answer is D. 3. Jan Ove practices ping pong for 5 hours every day. How many minutes does Jan Ove practice in a week? A. B. C. D.
35 300 420 2,100
You are looking for minutes per week. First, change hours per day to minutes per day. Since Jan Ove practices 5 hours every day, multiply the hours by 60 to get the minutes per day. He practices 300 minutes every day. Since he practices every day, he practices 7 days a week. To find out the number of minutes he practices per week, simply multiply the minutes per day by the number of days in a week. 7 days × 300 minutes = 2,100 minutes per week The correct answer is D. 4. If a car is traveling at 1 mile per minute, how many miles per hour is it traveling? A. B. C. D.
1 60 1 60 100
Note that you are looking for miles per hour. Since the car is going 1 mile per minute and there are 60 minutes in an hour, the car is going 60 miles per hour. The correct answer is C. Choices A and B can quickly be eliminated because they are not reasonable.
28
Strategies for the Math Test
5. Daniel read nine comic books at a rate of six comic books per hour. If each comic book was the same number of pages and took the same amount of time to read, how long did it take him to read all nine comic books? A. B. C. D.
30 minutes 54 minutes 90 minutes 150 minutes
First, note that you are looking for how long it took him to read nine comic books. Next, pull out important information. 9 comic books 6 comic books per hour Now you can simply divide the number of comic books by the rate (comic books per hour) and get: 9 =13 =11 2 6 6 Since 1 1 hours is 90 minutes, the correct answer is C. 2 Another method is as follows: If Daniel reads 6 books in an hour, that means he reads 1 book every 10 minutes (1 hour = 60 minutes). Therefore, if he is reading 9 books, it takes 9 × 10 minutes, or 90 minutes. 6. Use the following diagram to answer the question that follows. C
D r O
A
10
B
Circle O is inscribed in square ABCD as shown in the preceding figure. The area of the shaded region in square units is approximately— (A = πr2 and π ≈ 3.14) A. B. C. D.
10 25 30 50
Underline or circle the words area of the shaded region. There are several approaches to this problem. One solution is to first find the area of the square. 10 × 10 = 100
29
Part I: Working Toward Success
Then subtract the approximate area of the circle. The radius of the circle is half the length of one side of the square. So,
A = π(5)2 = 25π or about 75 (3 × 25)
Now,
100 – 75 = 25.
Therefore, the total area inside the square but outside the circle is approximately 25. One quarter of that area is shaded. Therefore, 25 is approximately the shaded area. The closest answer is 10, Answer A. 4 A more efficient method is to first find the area of the square. 10 × 10 = 100 Then divide the square into four equal sections as follows. D
C
A
10
B
Since a quarter of the square is 25, the only possible answer choice for the shaded area is 10. The correct answer is A.
C
7. In the preceding diagram, a square and a circle intersect. If C is the center of the circle, what percent of the circle remains unshaded? A. B. C. D.
25% 35% 45% 50%
You are looking for the percent unshaded. Because one angle of a square is 90˚ and the circle is 360˚, the percent is 90 or 1 = 25% . The correct answer is A. 4 360
30
Strategies for the Math Test
8. The two figures drawn below are similar. What is the height, h, of the second figure in inches?
h 4″ 8″
A. B. C. D.
24″
4 inches 8 inches 12 inches 16 inches
To find the height, h, of the second figure, you should set up a proportion. 4= h 8 24 Since 8 goes into 24 three times, the second figure is three times the size of the first. So 3 times 4 is 12. The correct answer is C. Or you can solve this by cross-multiplying 96 = 8h 4 8
h 24
Then divide by 8 96 = 8h 8 8 12 = h Or reduced Re duced 1= h 2 24 24 = h 2 12 = h 9. Genevieve has a fish tank custom made. This fish tank holds 48 gallons of water. She needs a tank that holds only 24 gallons of water. Which of the following changes should be made, based on the dimensions of the original tank, to satisfy Genevieve’s needs while she is custom making a new fish tank? A. B. C. D.
Divide length, width, and height by 2. Divide only the length by 0.5. Multiply the width and length by 0.5. Multiply only the height by 0.5.
31
Part I: Working Toward Success
Since the volume of the tank equals length times width times height (V = lwh), you only need to make one of those dimensions half the size to cut the entire volume in half. To make a dimension half of its original size, you can multiply by 0.5 or divide by 2. The correct answer is D. School
6 miles
First stop
8 miles
Second stop
10. A bus leaves from school and makes two stops. The first stop is 6 miles south of school, and the second stop is 8 miles east of the first stop. If the bus drives in a straight line back to school from the second stop, how many total miles is it traveling after school? A. B. C. D.
5 10 14 24
First, you are looking for total miles. Next, focus on the diagram given. Your plan should be to find the longest side (hypotenuse) of the right triangle, and then total the lengths of all the sides. You might recognize that this right triangle is in the ratio 3–4–5 (6–8–?), so the length of the longest side is 10. Now, simply add the lengths of the sides: 6 + 8 + 10 = 24 So the correct answer is D. If you don’t spot the 3–4–5 right triangle relationship, you can use the Pythagorean theorem to find the length of the hypotenuse: a2 + b2 = c2 62 + 82 = c2 36 + 64 = c2 100 = c2 10 = c Then add the other two sides to get 24.
32
Strategies for the Math Test
6
6
4
11. Which figure is similar but not congruent to the figure shown? A. 5
5
6
B. 3
3
2
C. 4
5
3
D.
6
6
4
The key here is that you are looking for similar figures. Since similar means that the figures have the same shape (same angle measurements and proportional corresponding sides), Choice B is the only correct answer. Notice that Choice D is exactly the same—congruent, not similar. The correct answer is B.
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Part I: Working Toward Success
12. In the following figure, the length of line segment AB is 8 centimeters (cm). B
C
8 cm
A
D
What is the radius of the circle inscribed in square ABCD? A. B. C. D.
4 cm 8 cm 4π cm 8π cm
Focus on the word radius. Because the circle is inscribed in the square, the diameter of the circle is the length of one side of the square, 8 cm. B
8 cm
A
C
8
D
Since the radius of a circle is half of the diameter of the circle, the radius is 4 cm. The correct answer is A. 13. Two-centimeter cubes are arranged as shown in the following figure.
What is the total surface area? A. B. C. D.
34
4 cm2 16 cm2 18 cm2 72 cm2
Strategies for the Math Test
This problem should be done in parts. First, you are looking for total surface area. After analyzing the figure, it is clear that there are four cubes that each have six faces. The three cubes that are on the outside (two on each side and one on top), each have five faces showing. Each face is 4 cm2 (2 × 2). Since there are three cubes with five faces showing, and each face is 4 cm2, part of the surface area is (3 × 5 × 4 cm2 = 60 cm2). Next, you need to include the forth cube that is the most central. This cube is the only cube touching all the other three, and this cube has only three faces showing, so the surface area that shows is 12 cm2 (4 cm2 × 3). The total surface area is: 60 cm2 + 12cm2 = 72 cm2 The correct answer is D. Another method to solve this problem is to count the number of faces that are showing: 3 bottoms, 12 sides, 3 tops (count carefully). Since each face is 4 cm2, you have 18 × 4 cm2, which equals 72 cm2.
6 3 4 8
14. What is the area of the shaded region in the preceding figure, in square units? A. B. C. D.
9 square units 12 square units 16 square units 18 square units
Note that you are looking for the area of the shaded region. Your plan should be to find the area of each triangle and then subtract the area of the smaller triangle from the larger triangle. First find the area of the larger triangle, triangle ABC. Since the area of a triangle is 1 bh , the area of the larger 2 triangle is: 1 6 8 = 3 8 = 24 2 ^ h^ h ^ h^ h Next, find the area of the smaller triangle. 1 3 4 = 1 12 = 6 2 ^ h^ h 2 ^ h Now subtract. 24 – 6 = 18 So the area of the shaded region is 18 square units. The correct answer is D.
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Part I: Working Toward Success
15. What is the maximum number of milk cartons, each 2 inches wide by 3 inches long by 4 inches tall, that can fit into a cardboard box with inside dimensions of 16 inches wide by 9 inches long by 8 inches tall? A. B. C. D.
18 20 24 48
Drawing a diagram, as shown in the following figure, might be helpful in envisioning the process of fitting the cartons into the box. If possible, whenever a figure is mentioned but not drawn, draw it. Notice that 8 cartons fit across the box, 3 cartons deep and 2 stacks high.
8″ 4″ 2″
3″
9″
16″
The correct answer is D. 16. As shown in the following figure, camp counselor Craig builds a footbridge from the summer camp to the lake so that the campers don’t have to crawl down a perpendicular 5-foot cliff and then trudge through 12 feet of swamp.
Summer Camp foo
5ft
t br idg e
swamp 12ft
lake
How long (in feet) is the footbridge? A. B. C. D.
17 15 14 13
To solve this, you need to find the hypotenuse (the footbridge) of the right triangle by using the Pythagorean theorem: a2 + b2 = c2 (5)2 + (12)2 = c2 25 + 144 = c2 169 = c2 13 = c
36
Strategies for the Math Test
So the correct answer is D. You might recognize the ratio 5:12:13 as a Pythagorean triple. This saves you the work of using the Pythagorean theorem. 17. The scale drawing of a football field follows. The scale is 1 centimeter (cm) = 8 meters (m).
6 18 cm
13 cm
What is the width of the football field in meters? A. B. C. D.
61 8 13 3 4 49 110
You must find the width of the football field. To find this width, you can set up a proportion. Set up the proportion like this: 1 1 cm = 6 8 cm x 8 meters To solve this problem, cross multiply: x = 8 c6 1 m 8 1
1 = 68 8 x x = 49 The correct answer is C.
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Part I: Working Toward Success
20 8 16
9
3
18. As shown in the preceding figure, two right triangles have been removed from the corners of the square. What is the area of the remaining shaded figure in square units? A. B. C. D.
280 340 360 380
You are looking for the area of the shaded figure. First, find the area of the square: 20 × 20 = 400. Next, use the dimensions given to find the dimensions of each triangle, and then find the area of each triangle. Since each side of the square is 20, by subtracting the dimensions given, you have the following: 20 8 16 20 12 4 9
3
The area of the larger triangle is: 1 bh = 1 ^ 9h^12 h = 1 ^108h = 54 2 2 2 The area of the smaller triangle is: 1 bh = 1 ^ 3h^ 4 h = 1 ^12 h = 6 2 2 2 Finally, subtract the sum of the areas of the triangles from the area of the square: 400 – (54 + 6) = 400 – 60 = 340 The correct answer is B.
38
Strategies for the Math Test
8 9 5
14
19. What is the area of the preceding figure in square units? A. B. C. D.
60 70 82 94
Underline or circle the words area of the figure. Since this is an irregular figure, use common shapes to work it out. Start by finding the area of the rectangle, which is length times width: 14 × 5 = 70 So, the area of the rectangle is 70 square units. At this point, you can eliminate Choices A and B. The area must be greater than 70. Now you need to find the dimensions of the triangle. By using the dimensions given and subtracting from the length, you get the following dimensions:
4 8 6 5
5
14
You only need the base and height of the triangle to find the area. The height is 4 and the base is 6. A = 1 bh 2 A = 1 ^ 6 h^ 4 h 2 A = 1 ^ 24 h = 12 2 Now add the areas of the rectangle and triangle, 70 + 12 = 82. The correct answer is C.
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Part I: Working Toward Success
y
B
C
A
D x
20. Which of the following rectangles can result from reflecting rectangle ABCD across the x-axis? A.
C.
y
y
x
B.
C
B
C
A
D
A
D
D.
y
x
40
x
B
B
C
A
D
y
B
C
A
D
x
Strategies for the Math Test
When you reflect a figure across the x-axis, the x-coordinates remain the same, but the y-coordinates become the negative of the original coordinates. So, the rectangle moves below the x-axis, but is still the same distance from the x-axis and the y-axis. You get a mirror image on the other side of the x-axis. The correct answer is C. y
B
C
A
D
A
D
B
C
x
Although Choice A shows the rectangle below the x-axis, it is not the same distance from the x- or y-axes. 21. Judith made a drawing of her school, using the scale of: 1 centimeter (cm) on the model = 12 feet (ft) in real life If the length of the main corridor of her school is 180 ft, how long should Judith have drawn the corridor on the scale drawing? A. B. C. D.
12 cm 15 cm 18 cm 20 cm
You must find how long the corridor should be on the scale drawing. To find this length, you need to set up a proportion. Set up the proportion like this: 1 cm = x 12 ft 180 ft To solve this problem, cross multiply: 180 = 12x 1 cm x = 12 ft 180 ft Then divide by 12.
180 = 12x 12 12 15 = x
So the corridor on the scale drawing should be 15 cm long. The correct answer is B.
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Part I: Working Toward Success
B A
C
22. Two small circles with the same radius, and with centers A and C, are inscribed in a large circle whose center is at B, as shown in the preceding diagram. If the distance from A to C is 10 cm, what is the radius of the large circle? A. B. C. D.
5 cm 10 cm 20 cm 25 cm
You are looking for the radius of the large circle. To solve this problem, you need to use your knowledge of radii and diameters. If AC = 10 cm, and the small circles have the same radius, then AB = 5. So the radius of circle A is 5 cm. Its diameter is double its radius, or 10 cm, which also happens to be the radius of the large circle. This can be more easily seen in the following diagram.
B A 5
The correct answer is B.
42
C 5
5
5
Strategies for the Math Test
23. The following diagram shows the dimensions of a piece of paper. 8 1/2″
11″
What is the radius (in inches) of the largest circle that can be drawn on the piece of paper? A. B. C. D.
11 81 2 51 2 41 4
You are looking for the radius. Without going off the paper, the diameter of the largest circle is 8.5 inches. 8 1/2″ 8 1/2″ 11″
Because the radius is half the diameter: 81 # 1 =4 1 2 2 4 17 1 17 OR ; # = =4 1E 2 2 4 4 The correct answer is D. The common mistake is B, which is the diameter.
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Part I: Working Toward Success
3″
4″ Cube A
Cube B
24. The two cubes shown in the preceding figure have edges of 3 in. and 4 in. What is Volume of cube A ? Volume of cube B 3 A. 4 B. 9 16 C. 27 64 81 D. 256 First find the volume of cube A, which is 3 × 3 × 3 = 27. Next, find the volume of cube B, which is 4 × 4 × 4 = 64. Finally, set up the ratio: Volume of Cube A = 27 Volume of Cube B = 64 The correct answer is C. 25. Which term best describes the polygon that is formed from the points (1,2), (1,4), (3,2), (3,4)? A. B. C. D.
pentagon triangle square rectangle
Here, you need to make a quick sketch of a grid and plot these points on the grid as follows:
4 3 2 1 1
2
3
4
From this sketch, it is easy to see that the figure is a 2 × 2 square. The correct answer is C. You can eliminate Choice A because you need 5 points for a pentagon (five sides). You can eliminate Choice B because you need three sides for a triangle (and three of these points aren’t on a straight line to give you three sides).
44
Strategies for the Math Test
26. The following diagram shows the dimensions of a cylinder.
10′ 12′
Which of the following is the best estimate of the volume (in cubic feet) of the cylinder? (π ≈ 3.14 and volume = πr2h) A. B. C. D.
3,600 4,500 7,700 14,400
To find the volume of a cylinder with its sides perpendicular to its base, first find the area (A) of its base (a circle): π times (radius)2
[A = πr2]
Then, multiply this number by the container’s height. So, Volume = π × (radius)2 × height = 3.14(10)2(12) = 3.14(100)(12), or approximately 3(100)(12) = 3,600 The correct answer is A.
Algebra and Functions There are 17 problems involving algebra and functions on the CAHSEE. These problems are grouped together. The areas covered include:
Standard Set 1.0: Using Algebraic Terminology, Expressions, Equations, Inequalities, and Graphs ■ ■ ■ ■ ■ ■
Use variables and appropriate operations. Write expressions, equations, and inequalities. Know the order of operations. Evaluate expressions. Represent relationships graphically. Interpret graphs and parts of graphs.
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Part I: Working Toward Success
Standard Set 2.0: Interpret and Evaluate Expressions Involving Powers and Simple Roots ■ ■ ■ ■ ■
Interpret positive whole-number powers. Interpret negative whole-number powers. Simplify and evaluate expressions with exponents. Multiply and divide monomials. Take powers and extract roots.
Standard Set 3.0: Graph and Interpret Linear and Nonlinear Functions ■ ■ ■ ■
Graph functions of the form y = nx2 and y = nx3. Graph linear functions. Know that slope is the ratio of rise over run. Plot the value of quantities with the same ratios.
Standard Set 4.0: Solve Simple Linear Equations and Inequalities (Rational Answers) ■ ■ ■ ■
Solve two-step linear equations and inequalities with one variable. Interpret solutions in context. Verify the reasonableness of the results. Solve multi-step problems involving rate, speed, distance, and time or direct variation.
Samples with Suggested Approaches 1. Which inequality expresses the following relationship: A number, n, multiplied by 3 is less than or equal to 12? A. B. C. D.
3n ≤ 12 3n ≥ 12 3n < 12 3n > 12
Spread out the information given, and then work carefully, step by step. A number, n, multiplied by 3, is less than or equal to 12. n
×
3
≤
12
≤
12
Rearranging slightly: 3n 3n shows n multiplied by 3. < means less than but “≤” means less than or equal to, which is what you are looking for. Therefore, 3n ≤ 12 expresses the relationship. The correct answer is A.
46
Strategies for the Math Test
2. A stamp collector has 3,000 foreign stamps. She then sells x number of her foreign stamps and purchases 4y number of foreign stamps. Which of the following gives an expression representing the number of foreign stamps she has following her purchase? A. B. C. D.
3000 x+y 3,000 – x + y 3,000 − 4x + y 3,000 − x + 4y
You are looking for the expression that represents the number of stamps she has following her purchase. Take the information given, and follow the steps that she takes. The stamp collector starts with 3,000 foreign stamps. She then sells x number of foreign stamps, which are represented by 3,000 – x foreign stamps. Then, she purchases 4y foreign stamps. Since she has purchased 4y foreign stamps, she is adding 4y foreign stamps to her collection, so she now has 3,000 – x + 4y foreign stamps. The correct answer is D. 3. This year Tina’s father is 39 years old. He is triple her age. Which equation can be used to determine Tina’s age? A. B. C. D.
39 = 3T 39 = 3 + T 39 = T – 3 39 = T ÷ 3
Triple means multiply by three, and the equation in Choice A describes this: 39 = 3T (39 is 3 times Tina’s age) The correct answer is A. 4. Tom is just 4 years older than Fran. The total of their ages is 24. What is the equation for finding Fran’s age? A. B. C. D.
x + 4x = 24 x + 4 = 24 4x + 4 = 24 x + (x + 4) = 24
If Tom is 4 years older than Fran, and we let x represent Fran’s age, Tom’s age must be 4 years more, or x + 4. Therefore, because the total of their ages is 24: Fran’s age + Tom’s age = 24 x
+
(x + 4) = 24
The correct answer is D. 5. If x = 3 and y = 5, then 6x2– 4y = A. B. C. D.
16 34 54 60
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Part I: Working Toward Success
To evaluate an expression, simply plug in the given numbers or values. These types of problems are usually easy to solve as long as you are careful in your calculations and understand the order of operations. Plugging in the values given for x and y: 6x2 – 4y = 6(3)2 – 4(5) = 6(9) − 4(5) = 54 – 20 = 34 The correct answer is B. Remember, the order of operations is Parentheses Exponents Multiplication or Division Addition or Subtraction A good tool for remembering the order of operations is PEMDAS. 6. Use the following graph to answer the next question.
% attendance drop
30 20 10 0 0
10
20
30
40
degree drop in temperature
According to the graph, if the temperature falls 35°, what percentage does the attendance drop? A. B. C. D.
10% 20% 30% 40%
When referring to a graph, be sure to understand the information given. Pay special attention to the labels on the graph itself. In this case, note that on the graph a 35° drop in temperature (horizontal line) correlates with a 20% attendance drop (the fourth slash up the vertical line). The correct answer is B.
48
Strategies for the Math Test
7. Annette does a school-wide survey and publishes her results, as shown in the following graph. Schoolwide Eye Color Survey
30% blue
40% brown
5%
20% hazel 5%
green other
If 62 people at Annette’s school have hazel eyes, how many have brown eyes? A. B. C. D.
62 93 124 155
First circle or underline the words brown eyes. Before doing any work, take a careful look at what information you are given. According to the graph, 20% of the people at Annette’s school have hazel eyes, while 40% have brown eyes. This means that there are twice as many brown-eyed people as there are hazel-eyed people. 62 people have hazel eyes. 2 × 62 people have brown eyes. So, 2 × 62 = 124 people who have brown eyes. The correct answer is C.
Distance in miles
8. The following graph shows the relationship between time and distance for four male distance runners. Runner A
4 3 2
Runner C Runner B
1
Runner D 10 20 30 40 Time (minutes)
50
According to the preceding graph, after 30 minutes of running, runner A is approximately how many miles ahead of his closest competitor? A. B. C. D.
1 2 2.5 3
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Part I: Working Toward Success
Distance in miles
Note what you are looking for and what information you are given. Since time is represented on the x-axis of this graph, you go right on the x-axis to the 30-minute mark. Then you find runner A at that time. After 30 minutes, he has run about 3.5 miles. Next you should find his closest competitor, runner C. After 30 minutes, he has run about 2.5 miles. Therefore, runner A is about 1 mile ahead of his closest competitor. The correct answer is A. Runner A
4 3 2
Runner C Runner B
1
Runner D 10 20 30 40 Time (minutes)
50
No. of moviegoers
70 60 50 40 30 20 10 1-10
11-20 21-30 31-40 41-50 51-60 Age (years)
9. The preceding graph shows the age of moviegoers. Approximately how many moviegoers are 30-years old or younger? A. B. C. D.
30 50 100 130
Underline or circle the words 30-years old or younger. To find how many moviegoers are 30-years old or younger, you need to read the graph carefully. You need to add up the number of moviegoers ages 1–10, 11–20, and 21–30. This gives you: 50 + 50 + 30 = 130 The correct answer is D.
50
Strategies for the Math Test
10. x4y4 = A. B. C. D.
4xy (xy)4 (xy4)4 xy8
This mechanical problem is testing your knowledge of the rules of exponents. Since x4y4 = x × x × x × x × y × y × y × y, which is the same as (xy) × (xy) × (xy) × (xy), you can write x4y4 as (xy)4. The correct answer is B. You can also work this problem from the answer choices. 11. a _ 9x 2 i k = 2
A. B. C. D.
3x 9x 9x2 81x4
If you take the square root of any number, n, and then square it, you are left with n. The action of squaring a term is cancelled out by taking the square root of that term. In this case you take the square root of the term 9x2 and then square it, leaving you with 9x2. The correct answer is C. 12.
25x 8 = A. B. C. D.
5x2 5x4 25x2 25x6
Since 25 = 5 and x 8 = x4, 25x 8 = 5x4. Another method of solving this problem is to note that 25x 8 = 5x 4 # 5x 4 , so 25x 8 = 5x4. The correct answer is B. 13. Simplify the expression shown below: (9s3t5)(3st4) A. B. C. D.
12s3t9 12s4t9 27s4t9 27s4t20
To answer this mechanical question, you must know the rules for multiplying monomials. First, multiply the numbers together, 9 × 3 = 27. Then, work with the variables: s3 × s = s4 (Keep the same base and add the exponents: s = s1.) Likewise, t5 × t4 = t9. +
+
(9s3t5)(3s1t4) = 27s4t9 ×
The correct answer is C.
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Part I: Working Toward Success
y 5 4 3 2
B
1 -5
-4
-3
-2
1
-1
2
3
4
5
x
-1
A
-2 -3 -4 -5
14. In the preceding graph, what is the slope of the line that passes through points A and B? A. B. C. D.
3 2 1 3 1 2 2 1
Probably the simplest method is to notice that you must go up three and across two to get from point A to point B. Since slope is rise/run, you get 3/2. The correct answer is D. y 5 4 3 2 2 -5
-4
-3
-2 3
A
B 1
2
3
4
5
-1 -2 -3 -4 -5
Another method to find the slope of the line is to use the slope formula: slope =
52
_ y 2 - y 1i ^ x 2 - x 1h
x
Strategies for the Math Test
Using the coordinates of A (–1,–2) and B (1,1), simply plug into the formula:
_ y 2 - y 1i ^1h - ^ - 2 h 1 + 2 3 = = = ^ x 2 - x 1h ^1h - ^ -1h 1 + 1 2 y . . . 4 3 2 1 -3
-2
-1
-1
1
2
3
4
. . .
. . .
x -4
-2 -3 -4 . . .
15. What is the slope of the line shown in the preceding graph? A. B. C. D.
–3 -1 3 1 3 3
First, you can eliminate Choices A and B because negative slopes go down to the right, and the line in the preceding graph goes up to the right. The simplest method of solving this problem is noticing that the line crosses the x-axis at –1, so you go to the 3 right one and up three (where the line crosses the y-axis). Since you want rise run you get 1 or just 3. The correct answer is D. Another method is to plug into the slope formula: slope =
_ y 2 - y 1i ^ x 2 - x 1h
as in the previous problem. slope =
^ 3h - ^ 0 h 3 = =3 ^ 0 h - ^ -1h 1
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Part I: Working Toward Success
16. The slope of the following line is - 3 . 4 y
a 8
x
What is the value of a? A. B. C. D.
2 4 6 10
Since the slope of the line is rise run , you can set up a proportion to find the value of a. - 3 = -a 4 8 (Cancel the negatives from each side.) You can multiply both sides by 8, and you get: ^8h^ 3h =a 4 24 = a 4
The correct answer is C.
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Strategies for the Math Test
17. What is the equation of the line on the following graph? y 4 3 2 1
x -4
-3
-2
-1
-1
1
2
3
4
-2 -3 -4
A. B. C. D.
y = 2x + 4 y = 2x – 4 y = 2x + 2 y = 2x – 2
A careful review of this graph can save you a lot of work. The line crosses the y-axis at –2, so the y-intercept is –2. The y-intercept form of the equation is y = mx + b, where b is the y-intercept. The equation of the preceding line must end in –2, y = mx–2 because –2 is the y-intercept. The only equation that has a –2y-intercept is D. So even though you didn’t work out the equation of the line, you do know that it can’t be Choices A, B, or C. Also, the slope of each equation is the same c 2 m ; the only difference is the y-intercept. The correct answer is D. 1
55
Part I: Working Toward Success
18. Which graph best represents the equation y = x2? y
y
x C.
A.
y
B.
x
y
x D.
x
A careful look at y = x2, let’s you know that this is not a linear equation (not a straight line). So you can immediately eliminate Choice C. Also, y can never be negative, so eliminate Choices A and D. You are left with Choice B, the correct answer. You can actually plot this equation from the beginning. You start by plugging in numbers.
y = x2 x
y
0 1 -1 2 -2
0 1 1 4 4
Finally, you plot these points and see the shape of the graph. Fortunately, with this particular problem, all this work was not necessary. The correct answer is B.
56
Strategies for the Math Test
19. The Make an Orphan Happy Foundation donates toys in groups of five with each toy valued at $2. The following table shows the number of toys donated and the value of the donations. Number of toys donated 5 10 15 20 25
Value ($) 10 20 30 40 50
Which of the following is a graph of the information given in the table? A.
C.
Value ($)
Value ($)
Number of toys donated
B.
Number of toys donated
D.
Value ($)
Value ($)
Number of toys donated
Number of toys donated
As the number of toys donated increases, the value increases in direct proportion. The only graph that shows this straight-line increase is graph B. The correct answer is B. 20. The number of students in a classroom is represented by x in the inequality 3x – 50 ≤ 100. Which phrase most accurately describes the number of students in this classroom? A. B. C. D.
at most 50 at least 50 more than 50 less than 50
To answer this question, you must solve the inequality. 3x – 50 ≤ 100
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Part I: Working Toward Success
First add 50 to both sides of the inequality. 3x - 50 # 100 + 50 + 50 3x # 150 Next, divide both sides by 3. 3x # 150 3 3 x ≤ 50
So,
This is read “x is less than or equal to 50.” This also means that x is at most 50. The correct answer is A. 21. If x + 6 = 9, then 3x + 1 = A. B. C. D.
3 9 10 34
You should first circle or underline the term 3x + 1 because this is what you are solving for. Solving for x leaves x = 3. Then substituting this value for x into 3x + 1 gives 3(3) + 1, or 10. The most common mistake made in solving this problem is to solve for x, which is 3, and mistakenly choose A as the answer. But, you are solving for 3x + 1, not just x. You should also notice that most of the other choices are possible answers if you made common or simple mistakes. Make sure that you are answering the right question. The correct answer is C. 22. Solve for x. 4x + 3 = 15 A. B. C. D.
3 4 3 4 5
First, note that you are solving for x. Now, follow the normal procedures for solving a simple equation. Subtract 3 from each side. 4x + 3 = 15 - 3 -3 4x = 12 Now divide both sides by 4. 4x = 12 4 4 So, The correct answer is B.
58
x=3
Strategies for the Math Test
You can also answer this question by plugging in the answer choices. Choice A: 3 4
4x + 3 = 15 4 c 3 m + 3 ? 15 4 3 + 3 ≠ 15
Choice B: 3
4(x) + 3 = 15 4(3) + 3 ? 15 12 + 3 = 15
Eliminate A.
The correct answer is B. 23. In a movie the weight of a monster varies directly with the height of the monster. A 10-foot tall monster weighs 280 pounds. How tall is a monster that weighs 322 pounds? A. B. C. D.
11 feet tall 11 1 feet tall 2 12 feet tall 13 feet tall
You are given a direct variation, so you can set up the following proportion: height 10 foot tall x = = weight 280 pounds 322 pounds Make the numerators the height of the monsters and the denominators the weights. Since you don’t know the height of the 322-pound monster, you should use x to represent that monster’s height. Multiply both sides by 322 to solve for x. ^10 h^ 322 h =x 280
3220 = x 280 Dividing by 280 gives: 11 1 = x 2 The correct answer is B. 24. Jennifer has 750 pounds of furniture to move into her new apartment. She has been working for 3 days and has moved 150 pounds of furniture. If she continues to move furniture at the same rate, how many days in total does it take her to move all her furniture into her new apartment? A. B. C. D.
5 12 15 18
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Part I: Working Toward Success
You are looking for how many days in total. If Jennifer moves 150 pounds of furniture in 3 days, she is moving 50 pounds of furniture a day (150 ÷ 3 = 50). Since she has a total of 750 pounds that she has to move at a rate of 50 pounds of furniture a day, you divide 750 by 50 and find that it takes her 15 days to move all her furniture into her new apartment. The correct answer is C. You can solve this problem using the following proportion: 3 = x 150 750 2250 = x 150 15 = x
Mathematical Reasoning (Grade 7) There are eight problems involving number sense on the CAHSEE. These problems are grouped together and incorporate skills from number sense, statistics, data analysis, probability, measurement and geometry, and algebra. The areas covered include:
Standard Set 1.0: Decide How to Approach Problems ■ ■ ■ ■ ■ ■
Analyze problems. Identify relationships. Distinguish relevant from irrelevant information. Identify missing information. Prioritize information. Observe patterns.
Standard Set 2.0: Use Strategies, Skills, and Concepts to Solve Problems ■ ■ ■ ■
Estimate to verify that an answer is reasonable. Estimate unknown values graphically. Solve problems using logical reasoning. Solve problems using arithmetic and algebraic techniques.
Standard Set 3.0: Determine That a Solution Is Complete and Generalize to Other Situations ■ ■ ■
60
Develop generalizations from results. Generalize from strategies used in other problems. Apply generalizations to new situations.
Strategies for the Math Test
Samples with Suggested Approaches 1. Andrea rode her bike for 3 hours at a rate of 15 miles per hour. What is the correct method to find the total miles that Andrea traveled? A. B. C. D.
Add 15 and 3. Divide 15 by 3. Multiply 15 by 3. Multiply 60 by 15.
Circle or underline the words total miles traveled. To find her total miles traveled, multiply 15 by 3. The actual total of miles traveled is 45 miles. Does this method give you a reasonable answer? Yes, if she travels 15 miles in one hour, then she travels 30 miles in 2 hours and 45 miles in 3 hours. The correct answer is C. 2. If x is an even number, then which of the following is true about x ? 2 A. It is an even number. B. It is an odd number. C. It is a multiple of 2. D. It can be odd or even. When given a situation using types of numbers (odd, even, negative, and so on), try some simple numbers. If x is 2 (which is an even number), then x is 1, which is an odd number. If x is 4, then x is 2, which is an even number. 2 2 The correct answer is D. 3. Which is the best estimate of 931 × 311? A. B. C. D.
2,700 27,000 270,000 2,700,000
First, check the answer choices to see how far apart they are. This gives you an indication of how close your approximation should be. In this case the choices are far apart, so your approximation does not need to be too accurate. As a matter of fact, these choices differ only by the number of zeros after 27. So be careful that you have the correct number of zeros in your estimate. Round each number to the nearest hundred.
932 × 311 900 × 300 = 270,000 So the best estimate is 270,000. The correct answer is C. 4. Which of the following is the best estimate of 595 ÷ 184? A. B. C. D.
2 3 4 5
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Part I: Working Toward Success
When you check the choices in this question, they seem close, but they still give you plenty of room to estimate.
595 ÷ 184 600 ÷ 200 = 3 So the best estimate is 3. The correct answer is B. 5. Andre is fighting the battle of the bulge. He is counting calories consumed and calories burned. Today, Andre consumed 280 calories from fruits and vegetables, 530 calories from meat, 125 calories from cereals, and 275 calories from dairy products. Andre burned 980 calories today. Which expression can be used to express the BEST estimate of his net calorie count for the day? A. B. C. D.
300 + 500 + 100 + 300 – 1,000 300 + 500 + 200 + 300 – 1,000 300 + 600 + 100 + 300 – 1,000 300 + 500 + 100 + 200 – 1,000
Each choice is rounded off to the nearest hundred. Also, you are looking for the best expression. To answer this question, round off each number and set up the expression as follows:
280 + 530 + 125 + 275 − 980 300 + 500 + 100 + 300 − 1,000 The correct answer is A. 6. Manuel purchases tickets to an amusement park for his family. Children’s tickets cost half price, and adult tickets cost the full price of $8.00 each. Which of the following expressions represents the total dollars spent for tickets if there are 2 adults and 6 children in Manuel’s family? A. B. C. D.
2(8) + 3(4) 2(4) + 6(6) 2(8) + 4(4) 2(8) + 6(4)
In this question you are being asked to set up an expression. Focus on key words and information given. From the information given you can figure out the total spent for adult and children’s tickets: 2 adults × $8 for each ticket 6 children × $4 for each ticket (Note: Half the price of $8 = $4.) So, the total spent for tickets was 2(8) + 6(4). The correct answer is D.
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Strategies for the Math Test
7. Union High School is planning to order yearbooks for the student body. The following chart shows the percent of students in each class that usually order yearbooks. Class
Percent
Freshman Sophomore Junior Senior
40% 60% 85% 95%
Considering the preceding information, if there are 500 sophomores and 600 juniors in the school, what additional information is necessary to find out how many more sophomores than freshman will probably order yearbooks? A. B. C. D.
the number of seniors in the school the number of juniors not ordering yearbooks the number of freshman in the school the number of sophomores not ordering yearbooks
Since the question asks for the additional information necessary, and focuses on the difference between freshman and sophomores, you need to know the number of freshman in the school. The correct answer is C. 8. A Zowie battery lasts 60 hours and costs $1.20. A Rayvox battery lasts 75 hours and is sold in a package at a cost of $4.99 per package. To determine whether a Zowie battery or a Rayvox battery is the better buy, which additional piece of information is needed? A. B. C. D.
Zowie batteries cost less than Rayvox batteries. Rayvox batteries last 25% longer than Zowie batteries. Rayvox batteries are sold only in packages containing three batteries. Zowie batteries are sold in packages containing only one battery.
To compare the cost of each battery, you must know how many Rayvox batteries are in the package that costs $4.99. You do not have to know which battery is the better buy. You only have to know what information you need to determine which is the better buy. The correct answer is C.
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Part I: Working Toward Success
9. The following graph shows the values of stocks and bonds for the first three quarters of the current year.
Values of Stocks and Bonds for Current Year
(In Thousands of Dollars)
70 60 50 Stocks Bonds
40 30 20 10 0
1st Qtr
2nd Qtr
3rd Qtr
4th Qtr
Of the following, which is the best prediction of the possible value of stocks in the fourth quarter? A. B. C. D.
$50,000 $60,000 $70,000 $80,000
The question is asking about the value of stocks, so make sure to underline the words prediction, value, stocks, and fourth quarter. Focus on stocks. Now, take a careful look at the value of stocks in each quarter—the first quarter is $20,000, the second quarter is $30,000, and the third quarter is $50,000. The increase is $10,000 from the first quarter to the second quarter, and $20,000 from the second quarter to the third quarter. It is reasonable to predict that the increase will be $30,000 from the third quarter to the fourth quarter (increases of $10,000, $20,000, and $30,000), so the value of stocks in the fourth quarter can be $30,000 more than the third quarter (50,000 + 30,000), or $80,000. Since not a lot of data is given (only three quarters), other predictions are possible, but D is the best choice given. The correct answer is D.
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Strategies for the Math Test
10. The following graph shows the average lamp prices for the years listed. Average Lamp Prices (and Projection) 40 35
Price (in dollars)
30 25 20 15 10 5 0
1970
1990
2010
Which of the following was the most probable average lamp price in 1970? A. B. C. D.
$5 $8 $14 $20
Underline the words average lamp price in 1970. Now, continue the slope of the line back to the left until you get to 1970. The most probable average price is $14. Using the edge of your answer sheet helps you follow the slope more accurately. The correct answer is C.
Verbal SAP Scores
11. According to the line of best fit shown on the following scatterplot, approximately how many hours of extra reading does a student have to do every month to score a 600 on the verbal part of the SAPs?
Line of best fit
800 600 400 200 10
20
30
40
50
60
No. of hours spent doing extra reading per month
A. B. C. D.
20 30 50 60
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Part I: Working Toward Success
First, focus on the line of best fit. Again, use the edge of your answer sheet to most accurately continue the line. (You can also use the side of your pencil.) The point that is on the line of best fit and is also across from the verbal score of 600 is directly above the 30 representing the number of extra hours of reading every month. The correct answer is B. 12. The following circle has a radius, r.
r
If r < 9, which of the following cannot be the area of the circle? A. B. C. D.
