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Authors Basich Whitney • Brown • Dawson • Gonsalves • Silbey • Vielhaber
Photo Credits coins: United States Mint, bills: Michael Houghton/StudiOhio; Cover Lori Adamski Peek/Getty Images; iv (tl bl br) File Photo, (tc tr) The McGraw-Hill Companies, (cl c) Doug Martin, (cr) Aaron Haupt; v (1 2 3 4 6 7 8 9 11 12) The McGraw-Hill Companies, (5 10 13 14) File Photo; vii Roy Ooms/Masterfile; viii Daryl Benson/ Masterfile; ix Jeremy Woodhouse/Masterfile; x Daryl Benson/Masterfile; 2–3 David Young-Wolff/PhotoEdit; 3 Michael Houghtory/StuiOhio; 11 Stockdisc/ Superstock; 19 Christine Balderas/iStockphoto; 45 Paul Whitten/Photo Researchers,Inc.; 54–55 Michael Newman/PhotoEdit; 60 David Young-Wolff/ PhotoEdit; 110 Suzannah Skelton/iStockphoto; 112 Jeff Greenberg/PhotoEdit; 117 Steve Cole/Getty Images; 119 GK Hart/Vikki Hart/Getty Images; 125 Comstock Images/Alamy; 126 StockTrek/Getty Images; 137 Getty Images
Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved. Except as permitted under the United States Copyright Act, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without prior permission of the publisher. Send all inquiries to: Glencoe/McGraw-Hill 8787 Orion Place Columbus, OH 43240-4027 ISBN: 978-0-07-878206 MHID: 0-07-878206-6 Printed in the United States of America. 1 2 3 4 5 6 7 8 9 10 055/027 16 15 14 13 12 11 10 09 08 07
California Math Triumphs Volume 2B
California Math Triumphs Volume 1 Place Value and Basic Number Skills 1A Chapter 1 Counting 1A Chapter 2 Place Value 1A Chapter 3 Addition and Subtraction 1B Chapter 4 Multiplication 1B Chapter 5 Division 1B Chapter 6 Integers Volume 2 Fractions and Decimals 2A Chapter 1 Parts of a Whole 2A Chapter 2 Equivalence of Fractions 2B Chapter 3 Operations with Fractions 2B Chapter 4 Positive and Negative Fractions and Decimals
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Volume 3 Ratios, Rates, and Percents 3A Chapter 1 Ratios and Rates 3A Chapter 2 Percents, Fractions, and Decimals 3B Chapter 3 Using Percents 3B Chapter 4 Rates and Proportional Reasoning Volume 4 The Core Processes of Mathematics 4A Chapter 1 Operations and Equality 4A Chapter 2 Math Fundamentals 4B Chapter 3 Math Expressions 4B Chapter 4 Linear Equations 4B Chapter 5 Inequalities Volume 5 Functions and Equations 5A Chapter 1 Patterns and Relationships 5A Chapter 2 Graphing 5B Chapter 3 Proportional Relationships 5B Chapter 4 The Relationship Between Graphs and Functions Volume 6 Measurement 6A Chapter 1 How Measurements Are Made 6A Chapter 2 Length and Area in the Real World 6B Chapter 3 Exact Measures in Geometry 6B Chapter 4 Angles and Circles iii
Authors and Consultants AUTHORS
Frances Basich Whitney
Kathleen M. Brown
Dixie Dawson
Project Director, Mathematics K–12 Santa Cruz County Office of Education Capitola, California
Math Curriculum Staff Developer Washington Middle School Long Beach, California
Math Curriculum Leader Long Beach Unified Long Beach, California
Philip Gonsalves
Robyn Silbey
Kathy Vielhaber
Mathematics Coordinator Alameda County Office of Education Hayward, California
Math Specialist Montgomery County Public Schools Gaithersburg, Maryland
Mathematics Consultant St. Louis, Missouri
Viken Hovsepian Professor of Mathematics Rio Hondo College Whittier, California
Dinah Zike Educational Consultant, Dinah-Might Activities, Inc. San Antonio, Texas
CONSULTANTS Assessment Donna M. Kopenski, Ed.D. Math Coordinator K–5 City Heights Educational Collaborative San Diego, California
iv
Instructional Planning and Support
ELL Support and Vocabulary
Beatrice Luchin
ReLeah Cossett Lent
Mathematics Consultant League City, Texas
Author/Educational Consultant Alford, Florida
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
CONTRIBUTING AUTHORS
California Advisory Board CALIFORNIA ADVISORY BOARD
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Glencoe wishes to thank the following professionals for their invaluable feedback during the development of the program. They reviewed the table of contents, the prototype of the Student Study Guide, the prototype of the Teacher Wraparound Edition, and the professional development plan.