3π 9π 18π 81π
The only way the area of the circle can be 81π is if r = 9, but r is less than 9; therefore the area cannot be 81π. The correct answer is D.
x
y
4 6 3 5 2
8 12 6 10 4
13. In the table above, showing corresponding values of x and y, which equation represents the relationship? A. B. C. D.
x=y+4 x= 1 y+6 2 x = 2y x= 1 y 2
A careful look at the table reveals that in each case, 2 × x gives y (4 and 8, 6 and 12, 3 and 6, and so on). So x is 1 y . The correct answer is D. If the x values are put in proper order (2, 3, 4, 5, 6), this relationship is much easier 2 to recognize. 14. How many numbers are less than 100 and are divisible by 2, 3, and 7? A. B. C. D.
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2 3 4 5
Strategies for the Math Test
For a number to be divisible by 2, 3, and 7, it must be a multiple of 2, 3, and 7. Simply multiply 2, 3, and 7, and you get 42, which is a multiple of these three numbers. Double 42 and you get 84, another multiple of 2, 3, and 7. So there are two numbers less than 100 that are divisible by 2, 3, and 7. The correct answer is A. 15. Tina is trying to find the y-intercept for the linear equation. x+y–6=0 Step 1: Add 6 to each side: x + y = 6 Step 2: Subtract x from each side: y = –x + 6 Step 3: Substitute 0 for x: y = –(0) + 6 Step 4: The y-intercept is 6: y = 6 Tina’s method shows that the y-intercept in the equation y = mx + b is which of the following? A. B. C. D.
m x y b
Tina’s method was to put the equation into the form y = mx + b, where m is the slope and b is the y-intercept. She substituted 0 for x, to find that the line represented by this equation crosses the y-axis at 6. So 6 is the y-intercept, represented by b. The correct answer is D. Adam eats 2 boxes of cookies every week. At this rate, how long does it take Adam to eat 18 boxes of cookies? 16. The same mathematical processes used to solve the preceding problem can be used to solve which of the following problems? A. B. C. D.
Right Says Fred records 2 hit records every month. How many hit records do they record in their 3-month career? Planet Jackson has saved 300 lives so far in her 18-month career as a super hero. How many lives does she save each week? Bill Plates eats 2 boxes of cookies every week. At this rate, how many boxes of cookies does he eat in 4 weeks? Lil’ Chill raps 2 hours every day. At this rate, how long does it take Lil’ Chill to rap for a total of 24 hours?
The process used to answer the original question involves dividing the total number of boxes of cookies that need to be eaten (18) by the rate that Adam eats cookies at (2 boxes per week). In Answer D, Lil’ Chill’s case, you also need to divide the total number of hours needed (24) by the rate at which he raps (2 hours a day). The correct answer is D. Eight out of ten dentists surveyed recommend Popsodent toothpaste. 17. Which of the following is a valid conclusion based on the information given? A. B. C. D.
Two out of ten dentists recommend another brand of toothpaste. Popsodent is the best-tasting toothpaste. More patients use Popsodent toothpaste than any other brand. At least one dentist surveyed could have recommended another brand.
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Part I: Working Toward Success
The fact that eight out of ten dentists surveyed recommend Popsodent toothpaste does not tell us anything about the other dentists. They might not recommend any toothpaste, so eliminate Choice A. Choice B, taste, is never addressed. For Choice C to be a valid conclusion, you have to assume that patients follow the recommendations of their dentists. Also, what about patients who go to dentists other than those surveyed. Since two dentists surveyed didn’t recommend Popsodent, they could have recommended another brand. The correct answer is D.
Algebra I There are 17 problems involving Algebra I on the CAHSEE. These problems are grouped together. The areas covered include:
Standard Set 2.0 ■
Understand opposites, reciprocals, and square roots.
Standard Set 3.0 ■
Solve equations and inequalities involving absolute values.
Standard Set 4.0 ■
Simplify expressions before solving linear equations and inequalities in one variable.
Standard Set 5.0 ■ ■
Solve problems with many steps including linear equations and linear inequalities in one variable. Provide justification for each step.
Standard Set 6.0 ■
Graph linear equations and compute x- and y-intercepts.
Standard Set 7.0 ■ ■
Verify that a point lies on the line of an equation given. Derive linear equations.
Standard Set 8.0 ■
Understand the concept of parallel lines and the slopes of those lines.
Standard Set 9.0 ■ ■ ■ ■
Solve a system of two linear equations in two variables algebraically. Interpret answers graphically. Solve a system of two linear inequalities in two variables. Sketch solution sets.
Standard Set 10.0 ■
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Use operations with monomials and polynomials—add, subtract, multiply, and divide.
Strategies for the Math Test
Standard Set 15.0 ■
Solve rate, work, and percent mixture problems algebraically.
Samples with Suggested Approaches 1. If x = –4, then –2x = A. B. C. D.
–8 -1 8 1 8 8
If x = –4, then substituting –4 into –2x gives –2(–4) = –(–8). The negative of a negative number is positive, so the answer is positive 8. The correct answer is D. 2. If –x = 3, then x = A. B. C. D.
–3 -1 3 1 3 3
Since –x means the opposite of x, and –x = 3, then x must be –3, the opposite of 3. Or you can simply multiply each side of the original equation by –1, as follows: –x = 3 (–1)(–x) = 3(–1) x = –3 The correct answer is A. 3. If the area of a rectangle with a width of 5 inches (in) is 40 in2, what is the perimeter of the rectangle? A. B. C. D.
26 in 40 in 80 in 100 in
If a geometric figure is described, but not drawn for you, draw the figure and label the information given.
5 in
40 in2
This gives you insight into how to work the problem. Be sure to underline or circle the word perimeter since that is what you are looking for.
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Part I: Working Toward Success
To find the perimeter of a rectangle, you must find the sum of the lengths of all the sides. You know that the area of the rectangle is 40 in2 and that the width is 5 in, so you can divide 40 by 5 to find the length of the rectangle (8 in). Now finish labeling the figure and total the sides. 8 in
5 in
40 in2
5 in
8 in Since 5 + 8 + 5 + 8 = 26, the correct answer is A. You can also work the problem as follows: Because you know that opposite sides are equal in a rectangle, the perimeter is 2(5) + 2(8) = 26. 4. If x is an integer, what is the solution to 5 x = 30 ? A. B. C. D.
{5, 6} {5, –6} {0, 6} {–6, 6}
Integers are positive and negative whole numbers and zero. One way to answer this question is to work from the choices given by plugging them in. Another method is to actually solve the problem as follows: 5 x = 30 Divide each side by 5
5 x = 30 5 5
Then
x =6
So
x = 6 or –6
The correct answer is D. 5. Given that x is an integer, which of the following is the solution set for x + 2 < 4 ? A. B. C. D.
{0} {0,1} {–2, –1, 0, 1} {–5, –4, –3, –2, –1, 0, 1}
Integers are positive and negative whole numbers and zero. To solve an inequality when you have an absolute value sign, you must form two equations. In this case your two inequalities are: x + 2 < 4 and x + 2 > –4
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Strategies for the Math Test
In the second inequality, when you take the negative, you change the direction of the sign. Now, solve each inequality by subtracting 2 from each side. x+2< 4 - 2 -2 x < 2 and x+2> -4 - 2 -2 x > -6 This can also be written –6 < x < 2; that is, x is between –6 and 2. Since x is an integer, x can be –5, –4, –3, –2, –1, 0, or 1. The correct answer is D. You can also find the solution set by working from the answers. Start by plugging in a number from Choice D (since it has the most members) that is different from numbers in the other choices. If the number works, then the answer is D; if not, then you can eliminate D. -5 + 2 < 4
Try –5
-3 < 4 3<4 This is true, so answer D is correct. Plugging in 0 is a waste of time since 0 is in each answer choice set. 6. Which of the following equations is equal to 5(x – 3) − 4(2x – 1) = 1? A. B. C. D.
5x – 15 – 8x + 4 = 1 5x – 15 – 8x – 4 = 1 5x – 3 – 8x + 1 = 1 5x – 3 – 8x – 1 = 1
Scan the answer choices. The answer choices are simplified only on the left side, and are not simplified completely. Like terms have not been added together. So do only as much work as is necessary. First, simplify the left side of the equation. 5(x – 3) − 4(2x – 1) = 1
5(x − 3) − 4(2x − 1) = 1 5(x − 3) − 4(2x − 1) = 1 You get: 5x – 15 – 8x + 4 = 1 Now, look at the choices again. The correct answer is A. When you multiply –4(2x –1), the negative sign with the 4 is also multiplied by each term inside the parenthesis.
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Part I: Working Toward Success
7 = 6x 9 x+3 7. Which of the following is equivalent to the preceding equation? A. B. C. D.
42x = 9x + 27 54x = 7(x + 3) 54 = 7 + (x + 3) 63 = 6x (x + 3)
First, take a quick look at the answer choices. They are equations that are not simplified. So your next step is to cross multiply.
7(x + 3) 9(6x) 7 = 6x 9 x+3 Simplifying only the right side leaves: 7(x + 3) = 54x which is the same as
54x = 7(x + 3)
The correct answer is B. Take a look at the choices as you simplify so that you don’t do more work than is necessary. 8. Which of the following is equivalent to 9x – 7 > –3(x – 2)? A. B. C. D.
6x > 13 8x > –12 12x > 13 12x > –13
Again, notice that the answer choices are simplified, but not completely solved. The x’s and numbers are sorted to each side of the inequality sign. So, do only as much work as is necessary. First, simplify the right side of the inequality. 9x – 7 > –3(x – 2)
9x − 7 > -3(x − 2) You get: 9x – 7 > –3x + 6
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Strategies for the Math Test
Next, get all the x’s on one side by adding 3x to both sides. 9x - 7 > - 3x + 6 + 3x + 3x 12x - 7 > 6 Now add 7 to both sides. 12x - 7 > + 6 +7 +7 12x > 13 The correct answer is C. 9. Solve for x. 4(2x + 3) < –2(3x – 4) A. B. C. D.
x < - 27 x<0 x>0 x> 7 2
First, simplify each side by multiplying through the parentheses. 4(2x + 3) < –2(3x–4) You get:
4(2x + 3) < -2(3x − 4) 8x + 12 < –6x + 8
Next, add 6x to each side. 8x + 12 < - 6x + 8 + 6x + 6x 14x + 12 < 8 Now subtract 12 from each side. 14x + 12 < 8 - 12 - 12 14x < -4 Finally, divide each side by 14. 14x < - 4 14 14 So
x < -4 14
This reduces to x < -72 . The correct answer is A.
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Part I: Working Toward Success
10. Sasha solved the inequality 5x – 5 ≥ 2x + 4 using the following steps. 5x – 5 ≥ 2x + 4 Step 1: 3x – 5 ≥ 4 Step 2: 3x ≥ 9 Step 3: x ≥ 3 Sasha did which of the following to get from step 1 to step 2? A. B. C. D.
subtracted 5 from each side added 5 to each side divided each side by 3 subtracted 2x from each side
Focus on the question. What did she do to get from step 1 to step 2. To get from step 1 to step 2, Sasha added 5 to each side. 3x - 5 $ 4 +5 +5 3x $ 9 The correct answer is B. 11. Which of the following statements describes the x-intercept? A. B. C. D.
the point where a line crosses the y-axis a line that is parallel to the x-axis the point where a line crosses the x-axis a line that is parallel to the y-axis
The x-intercept is the point where the line crosses the x-axis, the coordinates (x, 0). The correct answer is C.
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Strategies for the Math Test
y
x
12. Which of the following is the equation of the line shown in the preceding graph? A. B. C. D.
y = 2x + 1 y = 2x + 2 y = –2x – 1 y = –2x – 2
A quick look at the choices lets you eliminate some by looking for the y-intercept first. Since the line crosses the y-axis at –1, the y-intercept on the graph is –1. You can eliminate Choices A, B, and D. The correct answer is C. In this question, you don’t even need to deal with the slopes. But if you are working with the slopes, you can eliminate Choices A and B immediately because they are positive. Since the line goes down to the right, the slope is negative.
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Part I: Working Toward Success
13. Which of the following is a graph of y = 3x – 2? y
y
x C.
A.
y
B.
x
y
x D.
x
Since the equation given is in slope-intercept form (y = mx + b, where m is the slope and b is the y-intercept), pay special attention to the slope and y-intercept. In this equation, y = 3x–2, the slope is 3 and the y-intercept is 1 –2. The line must cross the y-axis at –2, so first eliminate any choices that do not cross the y-axis at –2. Eliminate Choices B and D. Next, eliminate Choice C because, upon careful examination, you can see that the slope is 1 , 1 not 3 . The correct answer is A. 1 14. Which of the following points lies on the line 3x – 2y = 12? A. B. C. D.
(0, –6) (0, 0) (4, 6) (6, 0)
Probably the best method to answer this question is to work from the choices given. Simply plug in the coordinates as values of x and y to see which make the equation true.
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Strategies for the Math Test
3x – 2y = 12 Choice A (0,–6)
3(0)−2(–6) = 12 0 + 12 = 12
Since (0, –6) satisfies the equation, this point must lie on the line. The correct answer is A. 15. What is the slope of the line parallel to the line y = 2x - 1 ? 2 A. –2 B. - 1 2 1 C. 2 D. 2 Underline or circle the words slope of the line parallel to. Your focus is first to find the slope. Because the equation is in slope-intercept form, y = mx + b, where m is the slope, it is easy to see that the slope is 2. Parallel lines have the same slope. Therefore, the slope of a line parallel to the given line is also 2. The correct answer is D. 16. What is the slope of the line identified by 3y = 4(x + 3)? A. B. C. D.
4 3 4 3 3 4
To answer this mechanical question, simply change the equation to slope-intercept form: y = mx + b 3y = 4(x + 3) 3y = 4x + 12 Next divide each side by 3, which leaves: y= c4m x+ 4 3 So the slope is 4 . The correct answer is C. 3 17. Which of the following points is the y-intercept of the line 3x + 2y = 6? A. B. C. D.
(2, 0) (2, 1) (0, 2) (0, 3)
Probably the fastest method to answer this question is working from the answers. First, note that you are looking for the y-intercept. The y-intercept is where the line crosses the y-axis, so x must be 0. Eliminate Choices A and B because the x coordinate is not 0. Next, plug in Answers C and D to see which is true for the equation.
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Part I: Working Toward Success
Choice C (0, 2): 3x + 2y = 6 3(0) + 2(2) ? 6 0+4≠6 So you can eliminate C. The correct answer is D. Choice D (0, 3): 3x + 2y = 6 3(0) + 2(3) ? 6 0+6=6 Another method for solving this problem is to change the equation of the line to slope-intercept form, y = mx + b, where b is the y-intercept. 3x + 2y = 6 Subtract 3x from each side. 3x - 2y = 6 - 3x - 3x 2y = - 3x + 6 Divide both sides by 2. 2y - 3x 6 = + 2 2 2 y = c -3 m x + 3 2 So the y-intercept is 3. x+y=6 ) 2x - y = 3
18.
Which of the following ordered pairs is the solution to the system of equations shown above? A. B. C. D.
(2, 4) (5, 1) (3, 3) It cannot be determined.
In this question you can either solve the system of equations or plug in the answer choices. Let’s solve the system of equations. First, combine the equations to eliminate a variable: x+y=6 (+) 2x - y = 3 3x =9
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Strategies for the Math Test
Now, dividing both sides by 3 leaves x = 3. Since x = 3, simply plug x into either equation and solve: x+y=6 (3) + y = 6 So y = 3. The correct answer is C, (3, 3). The other method of plugging in the answers goes like this: Since Choices A, B, and C each total 6, you only need to plug them into the second equation to find the right answer. Choice A, (2,4): 2(2) − (4) ? 3 4 − 4 ≠3 Eliminate Choice A. Choice B, (5,1): 2(5) − (1) ? 3 10 − 1 ≠ 3 Eliminate Choice B. Choice C, (3,3): 2(3) − (3) ? 3 6−3=3 Choice C is correct. Although it is not the correct answer for this problem, note that some problems have the answer choice of “It cannot be determined,” “No solution,” or “No intersection.” 7x + y = 8 ) x + 3y = 4
19.
What is the solution to the system of equations shown above? A. B. C. D.
(0, 0) (0, 8) (1, 1) (3, 4)
Again, to answer this question, you can work from the choices and plug in answers or solve algebraically. If you plug in the choices, simply take one of the equations and see if the solutions work.
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Part I: Working Toward Success
7x + y = 8 Choice A, (0, 0):
7(0) + 0 ? 8 0≠8
Eliminate Choice A. Choice B, (0, 8): 7(0) + (8) = 8 Choice B is possible. Choice C, (1, 1): 7(1) + (1) ? 8 7 + 1 =8 Choice C is possible. Choice D, (3, 4): 7(3) + (4) ? 8 21 + 4 ≠ 8 Eliminate Choice D. Now, take Choices B and C and plug them into the second equation. x + 3y = 4 Choice C, (1, 1): 1 + 3(1) ? 4 1+ 3 =4 The correct answer is C. If you decide to solve this problem algebraically, you solve for one variable first by eliminating the other one. Multiply the top equation by 3: 3(7x + y = 8) 21x + 3y = 24 Subtract the second equation from the first: (-)
80
21x + 3y = 24 x - 3y = 4 20x = 20
Strategies for the Math Test
Now, divide both sides by 20. 20x = 20 20 20 x=1 Now that you have your x value, plug it back into either original equation. 7x + y = 8 7(1) + y = 8 7+y=8 y=1 Now, you know that x = 1 and y = 1. You have the solution: (1, 1). 20. Simplify. (x2 + 7x – 7) − (4x2 – 6x – 1) A. B. C. D.
3x2 + x – 6 –3x2 + x – 8 –3x2 – 13x – 6 –3x2 + 13x – 6
First, take a look at the choices to see if they are simplified completely, that is, all operations are complete and all like terms are combined. In this case each choice is simplified completely. So simplify: (x2 + 7x – 7) − (4x2 – 6x – 1) First, multiply through the right parentheses. x2 + 7x – 7 – 4x2 + 6x + 1 Now, combine like terms.
x2 + 7x − 7 − 4x2 + 6x + 1
–3x2 + 13x – 6 The correct answer is D.
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Part I: Working Toward Success
x+2
21. The square shown above has a side of length x + 2 units. The area of the square can be represented by which of the following expressions? A. B. C. D.
x2 + 2 x2 + 4 x2 + 4x + 4 x2 + 2x + 4
To find the expression representing the area of the square, you must first note that all sides of a square are equal. So, if one side is x + 2 units, all sides are x + 2 units. Mark the diagram as follows: x+2 x+2
Now, simply multiply (x + 2) by (x + 2). You can use the FOIL method (first, outer, inner, last).
(x + 2) (x + 2) x2 + 2x + 2x + 4 x2 + 4x + 4 The correct answer is C. 22. Mr.Tuchman can paint 30 surfboards in an hour. Mr. Christianson can paint 60 surfboards in an hour. If they are both painting surfboards, how long does it take them to paint a total of 45 surfboards? A. B. C. D.
30 minutes 45 minutes 60 minutes 90 minutes
You are looking for how long it takes to paint 45 surfboards if both men are painting. You can work this problem from the answers. First, try Answer A, 30 minutes. If Mr. Tuchman can paint 30 surfboards in an hour, then he can paint 15 surfboards in half an hour. If Mr. Christianson can paint 60 surfboards in an hour, then he can paint 30 surfboards in half an hour. So, together they can paint 45 surfboards in 30 minutes. The correct answer is A.
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Strategies for the Math Test
You can also work this problem algebraically. If t is the number of hours, and Mr. Tuchman paints at a rate of 30 surfboards an hour, this can be expressed as 30t. If Mr. Christianson paints at a rate of 60 surfboards an hour, then this can be expressed as 60t. They both have to work to paint a total of 45 surfboards, so you can set up the equation 30t + 60t = 45. Now, solve as follows: 30t + 60t = 45 90t = 45 Dividing by 90 gives
t = 45 90
So, t = 1 hour, or 30 minutes. The correct answer is A. 2
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A Quick Review of Mathematics The following pages are designed to give you a quick review of some of the basic skills used on CAHSEE Math: arithmetic, algebra, measurement and geometry, properties of numbers, simple probability and statistics, and graphs. Before beginning the diagnostic review tests, it is wise to become familiar with basic mathematics terminology, formulas and general mathematical information. These topics are covered first in this chapter. Then proceed to the arithmetic diagnostic test, which you should take to spot your weak areas. Then use the arithmetic review that follows to strengthen those areas. After reviewing the arithmetic, take the algebra diagnostic test and again use the review that follows to strengthen your weak areas. Next, take the measurement and geometry diagnostic test and carefully read the complete measurement and geometry review. Even if you are strong in arithmetic, algebra, and measurement and geometry, you might wish to skim the topic headings in each area to refresh your memory about important concepts. If you are weak in math, you should read through the complete review.
Symbols, Terminology, Formulas and General Mathematical Information Common Math Symbols and Terms Symbol References: =
is equal to
≠
is not equal to
>
is greater than
<
is less than
≥
is greater than or equal to
≤
is less than or equal to
||
is parallel to
=
is perpendicular to
Terms: Natural numbers
The counting numbers: 1, 2, 3, . . .
Whole numbers
The counting numbers beginning with zero: 0, 1, 2, 3, . . .
Integers
Positive and negative whole numbers and zero: . . . –3, –2, –1, 0, 1, 2, . . .
Odd number
Number not divisible by 2: 1, 3, 5, 7, . . .
Even number
Number divisible by 2: 0, 2, 4, 6, . . .
Prime number
Number divisible by only 1 and itself: 2, 3, 5, 7, 11, 13, . . .
Composite number
Number divisible by more than just 1 and itself: 4, 6, 8, 9, 10, 12, 14, 15, . . . (0 and 1 are neither prime nor composite)
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Part I: Working Toward Success
Terms: Square
The results when a number is multiplied by itself: 2 × 2 = 4; 3 × 3 = 9. Examples of squares are 1, 4, 9, 16, 25, 36, . . .
Cube
The results when a number is multiplied by itself twice: 2 × 2 × 2 = 8; 3 × 3 × 3 = 27. Examples of cubes are 1, 8, 27, 64 . . .
Math Formulas Triangle
Perimeter = s1 + s2 + s3 Area = 1 bh 2
Square
Perimeter = 4s Area = s × s, or s2
Rectangle
Perimeter = 2(b + h), or 2b + 2h Area = bh, or lw
Parallelogram
Perimeter = 2(l + w), or 2l + 2w Area = bh
Trapezoid
Perimeter = b1 + b2 + s1 + s2 b + b2 Area = 1 h ^ b 1 + b 2 h , or h d 1 n 2 2
Circle
Circumference = 2πr, or πd Area = πr 2
Cube
Volume = s ⋅ s ⋅ s = s3 Surface area = s ⋅ s ⋅ 6
Rectangular Prism
Volume = l ⋅ w ⋅ h Surface area = 2(lw) + 2(lh) + 2(wh)
Pythagorean theorem (a2 + b2 = c2): The sum of the square of the legs of a right triangle equals the square of the hypotenuse.
Important Equivalents 1 = 0.1 = 1% 100 1 = .1 = .10 = 10% 10 1 = 2 = .2 = .20 = 20% 5 10 3 = .3 = .30 = 30% 10 2 = 4 = .4 = .40 = 40% 5 10 1 = 5 = .5 = .50 = 50% 2 10
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A Quick Review of Mathematics
3 = 6 = .6 = .60 = 60% 5 10 7 = .7 = .70 = 70% 10 4 = 8 = .8 = .80 = 80% 5 10 9 = .9 = .90 = 90% 10 1 = 25 = .25 = 25% 4 100 3 = 75 = .75 = 75% 4 100 1 = .33 1 = 33 1 % 3 3 3 2 = .66 2 = 66 2 % 3 3 3 1 = .125 = .12 1 = 12 1 % 8 2 2 3 = .375 = .37 1 = 37 1 % 8 2 2 5 = .625 = .62 1 = 62 1 % 8 2 2 7 = .875 = .87 1 = 87 1 % 8 2 2 1 = .16 2 = 16 2 % 3 3 6 5 = .83 1 = 83 1 % 3 3 6 1 = 1.00 = 100% 2 = 2.00 = 200% 3 1 = 3.5 = 3.50 = 350% 2
Math Words and Phrases Words that signal an operation: Addition ■ ■ ■ ■ ■ ■
Sum Total Plus Increase More than Greater than
Multiplication ■ ■ ■ ■ ■
Of Product Times At (sometimes) Total (sometimes)
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Subtraction ■ ■ ■ ■ ■ ■
Difference Less Decreased Reduced Fewer Have left
Division ■ ■ ■ ■ ■
Quotient Divisor Dividend Ratio Parts
Mathematical Properties Some Properties (Axioms) of Addition Commutative means that the order does not make any difference.
2+3=3+2 a+b=b+a Note: The commutative property does not hold for subtraction. 3–1≠1–3 a–b≠b–a Associative means that the grouping does not make any difference.
(2 + 3) + 4 = 2 + (3 + 4) (a + b) + c = a + (b + c) The grouping has changed (parentheses moved), but the sides are still equal. Note: The associative property does not hold for subtraction. 4 – (3 – 1) ≠ (4 – 3) – 1 a – (b – c) ≠ (a – b) – c The identity element for addition is 0. Any number added to 0 gives the original number.
3+0=3 a+0=a
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The additive inverse is the opposite (negative) of the number. Any number plus its additive inverse equals 0 (the identity).
3 + (–3) = 0; therefore, 3 and –3 are inverses. –2 + 2 = 0; therefore, –2 and 2 are inverses. a + (–a) = 0; therefore, a and –a are inverses.
Some Properties (Axioms) of Multiplication Commutative means that the order does not make any difference.
2×3=3×2 a×b=b×a Note: The commutative property does not hold for division. 2÷4≠4÷2 Associative means that the grouping does not make any difference.
(2 × 3) × 4 = 2 × (3 × 4) (a × b) × c = a × (b × c) The grouping has changed (parentheses moved), but the sides are still equal. Note: The associative property does not hold for division. (8 ÷ 4) ÷ 2 ≠ 8 ÷ (4 ÷ 2) The identity element for multiplication is 1. Any number multiplied by 1 gives the original number.
3×1=3 a×1=a The multiplicative inverse is the reciprocal of the number. Any number multiplied by its reciprocal equals 1.
2 # 1 = 1; therefore, 2 and 2 a # 1a = 1; therefore, a and
1 are inverses. 2 1 are inverses. a
A Property of Two Operations The distributive property is the process of distributing the number on the outside of a set of parentheses to each number on the inside.
2(3 + 4) = 2(3) + 2(4) a(b + c) = a(b) + a(c)
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Note: You cannot use the distributive property with only one operation. 3(4 × 5 × 6) ≠ 3(4) × 3(5) × 3(6) a(bcd) ≠ a(b) × a(c) × a(d) or (ab)(ac)(ad)
Arithmetic Diagnostic Test (Including Number Sense, Probability, Statistics and Graphs) Questions 1. 28 × 39 is approximately: 2. 6 = ? 4 3. Change 5 3 to an improper fraction. 4 32 4. Change to a whole number or mixed number in lowest terms. 6 5. 2 + 3 = 5 5 6. 1 3 + 2 5 = 8 6 7 5 - = 7. 9 9 8. 11 - 2 = 3 1 9. 6 - 3 3 = 4 4 1 1 # = 10. 6 6 11. 2 3 # 1 5 = 8 6 1 3 ' = 12. 4 2 13. 2 37 ' 1 1 = 4 14. .07 + 1.2 + .471 = 15. .45 – .003 = 16. $78.24 – $31.68 = 17. .5 × .5 = 18. 8.001 × 2.3 =
g
19. .7 .147 = 20. .002 g12 =
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A Quick Review of Mathematics
21. 1 of $7.20 = 3 22. Circle the larger number: 7.9 or 4.35. 23. 39 out of 100 means: 24. Change 4% to a decimal. 25. 46% of 58 = 26. Change .009 to a percent. 27. Change 12.5% to a fraction. 28. Change 3 to a percent. 8 29. Is 93 prime? 30. What is the percent increase in a rise in temperature from 80° to 100°? 31. –6 + 8 = 32. –7 × –9 = 33. - 9 = 34. 82 = 35. 32 × 35 = 36. The square root of 30 is approximately equal to: 37. 37,000,000 written in scientific notation is: 38. What is the probability of tossing a coin and getting heads twice in a row? 39. What is the median of the numbers 4, 3, 5, 7, 9, and 1? 40. Based on the following circle graph, how much time does Timmy spend doing homework?
School 33%
Eating 12.5%
Sleeping 22%
12.5% Misc. Home- 20% work
How Timmy Spends His 24-Hour Day
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Answers 1. 1,200 2. 24 3. 23 4 4. 5 2 or 5 1 3 6 5 or 1 5. 5 6. 4 5 24 2 7. 9 8. 10 1 3 2 9. 2 or 2 1 4 2 1 10. 36 11. 209 or 4 17 48 48 1 12. 6 13. 68 or 1 33 35 35 14. 1.741 15. .447 16. $46.56 17. .25 18. 18.4023 19. .21 20. 6,000 21. $2.40 22. 7.9 23. 39% or 39 100 24. .04 25. 26.68 26. .9% or 9 % 10 125 or 1 27. 1000 8 28. 37.5% or 37 1 % 2 29. No. 30. 25% 31. 2 32. 63 33. 9
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34. 64 35. 37 36. 5.5 or 5 1 2 37. 3.7 × 107 38. 1 4 39. 4 1 or 4.5 2 40. 3 hours
Arithmetic Review Place Value Each position in any number has place value. For instance, in the number 485, 4 is in the hundreds place, 8 is in the tens place, and 5 is in the ones place. Thus, place value is as follows: 8 7 6
.
4 3 6 2 9 7 0 2
and so on
,
tenths hundredths thousandths ten-thousandths hundred-thousandths millionths ten-millionths hundred-millionths
3 4 5
hundreds tens ones
,
hundred thousands ten thousands thousands
0 9 2
hundred millions ten millions millions
,
billions
3
Rounding Off To round off any number: 1. Underline the place value to which you’re rounding off. 2. Look to the immediate right (one place) of your underlined place value. 3. Identify the number (the one to the right). If it is 5 or higher, round your underlined place value up 1. If the number (the one to the right) is 4 or less, leave your underlined place value as it is and change all the other numbers to its right to zeros. For example: Round to the nearest thousand: 345,678 becomes 346,000 928,499 becomes 928,000 This works with decimals as well. Round to the nearest hundredth: 3.4678 becomes 3.47 298,435.083 becomes 298,435.08
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Estimating Sums, Differences, Products and Quotients Knowing how to approximate or estimate not only saves you time but can also help you check your answer to see whether it is reasonable.
Estimating Sums Use rounded numbers to estimate sums. For example, give an estimate for the sum 3,741 + 5,021 rounded to the nearest thousand. 3,741 + 5,021 ↓ ↓ 4,000 + 5,000 = 9,000 3,741 + 5,021 ≈ 9,000
So
Note: The symbol ≈ means is approximately equal to.
Estimating Differences Use rounded numbers to estimate differences. For example, give an estimate for the difference 317,753 – 115,522 rounded to the nearest hundred thousand. 317,753 – 115,522 ↓ ↓ 300,000 – 100,000 = 200,000 317,753 – 115,522 ≈ 200,000
So
Estimating Products Use rounded numbers to estimate products. For example, estimate the product of 722 × 489 by rounding to the nearest hundred.
So
722 × 489 ↓ ↓ 700 × 500 = 350,000 722 × 489 ≈ 350,000
If both multipliers end in 50 or are halfway numbers, then rounding one number up and one number down gives a better estimate of the product. For example, estimate the product of 650 × 350 by rounding to the nearest hundred.
So
650 × 350 ↓ ↓ Round one number up and one down. 700 × 300 = 210,000 650 × 300 ≈ 210,000
You can also round the first number down and the second number up and get this estimate:
So
650 × 350 ↓ ↓ 600 × 400 = 240,000 650 × 350 ≈ 240,000
In either case, this approximation is closer than if you round both numbers up, which is the standard rule.
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A Quick Review of Mathematics
Estimating Quotients Use rounded numbers to estimate quotients. For example, estimate the quotient of 891 ÷ 288 by rounding to the nearest hundred. 891 ÷ 288 ↓ ↓ 900 ÷ 300 = 3 891 ÷ 288 ≈ 3
So
Fractions Fractions consist of two numbers: a numerator (which is above the line) and a denominator (which is below the line). 1 numerator or numerator 1 denominator 2 2 denominator The denominator indicates the number of equal parts into which something is divided. The numerator indicates how many of these equal parts are contained in the fraction. Thus, if the fraction is 3 of a pie, then the denominator, 5, 5 indicates that the pie has been divided into 5 equal parts, of which 3 (the numerator) are in the fraction. Sometimes it helps to think of the dividing line (in the middle of a fraction) as meaning out of. In other words, 3 also 5 means 3 out of 5 equal pieces from the whole pie.
Common Fractions and Improper Fractions A fraction like 3 , where the numerator is smaller than the denominator, is less than one. This kind of fraction is called a 5 common fraction. Sometimes a fraction represents more than one. This is when the numerator is larger than the denominator. Thus, 12 7 is more than one. This is called an improper fraction.
Mixed Numbers When a term contains both a whole number (such as 3, 8 or 25) and a fraction (such as 1 , 1 or 3 ), it is called a mixed 2 4 4 number. For example, 5 1 and 290 3 are both mixed numbers. 4 4 To change an improper fraction to a mixed number, divide the denominator into the numerator. For example: 18 = 3 3 5 5
3 5 g18 15 3
To change a mixed number to an improper fraction, multiply the denominator by the whole number, add the numerator and put the total over the original denominator. For example: 41=9 2 2
2#4+1=9
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Reducing Fractions A fraction must be reduced to lowest terms. This is done by dividing both the numerator and the denominator by the largest number that divides evenly into both. For example, 14 is reduced by dividing both terms by 2, giving 7 . 8 16 Likewise, 20 is reduced to 4 by dividing both the numerator and denominator by 5. 25 5
Adding Fractions To add fractions, first change all denominators to their lowest common denominator (LCD)—the lowest number that can be divided evenly by all the denominators in the problem. When all the denominators are the same, add fractions by simply adding the numerators (the denominator remains the same). For example:
3 3 = 8 8 1 4 + = 2 8
one-half is changed to four-eighths
1 3 = 4 12 1 4 + = 3 12
7 8
change both fractions to LCD of 12
7 12
In the first example, we changed the 1 to 4 because 8 is the LCD, and then we added the numerators 3 and 4 to get 7 . 2 8 8 In the second example, we had to change both fractions to get the LCD of 12, and then we added the numerators to get 7 . Of course, if the denominators are already the same, just add the numerators. For example: 12 6 + 3 = 9 11 11 11
Adding Mixed Numbers To add mixed numbers, the same rule (find the LCD) applies, but always add the whole number to get your final answer. For example:
2 1⁄ 2 = 2 2 ⁄ 4 + 31⁄4 = 3 1⁄4
change one-half to two-fourths
5 3⁄ 4 remember to add the whole numbers
Subtracting Fractions To subtract fractions, the same rule (find the LCD) applies, except subtract the numerators. For example: 7= 8 -1 = 4
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7 8 2 8 5 8
3= 9 4 12 -1 = 4 3 12 5 12
A Quick Review of Mathematics
Subtracting Mixed Numbers When you subtract mixed numbers, sometimes you have to borrow from the whole number, just like you sometimes borrow from the next column when subtracting ordinary numbers. For example: 37 ⁄ 6
4 11
41 ⁄ 6 − 25 ⁄ 6 12 ⁄ 6 = 1 1 ⁄ 3
651 − 129 522
you borrowed one in the form 6 ⁄6 from the 1's column
you borrowed 1 from the 10's column
To subtract a mixed number from a whole number, you have to borrow from the whole number. For example:
6 − 31⁄5
borrow one in the form of 5 ⁄5 from the 6
55 ⁄ 5 = 3 1⁄5 =
24 ⁄5 remember to subtract the remaining whole numbers
Multiplying Fractions Simply multiply the numerators, and then multiply the denominators. Reduce to lowest terms if necessary. For example: 2 # 5 = 10 3 12 36
reduce 10 to 5 18 36
This answer had to be reduced because it wasn’t in lowest terms.
Canceling When Multiplying Fractions You could have canceled first. Canceling eliminates the need to reduce your answer. To cancel, find a number that divides evenly into one numerator and one denominator. In this case, 2 divides evenly into 2 in the numerator (it goes in one time) and 12 in the denominator (it goes in 6 times). Thus: 1
2 # 5 = 3 12 6 Now that you’ve canceled, you can multiply as before. 1
2 # 5 = 5 3 12 6 18 You can cancel only when multiplying fractions.
Multiplying Mixed Numbers To multiply mixed numbers, first change any mixed number to an improper fraction. Then multiply as previously shown. To change mixed numbers to improper fractions: 1. Multiply the whole number by the denominator of the fraction. 2. Add this to the numerator of the fraction.
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3. This is now your numerator. 4. The denominator remains the same. 3 1 # 2 1 = 10 # 9 = 90 = 7 6 = 7 1 3 4 3 4 12 12 2 Then, change the answer, if it is in improper form, back to a mixed number and reduce if necessary.
Dividing Fractions To divide fractions, invert (turn upside down) the second fraction and multiply. Then, reduce if necessary. For example: 1'1=1#5=5 6 5 6 1 6
1'1=1#3=1 6 3 6 1 2
Simplifying Fractions If either numerator or denominator consists of several numbers, these numbers must be combined into one number. Then, reduce if necessary. For example: 28 + 14 = 42 or 26 + 17 43 1+1 1 2 3 4 2 = 4 + 4 = 4 = 3 # 12 = 36 = 9 = 1 2 7 7 1+1 4 3 4 7 28 7 3 4 12 + 12 12
Decimals Fractions can also be written in decimal form by using a symbol called a decimal point. All numbers to the left of the decimal point are whole numbers. All numbers to the right of the decimal point are fractions with denominators of only 10, 100, 1,000, 10,000, and so on, as follows: .6 = 6 = 3 10 5 .7 = 7 10 .07 = 7 100 .007 = 7 1000 .0007 = 7 10000 25 .25 = =1 100 4
Adding and Subtracting Decimals To add or subtract decimals, just line up the decimal points and then add or subtract in the same manner as when adding or subtracting regular numbers. For example: 23.6 + 1.75 + 300.002 =
Adding zeros can make the problem easier to work:
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23.6 1.75 300.002 325.352
A Quick Review of Mathematics
23.600 1.750 300.002 325.352 54.26 - 1.10 53.16
and
54.26 - 1.1 =
8
and 78.9 - 37.43 =
1
78.9 0 - 37.43 41.47
Whole numbers can have decimal points to their right. For example: 6
91
17.0 0 - 8.43 8.57
17 – 8.43 =
Multiplying Decimals To multiply decimals, multiply as usual. Then, count the total number of digits above the line that are to the right of all decimal points. Place the decimal point in the answer so that the number of digits to the right of the decimal is the same as it is above the line. For example:
40.012 × 3.1 40012 120036 124.0372
3 digits 1 digit
4 digits
total of 4 digits above the line that are to the right of the decimal point decimal point placed so there is same number of digits to the right of the decimal point
Dividing Decimals Dividing decimals is the same as dividing other numbers, except that when the divisor (the number you’re dividing by) has a decimal, move it to the right as many places as necessary until it is a whole number. Then move the decimal point in the dividend (the number being divided into) the same number of places. Sometimes you have to add zeros to the dividend (the number inside the division sign). 4. 1.25 g 5. = 125 g 500. or 13000. 0.002 g 26. = 2 g 26000.