Linda Anderson
Cheryl L. Avalos
Bonnie Awes
Kathleen M. Brown
4th/5th Grade Teacher Oliveira Elementary School, Fremont, California
Mathematics Consultant Retired Teacher Hacienda Heights, California
Teacher, 6th Grade Math Monroe Clark Middle School San Diego, California
Math Curriculum Staff Developer Washington Middle School Long Beach, California
Carol Cronk
Audrey M. Day
Jill Fetters
Grant A. Fraser, Ph.D.
Mathematics Program Specialist San Bernardino City Unified School District San Bernardino, California
Classroom Teacher Rosa Parks Elementary School San Diego, California
Math Teacher Tevis Jr. High School Bakersfield, California
Professor of Mathematics California State University, Los Angeles Los Angeles, California
Eric Kimmel
Donna M. Kopenski, Ed.D.
Michael A. Pease
Chuck Podhorsky, Ph.D.
Mathematics Department Chair Frontier High School Bakersfield, California
Math Coordinator K–5 City Heights Educational Collaborative San Diego, California
Instructional Math Coach Aspire Public Schools Oakland, California
Math Director City Heights Educational Collaborative San Diego, California
Arthur K. Wayman, Ph.D.
Frances Basich Whitney
Mario Borrayo
Melissa Bray
Professor Emeritus California State University, Long Beach Long Beach, California
Project Director, Mathematics K–12 Santa Cruz County Office of Education Capitola, CA
Teacher Rosa Parks Elementary San Diego, California
K–8 Math Resource Teacher Modesto City Schools Modesto, California
v
California Reviewers CALIFORNIA REVIEWERS Each California Reviewer reviewed at least two chapters of the Student Study Guides, providing feedback and suggestions for improving the effectiveness of the mathematics instruction. Melody McGuire
Math Teacher California College Preparatory Academy Oakland, California
6th and 7th Grade Math Teacher McKinleyville Middle School McKinleyville, California
Eppie Leamy Chung
Monica S. Patterson
Teacher Modesto City Schools Modesto, California
Educator Aspire Public Schools Modesto, California
Judy Descoteaux
Rechelle Pearlman
Mathematics Teacher Thornton Junior High School Fremont, California
4th Grade Teacher Wanda Hirsch Elementary School Tracy, California
Paul J. Fogarty
Armida Picon
Mathematics Lead Aspire Public Schools Modesto, California
5th Grade Teacher Mineral King School Visalia, California
Lisa Majarian
Anthony J. Solina
Classroom Teacher Cottonwood Creek Elementary Visalia, California
Lead Educator Aspire Public Schools Stockton, California
vi
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Bobbi Anne Barnowsky
Volume 2A
Fractions and Decimals
Chapter
Parts of a Whole
1
1-1 Parts of a Whole and Parts of a Set ................................4. 2NS4.0, 4NS1.5
1-2 Recognize, Name, and Compare Unit Fractions ........11 2NS4.1
Progress Check.................................................................18 1-3 Representing Fractions....................................................19 2NS4.3, 4NS1.7
Assessment Study Guide .....................................................................26 Chapter Test .....................................................................28
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Standards Practice ...................................................30
Bixby Creek Bridge on Highway 1, south of Carmel
Chapters 1 and 2 are contained in Volume 2A. Chapters 3 and 4 are contained in Volume 2B.