Conversions Changing Decimals to Percents To change decimals to percents: 1. Move the decimal point two places to the right. 2. Insert a percent sign. .75 = 75% .05 = 5%
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Changing Percents to Decimals To change percents to decimals: 1. Eliminate the percent sign. 2. Move the decimal point two places to the left. (Sometimes adding zeros is necessary.) 75% = .75 5% = .05 23% = .23 .2% = .002
Changing Fractions to Percents To change a fraction to a percent: 1. Multiply by 100. 2. Insert a percent sign. 1 = 1 # 100 = 100 = 50% 2 c2m 2 2 = c 2 m # 100 = 200 = 40% 5 5 5
Changing Percents to Fractions To change percents to fractions: 1. Divide the percent by 100. 2. Eliminate the percent sign. 3. Reduce if necessary. 60% = 60 = 3 100 5
13% = 13 100
Changing Fractions to Decimals To change a fraction to a decimal, simply do what the operation says. In other words, 3 means 13 divided by 20. So, 20 do just that. (Insert decimal points and zeros accordingly.) .65 20 g13.00 = .65
5= 8
Changing Decimals to Fractions To change a decimal to a fraction: 1. Move the decimal point two places to the right. 2. Put that number over 100. 3. Reduce if necessary.
100
.65 = 65 = 13 100 20 .05 = 5 = 1 100 20 .75 = 75 = 3 100 4
.625 8 g 5.000 = .625
A Quick Review of Mathematics
Read it: .8 Write it: 8 10 Reduce it: 4 5
Using Percents Finding the Percent of a Number To determine the percent of a number, change the percent to a fraction or decimal (whichever is easier for you) and multiply. The word of means multiply. For example: 1. What is 20% of 80? c
20 # 80 = 1600 = 16 or .20 # 80 = 16.00 = 16 100 m 100
2. What is 12% of 50? c
3. What is 1 % of 18? 2
12 # 50 = 600 = 6 or .12 # 50 = 6.00 = 6 100 m 100
1 2 # 18 = c 1 m # 18 = 18 = 9 or .005 # 18 = .09 100 200 200 100
Other Applications of Percent Turn the question (word for word) into an equation. For what substitute the letter x; for is substitute an equal sign; for of substitute a multiplication sign. Change percents to decimals or fractions, whichever you find easier. Then solve the equation. For example: 1. 18 is what percent of 90?
18 = x ^ 90 h 18 = x 90 1 =x 5 20% = x
2. 10 is 50% of what number? 10 = .50 ^ x h 10 = x .50 20 = x 3. What is 15% of 60? x = c 15 m # 60 = 900 = 9 100 100 or .15 ^ 60 h = 9
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Percentage Increase or Decrease To find the percentage change (increase or decrease), use this formula: change # 100 = percentage change starting point For example: 1. What is the percentage decrease of a $500 item on sale for $400? Change: 500 - 400 = 100 change # 100 = 100 # 100 = 1 # 100 = 20% decrease starting point 500 5 2. What is the percentage increase of Jon’s salary if it goes from $150 a month to $200 a month? Change: 200 - 150 = 50 change # 100 = 50 # 100 = 1 # 100 = 33 1 % increase 3 3 starting point 150
Signed Numbers (Positive Numbers and Negative Numbers) On a number line, numbers to the right of 0 are positive. Numbers to the left of 0 are negative, as follows:
. . . −3
−2
−1
0
+1
+2
+3 . . .
Given any two numbers on a number line, the one on the right is always larger, regardless of its sign (positive or negative).
Adding Signed Numbers When adding two numbers with the same sign (either both positive or both negative), add the numbers and keep the same sign. For example: +5 ++ 7 + 12
-8 +- 3 - 11
When adding two numbers with different signs (one positive and one negative), subtract the numbers and keep the sign from the larger one. For example: +5 +- 7 -2
102
- 59 ++ 72 + 13
A Quick Review of Mathematics
Subtracting Signed Numbers To subtract positive and/or negative numbers, just change the sign of the number being subtracted and add. For example: +12 +12 -+ 4 +- 4 +8
-19 -19 -+ 6 +- 6 - 25
-14 -14 -- 4 ++ 4 -10
+ 20 + 20 -- 3 ++ 3 + 23
Multiplying and Dividing Signed Numbers To multiply or divide signed numbers, treat them just like regular numbers but remember this rule: An odd number of negative signs produces a negative answer; an even number of negative signs produces a positive answer. For example: ^ - 3h^ + 8h^ - 5h^ - 1h^ - 2 h =+ 240
Absolute Value The numerical value when direction or sign is not considered is called the absolute value. The value of a number is written 3 = 3 and - 4 = 4 . The absolute value of a number is always positive except when the number is 0. For example: -8 = 8 3 - 9 = -6 = 6 3 - - 6 = 3 - 6 = -3 Note: Absolute values must be taken first, or the work must be done first within the absolute value signs.
Powers and Exponents An exponent is a positive or negative number placed above and to the right of a quantity. It expresses the power to which the quantity is to be raised or lowered. In 43, 3 is the exponent. It shows that 4 is to be used as a factor three times. 4 × 4 × 4 (multiplied by itself twice). 43 is read four to the third power (or four cubed). For example: 24 = 2 × 2 × 2 × 2 = 16 32 = 3 × 3 = 9 Remember that x1 = x and x0 = 1 when x is any number (other than 0). For example: 21 = 2 20 = 1 31 = 3 30 = 1
Negative Exponents If an exponent is negative, such as 3–2, then the number and exponent can be dropped under the number 1 in a fraction to remove the negative sign. The number can be simplified as follows: 3 - 2 = 12 = 1 9 3
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Operations with Powers and Exponents To multiply two numbers with exponents, if the base numbers are the same, simply keep the base number and add the exponents. For example: 23 × 25 = 28 72 × 74 = 76
[(2 × 2 × 2) × (2 × 2 × 2 × 2 × 2) = 28]
[2(3 + 5) = 28]
To divide two numbers with exponents, if the base numbers are the same, simply keep the base number and subtract the second exponent from the first. For example: 9 3] 4 - 2 g = 3 2 C
34 ' 32 = 32
9 9] 6 - 2 g = 9 4 C
96 = 96 ' 92 = 94 92 Three Notes: ■
■
■
If the base numbers are different in multiplication or division, simplify each number with an exponent first, and then perform the operation. To add or subtract numbers with exponents, whether the base is the same or different, simplify each number with an exponent first, and then perform the indicated operation. If a number with an exponent is taken to another power (42)3, simply keep the original base number and multiply the exponents. For example: (42)3 = 46 (34)2 = 38
[4(2 x 3) = 46]
Squares and Square Roots To square a number, just multiply it by itself. For example, 6 squared (written 62) is 6 × 6, or 36. Thirty-six is called a perfect square (the square of a whole number). Following is a list of some perfect squares: 12 = 1 22 = 4 32 = 9 42 = 16 52 = 25 62 = 36 72 = 49 82 = 64 92 = 81 102 = 100 112 = 121 122 = 144 . . . Square roots of nonperfect squares can be approximated. Two approximations to remember are: 2 . 1.4 3 . 1.7
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To find the square root of a number, find some number that when multiplied by itself gives you the original number. In other words, to find the square root of 25, find the number that when multiplied by itself gives you 25. The square root of 25, then, is 5. The symbol for square root is . Following is a list of perfect (whole number) square roots: 1=1 4=2 9=3 16 = 4 25 = 5 36 = 6 49 = 7 64 = 8 81 = 9 100 = 10
Square Root Rules Two numbers multiplied under a radical (square root) sign equal the product of the two square roots. For example: ^ 4 h^ 25h = 4 # 25 = 2 # 5 = 10 or
100 = 10
Likewise with division: 64 = 4
64 8 = = 4 or 2 4
16 = 4
Addition and subtraction, however, are different. The numbers must be combined under the radical before any computation of square roots is done. For example: 10 + 6 = 16 = 4
10 + 6 does not equal 6Y = @ 10 + 6 93 - 12 = 81 = 9
Approximating Square Roots To find a square root that is not a whole number, you should approximate. For example: Approximate 57. Since 57 is between 49 and 64, it falls somewhere between 7 and 8. And because 57 is just about halfway between 49 and 64, 57 is approximately 7 1 . 2 Approximate 83. 9 10 81 < 83 < 100 Since 83 is slightly more than 81 (whose square root is 9), 83 is a little more than 9. Because 83 is only two steps up from the nearest perfect square (81) and 17 steps to the next perfect square (100), 83 is 2 of the way to 100. 19 2 . 2 . 1 = .1 19 20 10 Therefore, 83 . 9.1.
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Simplifying Square Roots To simplify numbers under a radical (square root sign): 1. Factor the number to two numbers, one (or more) of which is a perfect square. 2. Take the square root of the perfect square(s). 3. Leave the other factors under the . Simplify
75. 75 = 25 # 3 = 25 # 3 = 5 3
Simplify 200. 200 = 100 # 2 = 100 # 2 = 10 2 Simplify 900. 900 = 100 # 9 = 100 # 9 = 10 # 3 = 30
Scientific Notation Very large or very small numbers are sometimes written in scientific notation. A number written in scientific notation is a number equal to or greater than 1, but less than 10 and multiplied by a power of 10. For example: 1. 2,100,000 written in scientific notation is 2.1 × 106. Simply place the decimal point to get a number between 1 and 10, and then count the digits to the right of the decimal to get the power of 10.
2 100 000
moved 6 digits to the left
2. .0000004 written in scientific notation is 4 × 10–7. Simply place the decimal point to get a number between 1 and 10, and then count the digits from the original decimal point to the new one.
0000004.
moved 7 digits to the right
Whole numbers have positive exponents, and fractions have negative exponents. To change from scientific notation, simply move the decimal point according to the exponent of 10. For example: 1. 3.2 × 104 = 3.2000 = 32,000
In this case 4 places to the right.
2. 2.6 × 10-3 = 002.6 = .0026
In this case 3 places to the left.
Parentheses Parentheses are used to group numbers. Everything inside a set of parentheses must be done before any other operations. For example: 6 – (–3 + a – 2b + c) = 6 + (+3 – a + 2b – c) = 6 + 3 – a + 2b – c = 9 – a + 2b – c
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Order of Operations If addition, multiplication, division, powers, parentheses and so on are all contained in one problem, the order of operations is as follows:
1. parentheses 2. exponents 3. multiplication whichever comes first, left to right 4. division 5. addition 6. subtraction whichever comes first, left to right For example: 10 – 3 × 6 + 102 + (6 + 1) × 4 = 10 – 3 × 6 + 102 + (7) × 4 = (parentheses first) 10 – 3 × 6 + 100 + (7) × 4 = (exponents next) 10 – 18 × 100 + 28 = (multiplication) –8 + 100 + 28 = (addition/subtraction, left to right) 92 + 28 = 120 An easy way to remember the order of operations after parentheses is: Please Excuse My Dear Aunt Sally, or PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction).
Some Basic Probability and Statistics Probability Probability is the numerical measure of the chance of an outcome or event occurring. When all outcomes are equally likely to occur, the probability of the occurrence of a given outcome can be found by using the following formula: Probability= number of favorable outcomes number of possible outcomes For example: 1. Using the equally divided spinner shown in the following figure, what is the probability of spinning a 6 in one spin?
1
3
8
2
9
10
7
4
5
6
Because there is only one 6 on the spinner out of 10 numbers and all the numbers are equally spaced, the probability is 1 out of 10 or 1 . 10
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2. Again, using the spinner shown previously, what is the probability of spinning a 3 or a 5 in one spin? Because there are 2 favorable outcomes out of 10 possible outcomes, the probability is 2 out of 10 or 2 , or 1 . 10 5 When two events are independent of each other, multiply to find the favorable and/or possible outcomes. 3. What is the probability of tossing heads three consecutive times with a two-sided fair coin? Since each toss is independent and the odds are 1 for each toss, the probability is: 2 1#1#1=1 2 2 2 8 4. Three green marbles, two blue marbles, and five yellow marbles are placed in a jar. What is the probability of selecting at random, a green marble on the first draw? Since there are 10 marbles (total possible outcomes) and 3 green marbles (favorable outcomes), the probability is 3 out of 10, or 3 . 10
Statistics The study of numerical data and its distribution is called statistics. The three basic measures indicating the center of a distribution are: mean, median and mode.
Mean, Arithmetic Mean or Average To find the average of a group of numbers: 1. Add them up. 2. Divide by the number of items you added. For example: 1. What is the average of 10, 20, 35, 40 and 45? 10 + 20 + 35 + 40 + 45 = 150 150 ÷ 5 = 30 The average is 30. 2. What is the average of 0, 12, 18, 20, 31 and 45? 0 + 12 + 18 + 20 + 31 + 45 = 126 126 ÷ 6 = 21 The average is 21. 3. What is the average of 25, 27, 27 and 27?
The average is 26 1 . 2
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25 + 27 + 27 + 27 = 106 106 ' 4 = 26 1 2
A Quick Review of Mathematics
Median A median is simply the middle number in a list of numbers that have been written in numerical order. For example, in the following list—3, 4, 6, 9, 21, 24, 56—the number 9 is the median. If the list contains an even number of items, average the two middle numbers to get the median. For example, in the following list—5, 6, 7, 8, 9, 10—the median is 7 1 . Because there is an even number of items, the 2 average of the middle two, 7 and 8, gives the median. The list has to be in numerical order (or put in numerical order) first. The median is easy to calculate and is not influenced by extreme measures.
Mode A mode is simply the number most frequently listed in a group of numbers. For example, in the following group—5, 9, 7, 3, 9, 4, 6, 9, 7, 9, 2—the mode is 9 because it appears more often than any other number. There can be more than one mode. If there are two modes, the group is called bimodal.
Graphs Information can be displayed in many ways. The three basic types of graphs you should know about are bar graphs, line graphs and circle graphs (or pie charts). You should also know the scatter plot, which is similar to a coordinate graph. When answering questions related to a graph: ■ ■ ■ ■ ■ ■
Examine the entire graph—notice labels and headings. Focus on the information given. Look for major changes—high points, low points, trends. Do not memorize the graph; refer to it. Pay special attention to which part of the graph the question is referring to. Reread the headings and labels if you don’t understand.
Bar Graphs Bar graphs convert the information in a chart into separate bars or columns. Some graphs list numbers along one edge and places, dates, people, or things (individual categories) along the other edge. Always try to determine the relationship between the columns in a graph or chart. For example: 1. The following bar graph indicates that City W has approximately how many more billboards than City Y? City W City X City Y City Z 0
100
200
300
400
500
Billboards
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The graph shows the number of billboards in each city, with the numbers given along the bottom of the graph in increases of 100. The names are listed along the left side. City W has approximately 500 billboards. The bar graph for City Y stops about halfway between 100 and 200. Consider that halfway between 100 and 200 is 150. So City W (500) has approximately 350 more billboards than City Y (150). 500 – 150 = 350 2. Based on the following bar graph, answer these questions: A. B. C.
The number of books sold by Mystery Mystery from 1990 to 1992 exceeded the number of those sold by All Sports by approximately how many? From 1991 to 1992, the percent increase in number of books sold by All Sports exceeded the percent increase in number of books sold by Mystery Mystery by approximately how much? What caused the 1992 decline in Reference Unlimited’s number of books sold? 3.5
Number of Books in Millions
3.0
2.5
2.0
1.5
1.0
.5
1990 1991 1992
1990 1991 1992
1990 1991 1992
Mystery Mystery
Reference Unlimited
All Sports
The graph contains multiple bars representing each publisher. Each single bar stands for the number of books sold in a single year. You might be tempted to write out the numbers as you do your arithmetic (3.5 million = 3,500,000). Writing out the numbers is unnecessary, as it often is with graphs that use large numbers. Since all measurements are in millions, adding zeros does not add precision to the numbers. A. Referring to the Mystery Mystery bars, the number of books sold per year is as follows: 1990 = 2.5 1991 = 2.5 1992 = 3.4
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Use a piece of paper as a straightedge to determine this last number. Totaling the number of books sold for all three years gives 8.4. Referring to the All Sports bars, the number of books sold per year is as follows: 1990 = 1 1991 = 2.1 1992 = 3 Again, use a piece of paper as a straightedge, but don’t designate numbers beyond the nearest 10th because the graph numbers prescribe no greater accuracy than this. Totaling the number of books sold for all three years gives 6.1. So, the number of books sold by Mystery Mystery exceeded the number of books sold by All Sports by 2.3 million. B. Graph and chart questions might ask you to calculate percent increase or percent decrease. The formula for figuring either of these is the same. change = percent change starting point In this case, the percent increase in number of books sold by Mystery Mystery can be calculated first. Number of books sold in 1991 = 2.5 Number of books sold in 1992 = 3.4 Change = .9 The 1991 amount is the starting point, so: change = .9 = 36% starting point 2.5 The percent increase in number of books sold by All Sports can be calculated as follows: Number of books sold in 1991 = 2.1 Number of books sold in 1992 = 3 Change = .9 change = .9 = 4.28 . 43% starting po int 2.1 So the percent increase of All Sports exceeded that of Mystery Mystery by 7%: 43% – 36% = 7% C. This question cannot be answered based on the information in the graph. Never assume information that is not given. In this case, the multiple factors that could cause a decline in the number of books sold are not represented by the graph.
Line Graphs Line graphs convert data into points on a grid. These points are then connected to show a relationship among items, dates, times and so on. Notice the slopes of the lines connecting the points. These lines show increases and decreases— the sharper the slope upward, the greater the increase, the sharper the slope downward, the greater the decrease. Line graphs can show trends, or changes, in data over a period of time.
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For example: 1. Based on the following line graph, answer these questions: A. B.
In what year was the property value of Moose Lake Resort about $500,000? In which 10-year period was there the greatest decrease in the property value of Moose Lake Resort?
$800,000
Total Value
$700,000 $600,000 $500,000 $400,000 $300,000 $200,000 $100,000 1920 1930 1940 1950 1960 1970 1980 1990
Years
A. The information along the left side of the graph shows the property value of Moose Lake Resort in increments of $100,000. The bottom of the graph shows the years from 1920 to 1990. In 1970, the property value was about $500,000. Using a sheet of paper as a straightedge helps to see that the dot in the 1970 column lines up with $500,000 on the left. B. Since the slope of the line goes down from 1920 to 1930, there must have been a decrease in property value. If you read the actual numbers, you notice a decrease from $300,000 to about $250,000. 2. According to the following line graph, the tomato plant grew the most between which two weeks? 100 90 80
Centimeters
70 60 50 40 30 20 10 0 1
2
3
4
5
6
7
8
Weeks
The numbers at the bottom of the graph give the weeks of growth of the plant. The numbers on the left give the height of the plant in centimeters. The sharpest upward slope occurs between week 3 and week 4, when the plant grew from 40 centimeters to 80 centimeters, a total of 40 centimeters growth.
Circle Graphs or Pie Charts A circle graph, or pie chart, shows the relationship between the whole circle (100%) and the various slices that represent portions of that 100%—the larger the slice, the higher the percentage.
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For example: 1. Based on the following circle graph: A. B.
If Smithville Community Theater has $1,000 to spend this month, how much is spent on set construction? What is the ratio of the amount of money spent on advertising to the amount of money spent on set construction? Smithville Community Theater Expenditures
20% set construction 20% costumes and props 10% misc. items
25% staff salaries 15% advertising 10% physical plant
A. The theater spends 20% of its money on set construction. Twenty percent of $1,000 is $200, so $200 is spent on set construction. B. To answer this question, you must use the information in the graph to make a ratio. advertising = 15% of 1000 = 150 = 3 set construction 20% of 1000 200 4 Notice that 15% reduces to 3 . 20% 4 2. Based on the following circle graph: A. B.
If the Bell Canyon PTA spends the same percentage on dances every year, how much do they spend on dances in a year in which the total amount spent is $15,000? The amount of money spent on field trips in 1995 was approximately what percent of the total amount spent?
$2,900 field trips
$2,400 student store supplies
$1,100 misc. $1,400 student awards $2,200 dances
$10,000 Total Expenditures
A. To answer this question, must find a percent, and then apply this percent to a new total. In 1995, the PTA spent $2,200 on dances. This can be calculated as 22% of the total spent in 1995 by the following method: 2, 200 = 22 = 22% 10, 000 100
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Multiplying 22% by the new total amount spent ($15,000) gives the right answer. 22% = .22 .22 × 15,000 = 3,300 or $3,300 You could use another common-sense method. If $2,200 out of $10,000 is spent for dances, $1,100 out of every $5,000 is spent for dances. Since $15,000 is 3 × $5,000, 3 × $1,100 is $3,300. B. By carefully reading the information in the graph, you find that $2,900 was spent on field trips. The information describing the graph explains that the total expenditures were $10,000. Because $2,900 is approximately $3,000, the approximate percentage is worked out as follows: 3, 000 = 30 = 30% 10, 000 100
Scatter Plot A scatter plot is a graph representing a set of data and showing a relationship or connection between the two quantities given. The graph is typically placed in one part of a coordinate plane (the upper right quarter, called Quadrant I). When the data is placed on the scatter plot, usually a relationship can be seen. If the points appear to form a line, a linear relationship is suggested.
Biology test score (out of 100 possible)
If the line goes up to the right, that is, one quantity increases as another increases, then the relationship is called a positive correlation. For example: 100 90 80 70 60 50
1
2
3
4
5
6
7
Number of hours studying
Grade point average
If the line goes down to the right, that is, one quantity decreases as another increases, then the relationship is called a negative correlation. For example:
4.0 3.0 2.0 1.0 4
8
12
16
20
Days absent from school
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If the data does not appear to show any line or any relationship between the quantities, the scatter plot is said to show no correlation. For example:
Math test score (percent right)
100 90 80 70 60 30 60 90 120 150 180 210
Time spent listening to music (minutes)
Algebra Diagnostic Test Questions 1. The sum of a number and 9 can be written: 2. Evaluate: 3x2 + 5y + 7 if x = –2 and y = 3. 3. Solve for x: x + 5 = 17. 4. Solve for x: 4x + 9 = 21. 5. Solve for x: 5x + 7 = 3x – 9. 6. Solve for x: x - 4 = 8 . 7. Solve for x: mx – n = y. 8. Solve for x: rx = st . y 9. Solve for y: 37 = . 8 10. Simplify 8xy2 + 3xy + 4xy2 – 2xy. 11. Simplify 6x2(4x3y). 12. x–5 = 13. Simplify (5x + 2z) + (3x – 4z). 14. Simplify (4x – 7z) – (3x – 4z). 15. Factor ab + ac. 16. Factor x2 – 5x – 14. 17. Solve x2 + 7x = –10. 18. Solve for x: 2x + 3 ≤ 11.
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19. Solve for x: 3x + 4 ≥ 5x – 8. 20. Solve for x: x - 3 < 6 . 21. Solve for x: 2 x + 4 $ 8 . 22. Solve this system for x and y: 8x + 2y = 7 3x – 4y = 5 23. Graph x + y = 6 on the following graph: y 5 4 3 2 1 -5 -4 -3 -2 -1 0 -1
x 1
2
3
4
5
-2 -3 -4 -5
24. What is the slope and the y-intercept of the equation y = 6x + 2? 25. What is the equation of the line passing through the points (3,6) and (2,–3)?
Answers 1. n + 9 or 9 + n 2. 34 3. x = 12 4. x = 3 5. x = –8 6. x = 12 or –12
_ y + ni m 8. x = rts 3 9. y = 24 7 or 3 7 7. x =
10. 12xy2 + xy 11. 24x5y 12. 15 x
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13. 8x – 2z 14. x – 3z 15. a(b + c) 16. (x – 7)(x + 2) 17. x = –2 or x = –5 18. x ≤ 4 19. x ≤ 6 20. –3 < x < 9 could also be written x > –3 and x < 9 21. x ≥ 2 or x ≤ –2 22. x = 1, y = - 1 2 23. y 5 4 3 2 1 –5 –4 –3 –2 –1
0 –1
x 1
2
3
4
5
–2 –3 –4 –5
24. Slope = 6, y-intercept = 2 25. y = 9x – 21
Algebra Review Variables and Algebraic Expressions A variable is a symbol used to denote any element of a given set—often a letter used to stand for a number. Variables are used to change verbal expressions into algebraic expressions. For example: Give the algebraic expression: Verbal Expression
Algebraic Expression
(a) the sum of a number and 7
n + 7 or 7 + n
(b) the number diminished by 10
n – 10
(c) seven times a number
7n
(d) x divided by 4
x/4
(e) five more than the product of 2 and n
2n + 5 or 5 + 2n
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These words can be helpful in making algebraic expressions. Key Words Denoting Addition sum
larger than
enlarge
plus
gain
rise
more than
increase
grow
greater than Key Words Denoting Subtraction difference
smaller than
lower
minus
fewer than
diminish
lose
decrease
reduced
less than
drop
Key Words Denoting Multiplication product
times
multiplied by
twice
of
Key Words Denoting Division quotient
ratio
divided by
half
Evaluating Expressions To evaluate an expression, insert the value for the unknowns and do the arithmetic. For example: Evaluate each of the following. 1. ab + c if a = 5, b = 4 and c = 3 5(4) + 3 = 20 + 3 = 23 2. 2x2 + 3y + 6 if x = 2 and y = 9 2(22) + 3(9) + 6 = 2(4) + 27 + 6 = 8 + 27 + 6 = 41
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Equations An equation is a relationship between numbers and/or symbols. It helps to remember that an equation is like a balance scale, with the equal sign (=) being the fulcrum, or center. Thus, if you do the same thing to both sides of the equal sign (say, add 5 to each side), the equation is still balanced. To solve an equation, first you must get the variable you are looking for on one side of the equal sign and everything else on the other side. For example: 1. Solve for x: x – 5 = 23 To solve the equation x – 5 = 23, get x by itself on one side; therefore, add 5 to both sides: x - 5 = 23 +5 +5 x = 28 In the same manner, subtract, multiply, or divide both sides of an equation by the same (nonzero) number, and the equation does not change. Sometimes you have to use more than one step to solve for an unknown. For example: 2. Solve for x: 3x + 4 = 19 Subtract 4 from both sides to get the 3x by itself on one side: 3x + 4 = 19 -4 -4 3x = 15 Then divide both sides by 3 to get x: 3x = 15 3 3 x =5 3. Solve for x: 6x + 3 = 4x + 5 Add –3 to each side: 6x + 3 = 4x + 5 -3 -3 6x = 4x + 2 Add –4x to each side: 6x = 4x + 2 - 4x - 4x 2x = 2 Divide each side by 2: 2x = 2 2 2 x =1 Remember: Solving an equation requires using opposite operations until the variable is on one side by itself (for addition, subtract; for multiplication, divide; and so on).
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Solving Equations Containing Absolute Value To solve an equation containing absolute value, isolate the absolute value on one side of the equation. Then, set its contents equal to both + and – the other side of the equation and solve both equations. For example: 1. Solve for x: x + 2 = 5 Isolate the absolute value x +2= 5 -2 -2 x
= 3
Set the contents of the absolute value portion equal to +3 and –3. x=3 x = –3 Answer: 3, –3 2. Solve for x: 3 x - 1 - 1 = 11 Isolate the absolute value. 3 x - 1 - 1 = 11 +1 +1
Divide by 3.
3 x-1 3 x-1 3 x-1
= 12 = 12 3 =4
Set the contents of the absolute value portion equal to +4 and –4. Solving for x, x-1= 4 +1 +1 x = 5
x - 1 = -4 +1 +1 x = -3
Answer: 5, –3
Understood Multiplying When two or more variables, or a number and variables, are written next to each other, they are understood to be multiplied. Thus, 8x means 8 times x, ab means a times b, and 18ab means 18 times a times b. Parentheses also represent multiplication. Thus, (a)b means a times b. A raised dot also means multiplication. Thus, 6 ⋅ 5 means 6 times 5.
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Literal Equations Literal equations have no numbers, only symbols (variables). For example: Solve for Q: QP – X = Y First add X to both sides: QP - X = Y +X +X QP =Y+X Then divide both sides by P: QP Y + X P = P X Q= Y+ P Again opposite operations were used to isolate Q.
Cross Multiplying p Solve for x: bx = q To solve this equation quickly, cross multiply. To cross multiply: 1. Bring the denominators up next to the numerators on the opposite side. 2. Multiply. b= p x q bq = px Then, divide both sides by p to get x alone: bq px p = p bq bq p = x or x = p Cross multiplying can be used only when the format is two fractions separated by an equal sign.
Proportions Proportions are written as two fractions equal to each other. p Solve this proportion for x: q = xy This is read “p is to q as x is to y.” Cross multiply and solve: py = xq py xq q = q py py q = x or x = q
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Monomials and Polynomials A monomial is an algebraic expression that consists of only one term. For instance, 9x, 4a2, and 3mpxz2 are all monomials. A polynomial consists of two or more terms; x + y, y2 – x2, and x2 + 3x + 5y2 are all polynomials.
Adding and Subtracting Monomials To add or subtract monomials, follow the same rules as with regular signed numbers, provided that the terms are alike: 15x 2 yz - 18x 2 yz
3x + 2x = 5x
2
- 3x yz
Multiplying and Dividing Monomials To multiply monomials, add the exponents of the same terms: (x3)(x4) = x7 (x × x × x)(x × x × x × x) = x7 To divide monomials, subtract the exponents of like terms: y 15 = y 11 y4
x5 y2 = x2 y x3 y
36a 4 b 6 = - 4a 3 b 5 - 9ab
Remember: x is the same as x1.
Working with Negative Exponents If an exponent is negative, such as x–3, then the variable and exponent can be dropped under the number 1 in a fraction to remove the negative sign, as follows. x - 3 = 13 x Another example: a - 5 = 15 a
Adding and Subtracting Polynomials To add or subtract polynomials, just arrange like terms in columns, and then add or subtract: Add: Add:
a 2 + ab + b 2 3a 2 + 4ab - 2b 2 4a 2 + 5ab - b 2
Subtract: a2 + b2
Subtract: ^ - h 2a 2 - b 2
a2 + b2 +- 2a 2 + b 2 - a 2 + 2b 2
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Multiplying Polynomials To multiply polynomials, multiply each term in one polynomial by each term in the other polynomials. Then simplify if necessary: ^ 3x + a h^ 2x - 2a h = 2x - 2a
23 # 19
# 3x + a + 2ax - 2a 2 6x 2 - 6ax
similar to
6x 2 - 4ax - 2a 2
207 230 427
Factoring To factor means to find two or more quantities whose product equals the original quantity.
Factoring Out a Common Factor Factor 2y3 – 6y. 1. Find the largest common monomial factor of each term. 2. Divide the original polynomial by this factor to obtain the second factor. (The second factor is a polynomial.) 2y3 – 6y = 2y(y2 – 3) Another example:
x5 – 4x3 + x2 = x2(x3 – 4x + 1)
Factoring the Difference Between Two Squares Factor x2 – 144. 1. Find the square root of the first term and the square root of the second term. 2. Express your answer as the product of the sum of the quantities from step 1 times the difference of those quantities. x2 – 144 = (x + 12)(x – 12) Another example:
a2 – b2 = (a + b)(a – b)
Note: x2 + 144 is not factorable.
Factoring Polynomials That Have Three Terms: Ax2 + Bx + C To factor polynomials that have three terms, of the form Ax2 + Bx + C: 1. Check to see if you can monomial factor (factor out common terms). Then, if A = 1 (that is, the first term is simply x2), use double parentheses and factor the first term. Place these factors in the left sides of the parentheses. For example, (x )(x ). 2. Factor the last term, and place the factors in the right sides of the parentheses. To decide on the signs of the numbers, do the following:
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If the sign of the last term is negative: 1. Find two numbers whose product is the last term and whose difference is the coefficient (number in front) of the middle term. 2. Give the larger of these two numbers the sign of the middle term, and give the opposite sign to the other factor. If the sign of the last term is positive: 1. Find two numbers whose product is the last term and whose sum is the coefficient of the middle term. 2. Give both factors the sign of the middle term. For example, 1. Factor x2 – 3x – 10. First check to see if you can monomial factor (factor out common terms). Because this is not possible, use double parentheses and factor the first terms as follows: (x )(x ). Next, factor the last term (10) into 2 times 5. (Using the preceding information, 5 must take the negative sign and 2 must take the positive sign because they then total the coefficient of the middle term, which is –3.) Add the proper signs, leaving: (x – 5)(x + 2) Multiply the means (inner terms) and extremes (outer terms) to check your work.
(x − 5) (x + 2) − 5x + 2x − 3x (which is the middle term) To completely check, multiply the factors together.
x−5 × x +2 + 2x − 10 x2 − 5x x2 − 3x − 10 2. Factor x2 + 8x + 15. (x + 3)(x + 5) Notice that 3 × 5 = 15 and 3 + 5 = 8, the coefficient of the middle term. Also, the signs of both factors are +, the sign of the middle term. To check your work:
(x + 3) (x + 5) + 3x + 5x + 8x (the middle term) If, however, A ≠ 1 (that is, the first term has a coefficient—for example, 4x2 + 5x + 1), then additional trial and error is necessary.
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3. Factor 4x2 + 5x + 1. (2x + )(2x + ) might work for the first term. But when 1’s are used as factors to get the last term— (2x + 1)(2x + 1)—the middle term comes out as 4x instead of 5x.
(2x + 1) (2x + 1) + 2x + 2x + 4x Therefore, try (4x + )(x + ). This time, using 1’s as factors to get the last terms gives (4x + 1)(x + 1). Checking for the middle term:
(4x + 1) (x + 1) + 1x + 4x + 5x Therefore, 4x2 + 5x + 1 = (4x + 1)(x + 1). 4. Factor 5x3 + 6x2 + x. Factoring out an x leaves x(5x2 + 6x + 1). Now, factor as usual giving x(5x + 1)(x + 1). To check your work:
(5x + 1) (x + 1) + 1x
(the middle term after x was factored out)
+ 5x + 6x
Solving Quadratic Equations A quadratic equation is an equation that could be written in the form Ax2 + Bx + C = 0. To solve a quadratic equation: 1. 2. 3. 4. 5.
Put all terms on one side of the equal sign, leaving zero on the other side. Factor. Set each factor equal to zero. Solve each of these equations. Check by inserting your answer in the original equation.
For example, 1. Solve x2 – 6x = 16. Following the steps, x2 – 6x = 16 becomes x2 – 6x – 16 = 0. Factoring, (x – 8)(x – 2) = 0 x–8=0 x =8
or
x+2=0 x = –2
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To check: 82 – 6(8) = 16 64 – 48 = 16 16 = 16
(–2)2 – 6(–2) = 16 4 + 12 = 16 16 = 16
or or or
Both values 8 and –2 are solutions to the original equation. 2. Solve y2 = –6y – 5. Setting all terms equal to zero: y2 + 6y + 5 = 0 Factoring, (y + 5)(y + 1) = 0 Setting each factor to 0: y+5=0 y = –5
or or
(–5)2 = –6(–5) – 5 25 = 30 – 5 25 = 25
or or or
y+1=0 y = –1
To check: (–1)2 = –6(–1) – 5 1=6–5 1=1
A quadratic equation with a term missing is called an incomplete quadratic equation. 3. Solve x2 – 16 = 0. Factoring, (x + 4)(x – 4) = 0: x+4=0 or x = –4 or
x–4=0 x=4
To check: (–4)2 – 16 = 0 16 – 16 = 0 0=0
or or or
(4)2 – 16 = 0 16 – 16 = 0 0=0
4. Solve x2 + 6x = 0. Factoring, x(x + 6) = 0: x=0 x=0
or or
(0)2 + 6(0) = 0 0+0=0 0=0
or or or
x+6=0 x = –6
To check:
126
(–6)2 + 6(–6) = 0 36 + –36 = 0 0=0
A Quick Review of Mathematics
Inequalities An inequality is a statement in which the relationships are not equal. Instead of using an equal sign (=), as in an equation, we use > (greater than) and < (less than), or ≥ (greater than or equal to) and ≤ (less than or equal to). When working with inequalities, treat them exactly like equations, except, when you multiply or divide both sides by a negative number, reverse the direction of the sign. For example: 1. Solve for x: 2x + 4 > 6. 2x + 4 > 6 -4 -4 2x > 2 2x > 2 2 2 x >1 2. Solve for x: –7x > 14. Divide by –7 and reverse the sign. - 7x < 14 -7 -7 x < -2 3. Solve for x: 3x + 2 ≥ 5x – 10. 3x + 2 $ 5x - 10 -2 -2 3x $ 5x - 12 - 5x - 5x - 2x $ - 12 Divide both sides by –2 and reverse the sign. - 2x # - 12 -2 -2 x #6
Solving Inequalities Containing Absolute Value To solve an inequality containing absolute value, follow the same steps as solving equations with absolute value, except reverse the direction of the sign when setting the absolute value opposite a negative. For example, 1. Solve for x: x - 1 > 2 . Isolate the absolute value. x-1 >2 Set the contents of the absolute value portion to both 2 and –2. Be sure to change the direction of the sign when using –2. Solve for x. x-1> 2 +1 +1 x > 3
or
x - 1 < -2 +1 +1 x < -1
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2. Solve for x: 3 x - 2 # 1. Isolate the absolute value. 3 x -2# 1 +2 +2 3 x
#3
3 x 3
#3 3
x
#1
Set the contents of the absolute value portion to both 1 and –1. Be sure to change the direction of the sign when using –1. x < 1 and x > –1
Solving for Two Unknowns—Systems of Equations If you solve for two equations with the same two unknowns in each one, you can solve for both unknowns. There are three common methods for solving: addition/subtraction, substitution and graphing.
Addition/Subtraction Method To use the addition/subtraction method: 1. Multiply one or both equations by some number to make the number in front of one of the variables (the unknowns) the same in each equation. 2. Add or subtract the two equations to eliminate one variable. 3. Solve for the other unknown. 4. Insert the value of the first unknown in one of the original equations to solve for the second unknown. For example: 1. Solve for x and y: 3x + 3y = 24 2x + y = 13 First multiply the bottom equation by 3. Now the y is preceded by a 3 in each equation. 3x + 3y = 24 3(2x )+ 3(y) = 3(13)
3x + 3y = 24 6x + 3y = 39
Now you can subtract equations, eliminating the y terms. 3x + 3y = 24 - 6x + - 3y = - 39 - 3x = -15 - 3x = - 15 -3 -3 x =5
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Now insert x = 5 in one of the original equations to solve for y. 2x + y = 13
2 ^ 5h + y = 13 10 + y = 13 -10 -10 y=3 Answer: x = 5, y = 3 Of course, if the number in front of a variable is already the same in each equation, you do not have to change either equation. Simply add or subtract. 2. Solve for x and y: x+y=7 x–y=3
x+y= 7 x-y= 3 2x = 10 2x 2
= 10 2
x
= 5
Now, inserting 5 for x in the first equation gives: 5+y= 7 -5 -5 y= 2 Answer: x = 5, y = 2 You should note that this method does not work when the two equations are, in fact, the same. 3. Solve for a and b: 3a + 4b = 2 6a + 8b = 4 The second equation is actually the first equation multiplied by 2. In this instance, the system is unsolvable. 4. Solve for p and q: 3p + 4q = 9 2p + 2q = 6 Multiply the second equation by 2. (2)2p + (2)2q = (2)6 4p + 4q = 12 Now subtract the equations. 3p + 4q = 9 (-) 4p + 4q = 12 -p =- 3 p = 3
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Now that you know p = 3, you can plug in 3 for p in either of the two original equations to find q. 3p + 4q = 9 3(3) + 4q = 9 9 + 4q = 9 4q = 0 q=0 Answer: p = 3, q = 0
Substitution Method Sometimes a system is more easily solved by the substitution method. This method involves substituting one equation into another. For example, 1. Solve for x and y when x = y + 8 and x + 3y = 48. 1. From the first equation, substitute (y + 8) for x in the second equation. (y + 8) + 3y = 48 2. Now solve for y. Simplify by combining y’s. 4y + 8 -8 4y 4y 4 y
= 48 -8 = 40 = 40 4 = 10
3. Now insert y = 10 in one of the original equations. x=y+8 x = 10 + 8 x = 18 Answer: y = 10, x = 18
Graphing Method Another method of solving equations is by graphing each equation on a coordinate graph. The coordinates of the intersection are the solution to the system. If you are unfamiliar with coordinate graphing, carefully review the “Basic Coordinate Geometry” section before attempting this method. For example, solve the following system by graphing: x=4+y x – 3y = 4 1. First, find three values for x and y that satisfy each equation. (Although only two points are necessary to determine a straight line, finding a third point is a good way of checking.)
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A Quick Review of Mathematics
x=4+y
x − 3y = 4
x
y
x
y
4 2 5
0 -2 1
1 4 7
-1 0 1
2. Now graph the two lines on the coordinate plane, as shown in the following figure. y
(0,0) x (4,0) y=4
x
=
4
+
y
x−3
3. The point where the two lines cross (4, 0) is the solution of the system. 4. If the lines are parallel, they do not intersect, and there is no solution to the system.
Basic Coordinate Geometry Coordinate Graphs (x-y Graphs) A coordinate graph is formed by two perpendicular number lines. These lines are called coordinate axes. The horizontal axis is called the x-axis or the abscissa. The vertical line is called the y-axis or the ordinate. The point at which the two lines intersect is called the origin and is represented by the coordinates (0, 0), often marked simply 0.
•••
y 5 4 3 2 1
x 1
2
3
4
5
•••
0
−5 −4 −3 −2 −1 −1
•••
−2 −3 −4 •••
−5
Each point on a coordinate graph is located by an ordered pair of numbers called coordinates. Notice the placement of points on the following graph and the coordinates, or ordered pairs, that show their location. Numbers are not usually written on the x- and y-axes.
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+y • (3,5) (−3,3) • (−4,2) • • (1,1)
−
+x
0
(−3,−2) • • (2,−3) • (−5,−4) • (1,−5) −
On the x-axis, the numbers to the right of 0 are positive and to the left of 0 are negative. On the y-axis, numbers above 0 are positive and numbers below 0 are negative. The first number in the ordered pair is called the x-coordinate and shows how far to the right or left of 0 the point is. The second number is called the y-coordinate and shows how far up or down the point is from 0. The coordinates, or ordered pairs, are shown as (x, y). The order of these numbers is very important, as the point (3, 2) is different from the point (2, 3). Also, don’t combine the ordered pair of numbers, because they refer to different directions. The coordinate graph is divided into four quarters called quadrants. These quadrants are labeled as follows. y
Quadrant II
Quadrant I
x
Quadrant III
■ ■ ■ ■
Quadrant IV
In quadrant I, x is always positive and y is always positive. In quadrant II, x is always negative and y is always positive. In quadrant III, x is always negative and y is always negative. In quadrant IV, x is always positive and y is always negative.
Graphing Equations on the Coordinate Plane To graph an equation on the coordinate plane, find the solutions by giving a value to one variable and solving the resulting equation for the other variable. Repeat this process to find other solutions. (When giving a value for one variable, start with 0; then try 1, and so forth.) Then, graph the solutions.
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A Quick Review of Mathematics
For example: 1. Graph the equation x + y = 6. If x is 0, then y is 6. (0) + y = 6 y=6 If x is 1, then y is 5.
^1h + y = 6 -1 -1
y=5 If x is 2, then y is 4.
^2h + y = 6 -2 -2 y=4
Using a simple chart is helpful.
x
y
0 1 2
6 5 4
Now, plot these coordinates as shown in the following figure. y
(0,0)
x
These solutions, when plotted, form a straight line. Equations whose graphs of their solution sets form a straight line are called linear equations. Equations that have a variable raised to a power, show division by a variable, involve variables with square roots, or have variables multiplied together do not form a straight line when their solutions are graphed. These are called nonlinear equations.
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2. Graph the equation y = x2 + 4. If x is 0, then y is 4. y = (0)2 + 4 y=0+4 y=4 If x is 1, then y is 5. y = (1)2 + 4 y=1+4 y=5 If x is 2, then y is 8. y = (2)2 + 4 y=4+4 y=8 Use a simple chart.
x
y
0 1 2
4 5 8
Now, plot these coordinates as shown in the following figure. y
(0,0)
x
These solutions, when plotted, give a curved line (nonlinear). The more points plotted, the easier it is to see the curve and describe the solution set.
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A Quick Review of Mathematics
Slope and Intercept of Linear Equations Two relationships between the graph of a linear equation and the equation itself must be pointed out. One involves the slope of the line, and the other involves the point where the line crosses the y-axis. To see either of these relationships, the terms of the equation must be in a certain order. (+)(1)y = ( )x + ( ) When the terms are written in this order, the equation is said to be in y-form. y-form is written y = mx + b, and the two relationships involve m and b. For example: Write the equations in y-form. 1. x – y = 3 –y = –x + 3 y=x–3 2. y = –2x + l (already in y form) 3. x – 2y = 4 –2y = –x + 4 2y = x – 4 y= 1 x-2 2 As shown in the graphs of the three problems in the following figure, the lines cross the y-axis at –3, +1 and –2, the last term in each equation. y
y
(0,0)
(0,0) x
(1)
x
(2) y
(0,0) x
(3)
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If a linear equation is written in the form y = mx + b, b is the y-intercept. The slope of a line is defined as: the change in y the change in x The word change refers to the difference in the value of y (or x) between two points on a line. y at point A - y at point B yA- yB slope of line AB = x A - x B = x at point A - x at point B Note: Points A and B can be any two points on a line; there is no difference in the slope. 1. Find the slope of x – y = 3 using coordinates. To find the slope of the line, pick any two points on the line, such as A(3, 0) and B (5, 2), and calculate the slope. y A - y B ^ 0h - ^2h - 2 slope = x A - x B = = =1 ^ 3h - ^ 5h - 2 2. Find the slope of y = –2x – 1 using coordinates. Pick two points, such as A(1, –3) and B(–1, 1), and calculate the slope. y A - y B ^ - 3h - ^1h - 3 - 1 - 4 slope = x A - x B = = = = -2 1+1 2 ^1h - ^ - 1h Looking back at the equations for examples 1, 2 and 3 written in y-form, it should be evident that the slope of the line is the same as the numerical coefficient of the x term. 1.
y=x−3 slope = 1 y-intercept = −3
2.
y = -2x + 1 slope = -2 y-intercept = 1
3.
y = 1/2 x − 2 slope = 1/2 y-intercept = −2
Graphing Linear Equations Using Slope and Intercept Graphing an equation using its slope and y-intercept is usually quite easy. 1. 2. 3. 4.
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State the equation in y-form. Locate the y-intercept on the graph (that is, one of the points on the line). Write the slope as a ratio (fraction), and use it to locate other points on the line. Draw a line through the points.
A Quick Review of Mathematics
For example: 1. Graph the equation x – y = 2 using slope and y-intercept. x–y=2 –y = –x + 2 y=x–2 Locate –2 on the x-axis and, from this point, count as shown in the following figure: slope = 1
_ for every 1 up i or 1 1 _ go 1 to the righti _ for every 1 down i or - 1 - 1 _ go 1 to the lefti
y
(0,0) x
+1 +1 −1
y = −2
−1
2. Graph the equation 2x – y = –4 using slope and y-intercept. 2x – y = –4 –y = –2x – 4 y = 2x + 4 Locate +4 on the y-axis and, from this point, count as shown in the following figure: slope = 2
_ for every 2 up i or 2 1 _ go 1 to the righti _ for every 2 down i or - 2 - 1 _ go 1 to the lefti
y +1 +2 y=4 −2 −1 (0,0) x
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Finding the Equation of a Line To find the equation of a line when working with ordered pairs, slopes and intercepts, use the following approach. 1. 2. 3. 4.
Find the slope, m. Find the y-intercept, b. Substitute the slope and intercept into the slope-intercept form, y = mx + b. Change the slope-intercept form to standard form, Ax + By = C.
For example: 1. Find the equation of the line when m = –4 and b = 3. Find the slope, m. m = –4 (given) Find the y-intercept, b. b = 3 (given) Substitute the slope and intercept into the slope-intercept form y = mx + b. y = –4x + 3 Change the slope-intercept form to standard form Ax + By = C. Since y = –4x + 3 Adding 4x to each side gives: 4x + y = 3 2. Find the equation of the line passing through the point (6, 4) with a slope of –3. Find the slope, m. m = –3 (given) Find the y-intercept, b. Substitute m = –3 and the point (6, 4) into the slope-intercept form to find b. y = mx + b where y = 4, m = –3 and x = 6 4 = (–3)(6) + b 4 = –18 + b 18 + 4 = b 22 = b Substitute the slope and intercept into the slope-intercept form: y = mx + b. Since m = –3 and b = 22, y = –3x + 22. Change the slope-intercept form to standard form: Ax + By = C. Since y = –3x + 22, adding 3x to each side gives 3x + y = 22.
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A Quick Review of Mathematics
3. Find the equation of the line passing through points (5, –4) and (3, –2). Find the slope, m. change in y ^ - 4 h - ^ - 2 h - 4 + 2 - 2 m= = = = = -1 2 2 change in x ^ 5h - ^ 3h Find the y-intercept, b. Substitute the slope and either point into slope-intercept form. y = mx + b where m = –1, x = 5 and y = –4 –4 = (–1)(5) + b –4 = –5 + b 5 + –4 = b 1=b Substitute the slope and intercept into the slope-intercept form: y = mx + b. Since m = –1 and b = 1, y = –1x + 1 or y = –x + 1. Change the slope-intercept form to standard form: Ax + By = C. Since y = –x + 1, adding x to each side gives x + y = 1.
Measurement and Geometry Diagnostic Test Questions 1. How many inches are in 1 yard? 2. How many feet are in 1 mile? 3. How many square feet are in 1 square yard? 4. How many ounces are in 1 pound? 5. How many pounds are in 1 ton? 6. How many pints are in 1 quart? 7. How many quarts are in 1 gallon? 8. How many weeks are in 1 year? 9. One kilometer equals how many meters? 10. How many decimeters in 1 meter? 11. How many milligrams in 1 gram? 12. True or False: A meter is a little more than a yard. 13. If 2.2 pounds equal 1 kilogram, 20 pounds equals approximately how many kilograms? 14. Lines that stay the same distance apart and never meet are called _______ lines. 15. Lines that meet to form 90° angles are called _______ lines.
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Part I: Working Toward Success
B 18″
22″
A C
16. In the preceding triangle, AC must be smaller than _______ inches. A 8″ B
15″
C
17. What is the length of AC in the preceding figure? A 26″ 10″
B
18. What is the length of BC in the preceding figure? 19. Name each of the following polygons: A.
A
C
B
B.
A
B
D
C
AB = BC = CD = AD ∠A = ∠B = ∠C = ∠D = 90°
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C
A Quick Review of Mathematics
A
C.
D
B
C
AB DC AB = DC AD BC AD = BC ∠A = ∠C D.
A
B
D
C
AB = DC AD = BC ∠A = ∠B = ∠C = ∠D = 90° E.
A
B
D
C
AB DC D
B
R
C
S
A
20. Fill in the blanks for circle R in the preceding figure: A. B. C.
RS is called the _______. AB is called the _______. CD is called a _______.
7″
21. Find the area and circumference for the circle in the preceding figure (π ≈ 22 7 ): A. area = B. circumference =
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16″ 15″
12″
13″
30″
22. Find the area and perimeter of the preceding figure: A. B.
area = perimeter = B
A 3″
4″ D
C
6″
23. Find the area and perimeter of the preceding figure (ABCD is a parallelogram): A. B.
area = perimeter = 10″ 12″
24. Find the volume of the preceding figure if V = (πr2)h. (Use 3.14 for π):
4″ 4″
4″
25. What is the surface area and volume of the preceding cube? A. B.
surface area = volume = A
6
B
26. What is the area of ∆ABC in the preceding figure?
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4
C
A Quick Review of Mathematics
Answers 1. 36 inches = 1 yard 2. 5,280 feet = 1 mile 3. 9 square feet = 1 square yard 4. 16 ounces = 1 pound 5. 2,000 pounds = 1 ton 6. 2 pints = 1 quart 7. 4 quarts = 1 gallon 8. 52 weeks = 1 year 9. 1,000 meters = 1 kilometer 10. 1 meter = 10 decimeters 11. 1 gram = 1,000 milligrams 12. True 13. approximately 9 kilograms 14. parallel 15. perpendicular 16. 40 inches (Since AB + BC = 40 inches, AC < AB + BC and AC < 40 inches.) 17. AC = 17 inches 18. Since ∆ABC is a right triangle, use the Pythagorean theorem: a2 + b2 = c2 102 + b2 = 262 100 + b2 = 676 b2 = 576 b = 24” 19. A. B. C. D. E.
triangle square parallelogram rectangle trapezoid
A. B. C.
radius diameter chord
20.
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21. A.
B.
22. A.
B.
area = πr2 = π(72) = 22 7 (7)(7) = 154 square inches circumference = πd = π(14) (d = 14", because r = 7") = 22 7 (14) = 22(2) = 44 inches area = 1 ^ a + b h h 2 = 1 ^16 + 30 h 12 2 = 1 ^ 46 h 12 2 = 23(12) = 276 square inches perimeter = 16 + 13 + 30 + 15 = 74 inches
23. A.
B.
area = bh = 6(3) = 18 square inches perimeter = 6 + 4 + 6 + 4 = 20 inches
24. volume = (πr2)h = (π ⋅ 102)(12) = 3.14(100)(12) = 314(12) = 3,768 cubic inches 25. A. B.
All six surfaces have an area of 4 × 4, or 16 square inches because each surface is a square. Therefore, 16(6) = 96 square inches is the surface area. Volume = side × side × side, or 43 = 64 cubic inches.
26. 12 The area of a triangle is 1 × base × height. 2 A
6
B
144
4
C
A Quick Review of Mathematics
Base AB of the triangle is 4 units (because from A to the y-axis is 2 units and from the y-axis to B is another 2 units). Height BC of the triangle is 6 units (3 units from B to the x-axis and another 3 units to C). Note that ∠B is a right angle. area of triangle = 1 × 4 × 6 2 = 1 × 24 2 = 12
Measurement and Geometry Review Measures Customary System, or English System Length 12 inches (in) = 1 foot (ft) 3 ft = 1 yard (yd) 36 in = 1 yd 1,760 yd = 1 mile (mi) 5,280 ft = 1 mi
Area 144 square inches (sq in) = 1 square foot (sq ft) 9 sq ft = 1 square yard (sq yd)
Weight 16 ounces (oz) = 1 pound (lb) 2,000 lb = 1 ton (T)
Capacity 2 cups = 1 pint (pt) 2 pt = 1 quart (qt) 4 qt = 1 gallon (gal) 4 pecks = 1 bushel
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Time 365 days = 1 year 52 weeks = 1 year 10 years = 1 decade 100 years = 1 century
Metric System, or The International System of Units Length—Meter Kilometer (km) = 1,000 meters (m) Hectometer (hm) = 100 m Dekameter (dam) = 10 m 10 decimeters (dm) = 1 m 100 centimeters (cm) = 1 m 1,000 millimeters (mm) = 1 m
Volume—Liter 1,000 milliliters (ml or mL) = 1 liter (l or L) 1,000 liters = 1 kiloliter (kl or kL)
Mass—Gram 1,000 milligrams (mg) = 1 gram (g) 1,000 g = 1 kilogram (kg) 1,000 kg = 1 metric ton (t)
Some Approximations One meter is a little more than a yard. One kilometer is about .6 of a mile. One kilogram is about 2.2 pounds. One liter is slightly more than a quart.
Converting Units of Measure 1. If 36 inches equal 1 yard, then 3 yards equal how many inches? Intuitively, 3 × 36 = 108 inches By proportion, using yards over inches: 3= 1 x 36
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A Quick Review of Mathematics
Place the same units across from each other—inches across from inches, and so on. Then solve: 3= 1 x 36 Cross multiply: 108 = x x = 108 inches 2. Change three decades into weeks. Since 1 decade equals 10 years, and 1 year equals 52 weeks, 3 decades equal 30 years. 30 years × 52 weeks = 1,560 weeks in 30 years, or 3 decades This was converted step by step. It can be done in one step: 3 × 10 × 52 = 1,560 weeks 3. If 1,760 yards equal 1 mile, how many yards are in 5 miles? 1,760 × 5 = 8,800 yards in 5 miles
Geometry Types of Lines ■
Two or more lines that cross each other at a point are called intersecting lines. That point is on each of those lines. l Q m
■
In the diagram, lines l and m intersect at Q. Two lines that meet to form right angles (90° angles) are called perpendicular lines. The symbol = is used to denote perpendicular lines. l
m
In the diagram, l = m .
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Part I: Working Toward Success
■
Two or more lines that remain the same distance apart at all times are called parallel lines. Parallel lines never meet. The symbol is used to denote parallel lines. l m
In the diagram, l m.
Polygons Closed shapes or figures with three or more sides are called polygons. (Poly means many; gon means sides; thus, polygon means many sides.)
Triangles This section deals with those polygons having the fewest number of sides. A triangle is a three-sided polygon. It has three angles in its interior. The sum of these angles is always 180°. The symbol for triangle is ∆. A triangle is named by all three letters of its vertices. The following figure shows ∆ABC: B
A
C
There are various types of triangles: ■
■ ■ ■
A triangle having all three sides equal (meaning all three sides having the same length) is called an equilateral triangle. A triangle having two sides equal is called an isosceles triangle. A triangle having none of its sides equal is called a scalene triangle. A triangle having a right (90°) angle in its interior is called a right triangle.
Facts about triangles: ■
Every triangle has a base (bottom side) and a height (or altitude). Every height is the perpendicular (forming a right angle) distance from a vertex to its opposite side (the base). A
B
■
E
C
In this diagram of ∆ABC, BC is the base, and AE is the height. AE = BC . The sum of the lengths of any two sides of a triangle must be larger than the length of the third side. In the diagram of ∆ABC: A
AB + BC > AC AB + AC > BC AC + BC > AB B
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C
A Quick Review of Mathematics
Pythagorean theorem: ■
In any right triangle, the relationship between the lengths of the sides is stated by the Pythagorean theorem. A c
b
B
a
C
The parts of a right triangle are: ∠C is the right angle. ■
The side opposite the right angle is called the hypotenuse (side c). (The hypotenuse is always the longest side.) The other two sides are called the legs (sides a and b). The three lengths a, b and c are always numbered such that: a2 + b2 = c2 For example: If a = 3, b = 4 and c = 5: a2 + b2 = c2 32 + 42 = 52 9 + 16 = 25 25 = 25
■
■
5
4
3
Therefore, 3–4–5 is called a Pythagorean triple. There are other values for a, b and c that always work. Some are 1–1– 2, 5–12–13 and 8–15–17. Any multiple of one of these triples also works. For example, multiplying the 3–4–5 solution set shows that 6–8–10, 9–12–15 and 15–20–25 are also Pythagorean triples. If perfect squares are known, the lengths of these sides can be determined easily. A knowledge of the use of algebraic equations can also be used to determine the lengths of the sides. For example: Find the length of x in the triangle.
So,
a2 + b2 = c2 x2 + 102 = 152 x2 + 100 = 225 x2 = 125 x = 125 125 = 25 × 5 = 5 5 x=5 5
x
15
10
Quadrilaterals A polygon having four sides is called a quadrilateral. There are four angles in its interior. The sum of these interior angles is always 360°. A quadrilateral is named using the four letters of its vertices. The following figure shows quadrilateral ABCD. A
D
B
C
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Part I: Working Toward Success
Types of quadrilaterals: ■
A square has four equal sides and four right angles. A
a
B
a
a
D ■
C
a
A rectangle has opposite sides that are equal and four right angles. A
a
B
b
b
D ■
C
a
A parallelogram has opposite sides equal and parallel, opposite angles equal, and consecutive angles supplementary. Every parallelogram has a height. A
a
B
b
b
D
■
a
E
C
AE is the height of the parallelogram, AB CD, and AD BC. A rhombus is a parallelogram with four equal sides. A rhombus has a height. BE is the height. B
a
a
a
A ■
E
D
Other Polygons ■ ■ ■ ■
150
a
D
A trapezoid has only one pair of parallel sides. A trapezoid has a height. AE is the height. AB DC A
■
C
A pentagon is a 5-sided polygon. A hexagon is a 6-sided polygon. An octagon is an 8-sided polygon. A nonagon is a 9-sided polygon. A decagon is a 10-sided polygon.
E
B
C
A Quick Review of Mathematics
Perimeter Perimeter means the total distance all the way around the outside of any shape. The perimeter of any polygon can be determined by adding the lengths of all the sides. The total distance around is the sum of all sides of the polygon. No special formulas are really necessary.
Area Area (A) means the amount of space inside the polygon. The formulas for each area are as follows: Triangle: A = 1 bh 2
h
or
h b
b
For example: A = 1 bh 2
18" 24"
A = 1 ^ 24 h^18h = 216 sq in 2 Square or rectangle: A = lw w l
l l
or
w
w l
w For example: 4"
4"
A = l(w) = 4(4) = 16 sq in
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Part I: Working Toward Success
12" 5"
A = l(w) = 12(5) = 60 sq in Parallelogram: A = bh
h b
For example: A = b(h) 10" 6"
5"
6"
A = 10(5) = 50 sq in Trapezoid: A = 1 ^ b 1 + b 2 h h 2 b1 h b2 For example: 8"
A = 1 ^b1 + b 2h h 2 A = 1 ^8 + 12 h^ 7h 2 = 1 ^ 20 h^ 7h = 70 sq in 2
7" 12"
Circles A closed shape whose side is formed by one curved line, all points on which are equidistant from the center point, is called a circle. Circles are named by the letter of their center point.
M
This is circle M. M is the center point because it is the same distance away from all points on the circle.
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A Quick Review of Mathematics
Parts of a Circle ■
Radius is the distance from a center to any point on a circle. In any circle, all radii (plural) are the same length.
M
MA is a radius. MB is a radius.
B
A ■
Diameter is the distance across a circle, through the center. In any circle, all diameters are the same length. Each diameter is two radii. D
A
AB is a diameter. CD is a diameter.
M
B ■
C
The chord is a line segment whose end points lie on the circle itself. U
RS is a chord. UV is a chord.
R
S M
V
■
The diameter is the longest chord in any circle. An arc is the distance between any two points on the circle itself. An arc is a piece of the circle. The symbol ! is used to denote an arc. It is written on top of the two endpoints that form the arc. Arcs are measured in degrees. There are 360° around a circle. $ This is EF . $ Minor EF is the shorter distance between E and F. $ Major EF is the longer distance between E and F. $ When EF is written, the minor arc is assumed.
E
F
Circumference and Area ■
Circumference is the distance around a circle. Since there are no sides to add up, a formula is needed. π (pi) is a Greek letter that represents a specific number. In fractional or decimal form, the commonly used approximations are: π ≈ 3.14 or π ≈ 22 7. The formula for circumference is: C = πd or C = 2πr. For example: In circle M, d = 8 because r = 4. C = πd = π(8) = 3.14(8) = 25.12 inches
M
4"
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Part I: Working Toward Success
■
The area of a circle can be determined by: A = πr2. For example: In circle M, r = 5 because d = 10. A = π(r2) = π(52) = 3.14(25) = 78.5 sq in
10" M
Congruence and Similarity Two plane (flat) geometric figures are said to be congruent if they are identical in size and shape. They are said to be similar if they have the same shape, but are not identical in size. For example: All squares are similar.
The following triangles are congruent.
Volume In three dimensions, additional facts can be determined about shapes. Volume refers to the capacity to hold. The formula for volume of each shape is different. The volume of any prism (a three-dimensional shape having many sides, but two bases) can be determined by: Volume (V) = (area of base)(height of prism). Specifically, for a rectangular solid: V = (lw)(h) = lwh
h l w
Specifically, for a cylinder (circular bases): r
V = (πr2)h = πr2h
Volume is labeled in cubic units.
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h
A Quick Review of Mathematics
Surface Area The surface area of a three-dimensional solid is the area of all the surfaces that form the solid. Find the area of each surface, and then add those areas. The surface area of a rectangular solid can be found by adding the areas of all six surfaces. For example: The surface area of this prism is: top: bottom: left side: right side: front: back:
18 × 6 = 108 18 × 6 = 108 6 × 4 = 24 6 × 4 = 24 18 × 4 = 72 18 × 4 = 72 408 sq in
4" 18" 6"
Coordinate Geometry and Measurement Refer to “Basic Coordinate Geometry” in the “Algebra Review” section if you need to review coordinate graphs. Coordinate graphs can be used in measurement problems. For example: y A (-2,5)
B (3,5)
x 0
1. What is the length of AB in the preceding graph? Since the coordinates of the points are (–2, 5) and (3, 5), the first, or x-coordinate is the clue to the distance of each point from the y-axis. The distance to point B from the y-axis is 3, and the distance to point A from the y-axis is 2. (–2 is 2 in the negative direction.) So 3 + 2 gives a length of 5. y
D
C
(0,3)
(5,0) A 0
B
x
2. What is the area of rectangle ABCD in the preceding graph? The formula for the area of a rectangle is base × height. Since point A is at (0, 0) and point B is at (5, 0), the base is 5. Since point D is at (0, 3), the height is 3, so the area is 5 × 3 = 15.
155
PART II
F U LL-LE N G TH PR ACTI C E TE STS The CAHSEE Math has 92 multiple-choice questions, but only 80 actually count toward your score. The 12 additional questions, which can be scattered throughout the exam, are being tested for future exams. Each practice test in this book has 80 questions followed by complete explanations. On the actual test, the questions are grouped by strands, but the order could vary. The actual CAHSEE Math is given in two sessions—46 questions in each session. Remember there is no time limit on CAHSEE Math. But you should try to work at a steady pace, skipping problems that give you difficulty, taking a guess answer, and coming back to them later. The problems in these simulated practice exams follow the California standards and are similar in style and difficulty to the problems on the actual exam. The actual CAHSEE Math is copyrighted and may not be duplicated. These questions are not taken from the actual tests.
CAHSEE Practice Test #1 Directions: Mark only one answer to each question on your answer sheet. If you change an answer, make sure that you erase the previous mark completely. Notes: (1) Figures that accompany problems are drawn as accurately as possible EXCEPT when it is stated that a figure is not drawn to scale. All figures lie in a plane unless noted otherwise. (2) All numbers used on the exam are real numbers. All algebraic expressions represent real numbers unless stated otherwise.
1. The sun is approximately 93,000,000 miles from earth. How can that distance be expressed using scientific notation? A. B. C. D.
9.3 × 10–7 9.3 × 107 93 × 10 6 930 × 10 5
2. Joanna is buying food for a party. She purchases 6 pizzas for $9.75 each, 4 salads for $3.50 each, and 5 bottles of soda for $1.25 each. What is the total cost for all the food and drinks? A. B. C. D.
$14.50 $15.00 $58.50 $78.75
3. 2 ' 1 = 3 4 A. 2 12 B. 1 6 C. 11 12 D. 2 2 3
5. The Tigers have won 75% of their baseball games. How many of the 32 remaining games can they lose and still maintain their current winning percentage? A. B. C. D.
8 16 24 25
6. Approximately 4% of the automotive parts created in a factory are defective. If the factory creates a total of 27,000 parts one month, how many will be defective? A. B. C. D.
4 108 1,008 1,080
7. The price of gasoline in June is 20% higher than the price of gasoline in January. If the price in January is $1.50 per gallon, what will the price in June be? A. B. C. D.
$1.20 $1.70 $1.75 $1.80
4. Felix has an hour and a half to watch TV. If he spends two thirds of his time watching a football game, how much time will he have left to watch another show? A. B. C. D.
1 hour 3 1 hour 2 5 hour 6 1 hour
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Part II: Full-length Practice Tests
8. Michelle is buying a car, which is offered for sale at three dealerships. The prices are shown in the following table. The discounts shown have not been taken from the original price. Dealer
Original Price
Discount
Dealer A
$15,000
none
Dealer B
$16,500
10%
Dealer C
$18,000
15%
13. The square root of 200 is between A. B. C. D.
14. The difference between x and x is 15. What is the value of x? A. B. C. D.
Which dealer has the lowest price for the car? A. B. C. D.
Dealer A Dealer B Dealer C The three dealers are charging the same price.
9. Manjit deposited $1,000 in a savings account. The account pays 10% interest, compounded annually. What will be the value of the account in 4 years? A. B. C. D.
$1,040.00 $1,046.40 $1,400.00 $1,464.10
12 and 13 13 and 14 14 and 15 15 and 16
15.
–7.5 0 7.5 15
Janet has taken four tests in French class, scoring 62, 85, 90, and 75. What is her median score? A. B. C. D.
78 80 87.5 90
16. The rainfall for 5 months is shown in the following table. What was the mean rainfall for the period? Month
3 10. 3- 2 = 3 A. 3–6 B. 3–1.5 C. 3 D. 35
11.
4 + 2 = 15 27 6 A. 135 6 B. 42 C. 42 135 D. 46 135
12. (52)3 = A. B. C. D.
160
5 55 56 515
A. B. C. D.
3.0 3.1 3.6 15.0
Rainfall (in inches)
January
3.6
February
3.6
March
2.7
April
3.1
May
2.0
CAHSEE Practice Test #1
18. The following graph shows annual sales figures for three different cereals, brands A, B, and C. 18,000 16,000
Student
A. B. C. D.
Chocolate Sold
14,000 12,000
32 bars
Maria
15 bars
Hakim
27 bars
Paula
32 bars
4,000
Cheng
15 bars
2,000
15 27 32 15 and 32
Sales
Anselmo
10,000 8,000 6,000
0 1996
1997
1998
1999
Brand A Brand B Brand C
Which of the following statements is the most accurate interpretation of the data shown in the graph? A. B. C. D.
Sales of brand A increased at a higher rate than sales of brand B. Sales of brands A and B increased at the same rate. Sales of brand B increased at a higher rate than sales of brand A. Brand C was the most popular, but became the least popular.
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CAHSEE Practice Test #1
17. Five students are competing to win a prize for selling the most chocolate bars. The sales figures are shown in the following chart. What is the mode of the data set?
Part II: Full-length Practice Tests
19. A fair coin is tossed three times. The following diagram represents the possible outcomes. 3rd Flip H
HHH
T
HHT
H
HTH
21. What percentage of the time is the fair spinner shown in the following figure expected to land on green?
2nd Flip H 1st Flip Heads
Blue
T
Yellow T
HTT
H
THH
Green
Red H
Tails
T
THT
H
TTH
T
TTT
T
What is the probability of two or more heads? A. B. C. D.
3 8 1 2 3 5 4 4
20. Joe has a bag with six cookies in it, one of which is chocolate chip. Joe eats an oatmeal cookie and reaches into the bag to randomly choose another cookie. What is the probability that the cookie chosen is not chocolate chip? A. B. C. D.
162
1 6 1 5 4 5 5 6
A. B. C. D.
12.5% 25% 33.3% 50%
22. A coin flip experiment has produced a string of seven straight heads. What is the probability of heads on the next flip? A. B. C. D.
An outcome of heads is almost certain. An outcome of heads is somewhat more likely. An outcome of heads or tails is equally likely. An outcome of tails is almost certain.
CAHSEE Practice Test #1
Stock B
25
30
20
25
15
20 15
Price
10
Month
v. No
pt. Se
Ju ly
M ay
v. No
pt. Se
Ju ly
M ay
0 M ar.
0 Jan .
5 M ar.
10
5
Jan .
Price
Stock A
Month
Based on the graphs shown, which of the following statements is correct? A. B. C. D.
Mr. Smith made more money on stock A than on stock B. Mr. Smith made the same amount of money on both stocks. Mr. Smith made more money on stock B than on stock A. Mr. Smith lost money on both stocks.
24. Based on the following information, which of the following comparisons is true? Bonnie’s Budget: $2,000/month
Other 30% Rent 45%
Josie’s Budget: $4,000/month
Food 25%
A. B. C. D.
Other 35%
Rent 40% Food 25%
Bonnie and Josie spend the same amount of money on food. Bonnie spends more money on rent than Josie does. Bonnie spends half as much on rent as Josie does. Together the two women spend about $1,500 per month on food.
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CAHSEE Practice Test #1
23. On January 1, Mr. Smith invested equal amounts of money in two different stocks. Their price graphs are shown in the following figure.
Part II: Full-length Practice Tests
25. Which scatter plot shows a positive correlation? B. Monthly Charge for Service
Pounds Gained
A. 4 3 3 2 2 1 1 0 0
2,000
4,000
40 30 20 10 0 0
6,000
1,000
2,000
3,000
4,000
Minutes On-Line
C. 30
SAT Verbal Score
Amount Purchased
Calories Consumed
20 10 0 0.00
0.50
1.00
1.50
2.0
2.50
D. 800 700 600 500 400 300 200 0
Price Per Pound
Foot Size (in centimeters)
26. The following graph shows points representing the relationship of a person’s foot size to a person’s age.
60
80
A.
A.
B.
C. D.
300
No correlation exists between age and foot size. Some correlation exists between age and foot size, but the correlation decreases as age increases. Age and foot size are positively correlated throughout a person’s lifetime. As your feet get bigger, you get older.
2n + 5 = 27 2n = 27 + 5 2n – 5 = 27 + 5 2n = 27
28. You have 4 days to finish a book that contains a total number of pages represented by p. You have already read 50 pages. Which expression tells you how many pages you must average per day to finish the book?
Which of the following statements best describes the relationship shown on the graph?
B.
164
20 40 Age (in years)
200
27. Which of the following equations represents the statement, “Two times a number (n) is 5 more than 27”? A. B. C. D.
35 30 25 20 15 10 5 0 0
100
Daily Television Viewing (in Minutes)
C. D.
p + 50 4 p - 50 4 4 (p + 50) 4 (p – 50)
CAHSEE Practice Test #1
29. If x = –2, then 2x2 – x =
Celia's Homework Schedule: 3 hours total
Distance from home (miles)
6 10 16 18
180 Car A
120 Car B 60 0 0
1
2
3
4
Time (hours)
Spanish 1/5 Math 2/5 History 1/5
31. After 2 hours of driving, how far has car B traveled? A. B. C. D.
40 miles 80 miles 120 miles 160 miles
English 1/5
30. According to her schedule, how much time should Celia spend on her math homework? A. B. C. D.
24 minutes 36 minutes 72 minutes 96 minutes
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CAHSEE Practice Test #1
A. B. C. D.
240
Number of cars
Part II: Full-length Practice Tests
16 14 12 10 8 6 4 2 0 50 – 60
61 – 70
71 – 80
81 – 90
91 – 100
Speed (miles per hour)
32. The preceding graph shows the number of cars on a freeway traveling at the indicated speeds. How many cars are traveling faster than 70 miles per hour? A. B. C. D.
8 cars 10 cars 11 cars 26 cars
33. (a3b2)(ab–2) = A. B. C. D.
166
a4b a4b4 aaabb abb aaaabb bb
34.
9x 16 = A. B. C. D.
3 3x4 3x8 3x16
CAHSEE Practice Test #1
350
300
Area of a Circle (cm2)
250
200
150
100
50
5
10
15
20
Radius of a Circle (cm)
What is the approximate area of a circle with a radius of 8 cm? A. B. C. D.
2 cm2 50 cm2 200 cm2 300 cm2
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CAHSEE Practice Test #1
35. The area of a circle can be determined by the formula A = πr2, where π is a constant that equals approximately 3.14, and r is the radius of a circle. The function is demonstrated on the following graph.
Part II: Full-length Practice Tests
36. What is the slope of the line shown in the following graph? y
4 2
-4
x
0
-2
2
4
-2 -4
A. B. C. D.
–3 -1 3 1 3 3
37. The slope of the following line is 4 . 3 y
d
12
0
What is the value of d? A. B. C. D.
168
8 9 12 16
x
CAHSEE Practice Test #1
Number of CDs purchased
Cost
0
0
1
$10.50
2
$21
3
$31.50
4
$42
Which of the following could be part of a graph of the data on the table?