Standards Addressed in This Chapter 2NS4.0 Students understand that fractions and decimals may refer to parts of a set and parts of a whole. 2NS4.1
Recognize, name, and 1 1 compare unit fractions from ___ to __. 12 2 2NS4.3 Know that when all fractional parts are included, such as fourfourths, the result is equal to the whole and to one. 4NS1.5 Explain different interpretations of fractions, for example, parts of a whole, parts of a set, and division of whole numbers by whole numbers; explain equivalence of fractions (see Standard 4.0). 4NS1.7 Write the fraction represented by a drawing of parts of a figure; represent a given fraction by using drawings; and relate a fraction to a simple decimal on a number line.
vii
Contents Chapter
Equivalence of Fractions
2
Standards Addressed in This Chapter 2-1 Equivalent Fractions and Equivalent Forms of One ..............................................34 2NS4.3, 3NS3.1, 4NS1.5
2-2 Mixed Numbers and Improper Fractions....................41 2NS4.3, 4NS1.5, 5NS1.5
Progress Check 1 .............................................................50 2-3 Least Common Denominator and Greatest Common Factors ......................................51 4NS1.5
2-4 Compare and Order Fractions...................................... 59 3NS3.1, 6NS1.1
Progress Check 2 .............................................................68 2-5 Simplify Fractions ...........................................................69 3NS3.1, 4NS1.5
Assessment
Chapter Test .....................................................................82 Standards Practice ...................................................84 Alabama Hills, Owens Valley
viii
3NS3.1 Compare fractions represented by drawings or concrete materials to show equivalency and to add and subtract 1 simple fractions in context (e.g., __ of a 2 2 pizza is the same amount as __ of another 4 3 pizza that is the same size; show that __ is 8 1 larger than __). 4 4NS1.5 Explain different interpretations of fractions, for example, parts of a whole, parts of a set, and division of whole numbers by whole numbers; explain the equivalence of fractions (see Standard 4.0). 5NS1.5 Identify and represent on a number line decimals, fractions, mixed numbers, and positive and negative integers. 6NS1.1 Compare and order positive and negative fractions, decimals, and mixed numbers and place them on a number line.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Study Guide .....................................................................77
2NS4.3 Know that when all fractional parts are included, such as fourfourths, the result is equal to the whole and to one.
Contents Chapter
Operations with Fractions
3
3-1 Add Fractions with Like Denominators .......................4 3NS3.2, 6NS2.1
3-2 Subtract Fractions with Like Denominators ...............11 3NS3.2, 6NS2.1
Progress Check 1 .............................................................18 3-3 Multiply Fractions ...........................................................19 5NS2.0, 5NS2.5, 6NS2.1
3-4 Divide Fractions ............................................................. 25 5NS2.5, 6NS2.1
Progress Check 2 .............................................................32 3-5 Add Fractions with Unlike Denominators ..................33 3NS3.2, 5NS2.0, 6NS2.1
3-6 Subtract Fractions with Unlike Denominators ...........39 3NS3.2, 5NS2.0, 6NS2.1
Chapters 1 and 2 are contained in Volume 2A. Chapters 3 and 4 are contained in Volume 2B.