2y – 1 < –5 A. B. C. D.
y < –2 y > –2 y<2 y>2
42. The length of a shadow varies directly with the height of a vertical object. A pole that is 8-feet high casts a 5-foot shadow. If a pole casts a 40-foot shadow, how tall is the pole? A. B. C. D.
25 feet 37 feet 43 feet 64 feet
B.
43. An airplane has traveled 450 miles in an hour and a half. If it travels at the same rate, how long will it take to travel 1,100 more miles?
Cost
Cost
A.
41. Solve for y.
Number of CDs
A. B. C. D.
3 hours 3 hours, 20 minutes 3 hours, 40 minutes 4 hours
Cost
D.
Cost
C.
Number of CDs
44. Bernice is tiling a floor space as shown in the following figure. Number of CDs
Number of CDs
A. B. C. D.
at most $125 more than $125 less than $125 an amount greater than or equal to $125
40. Solve for x. x +4=6 2 A. B. C. D.
2 4 8 20
1 yard
39. In the inequality 6x + $250 ≤ $1,000, x represents the price of a chair to be purchased along with a dining room table. Which phrase most accurately describes how much money can be spent on each chair?
2 yards
The tiles she is using measure 1 foot by 1 foot. How many tiles will it take to cover the area? A. B. C. D.
2 6 9 18
45. How many minutes are in a day? A. B. C. D.
60 144 1,440 3,600
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CAHSEE Practice Test #1
38. The Good Buy Music Store is having a sale, and every CD is priced at $10.50. Part of a table representing the number of CDs and the cost to purchase them is shown.
Part II: Full-length Practice Tests
A straight line on the map measures 3.5 inches. What would be the actual distance the line represents? A. B. C. D.
3.5 kilometers 4 kilometers 11 kilometers 26.25 kilometers
49. A picture frame is built from wood that measures 2 inches in width, as shown in the following figure. 2 in.
2 in.
16 in.
46. The legend on a map reads “1 inch = 7.5 kilometers.”
47. It takes two men 7 hours to do a job. How long would it take four men to do the job if they worked at the same rate? A. B. C. D.
3 1 hours 2 5 hours 11 hours 14 hours
48. A bicycle rider traveled 3 miles in 15 minutes. How fast was she riding? A. B. C. D.
5 miles per hour 1 mile per minute 12 miles per hour 18 miles per hour
26 in.
How many square inches of wood are required to construct the frame? A. B. C. D.
152 sq. inches 168 sq. inches 264 sq. inches 416 sq. inches
50. Jackie is constructing the closed box shown in the following figure. How many square inches of cardboard does she need, not counting overlapping pieces?
8 in.
4 in. 2 in.
A. B. C. D.
170
14 sq. inches 56 sq. inches 64 sq. inches 112 sq. inches
CAHSEE Practice Test #1
53. All the angles shown in the following figure are right angles. What is the area of the figure? 3 m.
40 meters
6 m. 9 m. 6 m.
9 m.
6 m.
60 meters
A. B. C. D.
6 m.
100 meters 200 meters 240 meters 2,400 meters
9 m.
52. The following shape is made from 1-inch cubes. What is the total volume of the shape?
24 m.
A. B. C. D.
A. B. C. D.
387 square meters 468 square meters 504 square meters 576 square meters
20 cubic inches 84 cubic inches 120 cubic inches 180 cubic inches
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CAHSEE Practice Test #1
51. A fence is constructed around a rectangular playground as shown in the following figure. How many meters of fencing material are required for the perimeter of the playground?
Part II: Full-length Practice Tests
54. A company is designing a can that will hold twice as much as the can shown in the following figure.
55. The following box measures 1 foot by 1 foot by 1 foot. How many 1-inch cubes does it take to fill the box?
Tomato Sauce 16 oz. 1 ft.
Which of the following changes result in the desired volume? (V = h(π r2)) A. B. C. D.
Double the height of the can. Double the radius of the base. Double the diameter of the base. Double both the height and the radius.
1 ft.
A. B. C. D.
172
12 cubes 144 cubes 1,440 cubes 1,728 cubes
1 ft.
CAHSEE Practice Test #1
56. On the following coordinate grid, the distance between 0 and 1 is one unit.
CAHSEE Practice Test #1
y
6 4 2
–6
–4
0
–2 –2
x 2
4
6
A
–4 –6
Point A is a vertex on a square whose area is 9 square units. Which of the following points could be another vertex on the same square? A. B. C. D.
(2, 0) (2, 3) (3, –3) (6, –4)
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Part II: Full-length Practice Tests
57. On the following graph, which of the triangles represents the position of triangle T when it is reflected around the y-axis? y
6 4 A
–6
–4
2
0
–2 C
B x 2
4
–2
6
T
–4 –6
A. B. C. D.
D
A B C D
58. A school is constructing a fence around a triangular-shaped playground, as shown in the following figure.
59. Points A, B, and C lie on circle D, such that AB = 3" and AC = 5". B
3“ 3rd side 6 yds.
5“
A
D
8 yds.
How much fencing material is needed for the third side? A. B. C. D.
174
6 yards 8 yards 10 yards 14 yards
What is the length of BC? A. B. C. D.
3" 4" 8" 34 "
C
CAHSEE Practice Test #1
61. If x = –3, what is the reciprocal of x? B. C. 4 cm.
D.
60. Which of the following is congruent to the preceding figure? 4 cm.
A.
CAHSEE Practice Test #1
A.
5 cm. 3 cm.
–3 -1 3 1 3 3
62. What is the solution to x - 1 $ 2 - 4 ? A. B. C. D.
x > –1 x ≥ –1 x>3 x≥3
3 cm.
5 cm.
63. Which equation is the same as 3(x + 6) – 2(10 – 2x) = –7? A. B. C. D.
B. 3 cm.
3x + 18 – 20 + 4x = –7 3x + 6 – 20 + 4x = –7 3x + 18 – 20 –4x = –7 3x + 6 – 20 – 4x = –7
3 cm.
64. Which inequality is equivalent to 5x + 3 < 4(x + 4)? A. B. C. D.
3 cm.
C.
10 cm. 4 cm.
6 cm.
9x < 19 9x < 13 x < 13 x < 19
65. Solve for x. 2 cx - 1m $ 1 3 3 A. x $ - 1 3 1 B. x $ 4 1 C. x $ 2 D.
x≥2
D.
4.24 cm. 3 cm.
3 cm.
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Part II: Full-length Practice Tests
66. Which of the following is the graph of y = –3x + 3? y
–4
–4
4
4
2
2
0
–2
–2
y
x 2
4
–4
–2
–2
–4
–4
A.
B.
y
y
4
4
2
2
0
x 2
4
–4
C.
67. The point (5, 11) lies on a line that has a slope of 2. What is the y-intercept of the line? A. B. C. D.
(0, 1) (1, 0) (0, 6) (6, 0)
176
(0, –2) (2, 3) (3, 2) (3, 5)
–2
0
x 2
4
2
4
x
D.
69. Line A is parallel to a line that has a slope of 1 . 2 Which of the following could be the equation for line A? A. B. C.
68. Which of the following points lies on the line 2y = 3x – 4? A. B. C. D.
0
–2
D.
y = –2x + 4 y= 1 x+4 2 y=x+ 1 2 y = 2x + 4
CAHSEE Practice Test #1
6x - 2y = 4 ) 2x + y = 8
75. Which of the following is true for the statement a + b = b + a? A. B.
A. B. C. D.
(–4, 2) (–2, 4) (2, 4) (4, 2)
71. Which of the following is equivalent to _ 2ab 2 i_ 3a 2 bi ? ab A. B. C. D.
5ab 6ab 5a2b2 6a2b2
72. John can mow a lawn in 1 hour. Sarah can mow the same lawn in 40 minutes. How long does it take them to mow the lawn working together? A. B. C. D.
24 minutes 47 minutes 50 minutes 1 hour, 40 minutes
73. If Keisha pays $52 for 13 yards of fabric, which of the following expressions can be used to determine the cost per yard of fabric? A. B. C. D.
C. D.
The statement is true only for positive numbers. The statement is true only for negative numbers. The statement is true for all integers, but not for fractions. The statement is true for all numbers.
76. Salvador’s car payment is $265 per month for 36 months. To estimate the total amount he will pay, he rounds both figures and multiplies. He estimates that his payments will total about $12,000. Because of the method Salvador used, his estimate will be A. B. C. D.
less than the exact figure about the same as the exact figure more than the exact figure It cannot be determined.
77. Joe needs to buy 90 tiles; each tile costs $2.15. He estimates the total cost to be about $2,000. Which of the following statements is correct? A. B. C. D.
Joe’s estimate is about one tenth of the exact cost. Joe’s estimate is very close to the exact cost. Joe’s estimate is about 10 times the exact cost. Joe’s estimate is about 100 times the exact cost.
52 + 13 52 – 13 52 × 13 52 ÷ 13
74. Jason has a bag of marbles. All the marbles in the bag are either blue or red. There are 24 blue marbles. How many red marbles are there? Which of the following pieces of information is necessary to solve the preceding problem? A. B. C. D.
the price of the marbles the weight of the marbles the total number of marbles the volume of the marbles
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CAHSEE Practice Test #1
70. What is the solution to the following system of equations?
Part II: Full-length Practice Tests
78. The following graph shows sales data for QuickShop Market.
80. Jonathan read a 396-page book in 6 days. What was the average number of pages he read per day?
QuickShop Sales
Which of the following problems can be solved using the same arithmetic operations that are used to solve the preceding problem?
Non-Food 15%
A. Grocery 40%
B.
Produce 20%
C. Meat 15%
Deli 10%
D.
Terry read 6 books, each of which had 396 pages. How many pages did she read? Sanford has a collection of 396 baseball cards. National League players represent 241 of the cards, and the rest are for players from the American League. How many of his cards are for American League players? Grapefruit are selling at a price of 69 cents per pound. How much do 6 pounds of grapefruit cost? A bus holds 60 people. How many buses are needed for 360 people?
If sales figures for grocery items were $590,000, which of the following figures is the best approximation for produce sales? A. B. C. D.
$20,000 $300,000 $600,000 $1,200,000
79. Farryl baked cookies. The quantity she baked was less than 50, and could be divided evenly by 2, 3, and 5. How many cookies did Farryl bake? A. B. C. D.
6 10 30 60
STOP 178
Test #1—Answers and Explanations Reviewing Practice Test 1 Review your simulated CAHSEE Math practice examination by following these steps: 1. Check the answers you marked on your answer sheet against the answer key that follows. Put a check mark in the box following any wrong answer. 2. Fill out the Review chart (p. 180). 3. Read all the explanations (pp. 182–192). Go back to review any explanations that are not clear to you. 4. Fill out the Reasons for Mistakes chart on p. 180. 5. Go back to the “Math Review” section and review any basic skills necessary before taking the next practice test. Don’t leave out any of these steps. They are very important in learning to do your best on CAHSEE Math.
179
Part II: Full-length Practice Tests
Review Chart Use your marked answer key to fill in the following chart for the multiple-choice questions. Possible Number Sense (NS)
(1–14)
14
Statistics, Data Analysis, Probability (P)
(15–26)
12
Algebra and Functions (AF)
(27–43)
17
Measurement and Geometry (MG)
(44–60)
17
Algebra I (AI)
(61–72)
12
Mathematical Reasoning (MR)
(73–80)
8
Totals
Completed
Right
Wrong
80
Reasons for Mistakes Fill out the following chart only after you have read all the explanations that follow. This chart helps you spot your strengths, weaknesses, and repeated errors or trends in types of errors. Total Missed
Simple Mistake
Misread Problem
Lack of Knowledge
Number Sense (NS) Statistics, Data Analysis, Probability (P) Algebra and Functions (AF) Measurement and Geometry (MG) Algebra I (AI) Mathematical Reasoning (MR) Totals
Examine your results carefully. Reviewing the preceding information helps you pinpoint your common mistakes. Focus on avoiding your most common mistakes as you practice. The Lack of Knowledge column helps you focus your review in the “Math Review” section. If you are missing a lot of questions because of lack of knowledge, you should go back and spend extra time reviewing the basics.
180
Test #1—Answers and Explanations
Answer Key 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.
B (NS) D (NS) D (NS) B (NS) A (NS) D (NS) D (NS) B (NS) D (NS) D (NS) D (NS) C (NS) C (NS) A (NS) B (P) A (P) D (P) A (P) B (P) C (P)
21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40.
A (P) C (P) C (P) D (P) A (P) B (P) B (AF) B (AF) B (AF) C (AF) D (AF) C (AF) D (AF) C (AF) C (AF) A (AF) B (AF) D (AF) A (AF) B (AF)
Number Sense (NS)
14
Statistics, Data Analysis, Probability (P)
12
Algebra and Functions (AF)
17
Measurement and Geometry (MG)
17
Algebra I (AI)
12
Mathematical Reasoning (MR)
41. 42. 43. 44. 45. 46. 47. 48. 49. 50.
A (AF) D (AF) C (AF) D (MG) C (MG) D (MG) A (MG) C (MG) A (MG) D (MG)
51. 52. 53. 54. 55. 56. 57. 58. 59. 60.
B (MG) C (MG) A (MG) A (MG) D (MG) D (MG) C (MG) C (MG) B (MG) A (MG)
61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80.
B (AI) D (AI) A (AI) C (AI) C (AI) B (AI) A (AI) A (AI) B (AI) C (AI) D (AI) A (AI) D (MR) C (MR) D (MR) D (MR) C (MR) B (MR) C (MR) D (MR)
8
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Part II: Full-length Practice Tests
Answers and Explanations Number Sense (NS) 1. B. Note that Choices B, C, and D are equivalent; however C and D express 93,000,000 as the product of a power of 10 and a number greater than 10. To express a number correctly using scientific notation, multiply a positive or negative power of 10 by a number greater than or equal to 1 and less than 10. 2. D. The total cost for six pizzas is 6 × $9.75, or $58.50. Four salads cost $14.00, and five bottles of soda are $6.25. To find the total cost, add: $58.50 + $14.00 + $6.25 = $78.75 3. D. To divide fractions, invert the divisor (the second number) and multiply. However, if you forget this rule, you can find the answer by using other things you know about division and fractions. Just as 12 ÷ 5 means, “How many 5s are in 12?”, 2 ' 1 means, “How many fourths are in two thirds?” Two fourths are in 1 , and 2 is greater 3 4 2 3 than 1 , so the answer must be greater than 2. 2 4. B. This problem involves multiplication of fractions. Two thirds of one and a half hours means 2 # 1 1 = 2 # 3 = 1 hour . If he has used 1 hour watching the football game, then he has 1 an hour left. 3 2 3 2 2 75 3 = They must win three fourths of their games to maintain their percentage, so they can lose up to 5. A. 75% = 100 4 one fourth of their games. 1 # 32 = 8 games. 4 6. D. Four percent means 4 out of a hundred; out of 27,000, there are many more than 4 defective parts, so eliminate Choice A. To find 4% of 27,000, you can convert 4% to .04, then multiply: .04 × 27,000 = 1,080 7. D. Twenty percent of $1.50 = .2 × $1.50 = $.30. So, the price increases $.30 from January to June. Thirty cents more than $1.50 is $1.80. 8. B. Dealer A’s price is fixed at $15,000. To find the price at Dealer B, you must subtract 10% of the original price, which is $16,500. 10% of $16,500 = $1,650 $16,500 – $1,650 = $14,850 This is lower than Dealer A’s price, so you can eliminate Choices A and D. But you must now check the price at Dealer C. 15% of $18,000 = $2,700 $18,000 – $2,700 = $15,300 Both Dealer A and B are cheaper than Dealer C, and Dealer B is cheaper than Dealer A. 9. D. At the end of one year, Manjit has earned $100 in interest (10% of $1,000), so you can eliminate Choices A and B as much too low. Choice C is a calculation of simple interest at the rate of 10% for 4 years with no compounding. However, each year Manjit earns 10% of an increasing amount:
182
End of Year:
Interest Earned
Balance
Year One
$100
$1,100
Year Two
$110
$1,210
Year Three
$121
$1,331
Year Four
$133.10
$1,464.10
Test #1—Answers and Explanations
10. D. To divide common bases with exponents, you subtract the exponents: 33 = 33 -]- 2g = 35 3-2 11. D. The addition of fractions requires a common denominator. You can always find a common denominator by multiplying the two denominators; or, you can multiply the highest power of each prime factor: 4 + 2 = 15 27 15 = 3 × 5 27 = 33 So, the lowest common denominator for 15 and 27 is 33 × 5= 135. Then: 4 # 9 = 36 15 9 135 2 # 5 = 10 27 5 135 36 + 10 = 46 135 135 135 12. C. If you have a number to a power, to another power (that is a number with an exponent, to another exponent), keep the number and multiply the exponents. So (52)3 = 52 × 3 = 56 13. C. The square root of 144 is 12 because 12 × 12 = 144. Likewise, The square root of 169 is 13. The square root of 196 is 14. The square root of 225 is 15. The square root of 256 is 16. Two hundred is between 196 and 225; therefore, the square root of 200 is between 14 and 15. 14. A. The absolute value of a number is always positive. If x is also positive, then the difference between x and x is zero because they are the same number. If that difference is not zero, x must be a negative number. The exact value of x is half the difference between x and x , preceded by a minus sign.
Statistics, Data Analysis, Probability (P) Difference = Distance = 15
x = −7.5
0
|x| = 7.5
15. B. To find the median, rearrange the data values in ascending order: 62, 75, 85, 90 If an odd number of data values exists, one is in the middle, and it is the median. However, in this case, the median lies halfway between the two middle values, 75 and 85. 75 + 85 = 80 2 So, 80 is the median.
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Part II: Full-length Practice Tests
16. A. To find the mean, add the data values and divide by the number of data values: 3.6 + 3.6 + 2.7 + 3.1 + 2 = 3 5 So, the mean is 3 inches. 17. D. The mode of a set of data is the value that occurs most frequently. The possibilities are one mode, more than one mode, or no mode (if no value occurs more frequently than once). In this case, 32 and 15 each occur twice, while 27 occurs only once. 18. A. For the sales of Brand A to start at a very low level and rise to a level almost equal to the sales of the other two brands, they have to grow at a higher rate than the other two. Statement A is correct, and therefore, statements B and C are incorrect. Part of statement D is correct, but part is incorrect. If any part of an answer is false, then the whole answer is false. 19. B. The tree diagram shows 8 possible outcomes, each of which is equally likely. The probability of any one of the outcomes is 1 . Four of the outcomes include two or more heads, therefore the probability of two or more 8 heads = 4 = 1 . 8 2 20. C. After oatmeal is drawn from the bag, 5 cookies remain, only one of which is chocolate chip. Therefore, 4 cookies are not chocolate chip. Because the selection is random, the probability of not getting chocolate chip = 4 . 5 21. A. The green section is 1 of the spinner, so it can be expected to occur 1 of the time. 8 8 1 = 12.5% 8 22. C. Coin flips are independent events; that is to say, the next flip of a coin has nothing to do with the preceding flips. The chance of heads on any given flip is equal to the chance of tails. 23. C. Though the price graphs might look similar, significant differences exist. The main consideration is the starting price and ending price on each graph. Graph A begins at $5 and ends at $20, a difference of $15. Graph B begins at begins $10 and ends at about $27, a change of $17. Stock B was clearly much more profitable. 24. D. To solve this problem, you must recognize that the two women have a different amount of money to budget with: Josie has twice as much money to work with as Bonnie. Therefore, though both women budget 25% of their funds on food, they do not spend the same amount. Bonnie spends 25% of $2,000, which is $500. Josie spends 25% of $4,000, which is $1,000. Answer D is correct because the sum of $500 and $1,000 is $1,500. 25. A. Scatter plots demonstrate graphically how two variables are related. With a positive correlation, the points tend to fall in a line that slopes upward from left to right, indicating that as the variable indicated on the x-axis increases, the variable indicated on the y-axis also increases. With a negative correlation, the line slopes downward from left to right, as in Graph C. Graphs B and D show no definite correlation between the variables. 26. B. Aging from baby to adult, our feet grow. The string of upward sloping data points indicates this correlation. However, after reaching adulthood, the size of peoples’ feet vary greatly. In fact, growing older might result in your feet getting slightly shorter, as the toes tend to bend more, decreasing the length of your foot.
184
Test #1—Answers and Explanations
Algebra and Functions (AF) 27. B. The main relationship is between two times a number (2n) and 27. Which one is bigger? 2n is bigger. How much bigger is it? It’s 5 more than 27. So, to write an equation that makes them equal, you have a choice: Either you can give 5 more to 27: 2n = 27 + 5 or You can take 5 away from 2n: 2n – 5 = 27 Of course, the second equation does not appear in the answer choices, or the question would have two correct answers. 28. B. Since you have already read 50 pages, you have p – 50 pages left to read. To find an average number of pages you must read each day, you need to divide the remaining pages by the number of days, which is 4. If working with variables gives you trouble, choose a convenient number of pages and substitute that in the expressions to find which one makes sense. Suppose your book had 450 pages. If you had already read 50, you would have 400 pages left—so you would need to read an average of 100 pages a day. Only Answer B gives you that result. 29. B. To solve this problem, rewrite the expression by substituting –2 in place of x. 2x2 – x 2(–2)2 – (–2) Then perform the operations in the appropriate order. To do that, remember: Parentheses Exponents Multiplication Division Addition Subtraction
Left to right, whichever comes first Left to right, whichever comes first
So, you must first square –2: 2(4) – (–2) Next you must multiply: 8 – (–2) Finally, subtract –2: 8 – (–2) = 10
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Part II: Full-length Practice Tests
30. C. Celia spends 2 of her homework time on math, so you must find 2 of 3 hours. Since most of the answer 5 5 choices are expressed in minutes, convert 3 hours to 180 minutes, and find 2 of 180 minutes: 5 2 # 180 = 72 minutes 5 31. D. Car B is traveling faster than Car A, and gets home in 3 hours. That is an average speed of 80 miles per hour, so in 2 hours Car B has traveled 160 miles. 32. C. You are looking only for cars traveling faster than 70 miles per hour. These are represented in the graph by three bars. Eight cars are traveling between 71 and 80 mph; 2 cars are traveling between 81 and 90 mph; 1 car is traveling between 91 and 100 mph. 8 + 2 + 1 = 11 33. D. Raising a variable to a positive exponent can be represented as repeated multiplication. When the variable is raised to a negative exponent, that is the inverse of multiplication, which is division. This relationship is expressed in Answer D. The expression can be further simplified to aaaa = a4, but that answer is not shown, so the only correct answer is D. 34. C. The square root of 9 is 3 because 3 × 3 = 9. The square root of x16 is x8 because (x8)(x8) = x16. Therefore, 9x 16 = 3x 8 . 35. C. Radius is plotted on the x-axis of the graph. Find the mark that indicates a radius of 8cm. Move up the vertical line to the point where it intersects the curved function. Now move left on the horizontal line that intersects at the same point. The scale on the y-axis, which indicates the area of the circle, reads approximately 200cm2 at this point. 36. A. Since the line slopes downward from left to right, it has a negative slope; therefore, you can eliminate Answers C and D. Remember that slope is defined as: vertical shift horizontal shift In this case, the line goes down three units for every unit it moves to the right: - 3 = -3 1 37. B. The slope of the line given is 4 , which means it goes up 4 units for every 3 units it goes across. If it goes up 3 12 units, it must go across 9 units because 4 = 12 . 3 9 38. D. The relationship between the quantity of CDs purchased and the price charged is linear: It makes a straight line on a graph. If you understand that idea, you can immediately eliminate Graph B, which shows the price of CDs increasing with each CD that is purchased. Eliminate Graph A because it shows the cost decreasing as more CDs are purchased. Eliminate Graph C because it shows a constant cost no matter how many CDs are purchased. Only Graph D shows a linear relationship in which the cost increases at a rate proportional to the number of CDs purchased. 39. A. To solve the inequality: 6x + $250 ≤ $1,000 –$250 = –$250
Subtract $250 from both sides to isolate the variable.
6x ≤ $750 6x # $750 6 6
Divide both sides by 6.
x ≤ $125 The amount of money per chair must be less than or equal to $125—or, in other words, it is at most $125.
186
Test #1—Answers and Explanations
40. B. To solve this equation: x +4=6 2 –4 = –4 x =2 2 2 c x m = 2 ^2h 2 x=4
Subtract 4 from each side to isolate the variable.
Multiply each side by 2 to clear the fraction.
The correct answer could also be easily determined by substituting the answer choices into the equation. 41. A. To solve this inequality: 2y – 1 < –5 +1 = +1
Add 1 to each side to isolate the variable.
2y < –4 2y - 4 < 2 2
Divide each side by 2 to clear the coefficient.
y < –2 42. D. This problem can be solved by writing a proportion and solving for the unknown value: height of pole A height of pole B = length of shadow A length of shadow B Now write in values for what you know and a variable for the quantity you don’t know: 8= x 5 40 5x = (8)(40) = 320 5x = 320 5 5 x = 64
Take the cross-products to create an equation. Divide both sides by the coefficient (5).
43. C. This problem can also be solved by writing a proportion and solving for the unknown value: distance traveled = distance traveled time time Now write in values for what you know and a variable for the quantity you don’t know: 450 miles = 1100 miles x 1.5 hours If you convert hours to minutes, the problem gets easier: 450 miles = 1100 miles x 90 minutes Reduce to: 5 miles = 1100 miles x 1 minute Now the numbers in the equation are much simpler to work with: 5x = 1,100 x = 220 minutes = 3 hours 40 minutes
187
Part II: Full-length Practice Tests
Measurement and Geometry (MG) 1 yard = 3 feet
44. D. 1 yard = 3 feet; 1 square yard = 9 square feet.
2 yards = 6 feet
45. C. Each hour has 60 minutes, and each day has 24 hours, so there are 24 groups of 60: 24 × 60 = 1,440. 46. D. To calculate the distance represented by a 3.5-inch line, multiply by the miles/inch (7.5): 3.5 × 7.5 = 26.25. Even an estimation of 7 × 3 gets you to Answer D. 47. A. The trap here is Answer D, 14 hours. However, with twice as many people doing the job, the job takes half as long, not twice as long. 48. C. A good strategy is to glance at the answers, to see what terms time is expressed in. Most of the answers express the relationship in miles per hour, so it is a good idea to convert 3 miles in 15 minutes to a proportional amount in 1 hour: 3 miles = x 15 minutes 60 minutes This can be solved by cross-multiplying and solving for x, but you can see that 60 minutes is 4 times as much as 15 minutes; therefore, you travel 4 times as far—12 miles. 49. A. A simple method for solving problems like this is to calculate the area of the entire rectangle: 26 × 16 = 416 sq. inches. From this, subtract the area of the interior rectangle: 22 × 12 = 264 sq. inches. What is left is the area of the frame—152 sq. inches. 50. D. Jackie has to construct all six faces of the box. However, you can find the surface area of three faces and double that since front = back, top = bottom, and right side = left side. Right side = 2 in. × 8 in. = 16 sq. in. Front
= 4 in. × 8 in. = 32 sq. in.
Top
= 2 in. × 4 in. = 8 sq. in. 16 + 32 + 8 = 56 sq. inches for 3 faces; then double it to find the total surface area. 56 sq. in. × 2 = 112 sq. in.
51. B. The fence has four sides; two sides are each 40 meters long, and two sides are each 60 meters long. The perimeter is the sum of their measures: 40 + 40 + 60 + 60 = 200 meters of fencing. 52. C. You cannot use a simple formula for calculating the volume of the shape because it is not a prism. However, with a little manipulation, you can create a prism that has an equivalent volume:
4 cubes
6 cubes
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5 cubes
Test #1—Answers and Explanations
The volume can then be determined by multiplying the length by the width by the height: 6 × 5 × 4 = 120 cubic inches. Another method would be to calculate the volume in sections as follows: The shortest stack of blocks The middle stack of blocks The tallest stack of blocks
2 × 2 × 5 = 20 4 × 2 × 5 = 40 6 × 2 × 5 = 60 The total is 120 5
6 4 2
2
2
5
2
53. A. A simple way to solve this is to find the area of a larger rectangle and subtract the two pieces that are not part of the irregular figure. The dimensions of the larger rectangle are 24 meters by 21 meters; its area is 24 × 21 = 504 sq. meters. From this, subtract two smaller areas: 9 × 9 = 81 sq. meters, and 6 × 6 = 36 sq. meters. 504 – (81 + 36) = 387 square meters. 3 m. 6 m.
9 m. 6 m.
9 m.
6 m. 6 m. 9 m. 24 m.
54. A. The modifications suggested in Answers B, C, and D result in a much greater increase in volume than is desired. Doubling the height is like stacking another can on top of the current can, which would obviously be twice as much tomato sauce. 55. D. In essence, this question asks how many cubic inches are in 1 cubic foot. It is a surprisingly large number! Perhaps the easiest way to understand this is to re-label the drawing in terms of inches, rather than feet:
12 in. (1 ft.)
12 in. (1 ft.)
12 in. (1 ft.)
To find the number of cubic inches, multiply the length by the width by the height: 12 × 12 × 12 = 1,728 cubic inches.
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Part II: Full-length Practice Tests
56. D. If a square has an area of 9 square units, each side of the square must measure 3 units because A = S2. The coordinates of the point shown are (3, –1). To find another vertex, look for a point that is 3 units up, down, to the right, or to the left (or some combination of a change in x and a change in y if it is a vertex located diagonally from the given point). Only Answer D is possible: It is located 3 down and 3 to the right from Point A. 57. C. A reflection is also known as a flip, and is like a mirror image of the figure. If you fold the coordinate grid along the y-axis, Triangle T lies directly on top of Triangle C. 58. C. The playground is in the shape of a right triangle, with legs measuring 6 yards and 8 yards. The third side is the hypotenuse of the triangle, and its length is equal to the square root of the sum of the squares of the two legs: 3rd side = 6 2 + 8 2 = 36 + 64 = 100 = 10 yards 59. B. Triangle ABC is a right triangle with a hypotenuse that measures 5 inches and a leg that measures 3 inches. To solve for the missing leg: BC = 5 2 - 3 2 = 25 - 9 = 16 = 4 inches 60. A. A working definition of congruent is same shape, same size. Only Answer A shows a triangle with the same measurements as the given triangle. It is in a different position, but its position does not change its shape or its size.
Algebra I (AI) 61. B. To find the reciprocal of an integer, rewrite the number as a fraction with a numerator of 1 and a denominator of the given integer. Answer C is incorrect because the sign does not change when finding a reciprocal. 62. D. To solve this inequality: x-1 $ 2-4 x - 1 $ -2 x–1≥2 +1 = +1
Add 1 to each side to isolate the variable.
x≥3 63. A. To find the equivalent equation, multiply the expressions within the parentheses by the factors in front of them: 3(x + 6) = 3x + 18 –2(10 – 2x) = –20 + 4x
So, you can eliminate Answers B and D. So, you can eliminate Answer C.
64. C. To solve this inequality: 5x + 3 < 4(x + 4) 5x + 3 < 4x + 16 –3 = –3 5x < 4x + 13 –4x = –4x x < 13
190
Test #1—Answers and Explanations
65. C. To solve this inequality: 2 cx - 1m $ 1 3 3 2x - 2 $ 1 3 3
Expand the left side of the inequality.
+ 2 =+ 2 3 3
Add 2 to both sides to isolate the variable. 3
2x ≥ 1 x$ 1 2
Divide both sides by 2.
66. B. This equation is given in the slope-intercept form, where the coefficient (number in front) of x is the slope, and the second number is the y-intercept (or the point on the y-axis that the line passes through). Therefore, we know we are looking for a line that has a slope of –3 and a y-intercept of +3. All four choices have a y-intercept of +3. However, only Answer B shows a line that slopes downward from left to right (negative slope), dropping 3 units for each unit it moves to the right: 3 down = slope of - 3 1 to the right 67. A. To solve this, use the general equation for a line in the slope-intercept form, y = mx + b. Substitute what you know, using the given slope and point (5, 11), and solve for the y-intercept, b. y = mx + b 11 = (2)(5) + b 11 = 10 + b b=1 The y-intercept is the point where the line crosses the y-axis, and it always has an x-coordinate of 0. (Using only this piece of information, you can eliminate answers B and D.) So, the coordinates of the y-intercept are (0, 1). 68. A. If the point lies on the line, then it must satisfy the equation of the line. Substitute the answer choices into the given equation until you find the one that works. Fortunately, it might be the first one you try, Answer A: 2y = 3x – 4 2 (–2) = (3)(0) – 4 –4 = –4 That is true, so the point (0, –2) lies on the line. 69. B. Lines that are parallel have the same slope. All the answer choices are expressed in the slope-intercept form, y = mx + b, where m is the slope. Answer B is the only choice that has a slope of 1 . 2 70. C. If you are unfamiliar with techniques for solving a system of equations, the easiest way to solve this problem is to substitute the coordinates given as answer choices in the equations until you find a choice that works in both equations. That is Answer C: 6(2) – 2(4) = 4 2(2) + 4 = 8
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Part II: Full-length Practice Tests
71. D. To simplify this fractional expression, multiply the terms in the numerator and divide by the term in the denominator: _ 2ab 2 i_ 3a 2 b i
ab
=
^ 2 $ 3h^ a $ a $ a h^ b $ b $ b h = ab
a $ b $ ^ 2 $ 3h^ a $ a h^ b $ b h = a b 1 × 1 × 6a2b2 = 6a2b2 Just realizing that you are going to multiply 2 by 3 to get 6 allows you to eliminate Answers A and C. 72. A. The formula for solving this type of work problem is: 1 1 1 + = time it takes John time it takes Sarah time it takes together 1 + 1 =1 60 40 t 2 + 3 =1 120 120 t 5 =1 120 t 1 =1 24 t t = 24 minutes A simpler method, however, is to study the answers and eliminate those that are unreasonable. Logically you know that working together they can mow the lawn in less time than either can alone. By herself Sarah can mow the lawn in only 40 minutes, so answers B, C, and D are all absurd. Only Answer A is reasonable.
Mathematical Reasoning (MR) 73. D. To find unit cost, divide the total cost by the quantity purchased, as shown in Choice D. 74. C. This is a part-part-whole problem, in which you are given a part (blue marbles) and asked for the other part (red marbles). To solve this kind of problem, you need to know the whole (total number of marbles). whole – part = part 75. D. The statement given is a symbolic representation of the commutative property of addition, which is true for all numbers. 76. D. To estimate his payments, Salvador rounded the monthly payment from $265 to $300, and the number of months from 36 to 40. Then he multiplied $300 by 40 to get $12,000. Since he rounded both factors up, his estimate is greater than the exact product. 77. C. Perhaps Joe rounded 90 tiles to 100, and $2.15 to $2.00. This should produce an estimate of about $200, but Joe made a mistake with the decimal point. Two thousand dollars is 10 times as big as $200. 78. B. Grocery sales are 40%, and produce sales are 20%—half as much. So, the dollar value of produce sales should be half the value of grocery sales. Since $590,000 is approximately $600,000, and you are being asked for the best approximation, use $600,000 for any calculations. Half of $600,000 is $300,000. 79. C. Of the answer choices, only 30 and 60 are evenly divisible by 2, 3, and 5. However, 60 is greater than 50, so it must be eliminated, leaving 30 as the correct answer. 80. D. To find the average number of pages read per day, you must divide. Only Answer D involves division. To find the number of buses needed, divide 360 by 60.
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CAHSEE Practice Test #2 Directions: Mark only one answer to each question on your answer sheet. If you change an answer, make sure that you erase the previous mark completely. Notes: (1) Figures that accompany problems are drawn as accurately as possible EXCEPT when it is stated that a figure is not drawn to scale. All figures lie in a plane unless noted otherwise. (2) All numbers used on the exam are real numbers. All algebraic expressions represent real numbers unless stated otherwise.
1. What is the quotient of 3.7 × 107 divided by 102? A. B. C. D.
3.7 × 105 3.7 × 107 3.7 × 109 3.7 × 1014
2. A painter paints one third of a room. The next day he paints one third of the rest of the room. What fraction of the room still needs to be painted? A. B. C. D.
1 9 1 3 4 9 1 2
3. Tom counts the change in his pocket: He has 1 2 3 3 3 of a dollar, of a dollar, of a dollar, of 4 10 20 1 a dollar, and of a dollar. How much change 50 does Tom have? A. B. C. D.
4. Ana is making napkins. Each napkin requires 2 3 of a yard of fabric. How many napkins can she make from 9 yards of fabric? A. 6 81 B. 3 92 C. 3 D. 13 5. If Deshaun missed 1 problem out of 20, what percent of the problems did he solve correctly? A. B. C. D.
5% 19% 90% 95%
6. Three electronic stores, Big Buy, Good Deal and Save More, offer the same television before any discount for $200. They are all having a sale, and the discounts are shown in the following chart. Store
$1.42 $1.72 $2.00 $2.60
Discount
Big Buy
30% off
Good Deal
1⁄
Save More
$50 off
3 off
Which store’s sale price is lower? A. B. C. D.
Big Buy Good Deal Save More The three stores are charging the same price.
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Part II: Full-length Practice Tests
7. The price on a $10.00 T-shirt is lowered to $6.50. What is the percentage decrease in price? A. B. C. D.
10% 35% 50% 65%
8. A $100 item is being reduced in price by 50%. With a coupon a customer can save an additional 15% from the reduced price. What is the price of the shirt with a coupon? A. B. C. D.
$35.00 $42.50 $50.00 $65.00
9. Tran places $5,000 in a savings account that pays simple interest. He makes no deposits and no withdrawals. At the end of the year, the value of the account is $5,300. What rate of interest did the money earn? A. B. C. D.
3% 6% 15% 30%
10. 4 4 $ 13 = 4 A. 17 4 1 B. 4 C. 4 D. 47
194
11.
5 + 7 = 24 30 A. 12 120 B. 12 54 C. 53 120 D. 57 120 _5 3i
2
12.
52 A. B. C. D.
13. If A. B. C. D.
= 52 53 54 55
x = 10, then x2 = 1 10 100 10,000
14. If x = 2 , then x = 3 2 A. 3 B. - 3 2 2 C. or - 2 3 3 D. 3 or - 3 2 2
CAHSEE Practice Test #2
15. Some students are each given a small box of raisins. They open their boxes, count the raisins, and make a line plot of the data.
X 24
25
26
X X
X X X
X X X
27
28
29
X X 30
31
X 32
33
34
35
Each X represents the number of raisins in one box. What is the mode for the data? 28 28.5 29 28 and 29
16. The salary figures for a small company are shown in the following table: Employee
6
Salary $124,000
Salesman
$70,000
Accountant
$55,000
Secretary
$45,000
Technician
$45,000
Delivery Driver
$27,000
$45,000 $50,000 $55,000 $91,000
5 4 3 2 1 0 1
5
10
15
20
25
Age
A. B. C. D.
Gloria stopped growing after 10 years of age. Gloria’s greatest rate of growth was from age 10 to age 15. Gloria grew more than a foot from age 1 to age 5. By the time she was 5, Gloria was over 3 feet tall.
17. Juan has taken five math tests, scoring 75, 75, 85, 60, and 80. To calculate his grade, he has the option of choosing the mean, median, or the mode of his test scores. Which should Juan pick to get the best grade? A. B. C. D.
mean median mode It makes no difference.
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CAHSEE Practice Test #2
Owner
What is the median salary of the employees? A. B. C. D.
18. Based on the following bar graph, which of the following statements is true?
Height (in feet)
A. B. C. D.
Part II: Full-length Practice Tests
19. Rock-paper-scissors is a game children play to decide who goes first. Here are the rules: Rock beats scissors. Scissors cut paper. Paper covers rock. If both players show the same sign, the game is a tie. The following chart shows the possible outcomes with 2 players. Rock
Paper
Scissors
Rock
RR
RP
RS
Paper
PR
PP
PS
Scissors
SR
SP
SS
What is the probability that a game will not result in a tie? A. B. C. D.
1 3 1 2 2 3 1
20. Nine students are randomly drawing numbers from a bag to determine batting order for a softball game. Josie picks first and draws a 5, so she will bat fifth. Barbara picks next. What is the probability that she will pull out a 1 and bat first? A. B. C. D.
B. C. D.
196
C
1 9 1 8 7 8 1
21. A fair number cube has faces numbered 1 through 6. If the cube is rolled once, what is the probability that a number less than 4 is selected? A.
22. On the fair spinner shown in the following figure, what is the probability of landing on C?
1 6 1 3 1 2 4 6
A 36% B 42%
A. B. C. D.
22% 36% 42% 78%
CAHSEE Practice Test #2
23. The following graph shows the results of a recent election.
24. Sandy has a bag of 36 candies. She made the following graph to show the distribution of colors.
% of Votes Cast
Wisconsin Presidential Primary Sandy's Candies
40 30 20 10
Blue=9
0 Jones
Smith
Perry
Fox
Green=5
Others
Candidate Yellow=6
Which of the following statements about the data is correct? A. B. C.
Red=6
Orange=4
Which of the following statements about Sandy’s candies is correct? A. About 1 of her candies are blue. 4 B. More than half her candies are yellow or blue. C. She has fewer orange candies than any other color. D. She has 30 candies that are not green.
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CAHSEE Practice Test #2
D.
Jones got more than 30% of the votes. Smith got twice as many votes as Jones. The Other candidates combined to get less than 10% of the votes. Fox got less than a third as many votes as Smith got.
Brown=3
Part II: Full-length Practice Tests
25. Which scatter plot shows no correlation between the variables?
B.
200
Price Per Copy
Golf watched on TV (weekly, in minutes)
A.
150 100 50 0 0
250
500
1.5 1 0.5 0 0
750
500
1,000
1,500
Number of Copies
Golf Played (weekly, in minutes)
20
C.
Hair Length (in inches)
Number of Teachers
D.
200 150 100 50 0 0
1,000
2,000
3,000
4,000
15 10 5 0 48
58
Number of Students
Arm span (in inches)
26. The following graph shows points representing the relationship between a person’s height and a person’s arm span.
60 40 20 0 0
20
40
60
80
100
Height (in inches)
Which of the following statements best describes the relationship shown on the graph? A. B. C. D.
Arm span is negatively correlated with height. No correlation exists between arm span and height. Taller people tend to have a greater arm span. Shorter people tend to have smaller feet.
27. Which of the following inequalities represents the situation, “a number, x, is at least two more than 5”? x+2≥5 x+2≤5 x–2≥5 x–2≤5
28. John makes a gross salary of $325 each week. Taxes (t) are deducted from this amount, and he receives the balance in a weekly paycheck. Which of the following expressions represents John’s take-home pay for 4 weeks? A. B. C. D.
4($325 + t) (4 × $325) – t 4($325 – t) $325 - t 4
29. If x = 2, then 4(2x – 1)2 = A. B. C. D.
198
78
Height (in inches)
A. B. C. D.
100 80
68
16 36 64 144
CAHSEE Practice Test #2
30. Consider the following graph.
32. The following graph shows the trading price of a particular stock throughout a year, with the value on the y-axis representing the price of 1 share of stock.
70 Plan A 60
96
Plan B
94 92 90
40 Dollars
Dollars
50
30
88 86 84
20
82 80
10
78 76 Jan 2003
40
Number of Videos
Plan A is $20 cheaper than Plan B. Plan B is $20 cheaper than Plan A. Plan A is $60 cheaper than Plan B. Plan B is $40 cheaper than Plan A.
A. B. C. D.
A. B. C. D.
Floor space in the Freeman House (2,000 sq. ft. total)
10xy 25xy 5(xy) xxxxxyyyyy
4 ^ 2ah = 8a 2
Halls 11%
Kitchen 13%
$12 $120 $1,200 $12,000
33. x5y5 =
31. Consider the following graph.
Bath 7%
Jul 2003 Oct 2003 Jan 2004
If Beatrice buys 100 shares at the price shown for February and sells them at the price shown for November, how much money does she make?
Which statement correctly compares the cost of the two different plans for a quantity of 15 videos? A. B. C. D.
Apr 2003
34. Bedrooms 45%
Living Room 24%
A. B. C. D.
a 2a 2 8a
How many square feet of floor space are in the Freeman’s bathrooms? A. B. C. D.
7 sq. feet 70 sq. feet 140 sq. feet 1,400 sq. feet
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CAHSEE Practice Test #2
20
Part II: Full-length Practice Tests
35. The following graph relates the length of the edge of a cube to the volume of the cube. 600 550 500
Volume of Cube (in cm3)
450 400 350 300 250 200 150 100 50 5
10 Edge of Cube (in cm)
What is the length of the edge of a cube that has a volume of 512 cm3? A. B. C. D.
7 cm 8 cm 70 cm 80 cm y
14
12
10
8
6
4
2
x 5
36. What is the slope of the line shown in the preceding graph? A. B. C. D.
200
–4 -1 4 1 4 4
10
CAHSEE Practice Test #2
37. The slope of the following line is 3 . What is the value of n? 2 y
12
n
x
8 12 13 18
38. The price of apples at Shop ‘N’ Save is $1.29 per pound. Which of the following graphs correctly shows the relationship between the pounds of apples purchased and the cost?
2x – 100 ≤ 300 A. B. C. D.
Cost
B.
Cost
A.
39. Which phrase most accurately describes the solution to this inequality?
x can be no more than 200. x can be no less than 200. x must be at least 200. x must be equal to or greater than 200.
40. Solve for x. Pounds of apples D.
Pounds of apples
6 ≥ 3x – 3 A. B. C. D.
Cost
Cost
C.
Pounds of apples
x≥1 x≤1 x≥3 x≤3
Pounds of apples
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CAHSEE Practice Test #2
A. B. C. D.
Part II: Full-length Practice Tests
41. Solve for y. 5 = 3y + 1 4 4 1 A. y = 3 B. y = 1 2 C. y = 3 4 D. y = 3 42. A bullet train has completed 300 miles of a 750mile trip in 40 minutes. If it continues to travel at the same speed, how long will it take to complete the trip? A. B. C. D.
60 minutes 80 minutes 100 minutes 120 minutes
43. The gross salary of a part-time worker varies directly with the number of weekly hours worked. For 5 hours, the worker earns $41.25. How much does the worker earn for 13 hours of work? A. B. C. D.
$54.25 $66.00 $83.50 $107.25
44. The speed limit on many American freeways is 65 miles per hour. About how many kilometers per hour is that? (1 mile ≈ 1.6 kilometers) A. B. C. D.
202
41 kilometers per hour 67 kilometers per hour 85 kilometers per hour 104 kilometers per hour
45. The temperature in Brussels on Tuesday was 10°C. What is that temperature in °F? 9 c F = C + 32 m 5 A. B. C. D.
–12°F 32°F 42°F 50°F
46. An architect is creating a scale drawing of a house. His scale is 1 inch = 4.5 feet. If the actual length of a room is 18 feet, how many inches is it on the drawing? A. B. C. D.
4 19 22.5 81
47. The speed of sound traveling through air is about 750 miles per hour. How far does sound travel in a minute? A. B. C. D.
1.25 miles 12.5 miles 450 miles 4,500 miles
48. It takes 6 men 18 hours to do a job. At the same rate, how many men are needed to complete the job in 3 hours? A. B. C. D.
1 12 21 36
CAHSEE Practice Test #2
49. Paul cuts a rectangle apart and positions the pieces to make a parallelogram as shown in the following figure.
Which of the following statements correctly expresses the relationship between the two figures? A. B. C. D.
The two figures have the same perimeter, but not the same area. The two figures have the same area, but not the same perimeter. The two figures have the same area and the same perimeter. The two figures have neither the same area nor the same perimeter.
50. The area of a square is 64 sq. cm. What is the perimeter of the same square? 8 cm 16 cm 32 cm 64 cm
5 ft. 4 ft.
2 ft. 2 ft.
4 ft.
51. A rectangular prism is deconstructed to create the net shown in the following figure. 10 ft.
2 in.
8 ft.
2 in. A. B. C. D.
6 in.
200 ft.3 240 ft.3 360 ft.3 400 ft.3
2 in. 2 in. What is the surface area of the net? A. B. C. D.
8 in.2 40 in.2 48 in.2 56 in.2
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CAHSEE Practice Test #2
A. B. C. D.
52. What is the volume of the following shape?
Part II: Full-length Practice Tests
4 cm.
53. What is the area of the following figure?
55. The following figure is drawn to a scale of 1 in. = 3 ft.
5 cm. 1.5 in. 3 cm.
6 cm.
1 in.
5 cm.
3 cm.
6 cm. A. B. C. D.
4 cm.
36 cm2 42 cm2 48 cm2 100 cm2
54. The surface area of the following cube is 6 ft.2
1 ft.
1 ft.
1 ft.
If the length of each edge is increased from 1 foot to 3 feet, what is the resulting change in surface area? A. B. C. D.
204
The surface area would be 3 times as great. The surface area would be 6 times as great. The surface area would be 9 times as great. The surface area would be 27 times as great.
What is the actual area represented by the figure? A. B. C. D.
4.5 ft2 13.5 ft2 45 ft2 135 ft2
CAHSEE Practice Test #2
56. On the following graph, which of the figures is a translation of Figure E? y 6 4 B
C 2
-6
-4
-2
0
x 2
4
6
-2 A
E
D -4 -6
Figure A Figure B Figure C Figure D
57. The following drawing shows a rectangle on the coordinate grid. x
0 (2, -4)
58. A carpenter is measuring the corner of a doorway. He stretches a piece of string in the corner to create a triangle with the dimensions shown in the following figure. 3 ft.
P
A
(6, -6)
4 ft. 4 in.
C
5 ft., 4 in.
B y
What are the coordinates of Point P? A. B. C. D.
(2, –6) (–6, 2) (–4, 6) (6, –4) Which of the following statements about the measurement of ∠A is correct? A. B. C. D.
This gives no information about the measurement of ∠A. This indicates that ∠A is a right angle. This indicates that ∠A is not a right angle. This indicates that EA = EB = EC .
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CAHSEE Practice Test #2
A. B. C. D.
Part II: Full-length Practice Tests
59. What is the length (n) of rectangle ABCD shown in the following figure?
3"
A. B. C. D.
5"
n A. B. C. D.
2" 34" 4" 16"
Triangle A is an equilateral triangle with a side length of 4". Triangle B is an equilateral triangle with a side length of 4". 60. Which of the following statements about Triangles A and B is most correct? A. B. C. D.
62. If x is an integer, what is the solution to 2x + 1 # 5 ?
Triangle A and Triangle B are neither congruent nor similar. Triangle A and Triangle B are similar, but not congruent. Triangle A is congruent to Triangle B, but Triangle B is not congruent to Triangle A. Triangle A and Triangle B are congruent triangles.
63. Which of the following is equivalent to 2x + 6 = 10 ? 3 A. 2x + 6 = 30 B. 2x + 18 = 30 C. 6x + 6 = 30 D. 6x + 18 = 30 64. Which of the following is equivalent to 4 – 2(x – 3) < 3x + 5? A. B. C. D.
206
5 < 5x 5 > 5x –7 < 5x 15 < 5x
65. Solve for x. 9 + 2x < 4(3 – x) A. B. C. D.
61. If x = 1 , then - 1x = 8 A. - 1 8 1 B. 8 C. –8 D. 8
{0, 1, 2} {–3, –2, –1, 0} {–2, –1, 0, 1, 2} {–3, –2, –1, 0, 1, 2}
x< -7 2 x> -3 2 1 x< 2 x< 7 2
CAHSEE Practice Test #2
66. Which of the following is the graph of y = 1 x - 4 ? 4 y 6 y 4
2
-2
2
x
0
2
4
6
-2
0
-4
-2
x
-4 A.
-6
CAHSEE Practice Test #2
B.
y y
4 6 2
-4
-2
0
4 x 2
2
4
-2 -4
-2
0
2
4
6
-4 D. C.
67. The point (3,7) lies on a line that has a slope of 1. What is the y-intercept of the line? A. B. C. D.
(0, –4) (–4, 0) (0, 4) (4, 0)
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Part II: Full-length Practice Tests
68. The T table shown in the following figure gives the coordinates of four points on a line. What is the equation of the line?
A. B. C. D.
x
y
0 1 2 3
-1 2 5 8
y=x–1 y = 2x y = 2x + 1 y = 3x – 1
69. Line A is parallel to the line y = 2x – 3. What is the slope of Line A? A. B. C. D.
–3 1 2 2 3 y = 3x + 4 ) 5x - y = 8
70. What is the solution to the system of equations above? A. B. C. D.
(6, 22) (22, 6) (2, 10) (10, 2)
71. Which of the following is equivalent to (2a + 3a)2? A. B. C. D.
72. Sam can eat 10 hot dogs in a minute. Frank can eat 10 hot dogs in 5 minutes. How long does it take the two of them to eat a total of 18 hot dogs? A. B. C. D.
1 minute 1 minute, 30 seconds 3 minutes 6 minutes
A standard printer cartridge can make 5,000 copies. A high capacity cartridge can make 8,000 copies. 73. Which of the following expressions shows how many copies can be made from 10 standard cartridges and 5 high capacity cartridges? A. B. C. D.
(10 + 5)(5,000 + 8,000) (10 + 5,000)(5 + 8,000) (10 × 5,000)(5 × 8,000) (10 × 5,000) + (5 × 8,000)
74. X is the amount of money Brandon has saved, and y is the price of a car he wants to buy. Which of the following expressions tells how much more money he needs? A. B. C. D.
208
5a 5a2 25a 25a2
x+y x–y y–x y÷x
CAHSEE Practice Test #2
75. Consider Points A and B shown on the following number line. A -2
76. Jose is adding the fractions 1 , 1 , and 1 . 2 3 4 Which of the following is the closest estimate to the exact sum?
B
-1
0
1
A.
2
B. Which of the following could show the correct position of Point Z, which is the product of A and B? A
Z
C. D.
B
1 + 1 + 1 =11 2 2 2 2 1 + 1 + 1 =1 3 3 3 1+1+1=3 4 4 4 4 The three estimates are equally close to the exact sum.
A. -2
-1
0
1
2
0
1
2
AZ B B. -2
-1
77. Chris is balancing his checkbook with a calculator. His balance is $254.27. He subtracts $120.20, and the calculator reads “25306.8”. Which of the following statements best explains what happened? A.
BZ B.
C. -2
-1
0 A
1
2 C.
B Z
D. -2
-1
0
1
2 D.
Calculators don’t make mistakes; his new bank balance is $25,306.80. The new balance is somewhat too high; perhaps Chris added $120.20 to his previous balance instead of subtracting the amount. The new balance is much too high; perhaps Chris left the decimal point out of his previous balance and subtracted $120.20 from $25,427.00. The new balance is too high; the decimal point should be moved one place to the left to read $2,530.68.
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CAHSEE Practice Test #2
A
Part II: Full-length Practice Tests
78. The following graph shows the relationship between the diameter of a pizza and the square inches of pizza. It’s estimated that each person will eat about 50 square inches of pizza. What is the smallest diameter of a pizza adequate to feed 4 people? 240 220
Area (in square inches)
200 180 160 140 120 100 80 60 40 20 5
15
10
Diameter (in inches)
A. B. C. D.
8 inches 16 inches 18 inches 32 inches
79. A group of kindergarteners is playing a game with blocks. When two children share the blocks evenly, one block is left over. However, when three children share them evenly, none are left over. When four children share them, three are left over; but when five children share them, none are left over. Which of the following could be the number of blocks the children are using? A. B. C. D.
9 11 15 30
Joanne earns $6.00 an hour for babysitting. How much would she earn for 8 hours? 80. Which of the following problems can be solved using the same arithmetic operations that are used to solve the problem above? A. B. C. D.
Martin earns $350 in a five-day week. How much does he earn each day? Samantha earns $200 in regular pay and $64 for overtime. What is her total pay? Pamela earns $300, but $98 is deducted for taxes. What is Pamela’s take-home pay? Concert tickets are $35. How much does Patrick pay for 4 concert tickets?
STOP 210
Test #2—Answers and Explanations Reviewing Practice Test 2 Review your simulated CAHSEE Math practice examination by following these steps: 1. Check the answers you marked on your answer sheet against the answer key that follows. Put a check mark in the box following any wrong answer. 2. Fill out the Review chart (p. 212). 3. Read all the explanations (pp. 214–223). Go back to review any explanations that are not clear to you. 4. Fill out the Reasons for Mistakes chart on p. 212. 5. Go back to the “Math Review” section and review any basic skills necessary before taking the next practice test. Don’t leave out any of these steps. They are very important in learning to do your best on CAHSEE Math.
211
Part II: Full-length Practice Tests
Review Chart Use your marked answer key to fill in the following chart for the multiple-choice questions. Possible Number Sense (NS)
(1–14)
14
Statistics, Data Analysis, Probability (P)
(15–26)
12
Algebra and Functions (AF)
(27–43)
17
Measurement and Geometry (MG)
(44–60)
17
Algebra I (AI)
(61–72)
12
Mathematical Reasoning (MR)
(73–80)
8
Totals
Completed
Right
Wrong
80
Reasons for Mistakes Fill out the following chart only after you have read all the explanations that follow. This chart helps you spot your strengths, weaknesses, and your repeated errors or trends in types of errors. Total Missed
Simple Mistake
Misread Problem
Lack of Knowledge
Number Sense (NS) Statistics, Data Analysis, Probability (P) Algebra and Functions (AF) Measurement and Geometry (MG) Algebra I (AI) Mathematical Reasoning (MR) Totals
Examine your results carefully. Reviewing the preceding information helps you pinpoint your common mistakes. Focus on avoiding your most common mistakes as you practice. The Lack of Knowledge column helps you focus your review in the “Math Review” section. If you are missing a lot of questions because of lack of knowledge, you should go back and spend extra time reviewing the basics.
212
Test #2—Answers and Explanations
Answer Key 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.
A (NS) C (NS) B (NS) D (NS) D (NS) B (NS) B (NS) B (NS) B (NS) C (NS) C (NS) C (NS) D (NS) C (NS) D (P) B (P) D (P) C (P) C (P) B (P)
21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40.
41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60.
C (P) A (P) D (P) A (P) D (P) C (P) C (AF) C (AF) B (AF) B (AF) C (AF) C (AF) D (AF) B (AF) B (AF) C (AF) A (AF) A (AF) A (AF) D (AF)
Number Sense (NS)
14
Statistics, Data Analysis, Probability (P)
12
Algebra and Functions (AF)
17
Measurement and Geometry (MG)
17
Algebra I (AI)
12
Mathematical Reasoning (MR)
A (AF) A (AF) D (AF) D (MG) D (MG) A (MG) B (MG) D (MG) B (MG) C (MG) D (MG) A (MG) C (MG) C (MG) B (MG) A (MG) D (MG) C (MG) C (MG) D (MG)
61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80.
C (AI) D (AI) B (AI) A (AI) C (AI) A (AI) C (AI) D (AI) C (AI) A (AI) D (AI) B (AI) D (MR) C (MR) D (MR) B (MR) C (MR) B (MR) C (MR) D (MR)
8
213
Part II: Full-length Practice Tests
Answers and Explanations Number Sense 1. A. An easy way to understand this is to rewrite the problem: 3.7 × 107 divided by 102 means: 3.7 #
10, 000, 000 = 3.7 # 100, 000 = 3.7 # 10 5 100
2. C. On day one, the painter paints one-third of the room; so he doesn’t paint two-thirds of the room. The next day he paints one-third of the rest of the room—so he paints one-third of two-thirds—so he doesn’t paint two-thirds of two-thirds. 2 # 2 = 4 3 3 9 1 3 of a dollar = $.75 3. B. of a dollar = $.50 2 4 3 of a dollar = $.30 3 of a dollar = $.15 10 20 1 of a dollar = $.02 $.50 + $.75 + $.30 + $.15 + $.02 = $1.72 50 4. D. This is actually a division problem: It asks, “How many 2 are there in 9?” So this can be solved as follows: 3 9 ' 2 = 9 # 3 = 27 = 13 1 napkins 3 2 2 3 The half napkin remainder doesn’t really make sense in this context, so the correct answer is D, 13 napkins. 5. D. If Deshaun missed 1 problem out of 20, then he answered the remainder (19 problems) correctly. To convert this to a percentage, create a proportion and solve for the missing quantity: 19 = x 20 100 20x = 1,900 20x = 1, 900 20 20 x = 95 6. B. Big Buy is 30% off; 30% of $200 is $60; so Big Buy’s price is $140. Good Deal is one-third off; one-third of $200 is about $66.66, so Good Deal is cheaper than Big Buy. Save More takes off a straight $50, which is less of a discount than Good Deal offers; so Good Deal is the best deal. 7. B. The price went down $3.50, compared to the original price of $10.00, so: 3.50 = x 10 100 x = 35% 8. B. One hundred dollars reduced by 50% leaves a price of $50. Fifty dollars reduced by 15% is a reduction of $7.50; $50 – $7.50 = $42.50.
214
Test #2—Answers and Explanations
9. B. To calculate the interest rate, compare the change in the account to the original amount: $300 = x $5, 000 100 5,000x = 30,000 5, 000x 30, 000 = 5, 000 5, 000 x=6 10. C. 4 4 $ 13 = 4 $ 4 $ 4 $ 4 = 4 # 4 # 4 # 4 = 1 # 1 # 1 # 4 = 4 4$4$4 4 4 4 4 or 4 4 $ 13 = 4 4 $ 4 - 3 = 4 4 - 3 = 4 1 = 4 4 11. C. To add fractions they must have a common denominator. You can always find a common denominator by multiplying the two denominators; or, you can multiply the highest power of each prime factor: 5 + 7 = 24 30 24 = 23 × 3 30 = 2 × 3 × 5 So the lowest common denominator for 24 and 30 is 23 × 3 × 5 = 120. Then: 5 # 5 = 25 24 5 120 7 # 4 = 28 30 4 120 25 + 28 = 53 120 120 120 _ 53i
2
12. C. 13. D.
5
2
=
_ 5 3 i_ 5 3 i
52
= 5 $ 5 $ 5 $ 5 $ 5 $ 5 = 5 $ 5 $ 5 $ 5 = 54 5$5
x = 10 = 100, so x = 100, so x2 = 1002 = 100 × 100 = 10,000
14. C. The absolute value of a number is the distance of the number from zero on the number line; it is always a positive quantity. The numbers 2 and - 2 are both a distance of 2 from zero. Therefore, the absolute value of both 3 3 3 of those numbers is 2 . 3
Statistics, Data, Analysis, Probability (P) 15. D. The mode of a set of data is the value that occurs most frequently. There can be one mode, more than one mode, or no mode (if no value occurs more than once). In this case, 28 and 29 each occur three times, so they are both modes. 16. B. To find the median, arrange the data values in ascending order: $27,000
$45,000
$45,000
$55,000
$70,000
$124,000
If an odd number of data values exists, one is in the middle, and it is the median. However, in this case, the median lies halfway between the two middle values, $45,000 and $55,000. $45, 000 + $55, 000 = $50, 000 2 So $50,000 is the median.
215
Part II: Full-length Practice Tests
17. D. To solve this problem, find the mean, median, and mode of the data set, and determine which is the highest number. Mode = 75 Median = middle number when arranged in order = 75 At this point, the only answer you can choose is D. To confirm this, determine the mean: Mean = 75 + 75 + 85 + 60 + 80 = 75 5 No wonder the teacher gave Juan that choice. 18. C. Answer A is clearly false. You might not be sure about Answer B, but Answer C is clearly true, and Answer D is clearly false. Your best choice is Answer C. 19. C. There are nine equally likely outcomes, represented by the nine cells in the outcome grid. Three of the outcomes result in a tie, and six of the outcomes have a winner. Therefore, the probability of not getting a tie is: 6 = 2. 9 3 20. B. After Josie picks eight numbers are left in the bag, one of which is the number 1. The numbers are equally likely to be picked by Barbara; therefore, the probability of picking the number 1 is one out of eight. 21. C. The outcomes of 1 through 6 are equally likely to occur. 1, 2, and 3 are less than 4, so 3 outcomes out of 6 outcomes are less than 4. 3=1 6 2 22. A. The total area of the spinner equals 100%. Therefore, the area of C is the difference between 100 and the sum of the other two areas: C = 100 – (42 + 36) = 22% If C is 22% of the area of the spinner, the theoretical probability of landing on C is 22%. 23. D. Statement A is false; Jones got less than 30%. Statements B and C are also false. Statement D is true: Fox got 10%, whereas Smith got about 35%. Ten percent is less than a third of 35%. 24. A. The problem states that Sandy has 36 candies; the graph shows that 9 of the candies are blue. Nine is one-fourth of 36, so Statement A is true. 25. D. When there is a correlation between two variables, the points tend to fall into a line. When there is no correlation, they are randomly distributed. Graphs B and C clearly show a correlation. The correlation in Graph A is not as strong; however, the points in Graph D are more randomly distributed. 26. C. Statement C most clearly expresses the positive correlation between arm span and height. Statement D might be correct; however, it has nothing to do with the information displayed on the graph.
Algebra and Functions (AF) 27. C. Which is larger, 5 or x? X is larger; so to create a relationship of equality, either subtract 2 from x or add 2 to 5. Answers C and D express that relationship. However, the problem says that x is “at least two more than 5,” so x can be greater than 5. Answer C expresses that relationship: greater than or equal. 28. C. Taxes must first be deducted from John’s gross salary of $325. To indicate that an operation must be performed first, it is often enclosed in parentheses, as it is in Answer C. This gives the take-home pay for one week; to find the amount for 4 weeks, multiply by 4.
216
Test #2—Answers and Explanations
29. B. To solve this problem, rewrite the expression by substituting 2 for x. 4(2x – 1)2 4(2 × 2 – 1)2 Then perform the operations in the appropriate order. To do that, remember: Parentheses Exponents Multiplication Division
Left to right, whichever comes first
Addition Subtraction
Left to right, whichever comes first
So, first do the operations inside the parentheses, performing multiplication before subtraction. 4(3)2 Next, do the exponential part of the expression, 32: 4(3)2 = 4(9) Finally, multiply: 4(9) = 36 30. B. Plan A’s price for 15 videos is $60. Plan B’s price for 15 videos is $40. Statement B correctly expresses this relationship. 31. C. The bathrooms are 7% of 2,000 square feet. Seven percent of 200 is 14. Seven percent of 2,000 is 10 times as much, which is 140 square feet. 32. C. The price shown for February is about $78 per share, so the cost of 100 shares is about $7,800. The price for November is about $90, so the proceeds from the sale of 100 shares are about $9,000. Beatrice’s profit is the difference between the proceeds and the cost, which is $1,200. Answer choices A, B, and D aren’t even close. 33. D. The exponent tells how many times a base is multiplied by itself. x5y5 means, “the product of five x’s and five y’s”. This statement is expressed symbolically by Answer D. 34. B. To simplify this expression:
4 ^ 2a h 8a
2
First, square the quantity in parentheses:
4 _ 4a 2 i 8a
Then, multiply the factors in the numerator: 16a 2 8a Then, divide the numerator by the denominator: 16a 2 = 16 $ a 2 = 2a 8a 8 a
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Part II: Full-length Practice Tests
35. B. Find the place on the y-axis where volume equals about 512cm3. Move horizontally until you find that place on the curve. Then trace down to find the corresponding value on the x-axis. You should be at 8cm. 36. C. Since the line slopes upward from left to right, it has a positive slope; therefore, you can eliminate Answers A and B. Remember that slope is defined as: change in y change in x In this case, the line goes up one unit for every four units it moves to the right: 1 up =1 4 across 4 37. A. The slope of the line given is 3 , which means it goes up 3 units for every 2 units it goes across. If it goes up 2 12 units, it must go across 8 units because 3 = 12 . 2 8 38. A. The relationship between the pounds of apples purchased and the cost is linear: It makes a straight line on a graph. If you understand that idea, you can immediately eliminate Graph C, which shows the cost increasing at an increasing rate. Eliminate Graph D because it shows the cost decreasing as more pounds of apples are purchased. Eliminate Graph B because it shows a constant cost no matter how many pounds of apples are purchased. Only Graph A shows a linear relationship in which the cost increases at a rate proportional to the number of pounds of apples purchased. 39. A. To solve this inequality: 2x – 100 ≤ 300 +$100 = +$100
Add $100 to both sides to isolate the variable.
2x ≤ 400 2x # $400 2 2
Divide both sides by 2.
x ≤ $200 So x can be no more than $200. 40. D. To solve this inequality: 6 ≥ 3x –3 +3 = +3
Add 3 to each side to isolate the variable.
9 ≥ 3x 9 $ 3x 3 3
Divide both sides by 3.
3 ≥ x or x ≤ 3 41. A. To solve this equation: 5 = 3y + 1 4 4 -1 = -1 4 4 1 = 3y
And the fraction goes away, too!
1 = 3y 3 3
Divide both sides by 3.
y= 1 3
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Subtract 1 from each side to isolate the variable. 4
Test #2—Answers and Explanations
42. A. Be careful on this kind of problem! It’s not difficult, but you need to read it carefully. In this case, we’re not looking for the time required for the entire trip—just for the rest of it. The entire trip is 750 miles, and 300 miles have been completed—so 450 miles remain. Four hundred and fifty is one-and-a-half times as big as 300, so it takes one-and-a-half times as long as 40 minutes, which is an hour. Alternately, set up the following proportion and solve for x: 40 = x 300 450 43. D. This problem is easily solved by setting up and solving a proportion: 5 hours = 13 hours x $41.25 But this much work is not really required: Thirteen hours is about two-and-a-half times as long as five hours, so she makes two-and-a-half times as much money. Answer D is the only reasonable choice.
Measurement and Geometry (MG) 44. D. To solve this problem, multiply 65 by 1.6, or estimate: The result is a number that is a little more than oneand-a-half times as large as 65. The only reasonable choice is Answer D. 45. D. Substitute the given temperature in the equation, and solve for F: F = 9 C + 32 5 9 F = ^10 h + 32 = ^ 9 $ 2 h + 32 = 18 + 32 = 50 % 5 46. A. This problem is easily solved by setting up and solving a proportion: 1 inch = x inches 4.5 feet 18 feet At this point you might realize that 18 is four times as much as 4.5; therefore, the line representing it is four times as long. Four times as long as 1 inch is 4 inches. 47. B. Sixty minutes are in an hour; so in 1 minute sound travels one-sixtieth of the distance it travels in 1 hour. 750 ÷ 60 = 12.5 miles 48. D. Three hours is one-sixth as long as 18 hours. To do the job in one-sixth of the time, six times as many workers are needed. 6 × 6 workers = 36 workers 49. B. Since no pieces of the original shape have been lost or are overlapping, the area is unchanged. Careful observation reveals, however, that the perimeter has changed: Two widths of the rectangle have been replaced by the diagonal sides of the parallelogram, which are longer. 50. C. A = S2, so 64cm2 = S2, so S = 8cm. P = 4S, so P = 4(8cm) = 32cm. 51. D. The net consists of four congruent rectangles and two congruent squares; the surface area is the sum of their areas. Rectangles = 4(6in. × 2in.) = 48 in.2 Squares = 2(2in. × 2in.) = 8 in.2 48 in.2 + 8 in.2 = 56 in.2
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Part II: Full-length Practice Tests
52. A. The shape is formed by two rectangular prisms; to find its volume, find the volume of each prism and add them: Bottom prism: 8ft. × 10ft. × 2 ft. = 160 ft.3 Top prism: 5ft. × 4ft. × 2ft. = 40 ft.3 160 ft.3 + 40 ft.3 = 200 ft.3 53. C. The figure is formed by one square and two congruent triangles; the triangles can be combined to form a rectangle. The area of the figure, then, is the sum of the areas of the square and the rectangle: Square = 6cm × 6cm = 36 cm2 Rectangle = 4cm × 3cm = 12 cm2 36 cm2 + 12 cm2 = 48 cm2 54. C. The surface area of each face of the original cube is 1 ft2. However, if the edge is increased to 3 ft., the surface area of each face becomes 3 ft. × 3 ft., or 9 ft. So the surface area is increased by a factor of 9. 55. B. To solve this problem, replace the dimensions of the drawn figure with the actual dimensions: 1in. = 3ft., and 1.5in. = 4.5ft., so the area of the figure is 3ft. × 4.5ft. = 13.5 ft2. 56. A. When a figure is translated, its vertices and sides remain in the same orientation in relation to the original figure—it is simply slid in some direction, but not flipped or turned. Only Figure A maintains this relationship with respect to Figure E. 57. D. Point P is perpendicular to the point below it—so it has the same x-coordinate as that point, which is 6. Along with the point to the left of it, it lies on a line parallel to the x-axis—so it has the same y-coordinate as that point, which is –4. Its coordinates therefore are (6, –4). 58. C. This figure tests your understanding of the Pythagorean theorem and its converse. The Pythagorean theorem states that in a right triangle, the sum of the squares of the legs is equal to the square of the hypotenuse. This applies to each and every right triangle. If this relationships is not true, then the triangle is not a right triangle. In the triangle created by the rope, 32 + 42 ≠ 5.332. Therefore, ∠A is not a right angle. 59. C. n is the missing leg of a right triangle; its other leg measures 3in., and the hypotenuse measures 5in. Therefore, n = _ 5 2 - 3 2 i = ^ 25 - 9h = 16 = 4 . 60. D. Triangles A and B have the same shape, so their angles are congruent, and their sides are also congruent; therefore, they are congruent triangles. Statement C cannot be true: It is impossible for A to be congruent with B without B being congruent to A.
Algebra I (AI) 61. C. 1x means the reciprocal of x; the reciprocal of 1 is 8. However, the question asks for the negative reciprocal 8 of x; therefore, the correct answer is –8. 62. D. A good technique for solving this problem is to substitute the answer choices into the inequality. Start with the most inclusive set, which is the set represented in Answer D. Then start with the most extreme values, to see if they are solutions to the equation. In this case, they are, which saves some work: 2x + 1 # 5 2 ^ - 3h + 1 # 5
Test with the most extreme values: –3 and 2. Substitute –3 for x.
-6 + 1 # 5 -5 # 5
The absolute value of a number is always positive, so:
5≤5 Positive 2 satisfies the equation as well. The other integers in this set are also solutions to the inequality.
220
Test #2—Answers and Explanations
63. B. A glance at the answer choices suggests that the main change from the original equation is that the fraction has been removed from the left side of the equation. This has been done by multiplying both sides of the equation by 3, resulting in 30 on the right side of the equation. Both terms in the expression must be multiplied by 3 on the left side: 2x + 6 = 10 3 3 c 2x + 6 m = 3 ^10 h 3 3 c 2x m + 3 ^ 6 h = 30 3 2x + 18 = 30 64. A. To solve this, you must first expand the expression on the left side, combine like terms, then isolate the variable: 4 – 2(x – 3) < 3x + 5 4 – 2x + 6 < 3x + 5 10 < 5x + 5 5 < 5x
Add 2x to both sides. Subtract 5 from both sides.
65. C. To solve this, you must first expand the expression on the right side, isolate the variable, and then divide by the coefficient: 9 + 2x < 4(3 – x) 9 + 2x < 12 – 4x
Add 4x to both sides.
9 + 6x < 12
Subtract 9 from both sides.
6x < 3
Divide both sides by 6.
x< 1 2 66. A. This equation is in the slope-intercept form: y = 1 x - 4 . The number in front of x c 1 m gives the slope, and 4 4 the constant (–4) gives the y-intercept. What we’re looking for is a line with a positive slope—sloping upward from left to right—rising 1 unit for every 4 units it moves to the right. It cuts the y-axis at –4. This is the line shown in Graph A. Eliminate Graphs B and D immediately because their slopes are negative. Graph C is incorrect because the line has a slope of 4, not 1 . 4 67. C. The y-intercept is the place where the line cuts the y-axis, so the x-coordinate of that point is always 0. For that reason, immediately eliminate Answers B and D. Since you are given a slope and a point, you can plug that information into the slope-intercept equation of the line and solve for the y-intercept: y = mx + b 7 = 1(3) + b 7=3+4 So the y-intercept is positive 4, and its coordinates are (0, 4).
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Part II: Full-length Practice Tests
68. D. If you aren’t familiar with techniques for solving this problem, the easiest way is to substitute x and y values from the table in the answer choices to see which equation works with all the x and y pairs.
y
x
−1 2 5 8
0 1 2 3 y=x–1
This works for (0, –1), but not for the other values.
y = 2x
This works for (1, 2), but not for the other values.
y = 2x + 1
This works for (2, 5), but not for the other values.
y = 3x – 1
This works for all given (x, y) pairs.