Standards Addressed in This Chapter 3NS3.2 Add and subtract simple 1 3 fractions (e.g., determine that __ + __ is the 8 8 1 same as __). 2 5NS2.0 Students perform calculations and solve problems involving addition, subtraction, and simple multiplication and division of fractions and decimals. 5NS2.5 Compute and perform simple multiplication and division of fractions and apply these procedures to solving problems. 6NS2.1 Solve problems involving addition, subtraction, multiplication, and division of positive fractions and explain why a particular operation was used for a given situation.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Progress Check 3 .............................................................45 Assessment Study Guide .....................................................................46 Chapter Test .....................................................................50
San Diego Harbor
Standards Practice ...................................................52
ix
Contents Chapter
Positive and Negative Fractions and Decimals
4
4-1 Introduction to Decimals ...............................................56 3NS3.4, 4NS1.6, 4NS1.7
4-2 Decimals and Money ......................................................63 2NS5.1, 2NS5.2
Progress Check 1 .............................................................72 4-3 Compare and Order Decimals ......................................73 5NS1.5, 6NS1.1
4-4 Compare and Order Fractions and Decimals ............ 81 5NS1.5, 6NS1.1, 4NS1.7
Progress Check 2 .............................................................88 4-5 Add Decimals ..................................................................89 4NS2.0, 5NS2.0, 5NS2.1, 7NS1.2
4-6 Subtract Decimals........................................................... 97 4NS2.0, 5NS2.0, 5NS2.1, 7NS1.2
Progress Check 3 ...........................................................104 5NS2.0, 5NS2.1, 7NS1.2
4-8 Divide Decimals ........................................................... 113 5NS2.0, 5NS2.1, 7NS1.2
Progress Check 4 ...........................................................120 4-9 Operations with Positive and Negative Numbers ............................................... 121 4NS1.8, 5NS2.1, 6NS2.3, 7NS1.2
Assessment Study Guide ...................................................................128 Chapter Test ...................................................................134 Standards Practice .................................................136 Antelope Valley
x
3NS3.4 Know and understand that fractions and decimals are two different representations of the 1 same concept (e.g., 50 cents is __ of a dollar, 2 3 __ 75 cents is of a dollar). 4 4NS1.6 Write tenths and hundredths in decimal and fraction notations and know the fraction and decimal equivalents for halves and fourths 3 1 2 (e.g., _ = 0.5 or 0.50; __ = 1_ = 1.75). 4 2 4 4NS1.7 Write the fraction represented by a drawing of parts of a figure; represent a given fraction by using drawings; and relate a fraction to a simple decimal on a number line. 2NS5.1 Solve problems using combinations of coins and bills. 2NS5.2 Know and use the decimal notation and the dollar and cent symbols for money. 5NS1.5 Identify and represent on a number line decimals, fractions, mixed numbers, and positive and negative integers. 6NS1.1 Compare and order positive and negative fractions, decimals, and mixed numbers and place them on a number line. 4NS2.0 Students extend their use and understanding of whole numbers to the addition and subtraction of simple decimals. 5NS2.0 Students perform calculations and solve problems involving addition, subtraction, and simple multiplication and division of fractions and decimals. 7NS1.2 Add, subtract, multiply, and divide rational numbers (integers, fractions, and terminating decimals) and take positive rational numbers to whole-number powers. 5NS2.1 Add, subtract, multiply, and divide with decimals; add with negative integers; subtract positive integers from negative integers; and verify the reasonableness of the results. 4NS1.8 Use concepts of negative numbers (e.g., on a number line, in counting, in temperature, in “owing”). 6NS2.3 Solve addition, subtraction, multiplication, and division problems, including those arising in concrete situations, that use positive and negative integers and combinations of these operations.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
4-7 Multiply Decimals.........................................................105
Standards Addressed in This Chapter
R E G N E V A SC HUNT Let’s Get Started Use the Scavenger Hunt below to learn where things are located in each chapter. 1 What is the title of Chapter 4?
Positive and Negative Fractions and Decimals 2
What is the Key Concept of Lesson 3-6?
Subtract Fractions with Unlike Denominators 3
What figure is used as a model in Example 1 on page 39?
circle 4
What are the vocabulary words for Lesson 4-5?
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
tenths, hundredths 5
How many Examples are presented in Lesson 3-4? 3
6
What are the California Standards covered in Lesson 4-6?