69. C. Parallel lines have the same slope. When the equation for a line is given in the slope-intercept form, y = mx + b, the number written in front of x is the slope of the line. In this case, the given line’s slope is 2. Therefore, a line parallel to it also has a slope of 2. 70. A. If you are unfamiliar with techniques for solving a system of equations, the easiest way to solve this problem is to substitute the coordinates given as answer choices into the equations until you find a choice that works in both equations. Fortunately, that happens with the first answer, A (6, 22): y = 3x + 4
22 = 3(6) + 4
5x – y = 8
5(6) – 22 = 8 2
71. D. First, add the terms in the parentheses: (2a + 3a) = (5a)2. Then, multiply 5a by 5a = (5a)(5a) = (5 × 5)(a × a) = 25a2. 72. B. Instead of trying to solve this problem algebraically, solve it logically. If Frank can eat 10 hot dogs in 5 minutes, he can eat 2 hot dogs in 1 minute. So, at the end of 1 minute, Sam eats 10 hot dogs and Frank eats 2, for a total of 12 hot dogs—not enough. But, at the end of 2 minutes, they eat 24—too many. The correct answer has to be more than 1 minute and less than 2 minutes. Only Answer B is in that range.
Mathematical Reasoning (MR) 73. D. This kind of problem can look confusing, but we do it every day. How much money do you have if you have 3 dimes and 4 nickels? You multiply 3 by the value of a dime and 4 by the value of a nickel, then you add the two products. This relationship is expressed in Answer D. 74. C. If problems with variables confuse you, replace them with small numbers, determine the appropriate relationship, and then put the variables back in. Imagine that the price of the car is $10, and Brandon has $3. How much more does he need? Seven dollars. How do you get that? $10 – $3 = $7
So you wind up with price minus savings, which is y – x.
75. D. A and B are negative numbers, so their product has to be positive. Only Answer D shows P as a positive number. 76. B. 1 > 1 > 1 , so in estimating the sum, Answer A is too high, and C is too low. 1 is a little smaller than 1 and a 2 3 4 3 2 little larger than 1 , so Answer B is a good estimate. Answer D is incorrect because the three sums are not equally 4 close to the exact sum, 1 1 . 12
222
Test #2—Answers and Explanations
77. C. Calculators don’t make mistakes, but people do, and Chris certainly did. Answer A is incorrect. Answer B is incorrect because adding the balance with $120.20 does not make such a large amount. Answers C and D both start out good, but D’s explanation for the mistake does not account for the error. 78. B. If each person eats 50 square inches, then four people eat 200 square inches. Find that point on the y-axis and trace horizontally until you run into the curve. Then trace vertically down to the x-axis. You should be at the point that represents a diameter of 16 inches. 79. C. If one block is left over when two children share the blocks, the number of blocks must be odd. If none are left over when three or five children share them, the number must be a multiple of 3 and 5. Answers C and D (15 and 30) are both multiples of 3 and 5; however, 30 is an even number and must be eliminated. 80. D. The example given requires the addition of equal groups, which is (more simply) multiplication. Answer A requires division, Answer B requires addition of unequal groups, and Answer C requires subtraction. Only Answer D also involves multiplication.
223
CAHSEE Practice Test #3 Directions: Mark only one answer to each question on your answer sheet. If you change an answer, make sure that you erase the previous mark completely. Notes: (1) Figures that accompany problems are drawn as accurately as possible EXCEPT when it is stated that a figure is not drawn to scale. All figures lie in a plane unless noted otherwise. (2) All numbers used on the exam are real numbers. All algebraic expressions represent real numbers unless stated otherwise.
1. Which of the following has the greatest value? A. B. C. D. 2.
1.99 × 10–1 9.19 × 10 2 1.19 × 10 3 9.91 × 10–4
3 + 3-2 = 14 c 8 7 m A. B. C. D.
4 15 17 56 9 28 17 14
3. At 6 p.m. in Alaska the thermometer read –3°F. By midnight the temperature had dropped 18°. What was the reading on the thermometer at that time? A. B. C. D.
–21° –18° –15° 15°
4. Sonia earns $300, pays 1 of the money in taxes, 5 and keeps the rest. How much money does Sonia keep? A. B. C. D.
$50 $60 $240 $250
One dollar is worth about .81 of a Euro (European Economic Unit). 5. Which of the following statements is consistent with the information in the box? A. B. C. D.
A dollar is worth less than half a Euro. A dollar is worth about four-fifths of a Euro. Eighty-one dollars are worth about 100 Euros. A Euro is worth about 81 cents.
6. By halftime the Wolverines had scored 49 points, 18 by their center. Approximately what percent of the team’s points were scored by other players? A. B. C. D.
17% 37% 63% 83%
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Part II: Full-length Practice Tests
7. In 30 years the price of premium gasoline has increased from about $.25 a gallon to about $2.75 a gallon. What has been the percent increase in the price of premium gasoline? A. B. C. D.
100% 250% 1,000% 2,500%
8. An $80 jacket is on sale for 40% off. With a coupon, John saves an additional 15% off the reduced price. How much does John save off the original price of the jacket? A. B. C. D.
$36.00 $39.20 $40.80 $44.00
9. A movie star is paid 5% of the gross ticket sales above $30,000,000. If the movie sells $50,000,000 worth of tickets, how much does the actor earn? A. B. C. D.
$1,000,000 $1,500,000 $2,500,000 $20,000,000
_3 4i
3
12.
34 # 32 A. B. C. D.
B. C. D.
–9 - 1 27 1 27 1 9
11. Which of the following is the prime-factored form of the lowest common denominator of 5 + 3? 6 8 A. B. C. D.
226
2×1 2×2×2×3 2×3×2×2×2 6×8
1 2 32 36
13. The square of a whole number is between 2,500 and 3,600. The number is between A. B. C. D.
30 and 40 40 and 50 50 and 60 60 and 70
14. If x = 5, then x = A. B. C. D.
–5 5 –5 or 5 1 or 5 5
15. The following table shows sales figures for five salesmen:
10. Which of the following is equivalent to (3)–3? A.
=
Salesman
Sales
Salesman #1
$20,000
Salesman #2
$21,000
Salesman #3
$19,000
Salesman #4
$21,000
Salesman #5
$25,000
A sixth salesman joins the company and records $55,000 in sales. Which of the measures of central tendency does not change when the new salesman’s figures are added to the data? A. B. C. D.
the mean the median the mode neither the median nor the mode
CAHSEE Practice Test #3
16. Tom has an average (mean) of 80% for six history tests. His average for the first three of those tests was 72%. He scored 90% on his fourth test, and 88% on his fifth test. What was his score on the sixth test? A. B. C. D.
17. The sixth grade is having a raffle ticket sale to raise money. The following table shows sales figures for the five classes: Teacher
70% 76% 80% 86%
Ticket Sales
Mr. Navarete
480 tickets
Ms. Green
650 tickets
Ms. Schuman
520 tickets
Ms. Petrie
290 tickets
Mr. Frankl
610 tickets
Mr. Peters
140 tickets
What is the median number of tickets sold? A. B. C. D.
405 480 500 520
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18. Researchers surveyed algebra students to determine how much time they spent on daily homework. No time 25%
1 hour or more 43%
Some time, but less than 15 minutes 13%
15 minutes to half an hour 19%
According to the preceding circle graph A. B. C. D.
More than half the students worked less than half an hour. More than half the students either did no homework or spent an hour or more on homework. Students are not spending enough time on math homework. Less than half the students worked half an hour or more.
19. Karl rolls a fair number cube and flips a fair coin. The possible outcomes are shown in the following figure. 1
Heads Tails
2
Heads Tails
3
Heads Tails
4
Heads Tails
5
Heads Tails
6
Heads Tails
What is the probability that he rolls an odd number and flips heads? A. 1 6 B. 1 4 C. 1 2 D. 1
228
20. Mr. Garvitch buys a dozen donuts. There are 3 jelly donuts, 2 chocolate donuts, 2 crumb donuts, 2 sugar donuts, a buttermilk donut, and 2 glazed donuts. If he randomly selects a donut, what is the probability that it is not a jelly donut or a buttermilk donut? A. B. C. D.
1 3 2 3 3 4 11 12
CAHSEE Practice Test #3
21. Pancho spins the following spinner one time:
23. The following graph represents information on the siblings of 500 students at an elementary school.
Red Green
4 or more siblings 12%
Green
no siblings 18%
Blue Red
3 siblings 17%
What is the probability that the pointer lands on a green section? A. B. C. D.
33.3% 40% 50% 60%
2 siblings 23%
How many students had no siblings or only one sibling?
22. The fair spinner shown in the following figure has landed on A four times in a row. How does this affect the theoretical probability of an outcome of A on the next spin?
A
A.
B.
D.
A. B. C. D.
18 students 30 students 48 students 240 students
B
To answer this question, more needs to be known about the spins that preceded the last four spins. Because A has come up several times in a row, an outcome of A is more likely than an outcome of B. Because A has come up several times in a row, an outcome of A is less likely than an outcome of B. Outcomes of the previous spins have nothing to do with the outcome of the next spin.
CAHSEE Practice Test #3
C.
1 sibling 30%
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Part II: Full-length Practice Tests
Biology scores
24. The following graph shows Dontrey’s scores on biology tests. 100 90 80 70 60 50 40 30 20 10 0 1st test
2nd test
3rd test
4th test
5th test
6th test
Biology tests
Dontrey’s scores decreased the most from A. B. C. D.
the 2nd to the 3rd test the 3rd to the 4th test the 4th to the 5th test the 5th to the 6th test
25. Which scatter plot shows a negative correlation between the two variables? B.
0
Days of Training
Price per Day
A. 7 6 5 4 3 2 1
20 15 10 5 0
20
40
20
40
60
80
Number of Animals
60
Number of Days C.
D. 60
20
Miles Run
Price per Call
30
10
0
5
10
Number of Phone Calls
40
20
15 0
10
20
Number of Students
230
30
CAHSEE Practice Test #3
26. Which statement best describes the relationship between the price of cherries and the pounds of cherries purchased by shoppers?
Number of pounds sold (in 100,000s)
Cherry Price and Cherry Purchases 200
27. Which of the following expressions represents the statement, “Four less than one-third the product of 10 and n”? A. B.
150
C.
100
D.
50 0 $0.00
28. Which equation represents the statement, “The sum of five and y is seven more than x”? $1.00
$2.00
$3.00
$4.00
$5.00
Price per pound
A. B. C. D.
10 + n - 4 3 10n - 4 3 1 ^10n - 4 h 3 1 ^10 + n - 4 h 3
The number of pounds purchased increases as the price decreases. The number of pounds purchased increases as the price increases. The number of pounds purchased decreases as the price decreases. There is no relationship between the price and the pounds purchased.
A. B. C. D.
y=2+x 2+y=x 5+x=7+y y=2+x
29. If x = 2 and y = –3, then xy(10 – y) = 3 A. B. C. D.
–26 –14 14 26
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Part II: Full-length Practice Tests
Palm Springs And Los Angeles– Average Monthly High Temperatures 110 100 Los Angeles Palm Springs
90 80 70
Ja n Fe uary br ua M ry ar ch Ap ril M ay Ju ne Ju A ly Se ug pt us em t O ber N cto b ov e e r De mb ce er m be r
60
30. According to the preceding graph, which month shows the least difference between high temperatures in Los Angeles and Palm Springs? A. B. C. D.
January July August December
31. The following table shows the distribution of grades for students who take algebra during first period versus students who take algebra during sixth period. Grades
1st period
6th period
A
10
2
B
13
7
C
25
9
D
30
10
F
22
22
According to the table, what is the difference for the two periods in the percentage of students receiving a grade of F? A. B. C. D.
232
no difference 11% difference 22% difference 44% difference
CAHSEE Practice Test #3
Number of Models
Gas Barbecue Grills 25 20 15 10 5 0 $50–100
$101–200 $201–300 $301–400 $401–5,000
Price Range
32. The preceding graph shows how many different models of gas barbecues are marketed for various price ranges. How many models are available for more than $300? A. B. C. D.
33.
15 30 44 73
x 3 y-2 = x 2 y-3 A. B. C. D.
xy x y xxxyy xxyyy y x
34. Simplify the expression (2a2b)2(3ab5). A. B. C. D.
6a3b6 12ab5 12a5b7 36a6b12
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Part II: Full-length Practice Tests
35. Which of the following could be the graph of y = 1 x 2 ? 2 y A.
y B.
4
4
2
–4
2
0
–2
x 2
4
–4
0
–2
–2
–2
–4
–4
y C.
D.
234
0
2
4
4
2
–2
4
y
4
–4
x 2
2
x 2
4
–4
0
–2
–2
–2
–4
–4
x
CAHSEE Practice Test #3
36. What is the slope of the line shown in the following graph? y 6
4
2
–4
x
0
–2
2
4
6
–2
A. B. C. D.
-3 2 -2 3 2 3 3 2
37. What is the equation of the following graph? y 6
4
2
–6
–4
0
–2
x 2
4
6
–2
CAHSEE Practice Test #3
–4
–6
A. B. C. D.
y= 3 x-5 4 y= 4 x-5 3 y = - 5x - 3 4 4 y = 5x 3
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Part II: Full-length Practice Tests
38. According to the following graph, how fast is the airplane flying in miles per hour? Airspeed
Distance (miles)
2500
42. Joe drives a total of 20 miles to get to work. He drives 3 miles to get to the freeway in 9 minutes, 16 miles on the freeway in 18 minutes, and the last mile in 3 minutes. What is his average rate of speed for the whole trip?
2000
A.
1500
B. C. D.
1000 500 0 0
1
2
3
4
5
Time (hours)
A. B. C. D.
300 400 500 600
43. The Valdez family drives 720 miles on a trip. Their car averages 24 miles per gallon of gas, and they pay an average of $2.40 per gallon. How much do they spend on gas? A. B. C. D.
39. Solve for x. 8- x <5 3 A. B. C. D.
x < –7 x > –7 x<9 x>9
44. If a package weighs 7 pounds, 8 ounces, which would be the closest to its metric weight?
5 13 15 20
A. B. C. D.
3.405 kilograms 340.5 grams 7500 grams 7500 kilograms
On the Fahrenheit thermometer, water freezes at 32º and boils at 212º. On the Celsius thermometer, water freezes at Oº and boils at 100º. 45. Based on the information, which of the following statements is correct?
41. Which number line shows the solution to the inequality 3 + 3x ≤ –6? A.
A. B.
–6 –5 –4 –3 –2 –1 0
1
2
3
4
5
6
C.
–6 –5 –4 –3 –2 –1 0
1
2
3
4
5
6
D.
–6 –5 –4 –3 –2 –1 0
1
2
3
4
5
6
–6 –5 –4 –3 –2 –1 0
1
2
3
4
5
6
B. C. D.
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$30.00 $57.60 $72.00 $720.00 454 grams ≈ 1 pound 1000 grams = 1 kilogram 16 ounces = 1 pound
40. Tania is saving money for a CD player that costs $100. She has $35, and each week she saves her $5 allowance. She calculates her total savings by using the equation S = 5W + 35, where S is her total savings, and W is the number of weeks she saves her allowance. According to her equation, how many weeks must she save before she has enough money for the CD player? A. B. C. D.
2 of a mile per minute 3 1.5 miles per minute 50 miles per hour 60 miles per hour
Each degree Fahrenheit is equivalent to about 2 degrees Celsius. Each degree Celsius is equivalent to about 2 degrees Fahrenheit. Degrees Fahrenheit and degrees Celsius are roughly equivalent. Based on this information, no relationship can be established between the two systems.
CAHSEE Practice Test #3
46. John is making a drawing of a tennis court, using a scale of 1 inch = 12 feet. On the drawing, how long should he make the 78-foot length of the court? 21‘
21‘
18‘ 12‘
POST
SIDE LINE
4‘6“
SERVICE LINE
ALLEY LINE
CENTER SERVICE LINE
13‘6“
13‘6“
NET
SINGLES
DOUBLES
36‘ 27‘
21‘
BACK SCREEN
18‘
BASE LINE
21‘
42‘ 78‘ SIDE SCREEN
A. B. C. D.
936 inches 65 inches 6.5 inches 1 inch
47. If it takes 8 days for one person to build a fence, how long does it take with three people on the job? A. B.
A. B. C. D.
4 hours 4 hours, 6 minutes 4 hours, 10 minutes 4 hours, 30 minutes
CAHSEE Practice Test #3
C. D.
3 day 8 2 2 days 3 4 days 24 days
48. Sam walks 12 1 miles at a speed of 3 miles per 2 hour. How long does it take him to walk that distance?
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Part II: Full-length Practice Tests
15 inches 9 inches 7.2 inches
12 inches
49. What is the area of the preceding triangle? A. B. C. D.
36 sq. inches 43.2 sq. inches 54 sq. inches 108 sq. inches
50. Moira is constructing a cylinder from the net shown. What is the approximate volume of the cylinder?
51. A circle with a radius of 5 centimeters is inscribed inside a square as shown in the following figure.
6 in.
3 in.
What is the perimeter of the square?
A. B. C. D.
238
18 cubic inches 54 cubic inches 113 cubic inches 169 cubic inches
A. B. C. D.
20 centimeters 25 centimeters 40 centimeters 100 centimeters
CAHSEE Practice Test #3
52. What is the volume of the following solid figure? (Volume of rectangular solid = lwh) (Volume of triangular pyramid = 1 × base area 2 × height)
53. The area of the shaded triangle is 4 sq. units. What is the area of square ABCD?
8 ft. 6 ft.
A. B. C. D.
A
B
D
C
32 square units 48 square units 56 square units 64 square units
54. Jeremy built the following shape from cubes. 3 ft.
A. B. C. D.
4 ft.
72 cubic feet 84 cubic feet 96 cubic feet 576 cubic feet Ron wants to make the same shape, but twice as high, twice as wide, and twice as long. How many cubes does Ron need to make the shape? A. B. C. D.
54 cubes 108 cubes 162 cubes 216 cubes
CAHSEE Practice Test #3 GO ON TO THE NEXT PAGE 239
Part II: Full-length Practice Tests
55. The following piece of wood measures 8ft. by 6in.
56. On the following coordinate grid, the distance between 0 and 1 is one unit. y
6 in.
B
8 ft. 6
What is the area of the piece of wood in square feet? A. B. C. D.
4
4 sq. ft. 8.5 sq. ft. 48 sq. ft. 576 sq. ft.
C A
2
0
2
D 4
x 6
What is the area of square ABCD? A. B. C. D.
240
25 sq. units 36 sq. units 49 sq. units 50 sq. units
CAHSEE Practice Test #3
57. Rectangle ABCD is drawn on the coordinate grid as shown in the following figure. y
A
D
B
C
(2,2)
(10,2) x
0
If the area of rectangle ABCD is 24 sq. units, what are the coordinates of point A? A. B. C. D.
(5, 5) (2, 3) (2, 5) (2, 8)
58. Triangle ABC is drawn on the following coordinate grid. y
8 B 6 4 2 A 0
–2
C
x 2
4
6
8
10
12
14
CAHSEE Practice Test #3
–2
What is the length of side BC? A. B. C. D.
8 units 10 units 12 units 14 units
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Part II: Full-length Practice Tests
59. Right triangle ABC is created by positioning three squares as shown in the following figure. D
F
A
Area = 16 sq. units
G
A. B. C. D.
242
1 unit 3 units 4 units 9 units
c
b
C
H
What is the length of triangle side a?
Area = 25 sq. units
a
B
I
E
CAHSEE Practice Test #3
60. Which figure contains two congruent triangles?
62. What is the solution set for the equation 3x - 3 = 15 ? A. B. C. D.
A.
{–4} {6} {–4, 6} {–6, 6}
63. Which equation is the same as x - 2 = 6 ? 5 3 A. 3x – 10 = 90 B. 5x – 6 = 90 C. 3x – 6 = 6 D. 5x – 6 = 6
B.
64. Which equation is equivalent to 1 ^ 4x - 12 h - 1 ^12 - 3x h = 5 ? 2 3 A. B. C. D.
C.
–x – 2 = 5 –x – 10 = 5 3x – 2 = 5 3x – 10 = 5
65. Francisco solved the equation 2 x - 6 = 4 using 3 the following steps: Given: 2 x - 6 = 4 3 Step 1: 2 x = 10 3
D.
Step 2: x = 15 2
-1 4 61. What number is equivalent to ^ 4 h ^ 2 h c 1 m ? 2
A. B. C. D.
–32 –16 1 4
To get from Step 1 to Step 2, Francisco— A. B. C.
CAHSEE Practice Test #3
D.
multiplied both sides by 2 3 multiplied both sides by 3 2 3 divided both sides by 2 1 added to both sides 3
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Part II: Full-length Practice Tests
y 6
4
2
–8
–6
–4
0
–2
x 2
4
6
8
–2
–4
–6
66. What is the equation of the line shown in the preceding graph? A. y = 3 x - 3 4 B. y = - 3 x - 3 4 4 C. y = x + 4 3 D. y = - 4 x + 4 3 67. What are the x- and y-intercepts for –2x + 3y = 6? A. B. C. D.
x-intercept: (2, 0); y-intercept: (0, –3) x-intercept: (0, –3); y-intercept: (2, 0) x-intercept: (0, 2); y-intercept: (–3, 0) x-intercept: (–3, 0); y-intercept: (0, 2)
69. What is the slope of a line that is parallel to the graph of 5x + 10y = 7? A. B. C.
68. Which of the following points lies on the line y = - 1 x? 2 A. B. C. D.
244
(–6, –3) (–3, –6) (6, –3) (–3, 6)
D.
–2 -1 2 1 2 2
CAHSEE Practice Test #3
70. Which graph represents the solution to the following system of equations? -x+y=3 ) - 4x + 2y = 6 y
y B.
8
8
A.
–4
–2
6
6
4
4
2
2
0
x 2
4
–4
–2
0
y C.
x 2
4
2
4
y
8
8
D.
–4
–2
6
6
4
4
2
2
0
x 2
A. B. C. D.
–x 4x2 – x 4x2 + 7x 7x
–4
–2
0
x
72. How much more time does it take a car traveling at an average of 60 miles per hour to cover a distance of 200 miles than it does for a car traveling at a rate of 75 miles per hour? A. B. C. D.
40 minutes 1 hour, 20 minutes 2 hours, 40 minutes 3 hours, 20 minutes
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CAHSEE Practice Test #3
71. Which of the following is equivalent to 3x – 2x(x – 2) + 2x2?
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Part II: Full-length Practice Tests
Paula needs 180 units to graduate from college. She plans to earn 45 units each year to graduate in 4 years. How many classes should Paula take each year to stick to her plan?
76. Jennifer is keeping a running total of her purchases at a store to estimate her bill. So far her cart contains:
73. What other information is needed in order to solve this problem? A. B. C. D.
Paula’s grade point average the cost of each class the number of units earned per class Paula’s academic major
Three men go fishing, then divide the day’s catch. Each man takes one third of the fish. One fish is left over, which they toss to a hungry cat. 74. If each man takes x fish, which expression can be used to find the total number of fish the men caught? A. B. C. D.
x = b, 3
246
CDs
$12.39
Chicken
$19.75
Computer Printer
$145.99
Which of the following expressions gives the best estimate of Jennifer’s checkout bill? A. B. C. D.
30 + 10 + 20 + 200 30 + 12 + 20 + 100 30 + 10 + 20 + 150 40 + 10 + 20 + 100
a and b are positive integers.
Which of the following conclusions can be made from the preceding information? A. B. C. D.
Price $34.27
77. Jones converts fractions to decimals to do problems on the calculator. To add 1 + 1 + 1 , 3 3 3 he first enters 1 ÷ 3 on the calculator, then rounds the decimal quotient to .3. He then adds .3 + .3 + .3 and determines that the sum of the 3 fractions is .9, or 9 10
3x +1 3(x + 1) x -1 3 x-1 3
75. x = a, 2
Item Books
x is a multiple of 4. x is a fraction. x is a negative number. x is evenly divisible by 2 and by 3.
Which of the following best explains the mistake Jones made? A. B. C. D.
He misplaced the decimal point in the sum. Rounding introduced a serious error into what should be an exact computation. He should have multiplied .3 by 3. 1 + 1 + 1 = 3 , not 3 3 3 3 9 10
CAHSEE Practice Test #3
78. A market research company sent out 500 questionnaires to determine customers’ movie preferences. The following graph shows data from the questionnaires that were returned.
79. The following table shows values for x and the corresponding values for y.
Movie Preferences Other Drama (10) (20) Horror (20) Comedy (130)
x
y
0
3
1
5
5
13
13
29
Which of the following represents the relationship between x and y? A. B. C. D.
y = 2x + 3 y=x+3 y = 5x y=x+4
Action (140)
How many people did not return the questionnaire? A. B. C. D.
It cannot be determined from the graph. 180 320 500
80. A combination of 10 nickels and dimes has total value of $.75. To determine the number of dimes, Juliet writes the equation: .10x + .05(10 – x) = .75 If, instead, there were a total of 13 coins, which of the following equations shows the correct adjustment to Juliet’s equation? A. B. C. D.
.13x + .05(10 – x) = .75 .10x + .05(13 – x) = .75 .13x + .05(13 – x) = .75 .10x + .05(13) = .75
CAHSEE Practice Test #3
STOP 247
Test #3—Answers and Explanations Reviewing Practice Test 3 Review your simulated CAHSEE Math practice examination by following these steps: 1. Check the answers you marked on your answer sheet against the answer key that follows. Put a check mark in the box following any wrong answer. 2. Fill out the Review chart (p. 250). 3. Read all the explanations (pp. 252–264). Go back to review any explanations that are not clear to you. 4. Fill out the Reasons for Mistakes chart on p. 250. 5. Go back to the “Math Review” section and review any basic skills necessary before taking the next practice test. Don’t leave out any of these steps. They are very important in learning to do your best on CAHSEE Math.
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Part II: Full-length Practice Tests
Review Chart Use your marked answer key to fill in the following chart for the multiple-choice questions. Possible Number Sense (NS)
(1–14)
14
Statistics, Data Analysis, Probability (P)
(15–26)
12
Algebra and Functions (AF)
(27–43)
17
Measurement and Geometry (MG)
(44–66)
17
Algebra I (AI)
(67–72)
12
Mathematical Reasoning (MR)
(73–80)
8
Totals
Completed
Right
Wrong
80
Reasons for Mistakes Fill out the following chart only after you have read all the explanations that follow. This chart helps you spot your strengths, weaknesses, and your repeated errors or trends in types of errors. Total Missed
Simple Mistake
Misread Problem
Lack of Knowledge
Number Sense (NS) Statistics, Data Analysis, Probability (P) Algebra and Functions (AF) Measurement and Geometry (MG) Algebra I (AI) Mathematical Reasoning (MR) Totals
Examine your results carefully. Reviewing the preceding information helps you pinpoint your common mistakes. Focus on avoiding your most common mistakes as you practice. The Lack of Knowledge column helps you focus your review in the “Math Review” section. If you are missing a lot of questions because of lack of knowledge, you should go back and spend extra time reviewing the basics.
250
Test #3—Answers and Explanations
Answer Key 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.
C (NS) B (NS) A (NS) C (NS) B (NS) C (NS) C (NS) B (NS) A (NS) C (NS) B (NS) D (NS) C (NS) C (NS) D (P) D (P) C (P) B (P) B (P) B (P)
21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40.
41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60.
C (P) D (P) D (P) D (P) C (P) A (P) B (AF) D (AF) A (AF) D (AF) C (AF) B (AF) A (AF) C (AF) B (AF) B (AF) A (AF) B (AF) D (AF) B (AF)
Number Sense (NS)
14
Statistics, Data Analysis, Probability (P)
12
Algebra and Functions (AF)
17
Measurement and Geometry (MG)
17
Algebra I (AI)
12
Mathematical Reasoning (MR)
A (AF) A (AF) C (AF) A (MG) B (MG) C (MG) B (MG) C (MG) C (MG) D (MG) C (MG) B (MG) D (MG) D (MG) A (MG) A (MG) C (MG) B (MG) B (MG) B (MG)
61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80.
C (AI) C (AI) A (AI) D (AI) B (AI) A (AI) D (AI) C (AI) B (AI) A (AI) D (AI) A (AI) C (MR) A (MR) D (MR) C (MR) B (MR) B (MR) A (MR) B (MR)
8
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Answers and Explanations Number Sense 1. C. Since the first factor in each of these expressions is a number between 1 and 10, concentrate on the second factor (10 raised to some power). When 10 is raised to a negative exponent, it becomes a fraction: 10 -1 = 1 10 10 - 2 = 1 100 Therefore, the expressions in Answers A and D are both very small. B is equivalent to 919, but C is equivalent to 1,190, so it is the largest. 2. B. The least common denominator for 8 and 7 is 56, so: 3 + 3-2 = 14 c 8 7 m 3 + 21 - 16 = 3 + 5 14 c 56 56 m 14 56 Fourteenths can be converted into fifty-sixths: 12 + 5 = 17 56 56 56 3. A. When temperature drops, it is like subtracting a positive, so we can write an equation to solve the problem: –3˚ – 18˚ = –21˚ 4. C. Taxes = 1 × $300 = $60. The “rest of the money” = $300 – $60 = $240 5 5. B. It’s easy to get confused when evaluating these statements because several of them are merely the reverse of the correct relationship between the dollar and the Euro. The key thing to remember is that the statement says the Euro is worth more than the dollar; therefore, it’s going to take more dollars to equal fewer Euros. Rule out Answers C and D on that basis. Rule out Answer A since .81 is more than half (.50). 4 is equal to .8, which is 5 close to .81, so Answer B is the most accurate. 6. C. The rest of the team scored 49 – 18 = 31 points, so we can set up a proportion to solve the problem: 31 = x 49 100 Since 49 is close to half of 100, to estimate, just double 31; so 31 is close to 62% of 49. The only answer close to that is C. 7. C. When calculating percent change, use the formula: change in price percent = 100 starting price The change in price = $2.75 – $.25 = $2.50; the starting price = $.25, so: $2.50 = x $.25 100 To solve a proportion, you can cross-multiply and set the products equal:
Then:
252
$250 = $.25x $250 $.25x = $.25 $.25 x = 1,000
Test #3—Answers and Explanations
8. B. Forty percent of 80 is $32. John saves 15% of the reduced price. The reduced price is $80 – $32 = $48. Fifteen percent of $48 is $7.20. So, John saves $32 + $7.20 = $39.20. 9. A. The star makes 5% of the sales above $30,000,000. Those sales are $50,000,000 – $30,000,000 = $20,000,000. Five percent of $20,000,000 is $1,000,000. 10. C. This problem can be rewritten: -3 1 1 1 ^ 3h = 3 = 3 $ 3 $ 3 = 27 3
11. B. The lowest common denominator of 6 and 8 is 24: It is the smallest number that is divisible by both 6 and 8. Twenty-four can be factored as 4 × 6. Because neither 4 nor 6 are prime numbers, they must be factored as well: 4 × 6 = (2 × 2)(2 × 3) = 2 × 2 × 2 × 3 Answer D is incorrect for two reasons: Neither 6 nor 8 are prime numbers; therefore, this cannot be the prime factored form. Moreover, 48 (the product of 6 and 8) is a common denominator, but not the lowest common denominator. 12. D. If you forget the rules for exponents c 3 ' 3 = 3 m , you can rewrite the problem to find the solution: 12 6 6 _3 4 i
3
34 # 32
=
34 # 34 # 34 = 3 # 3 # 3 # 3 # 3 # 3 # 3 # 3 # 3 # 3 # 3 # 3 = 3 # 3 # 3 # 3 # 3 # 3 = 36 3#3#3#3#3#3 3#3#3#3#3#3 1 13. C. Both 2,500 and 3,600 are perfect squares. 2500 = 25 × 100 = 5 × 10 = 50 3600 = 36 × 100 = 6 × 10 = 60 Pick any number between 50 and 60, multiply it by itself, and that product falls between 2,500 and 3,600. 14. C. The absolute value of a number is always positive: It is the distance between a particular number and zero on the number line. From 0 to 5 is a distance of 5 units; however, from –5 to 0 is also a distance of 5 units. Therefore, x could be either –5 or 5.
Statistics, Data Analysis, Probability (P) 15. D. To solve this problem, first arrange the sales figures in numerical order: $19,000, $20,000, $21,000, $21,000, $25,000 Now determine the mean, median, and mode for this data set: Mean: $21,200 Median: $21,000 Mode: $21,000 Now add the sixth value to the data set: $19,000, $20,000, $21,000, $21,000, $25,000, $55,000 The mode does not change because $21,000 is the only repeated data value. The median is the mean of the two middle terms; however, because the two middle terms are the same value, the median is still $21,000. Only the mean is affected in this case by the addition of a relatively extreme data value.
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16. D. Since Tom had a mean of 80% after six tests, he must have had a total of 480 points because 480 ÷ 6 = 80. Tom’s average after three tests was 72%, so after three tests he had 216 points because 3 × 72 = 216. When his fourth and fifth tests are added to this, the result is 394. So his sixth test had to be the difference between the total points he needed (480) and the points he had after five tests (394). 480 – 394 = 86 17. C. To find the median, first arrange the data values in numerical order: 140, 290, 480, 520, 610, 650 When there are an even number of data values, the median is the mean of the middle two values: 480 + 520 = 500 2 18. B. Answer A is false; less than half the students worked less than half an hour. Answer B is correct; 20% of the students did no homework, and 35% of the students worked for an hour or more. That is a total of 55%, which is more than half. Answer C is a subjective statement, not a mathematical observation. Answer D is backward: More than half the students worked more than half an hour. 20% + 35% = 55% 19. B. Half the time an odd number is rolled, and half the time heads are flipped. 1 # 1 = 1 . This can also be seen 2 2 4 on the tree diagram, which shows 12 possible outcomes, each of which is equally likely. The favorable outcomes are 1 with heads, 3 with heads, and 5 with heads. Three favorable outcomes out of a total of 12 outcomes is one-fourth. 20. B. Each of the 12 donuts is equally likely to be picked. Eight of the 12 donuts are neither a jelly donut nor a buttermilk donut. 8 =2 12 3 21. C. The trick to this problem is that the sections are not all the same size. The small green section is 1 of the 8 circle’s area, whereas the large green section is 3 of the circle’s area. 1 + 3 = 4 = 1 Because green represents 8 8 8 8 2 half the area of the circle, it can be expected to occur half the time. 22. D. Spins are independent events; that is to say, one spin has nothing to do with the preceding spins. Answers A, B, and C either imply or directly state that the probability of the outcome of the next spin depends on the outcomes of the previous spins, which is false. Only Answer D correctly explains the independent relationship of the spins. 23. D. Siblings are brothers and sisters, but even if the word is unfamiliar to you, don’t let it throw you: You can still solve the problem. Eighteen percent of the students have no siblings, and 30% have one sibling, so 48% of the students have no siblings or one sibling. That means that out of 100 children, 48 are in this situation. Out of 500 children, five times as many are in this situation. 48 × 5 = 240 children 24. D. To find the greatest decrease, look for the portion of the line graph that shows the greatest (steepest) negative slope. In this case, it occurs when Dontrey’s score falls from 90 to 70. Though 70 is not his lowest score, the change from 90 to 70 is greater than the change from 70 to 60. 25. C. What might make this question tricky is that no information is given on the variables to help you puzzle out the relationship. A scatter plot that shows a negative correlation presents an array of points that fall into a line sloping downward from left to right. This represents an inverse relationship between the two variables: When one increases, the other decreases, and vice versa.
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Test #3—Answers and Explanations
26. A. The amount of cherries purchased decreases as the price increases; however, none of the answer choices express this exact relationship. However, the converse of the statement is also true: The number of pounds purchased increases as the price decreases.
Algebra and Functions (AF) 27. B. Frequently the relationship expressed first in a verbal description is the operation that must be performed last. In this case, the first relationship to establish is “the product of 10 and n.” Product means the result of multiplication, which can be expressed as 10n. One-third of 10n can either be expressed as 10n or 1 (10n) . From this quantity, we 3 3 must then subtract 4. Though Answers B and C appear similar, the order of the operations is different, and the result is different. To appreciate this, substitute the value of 1 for n. 10n - 4 = 10 $ 1 - 4 = 3 1 - 4 = - 2 3 3 3 3 1 ^10n - 4 h = 1 ^10 $ 1 - 4 h = 1 ^ 6 h = 2 3 3 3 28. D. The “sum of five and y” is 5 + y. “Seven more than x” is 7 + x. Because they are equal, we can say that 5 + y = 7 + x. However, that answer choice is not available. By subtracting five from each side, we can simplify the relationship. y=2+x 29. A. To evaluate this expression, plug the given values in for the variables, and perform the operations: xy _10 - y i = c 2 m^ - 3h^10 - - 3h = ^ - 2 h^13h = - 26 3 30. D. To find the least difference, look for the place where the two lines are the closest, which has to be January or December. Because they are slightly closer in December, the correct answer is D. Answers B and C are the months during which the highs have the greatest difference. 31. C. A total of 100 students are in first period, and 22 of them received an F, which is 22% of those students. However, only 50 students are in sixth period. Twenty-two of those students represents 44% of the sixth-period students. The difference in the failure rate is: 44% – 22% = 22% 32. B. Gas grills selling for more than $300 are shown by the two far-right columns, each of which represents a quantity of 15 models. Therefore, the total number of models available at this price is 30. 33. A. Raising a variable to a positive exponent can be represented as repeated multiplication. When the variable is raised to a negative exponent, that is the inverse of multiplication, which is division. We can rewrite the expression: x 3 y - 2 xxxyyy = xxyy = xy x 2 y -3 34. C. The trick is to note that the first term is squared. It might be helpful to rewrite the expression: (2a2b)2(3ab5) = (2a2b)(2a2b)(3ab5) = (2 × 2 × 3)(aaaaa)(bbbbbbb) = 12a5b7 35. B. When x is raised to a power of two, its graph is a parabola, which is the form of the line in Graphs A and B. However, when the coefficient of x is negative, the parabola opens downward, as in Graph A, so you must eliminate it. Graphs C and D are linear functions and do not contain a squared variable.
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36. B. Because the line slopes downward from left to right, it has a negative slope—therefore you can eliminate Answers C and D. Slope is defined as: vertical shift horizontal shift In this case, the line goes down two units for every three units it moves to the right: -2 = - 2 3 3 37. A. This line goes up 3 units for every 4 units it moves to the right, which gives it a slope of 3 . It crosses the 4 y-axis at –5, which gives it a y-intercept of –5. Its equation is then written in the slope-intercept form as: y= 3 x-5 4 Once you spot that the y-intercept is –5, you could eliminate Choices C and D. Or, if you spot that the slope is 3 4 the correct answer is A. It is the only choice with a slope of 3 . 4 38. B. The easiest way to solve this problem is to look for a place where the graphed line crosses an intersection on the grid. This happens at 2.5 hours, and at 5 hours. At the 5 hours mark, the airplane has traveled 2,000 miles. 2,000 ÷ 5 = 400 39. D. To solve the inequality: 8- x <5 3 3 c 8 - x m < 15 3 24 – x < 15 +x = +x
Multiply both sides by 3, to clear the fraction. Multiply both terms of the expression in parentheses. Add x to both sides.