4NS2.0, 5NS2.0, 5NS2.1, 7NS1.2 7
What is the average temperature on Saturn as given on page 126? -218° F
8
What do you think is the purpose of Progress Check 2 on page 32? It allows you to practice problems from the
last two lessons. 9
On what pages will you find the Study Guide for Chapter 3?
pages 46–49 10
In Chapter 4, find the logo and Internet address that tells you where you can take the Online Readiness Quiz. It is
found on page 55. The URL is ca.mathtriumphs.com. 1
Chapter
3
Operations with Fractions You add, subtract, multiply, and divide fractions every day. You do this mostly with money. How many quarters are in $3? What is a dollar and a half plus a dollar and a half? If you and your friend must evenly split $5, how much do you each get?
Copyright © by The McGraw-Hill Companies, Inc.
2
Chapter 3 Operations with Fractions
David Young-Wolff/PhotoEdit
STEP
STEP
1 Quiz
Are you ready for Chapter 3? Take the Online Readiness Quiz at ca.mathtriumphs.com to find out.
2 Preview
Get ready for Chapter 3. Review these skills and compare them with what you’ll learn in this chapter.
What You Know
What You Will Learn
You add and subtract fractions when you count money.
Lessons 3-1 and 3-2 = __1 of 4
=
You know how to add and subtract. Examples:
1+3=4 5-4=1
Copyright © by The McGraw-Hill Companies, Inc.
TRY IT 1
2 + 12 =
14
2
4+3=
7
3
33 - 5 =
28
4
42 - 6 = 36
You know how to multiply. Examples:
4 × 6 = 24 5 × 7 = 35
TRY IT
36
5
6×6=
6
10 × 2 =
7
8×7=
56
8
3×3=
9
+
2 , or __ 1 , of = __ 2 4
+
+
3 = __ of 4
Lessons 3-1 and 3-2 To add fractions with the same denominators, add the numerators. 1+3 4 Example: _____ = __ 5 5
Numerator Denominator
To subtract fractions with the same denominators, subtract the numerators. 5-4 1 Example: _____ = __ 7 7
Numerator Denominator
Lessons 3-3 and 3-4 To multiply fractions, multiply the numerators and the denominators. × 6 ___ 6 4_____ 4 × __ __ = 24 = 5
7
5×7
35
Numerator Denominator
20
3 coins: United States Mint, bill: Michael Houghton/StudiOhio
Lesson
3-1 Add Fractions with Like Denominators KEY Concept To add fractions, the fractions must have common denominators. MJLF GSBDUJPOT
4VNPGUIFOVNFSBUPST PGUIFMJLFGSBDUJPOT
VOCABULARY
$PNNPOEFOPNJOBUPS
To add like fractions, add the numerators. The demoninators stay the same.
_
A quarter is one-fourth of a dollar.
common denominator
4
PS
Always write the sum of like fractions in simplest form. A fraction is in simplest form , or lowest terms, when the numerator and the denominator have no common factor other than 1. 4
Chapter 3 Operations with Fractions
United States Mint
Copyright © by The McGraw-Hill Companies, Inc.
like fractions fractions that have the same denominator 1 2 Example: __ and __ 5 5 simplest form (Lesson 2-5, p. 69) a fraction in which the numerator and the denominator have no common factor greater than 1 3 Example: __ is the 5 6 simplest form of ___. 10 mixed number a number that has a whole part and a fraction part 3 Example: 6 __ 4
1+_ 1 =_ 4 4 1+1 2 _ =_=_ =1 4 2 4 1+_ 1+_ 1 =_ 4 4 4 1+1+1 _ _ = =3 4 4 1 1 1+_ 1 _ _ = + +_ 4 4 4 4 1 + 1_ +1+1 = __ 4 4 , or 1 =_
common denominator the same denominator (bottom number) used in two or more fractions, also called like denominator 2 1 Example: __ and __ 4 4
⎧ ⎨ ⎩
You can use money to help you understand the rule for adding fractions. =1 4
3NS3.2 Add and subtract simple fractions. 5NS2.0 Students perform calculations and solve problems involving addition, subtraction, and simple multiplication and division of fractions and decimals. 6NS2.1 Solve problems involving addition, subtraction, multiplication, and division of positive fractions and explain why a particular operation was used for a given situation.
Example 1 Add
YOUR TURN!
_1 + _3 using a drawing. Write the sum
8 8 in simplest form.
Add
_1 + _2 using a drawing. Write the sum
6 6 in simplest form.
1. Draw a circle with 8 equal parts.
1. Draw a circle with
2. Color 1 part.
2. Color
1
part.
3. Color 3 parts.
3. Color
2
parts.
6
equal parts.
4. How many colored parts are there?
3
_1
5. Write the sum in simplest form.
2
4. Count the total number of colored parts. 5. Write the sum in simplest form. 5IF($'PGBOEJT 3FDBMMUIBUUIF($'JTUIFMBSHFTU OVNCFSUIBUEJWJEFTFWFOMZ JOUPUXPPSNPSFOVNCFST
@@@ @@@
@@@@@@@@@@ µ @@@ @@@@@@@@@@ @@@
Copyright © by The McGraw-Hill Companies, Inc.
µ
Example 2 Add
YOUR TURN!
_3 + _3. Write the sum in simplest form. 4
4
1. The common denominator is 4. 2. Add the numerators.
_ _
3 3 simplest form.
1. The common denominator is
5
3. Write in simplest form. ÷2 _ _6 = 6_ =3 4÷2
_5 + _2. Write the sum in 3
.
2. Add the numerators.
3+3 6 = 4 4
4
Add
2
+
2
7
______________ = ______
3
3
_7
3. What is the sum in simplest form?
3
GO ON Lesson 3-1 Add Fractions with Like Denominators
5
Example 3 Add
_1 + _4. Write the sum in simplest form. 5
5
1. 1. Mark off a segment length of __ 5 1 4 2. Add __ to the __ by extending your arrow. 5 5
5 5 __ ÷ __ = 1
3. Write the fraction in simplest form.
5
5
YOUR TURN! 2 1 Add + . Write the sum in simplest form. 6 6 2. 1. Mark off a segment length of __ 6 1 to the __ 2 by extending your arrow. 2. Add __ 6 6
_ _
3. Write the fraction in simplest form.
_1
2
Who is Correct? Add
_4 + _3. Write the sum in simplest form. 5
5
_ _ _ _
7 4+3 3 4 + = 5 + 5 = 10 5 5
Brighton
_7 _2 +3 _4 + _3 = 4_ = = 5 5 5 5
5
Circle correct answer(s). Cross out incorrect answer(s).
Guided Practice Add using drawings. Write each sum in simplest form. 1
6
5 6
4 = ______ 1 + __ __ 6
6
Chapter 3 Operations with Fractions
Charo
_7 +3 _4 + _3 = 4_ = 5 5 5
5
Copyright © by The McGraw-Hill Companies, Inc.
Mia
2
5 4
3 __ __ + 2 = ______ 4
4
@@
@@
Step by Step Practice 3
3 1 + ___ . Write the sum in simplest form. Find ___ 12 12 Step 1 Check to see if the denominators are the same or different. 1 3 ___ + ___ 12
like denominators
12
Step 2 Add the numerators.
1
+
3
______________
12
Step 3 Write the sum of the numerators over the like denominator.
1
+
3
4 12
______________ = ______
12
Copyright © by The McGraw-Hill Companies, Inc.
Step 4 Write the fraction in simplest terms. 4÷4 4 ___ = ______ 12
12 ÷ 4
1 3
= ______
Add. Write each sum in simplest form. 4
5
6
2
8 9
+ 6 __ __ + 2 = _____________ = ______ 9
9
12 ___ 18 3 + ___ 18 −−−−−
15 18
9
6
5 6
______ = ______
1 __
7
2 1 + __ 2 −−−−−
2 2
______ =
1
2 __ 3 2 + __ 3 −−−−−
4 3
______
GO ON Lesson 3-1 Add Fractions with Like Denominators
7
Step by Step Problem-Solving Practice
Problem-Solving Strategies ✓ Draw a diagram.