24 < x + 15 –15 = –15
Subtract 15 from both sides.
9 < x or x > 9 40. B. To solve this equation: S = 5W + 35 100 = 5W + 35 –35 = –35
Subtract 35 from each side to isolate the variable.
65 = 5W 65 = 5 W 5 5
Divide each side by 5.
W = 13 The correct answer could also be easily determined by substituting the answer choices into the equation.
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Test #3—Answers and Explanations
41. A. To solve this inequality: 3 + 3x ≤ –6 –3 = –3
Subtract 3 from each side to isolate the variable.
3x ≤ –9 3x # - 9 3 3
Divide each side by 3 to clear the coefficient.
x ≤ –3 So, we are looking for the number line that includes the point –3 and all points that are smaller than –3, which are the points to the left of that number. That is Answer A. 42. A. We are comparing miles to time, either minutes or hours. Since the information is given in minutes, it’s best to start with minutes: 3 + 16 + 1 = 20 = 2 9 + 18 + 3 30 3 So Joe drives 2 miles in every 3 minutes. Or, Joe drives 2 of a mile per minute. Since some of the answer choices 3 are given in terms of miles per hour, perhaps we’d better convert this to mph to be sure they are incorrect. 2 # 20 = 40 = 40 miles for 60 minutes, or 40 miles per hour 3 20 60 43. C. Dividing the distance traveled by the miles per gallon tells how many gallons of gasoline the family uses: 720 ÷ 24 = 30 gallons Now, multiplying the number of gallons by the price per gallon tells how much money they spend on gas: $2.40 × 30 = $72.00 This might seem like going in circles because of the numerical similarity between miles per gallon (24) and the price of gasoline ($2.40).
Measurement and Geometry (MG) 44. A. Eight ounces = 1 of a pound because 8 = 1 . So 7 pounds, 8 ounces equals 7.5 pounds. To convert pounds to 2 16 2 grams, multiply by 454: 7.5 pounds × 454 grams per pound = 3,405 grams However, this answer is not one of the choices that are available. Answers B and C, which are both expressed in terms of grams, are clearly incorrect and can be eliminated. That leaves Answers A and D to consider, both of which are expressed in terms of kilograms. To convert grams to kilograms, divide by 1,000: 3,405 grams ÷ 1,000 grams per kilogram = 3.405 kilograms
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45. B. The change in temperature necessary to cause a state change in water from boiling to freezing is the same, no matter which system is used to measure it. In Fahrenheit, that change in temperature is equal to the difference in the two temperatures: 212˚ – 32˚ = 180˚F In Celsius: 100˚ – 0˚ = 100˚C So: 180˚F = 100˚C If it takes fewer degrees Celsius to equal the degrees Fahrenheit, a Celsius degree must be bigger than a Fahrenheit degree. 46. C. The length of the court is 78 feet. If 1 inch = 12 feet, then divide 78 by 12 to see how long to make the length: 78 feet ÷ 12 feet/inch = 6.5 inches 47. B. A job that can be done in 8 “man-days” can be completed by 8 men in one day, or 1 man in eight days. The job is a product of the number of men and the number of days. So we can ask: 3 men × how many days = 8 man days To solve for days, divide both sides by 3 men: how many days = 8 days = 2 2 days 3 3 48. C. All the answers include 4 hours since it takes 4 hours to walk 12 miles. The question is, how long does it take to walk the extra half mile? Eliminate Answer A since it allows no time whatsoever. Three miles per hour equals 6 half-miles per hour; so to walk half a mile takes one-sixth of an hour. One-sixth of an hour is 10 minutes. 49. C. The area of a triangle = 1 base × height. 2 Any side of a triangle can be used as the base; the important consideration is that the height is a line drawn perpendicular to the base from the opposite vertex. So in this drawing, use 15 inches as the base and 7.2 inches as the height: 1 15 inchesh^ 7.2 inchesh = 54 square inches 2^ 50. D. The formula for the volume of a cylinder is base × height. The base of a cylinder is a circle; the formula for the area of a circle is πr2. The radius of the circle shown is 3 inches, and the height of the cylinder is 6 inches, so to calculate the volume: base × height = πr2 × 6 in. = (3.14)(32)(6) ≈ 169.56 in.3 51. C. If the radius of the circle is 5 inches, the diameter of the circle is twice that, or 10 inches. The diameter of the circle is equal to the length of one of the sides of the square. To find the perimeter of the square, multiply the length of one side by 4: 10 in. × 4 = 40 in.
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52. B. The solid figure is formed by combining a rectangular prism with a triangular prism.
2
4
3 8 ft.
6 ft.
4 ft.
3 ft.
Volumerectangular prism = length × width × height = (3 ft.)(4 ft.)(6 ft.) = 72 ft.3 Volumetriangular prism = 1 × base area × height = 1 (2)(3)(4) = 12 ft.3 2 2 The total volume of the figure is the sum of the two volumes: 72 ft.3 + 12 ft.3 = 84 ft.3 53. D. The large square, ABCD, is subdivided into several different polygons whose areas are related. The area of the small, shaded triangle is one-half the area of the small square, the parallelogram, and the medium-sized triangle. So: 2 small triangles + small square + parallelogram + medium triangle = 8(area of small triangle) = 8(4 sq. units) = 32 sq. units This is half the area of the large square, so: Arealarge square ABCD = 2(32 sq. units) = 64 sq. units Another method would be to break the figures in half of the triangle into small congruent triangles and count them as follows: A
B 1
4 2
3 7
5 6 8
D
C
8 small triangles times 4 sq. units = 32 sq. units. Doubling this would give 64 sq. units.
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54. D. The length, width, and height as shown are each 3; so if Ron doubles each of the dimensions, he needs 6 × 6 × 6 = 216 cubes. 55. A. Six inches is half a foot, so the area of the board is 8 ft. × .5 ft. = 4 ft.2 56. A. To find the area of the square you will need to find the side and then square it. By using the coordinate grid you can make a right triangle with the origin at (0,0); one leg goes to point D (4,0), and the other goes to point A (0,3).
B 6
4
C A
2
0
2
D 4
6
So the lengths of the legs of the right triangle are 3 and 4. If you know Pythagorean Triples you would know that the hypotenuse is then 5, as this is a 3-4-5 right triangle. Otherwise use the Pythagorean theorem: a2 + b2 = c2 (3)2 + (4)2 = c2 9 + 16 = c2 25 = c2 5 = c2 Since one side of the square is 5, the area is 5 × 5 or 25 square units. 57. C. The length of Side DC is equal to the difference in x-coordinates, or: 10 – 2 = 8 units Therefore, to have an area of 24 units, the length of Width AD has to be 3 units. The x-coordinate of Point A is the same as the x-coordinate of Point D since AD is parallel to the y-axis. Its y-coordinate has to be 3 units above that of Point D. That location is (2,5) on the coordinate grid. 58. B. ABC is a right triangle with legs measuring 6 units and 8 units. Side BC is the hypotenuse of the triangle, and its length is equal to the square root of the sum of the squares of the two legs: BC = 6 2 + 8 2 = 36 + 64 = 100 = 10 units 59. B. By the Pythagorean theorem: AreaBCHI = AreaABED – AreaACGF = 25 – 16 = 9 units2 Since BCHI is a square, the length of one of its sides is the square root of its area: 9 units2 = 3 units = a 60. B. Segments marked with the same number of hash marks are congruent to each other. Figure B is a parallelogram divided into two triangles by its diagonal. The hash marks show that each triangle has two sides that are congruent to two sides in the other triangle. The diagonal forms the third side of each triangle.
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Test #3—Answers and Explanations
Algebra I (AI) 61. C. Rewriting this equation might help to simplify it. The base gives the factor, the exponent tells how many times to use it as a factor. A positive number raised to a negative exponent becomes a fraction, not a negative number. 2
-1 4 1 1 1 1 1 1 16 ^ 4 h ^ 2 h c 2 m = c 4 m^ 2 $ 2 $ 2 $ 2 hc 2 $ 2 m = 4 $ 16 $ 4 = 16 = 1
62. C. The easiest way to deal with an absolute value equation like this might be to substitute the answer choices into the equation and see which ones work. Absolute value is always positive, so the total quantity within the absolute value bars can be positive or negative. What makes this equation tricky is that the entire left side of the equation is enclosed within absolute value bars. In essence, what it’s saying is that 3x – 3 is equal to either 15, or to –15. So you need to solve two equations: 3x – 3 = 15
3x – 3 = –15
+3 = +3
+3 = +3
3x = 18
3x = –12
3x = 18 3 3
3x = - 12 3 3
x=6
x = –4
63. A. To clear the fractions on the left side of the equation, multiply each side of the equation by the least common denominator of the fractions. The least common denominator of 5 and 3 is 15, so: x - 2 =6 5 3 15 c x - 2 m = 15 ^ 6 h 5 3 15 $ x - 15 $ 2 = 90 3 5 3x – 10 = 90 64. D. To simplify the left side of the equation, you must distribute the multiplication by each factor outside the parentheses to the terms within the parentheses, then combine like terms: 1 4x - 12 h - 1 ^12 - 3x h = 5 2^ 3 1 1 1 1 c m^ 4x h - c m^12 h + c - m^12 h + c - m^ - 3x h = 5 2 2 3 3 2x – 6 – 4 + x = 5 3x – 10 = 5 65. B. At the end of Step 1, Francisco has isolated the variable. His objective now is to get a coefficient of 1. To get a coefficient of 1, divide both sides by the current coefficient. However, none of the answer choices say, “Divided both sides by 2 .” Multiplication and division are inverse operations, so instead of dividing, multiply by the 3 inverse of 2 , which is known as the reciprocal. Multiplying both sides by 3 (the correct answer) is the same as 3 2 dividing both sides by 2 . 3
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66. A. The answer choices are all given in the slope-intercept form, where the coefficient (number in front) of x is the slope, and the second number is the y-intercept (or the point on the y-axis that the line passes through). Lines that slope upward from left to right have a positive slope, so eliminate Answers B and D, which show a negative slope. This line passes through the y-axis at –3, so it has a y-intercept of –3. With this piece of information, you can eliminate Answer C, leaving Answer A as the only possibility. To confirm the exact slope, observe that the line rises 3 units for every 4 units it moves to the right, which is a slope of 3 . 4 67. D. The y-intercept is the point where the line crosses the y-axis, and it always has an x-coordinate of 0. Likewise, the x-intercept is the point where the line crosses the x-axis, and it always has a y-coordinate of 0. Knowing this eliminates Answers B and C, which have this relationship backward. To solve for the y-intercept, set x = 0 and solve the equation: –2x + 3y = 6 –2(0) + 3y = 6 3y = 6 y=2 So the y-intercept is (0, 2). To solve for the x-intercept, set y = 0 and solve the equation: –2x + 3y = 6 –2x + 3(0) = 6 –2x = 6 x = –3 So the x-intercept is (–3, 0). 68. C. It’s a good idea to visually inspect the equation to see whether any relationships can be taken advantage of to eliminate answers. In this case, there are: y = - 1 x . The signs for x and y are opposite, so whenever x is positive, 2 y is negative, and vice versa. On that basis, eliminate Answers A and B, which have both x- and y-coordinates as negative. At this point, perhaps substitute the coordinates into the equation to see which pair satisfies the equation. Using the coordinates from Answer C: - 3 = - 1 ^6h 2 This is true, so those coordinates do represent a point on the line. 69. B. Lines that are parallel have the same slope. The difficulty is that, in this problem, the equation of a line is given in the standard form, not the slope-intercept form (which is y = mx + b), where m, the coefficient of x, is the slope of the line. The easiest way to get the slope is to just rework the equation into the slope-intercept form: 5x + 10y = 7 10y = –5x + 7 y= -1 x+ 7 2 10 So the slope is - 1 , which is Answer B. 2
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70. A. The answer is the graph that represents these two equations. The easiest way to determine that is to convert these two equations to the slope-intercept form, y = mx + b. –x + y = 3 → y = x + 3 –4x + 2y = 6 → y = 2x + 3 The lines in the answer both have a positive slope and a y-intercept of +3. Lines with a positive slope travel upward from left to right; the y-intercept is the point on the y-axis that the line passes through. All four answers show graphs with correct y-intercepts. However, only Graph A shows two lines with a positive slope, so we can identify it as the only possible answer without going any further. 71. D. The trick to simplifying this expression is to correctly distribute –2x to both terms within the parentheses: 3x – 2x(x – 2) + 2x2 = 3x + (–2x)(x) + (–2x)(–2) + 2x2 = 3x – 2x2 + 4x + 2x2 = 7x 72. A. The formula for solving this type of time problem is: distance = time rate So, to drive 200 miles at 60 mph takes: 200 miles = 10 = 3 1 hours 3 3 60 mph To drive 200 miles at 75 mph takes: 200 miles = 8 = 2 2 hours 3 3 75 mph The difference in time is 3 1 - 2 2 = 2 of an hour, which equals 40 minutes. 3 3 3
Mathematical Reasoning (MR) 73. C. Though Answers A, B, and D all mention factors that could affect Paula’s plans to graduate in 4 years, Answer C identifies the specific information needed to formulate her initial class schedule. For instance, if each class is worth 5 units, Paula only needs to take 9 classes a year to earn 45 units. However, if each class is worth only 3 units, Paula needs to take 15 classes to earn the same number of units. 74. A. An easy way to solve this kind of problem is to consider a specific case of this situation, and then substitute the numbers into the expressions to see which expression yields the desired total. If each man gets only 1 fish, together the men have 3 fish, plus one they gave to the cat, for an initial total of 4 fish. So with x = 1, which of these expressions is worth 4? Only the expression in Answer A: 3x +1 = 3(1) + 1 = 3 + 1 = 4 75. D. Positive integers are whole numbers, like 3 or 7. If division by a particular number results in a quotient that is a whole number, the number is said to be divisible by that divisor. The symbolic information presented in the box gives the same information as Answer D. Answer A might be true, but we do not have enough information to be sure: Many numbers that are divisible by both 2 and 3 are not multiples of 4, such as the number 6. 76. C. The expression for this answer rounds each item to the closest $10. Answers B and C round the computer printer to the closest $100, which introduces a significant error of about $46. Answer A incorrectly rounds the printer to $200, introducing an even bigger error.
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77. B. The decimal equivalent for 1 is a repeating decimal, .3, which is significantly larger than .3. By rounding 3 off so early in his computation, and then rounding off the other addends in the same way, Jones introduces a significant error that produces an incorrect sum. This is a common problem when converting fractions to decimals to do computations. 78. B. To determine the number of people who did not send back responses, add the number of responses reported from each category and subtract the total from 500: 130 + 140 + 20 + 20 + 10 = 320 500 – 320 = 180 79. A. For each of the x/y pairs, multiplying the x by 2, then adding 3, gives the corresponding y value. The other answer choices might be true for a single x/y pair, but not for all pairs. 80. B. In this equation, x represents the number of dimes. When there are 13 coins, 13 – x represents the number of nickels. The value of one dime multiplied by the number of dimes is .10x. This gives the total value of the dimes in the combination. Since the value of one dime does not change, this part of the equation should not be changed.
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Arithmetic/Statistics and Probability Glossary of Terms ABSOLUTE VALUE: The distance a number is from zero on the number line. Considering + and − as directions on the number line, absolute refers to distance, not direction. For example, the absolute value of −3 is 3. Also written as |−3| = 3. The absolute value of + 3 is also 3. ADDITIVE INVERSE: The opposite (negative) of a number. Any number plus its additive inverse equals 0. For example, 3 + (−3) = 0. ASSOCIATIVE PROPERTY: A property stating that the grouping of elements does not make any difference in the outcome. This is only true for multiplication and addition. For example, (2 + 3) + 4 = 2 + (3 + 4). BAR GRAPH: A graph using horizontal or vertical bars to display the data. The longer the bar, the greater the quantity. BRACES: Grouping symbols used after the use of brackets. Also used to represent a set. { } BRACKETS: Grouping symbols used after the use of parentheses. [ ] CANCELING: In multiplication of fractions, dividing the same number into both a numerator and a denominator. CIRCLE GRAPH (or pie chart): Displaying data on a circular graph by dividing the circle into sections. COMBINATIONS: The total number of independent possible choices. COMMON DENOMINATOR: A number that can be divided evenly by all denominators in the problem. For example, the common denominator of 1 and 1 is 6. 2 3 COMMON FACTORS: Factors that are the same for two or more numbers. For example, 3 is a common factor of 6 and 9. COMMON MULTIPLES: Multiples that are the same for two or more numbers. For example, 10 is a common multiple of 2 and 5. COMMUTATIVE PROPERTY: A property stating that the order of elements does not make any difference in the outcome. This is only true for multiplication and addition. For example, 2 + 3 = 3 + 2. COMPLEX FRACTION: A fraction having a fraction in the numerator and/or denominator. COMPOSITE NUMBER: A number divisible by more than just 1 and itself. {4, 6, 8, 9, . . .}. 0 and 1 are not composite numbers. CORRELATION: Comparing the relationship of two pairs of data. CUBE: The result when a number is multiplied by itself twice. For example, 8 is a cube because 2 × 2 × 2 = 8. CUBE ROOT: The number that is multiplied by itself twice to get the resulting cubed number. For example, 5 is the cube root of 125 because 5 × 5 × 5 = 125. Its symbol is 3 . 3 125 = 5 . DECIMAL FRACTION: A fraction with a denominator of 10, 100, 1000, or any multiple of 10, written using a decimal point (for example, .3, .275). DECIMAL POINT: A point used to distinguish decimal fractions from whole numbers. DECREASED BY: To make a quantity smaller by a certain value. DENOMINATOR: The bottom symbol or number in a fraction.
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DEPENDENT EVENT: When the outcome of one event has a bearing or effect on the outcome of another event. DIFFERENCE: The result of subtraction. DISTRIBUTIVE PROPERTY: The process of distributing a number on the outside of a set of parentheses to each number on the inside, for example, 2(3 + 4) = 2(3) + 2(4). This can also be written algebraically as a(b + c) = ab + ac. EVEN NUMBER: An integer (positive whole numbers, zero, and negative whole numbers) divisible by 2 (which means with no remainder). EXPANDED NOTATION: Pointing out the place value of a digit by writing the number as the digit multiplied by its place value. For example, 342 = (3 × 102) + (4 × 101) + (2 × 100). EXPONENT: A small number placed above and to the right of a number. It expresses the power to which the quantity is to be raised or lowered. For example, in the number 32, 2 is the exponent. FACTOR (noun): A number or symbol that divides evenly into a larger number. For example, 6 is a factor of 24. FACTOR (verb): To find two or more quantities whose product equals an original quantity. For example, 15 can be factored into 3 × 5. FRACTION: A symbol expressing part of a whole. It consists of a numerator and a denominator (for example, 3 , 9 ). 5 4 GREATEST COMMON FACTOR: The largest factor common to two or more numbers (that is, the largest number that divides into two or more numbers evenly). For example, 6 is the greatest common factor of 18 and 24. HUNDREDTH: The second decimal place to the right of the decimal point. For example, .08 is eight hundredths. IDENTITY ELEMENT FOR ADDITION: 0. Any number added to 0 gives the original number. For example, 2 + 0 = 2. IDENTITY ELEMENT FOR MULTIPLICATION: 1. Any number multiplied by 1 gives the original number. For example, 3 × 1 = 3. IMPROPER FRACTION: A fraction in which the numerator is greater than the denominator. For example, 3 . 2 INDEPENDENT EVENT: When the outcome of one event has no bearing or effect on the outcome of another event. INTEGER: A whole number, either positive, negative, or zero. {. . .−3, −2, −1, 0, 1, 2, 3 . . .} INVERT: To turn upside down, as in “invert 2 ” = 3 . 3 2 IRRATIONAL NUMBER: A number that is not rational (that is, it cannot be written as a fraction xy , with x as an integer and y as a natural number). For example, 3 or π. LEAST COMMON MULTIPLE: The smallest multiple that is common to two or more numbers. For example, 6 is the least common multiple of 2 and 3. LINE GRAPH: Graphing on an x-y graph by placing points on the graph and connecting them to show relationships in the data. LOWEST COMMON DENOMINATOR: The smallest number that can be divided evenly by all denominators in the problem. For example, in the problem 2 + 1 , the lowest common denominator is 12. 3 4 MEAN (arithmetic): The average of a number of items in a group (found by totaling the items and dividing by the number of items). MEDIAN: The middle item in an ordered group. If the group has an even number of items, the median is the average of the two middle items. The items in the group have to be in placed in consecutive order.
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Arithmetic/Statistics and Probability Glossary of Terms
MIXED NUMBER: A number containing both a whole number and a fraction (for example, 5 1 ). 2 MODE: The number appearing most frequently in a group. MULTIPLES: Numbers found by multiplying a number by 2, by 3, by 4, and so on. MULTIPLICATIVE INVERSE: The reciprocal of a number. Any number multiplied by its multiplicative inverse equals 1 (for example, 1 # 3 = 1). 3 NATURAL NUMBER: A counting number. {1, 2, 3, 4, . . .} NEGATIVE CORRELATION: In a scatterplot, when one set of data increases while another decreases. NEGATIVE NUMBER: A number less than 0. NUMBER LINE: A visual representation of the positive and negative numbers and zero. The line can be thought of as an infinitely long ruler with negative numbers to the left of zero and positive numbers to the right of zero. NUMBER SERIES: A sequence of numbers with some pattern. One number follows another in some defined manner. NUMERATOR: The top symbol or number in a fraction. ODD NUMBER: An integer that is not divisible by 2. OPERATION: Multiplication, addition, subtraction, or division. ORDER OF OPERATIONS: The priority given to an operation relative to other operations. For example, multiplication takes precedence (is performed before) addition. PARENTHESES: Grouping symbols. ( ) PERCENT OR PERCENTAGE: A common fraction with 100 as its denominator (for example, 37% is 37 ). 100 PIE CHART (or circle graph): Displaying data on a circular graph by dividing the circle into sections. PLACE VALUE: The value given to a digit by the position of that digit in a number. For example, in the number 37, 3 is in the tens place (meaning 3 tens). POSITIVE CORRELATION: In a scatterplot, when one set of data increases while another also increases. POSITIVE NUMBER: A number greater than 0. POWER: A product of equal factors. 4 × 4 × 4 = 43, read “four to the third power” or “the third power of four.” Power and exponent are sometimes used interchangeably. PRIME NUMBER: A number that can be divided by only itself and one. A number that has exactly two factors (one and itself), for example, 2, 3, 5, 7, and so on. 0 and 1 are not prime numbers. PROBABILITY: The numerical measure of the chance of an outcome or event occurring. PRODUCT: The result of multiplication. PROPER FRACTION: A fraction in which the numerator is less than the denominator (for example, 2 ). 3 5 PROPORTION: An expression written as two equal ratios (for example, 5 is to 4 as 10 is to 8, or = 10 ). 4 8 QUOTIENT: The result of division. RANDOM: When all possible choices have an equal probability of being selected.
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RANGE: The difference between the largest and the smallest number in a set of numbers. RATIO: A comparison between two numbers or symbols. It can be written x:y, xy , or x is to y. For example, 1:2 is the same as 1 , which is the same as 1 is to 2. 2 RATIONAL NUMBER: An integer or fraction such as 7 , 9 , or 5 . Any number that can be written as a fraction 2 8 4 1 3 (where x is an integer and y is a natural number). REAL NUMBER: Any rational or irrational number. RECIPROCAL: The multiplicative inverse of a number. For example, 2 is the reciprocal of 3 . 3 2 2 1 REDUCING: Changing a fraction into its lowest terms. For example, can be reduced to . 4 2 ROUNDING OFF: Changing a number to a nearest place value as specified; it is a method of approximating. For example, 56 rounded off to the nearest ten is 60. SCATTERPLOT: Plotting points on an x-y graph to find a correlation between data. SCIENTIFIC NOTATION: A number equal to or greater than 1 and less than 10 that is multiplied by a power of 10. It is used for writing very large or very small numbers (for example, 2.5 × 104). SIMPLE INTEREST: Interest calculated by multiplying an amount of money by an interest rate and an amount of time. SQUARE: The result when a number is multiplied by itself. For example, 16 is a square number because 4 × 4 = 16. SQUARE ROOT: The number that is multiplied by itself to get the resulting square number. For example, 5 is the square root of 25 because 5 × 5 = 25. Its symbol is . 25 = 5 . SUM: The result of addition. TENTH: The first decimal place to the right of the decimal point. For example, .7 is seven tenths. WEIGHTED MEAN: The mean of a set of numbers that has been weighted (that is, multiplied by the relative importance or frequency of occurrence of the numbers). WHOLE NUMBERS: 0, 1, 2, 3, and so on. ZERO CORRELATION: In a scatterplot, when no apparent pattern or relationship exists between data.
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Algebra Glossary of Terms ABSCISSA: The distance along the horizontal axis in a coordinate graph. ABSOLUTE VALUE: The numerical value when direction or sign is not considered. The symbol for absolute value is | |. ALGEBRA: Arithmetic operations using letters and/or symbols in place of numbers. ALGEBRAIC FRACTIONS: Fractions using a variable in the numerator and/or denominator. ASCENDING ORDER: Basically, when working with polynomials, the power of a term increases for each succeeding term (for example, x2 + x3 + x4). BINOMIAL: An algebraic expression consisting of two terms (for example, x − 3 or 2x + 3y). CARTESIAN COORDINATES: A system of assigning ordered number pairs to points on a plane. COEFFICIENT: The number in front of a variable. For example, in 9x, 9 is the coefficient. COORDINATE AXES: Two perpendicular number lines used in a coordinate graph. COORDINATE GRAPH: Two perpendicular number lines, the x-axis and the y-axis, creating a plane on which each point is assigned a pair of numbers. These numbers are referred to as ordered pairs. CUBE: The result when a number is multiplied by itself twice. This is designated by the exponent 3, as in x3. For example, 2 × 2 × 2 = 23, and y × y × y = y3. CUBE ROOT: The number that is multiplied by itself twice to get the resulting cubed number. For example, 5 is the cube root of 125 because 5 × 5 × 5 = 125. Its symbol is 3 . 3 125 = 5 . DENOMINATOR: Everything below the fraction bar in a fraction. DESCENDING ORDER: Basically, when working with polynomials, the power of a term decreases for each succeeding term. For example, x4 + x 3 + x2. EQUATION: A balanced relationship between numbers and/or symbols, that is, a mathematical sentence. EQUIVALENT EXPRESSIONS: Numerical expressions that have the same value, that is, variable expressions (expressions containing variables) that have the same values for every value of the variable. EVALUATE: To determine the value or numerical amount. EXPONENT: A numeral or symbol used to indicate the power of a number. EXPRESSION: A number, a variable, or a combination of numbers, variables, and symbols (for example, 3x2, 2x + 3y). EXTREMES: Outer terms. For example, in the multiplication problem (x + 3)(3x + 2), the extremes would be x and 2. FACTOR: To find two or more quantities whose product equals the original quantity. For example, factoring 4x + 2 gives 2(2x + 1). A number of forms of factoring exist in Algebra. FINITE: Countable; having a definite ending. F.O.I.L. METHOD: A method of multiplying binomials in which first terms, outside terms, inside terms, and last terms are multiplied. IMAGINARY NUMBERS: Square roots of negative numbers. The imaginary unit is i.
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INEQUALITY: A statement in which the relationships are not equal; the opposite of an equation. For example, 3x + 2 > 4 is an inequality. INFINITE: Uncountable; continues forever. LINEAR EQUATION: An equation whose solution set forms a straight line when plotted on a coordinate graph. LITERAL EQUATION: An equation having mostly variables. MEANS: Inner terms. For example, in the multiplication problem (x + 3)(3x + 2), the means are 3x and 3. MONOMIAL: An algebraic expression consisting of only one term (for example, x2, 3x, or 2xy2). NONLINEAR EQUATION: An equation whose solution set does not form a straight line when plotted on a coordinate graph. NUMBER LINE: A graphic representation of integers and real numbers. The point on this line associated with each number is called the graph of the number. NUMERATOR: Everything above the fraction bar in a fraction. ORDERED PAIR: Any pair of elements (x, y) having a first element x and a second element y. This is used to identify or plot points on a coordinate graph. ORDINATE: The distance along the vertical axis on a coordinate graph. ORIGIN: The point of intersection of the two number lines on a coordinate graph. This is represented by the coordinates (0,0). POLYNOMIAL: An algebraic expression consisting of two or more terms (for example, 2x + 3, 3xy + 2x + 4, or x2 + 5x − 1). PROPORTION: Two ratios equal to each other. For example, a is to c as b is to d. QUADRANTS: The four quarters or divisions of a coordinate graph. The quadrants are numbered I, II, III, and IV; starting in the upper right quarter and moving counterclockwise. QUADRATIC EQUATION: An equation that can be written in the form Ax2 + Bx + C = 0. RADICAL SIGN: The symbol used to designate a square root. RATIO: A method of comparing two or more numbers. For example, a:b. It is often written as a fraction. REAL NUMBERS: The set consisting of all rational and irrational numbers. SIMPLIFY: To combine several (or many) terms into fewer terms. SLOPE OF A LINE: On a graph, the ratio of the horizontal change in a line, calculated as the rise over the run. The change in y over the change in x. Slope = rise run . SOLUTION SET (or solution): All the answers that satisfy an equation. SQUARE: The result when a number is multiplied by itself. It is designated by the exponent 2 (for example, x2). SQUARE ROOT: The number that is multiplied by itself to get the resulting squared number. For example, 5 is the square root of 25 because 5 × 5 = 25. Its symbol is . 25 = 5 . SYSTEM OF EQUATIONS: Simultaneous equations. Two equations that when plotted on a coordinate graph either intersect (giving an ordered pair that is a solution) or are parallel (having no solution).
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Algebra Glossary of Terms
TERM: A numerical or literal expression with its own sign. TRINOMIAL: An algebraic expression consisting of three terms (for example, x2 + 2x + 1 or 3x + 2y − 4). UNKNOWN: A letter or symbol whose value is not known. VALUE: A numerical amount. VARIABLE: A symbol used to stand for a number. X-AXIS: The horizontal axis in a coordinate graph. X-COORDINATE: The first number in an ordered pair. It refers to the distance on the x-axis (abscissa). X-INTERCEPT: The value of x in an ordered pair where the graph of the line crosses the x-axis. In an ordered pair, if y is 0, x is the x-intercept. For example, (3,0) indicates an x-intercept of 3. Y-AXIS: The vertical axis in a coordinate graph. Y-COORDINATE: The second number in an ordered pair. It refers to the distance on the y-axis (ordinate). Y-INTERCEPT: The value of y in an ordered pair where the graph of the line crosses the y-axis. In an ordered pair, if x is 0, y is the y-intercept. For example, (0,3) indicates a y-intercept of 3.
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Measurement and Geometry Glossary of Terms ALTITUDE OF A TRIANGLE (height of a triangle): The perpendicular line segment from a vertex (point) to the opposite side (or an extension of the opposite side). ARC OF A CIRCLE: A connected portion of a circle. AREA: A measure of the interior of a flat (two-dimensional) figure. It is expressed in square units such as square inches (in2) or square centimeters (cm2) or in special units such as acres. BASE OF A TRIANGLE: Any side of a triangle. BASES OF A PARALLELOGRAM: Each pair of parallel sides in a parallelogram can serve as bases. BASES OF A TRAPEZOID: The parallel sides of a trapezoid. CENTER OF A CIRCLE: The (fixed) interior point that is equidistant from all points on a circle. CIRCLE: A flat figure with all its points equidistant from a fixed point (the center). CIRCUMFERENCE: The distance around a circle. CONGRUENT TRIANGLES (or figures): Triangles (or figures) that have exactly the same size and shape. COORDINATES OF A POINT: The ordered pair of numbers assigned to a point in a plane. COORDINATE PLANE: The x-axis, the y-axis, and all the points in the plane they determine. CORRESPONDING PARTS OF TRIANGLES: The parts of two (usually) congruent or similar triangles that are in the same relative position. CUBE: A six-sided solid where all sides are equal squares, and all edges are equal. CYLINDER: A prism-like solid whose bases are circles. DECAGON: A 10-sided polygon. DEGREE MEASURE OF A SEMICIRCLE: 180°. DIAGONAL OF A POLYGON: Any line segment that joins two nonconsecutive vertices of a polygon. DIAMETER OF A CIRCLE: A line segment that passes through the center of a circle, starting and ending on the circle. EXTREMES OF A PROPORTION: When a proportion is written in the form a:b = c:d, a and d are referred to as extremes of the proportion. GRAPH OF AN ORDERED PAIR: The point (in a coordinate plane) associated with an ordered pair of real numbers. HEIGHT OF A PARALLELOGRAM (or trapezoid): Any perpendicular segment connecting two bases of a parallelogram (or trapezoid). HEIGHT OF A TRIANGLE (altitude of a triangle): The perpendicular line segment from a vertex (point) to the opposite side (or an extension of the opposite side) of a triangle. HEPTAGON: A seven-sided polygon. HEXAGON: A six-sided polygon.
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CliffsTestPrep California High School Exit Exam: Math
HYPOTENUSE: The side opposite the right angle in a right triangle. INTERSECTING LINES: Two or more lines that meet at a single point. LEGS OF A RIGHT TRIANGLE: The two sides other than the hypotenuse in a triangle. LEGS OF A TRAPEZOID: The nonparallel sides of a trapezoid. LINEAR EQUATION: An equation whose graph is a straight line. MEANS OF A PROPORTION: When a proportion is written in the form a:b = c:d, b and c are referred to as means of the proportion. MIDPOINT OF A LINE SEGMENT: The point on a line segment that is equidistant from the endpoints; the halfway point. NONAGON: A nine-sided polygon. OCTAGAN: An eight-sided polygon. ORDERED PAIR: A pair of numbers whose order is important; these are used to locate points in a plane. ORIGIN: In two dimensions, the point (0,0); it is the intersection of the x-axis and the y-axis. PARALLEL LINES: Two lines that lie in the same plane and never intersect. PARALLELOGRAM: Any quadrilateral with both pairs of opposite sides parallel. PENTAGON: A five-sided polygon. PERIMETER: The distance around a figure. PERPENDICULAR LINES: Lines that intersect and form right angles. POINT-SLOPE FORM OF THE EQUATION OF A LINE: The form y − y1 = m (x − x1), where m is the slope of a line, and (x, y) (x1, y1) are specific points on that line. POLYGON: A plane closed figure with three or more sides. PROPORTION: An equation stating that two ratios are equal. PYTHAGOREAN THEOREM: In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides of the triangle: a2 + b2 = c2 where c is the longest side (the hypotenuse). PYTHAGOREAN TRIPLE: Three positive integers (a, b, c) that satisfy the equation a2 + b2 = c2 (for example, 3, 4, 5 or 5, 12, 13). QUADRANTS: The four regions of a coordinate plane separated by the x-axis and the y-axis. QUADRILATERAL: A four-sided polygon. RADIUS OF A CIRCLE: A line segment with the center of a circle and a point on that circle as endpoints (plural: radii). RATIO OF TWO NUMBERS a AND b: The fraction a/b, usually expressed in simplest form; also denoted a:b. RECTANGLE: A quadrilateral in which all the angles are right angles. Opposite sides are parallel and equal. REFLECTION: Transforming each point in a plane by mapping it to its mirror image across a line. When the x-coordinate of each point is changed to its opposite sign, the image is reflected across the y-axis. When the y-coordinate of each point is changed to its opposite sign, the image is reflected across the x-axis.
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Measurement and Geometry Glossary of Terms
RHOMBUS: A quadrilateral with all four sides equal. RIGHT ANGLE: A 90° angle. RIGHT CIRCULAR CYLINDER: A cylinder with the property that the segment joining the centers of the circular bases is perpendicular to the planes of the bases. SECTOR OF A CIRCLE: A region bounded by two radii and an arc of the circle intercepted by (an angle formed by) those two radii. SEMICIRCLE: An arc whose endpoints are the endpoints of a diameter of the circle. A semicircle measures 180°. SIMILAR POLYGONS: Polygons with the same shape; all their corresponding angles have the same measure. SLOPE-INTERCEPT FORM OF A LINE: The form y = mx + b where m is the slope of the line, and b is the y-intercept. SLOPE OF A LINE: On a graph, the ratio of the horizontal change in a line, calculated as the rise over the run. The change in y over the change in x. Slope = rise run . SQUARE: A quadrilateral in which all the angles are right angles, and all the sides are equal. STANDARD FORM FOR THE EQUATION OF A LINE: The form Ax + By = C, where A, B, and C are real numbers, and A and B are not both zero. TRANSLATION: Transforming each point in a plane by mapping it to a different location by sliding the image a specific distance and direction. TRAPEZOID: A quadrilateral with only one pair of opposite sides parallel. TRIANGLE: A three-sided plane figure (polygon). VERTEX: An endpoint of a side of a polygon. VOLUME: The measure of the interior of a solid; the number of unit cubes necessary to fill the interior of such a solid. X-AXIS: A horizontal line used to help locate points in a plane, also called the abscissa. X-COORDINATE: The first term of an ordered pair; it appears to the left of the comma and indicates movement along the x-axis. X-INTERCEPT: The point where a line crosses or intersects the x-axis. Y-AXIS: A vertical line used to help locate points in a plane, also called the ordinate. Y-COORDINATE: The second term of an ordered pair; it appears to the right of the comma and indicates movement along the y-axis. Y-INTERCEPT: The point where a line crosses or intersects the y-axis.
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Final Preparations Final Preparation and Sources Finishing Touches 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
Make sure that you are familiar with the areas covered on the test. Spend the last week of preparation on a general review of key concepts and strengthening weak areas. Don’t cram the night before the exam. It is a waste of time! Start off crisply, working the questions you know first, then going back and trying to answer the others. Try to eliminate one or more choices before you guess, but make sure that you fill in all the answers. There is no penalty for guessing! Underline key words in questions. Write out important information, and make notations on diagrams. Take advantage of being permitted to write in the test booklet. Make sure that you answer what is being asked and that your answer is reasonable. Cross out incorrect choices immediately: This can keep you from reconsidering a choice that you have already eliminated. Don’t get stuck on any one question. They are all of equal value. The key to getting a good score on CAHSEE Math is reviewing properly, practicing, and getting the questions right that you can and should get right. A careful review of Parts I and II of this book helps you focus during the final week before the exam.
Sources If you need additional review or practice, the following books can be very helpful: CliffsQuickReview Algebra I CliffsQuickReview Basic Math and Pre-Algebra Cliffs Math Review for Standardized Tests
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