Solve. Write your answer in simplest form. 3 8 WEATHER Yesterday it rained __ of an inch. Today it 8 1 of an inch. How much did it rain altogether? rained __ 8
_3
Look for a pattern. Guess and check. Solve a simpler problem. Work backward.
Read the problem. Write what you know.
Understand
It rained
_1 8
8
of an inch yesterday. It rained
of an inch today.
Pick a strategy. One strategy is to draw a diagram.
Plan
Draw 8 equal parts. Shade 3 to show 3 the __ of an inch it rained yesterday. 8 1 of an inch it rained today. Shade 1 part to show __ 8
3 8
Write in simplest form.
4 4
1 2
✔
Understand
✔
Plan
✔
Solve
✔
Check
_2 3
7 of the shelves. BOOKS Ling has a bookcase. Science fiction takes up ___ 12 2 of the shelves, and comic books take up ___ 1 of Mysteries take up ___ 12 12 the shelves. What fraction of the bookcase is filled?
12
12
12
12
Chapter 3 Operations with Fractions
6
Copyright © by The McGraw-Hill Companies, Inc.
1 of a pizza. Colin ate __ 1 of the pizza, and Logan ate __ 6 6 2 of the pizza. What fraction of the pizza was eaten? Landon ate __ 6 Check off each step. FAMILY
10 = _ 5 _7 + _2 + _1 = _
8
4 8
÷ = _____________ = ______ ÷
Does the answer make sense? Review your drawing. Look at your drawing as two equal parts. Is one of those parts shaded?
Check
10
4 8
Count the shaded parts. ______ + ______ = ______
Solve
9
1 8
How is adding like fractions the same as adding whole numbers?
11
See TWE margin.
Skills, Concepts, and Problem Solving Add using drawings. Write each sum in simplest form. 4=1 3 1 2 1 12 __ + __ = 13 __ + __ = 8 8 5 5 8 2
_ _
14
_7
3 4 + __ __ = 6
Add. Write each sum in simplest form. 5 3 7 16 ___ + ___ = 12 12 6
5
_8
4 + __ 4= __
15
6 See TWE margin for drawings. 6
_3
5
5
5
_
Copyright © by The McGraw-Hill Companies, Inc.
17
3
9
18
3 __ 1 + __ __ +1=
20
5 1 + __ __ =
7
7
6
6
_1
1= 2 + __ __ 9
_5 7
7
1
19
5 __ 2= __ + 1 + __
21
3 __ 6 __ + =
9
7
9
_8 9
9
7
_9 7
Solve. Write your answer in simplest form. 22
NUTRITION Shontelle sliced some cheese into 8 slices. She ate 3 slices. Her sister ate 2 slices. What fraction of the slices was left?
_3
23
8
HOBBIES Jade has 15 miniature tea sets. Last year she added 4 sets to her collection. This year she added 3 sets. What fraction of the sets did she collect in the past two years?
_7 15
GO ON
Lesson 3-1 Add Fractions with Like Denominators
9
24
FOOD The school cafeteria serves five different types of fruit at lunch. What fraction represents the top three favorite fruits? Explain how to solve this problem.
'SVJU 4USBXCFSSJFT @@
_8 ; find the 3 largest fractions, then add the numerators
1MVNT @@
10 and keep the denominator 10.
Vocabulary Check Write the vocabulary word that completes each sentence. 25
"QQMFT @@
.
Fractions that have the same denominators have
common (or like) denominators 27
.FMPO @@
Fractions that have the same denominator are called
like fractions 26
0SBOHFT @@
.
Writing in Math Berta is mixing sand and water to make cement. She mixes 3 half-cups of water to 6 half-cups of sand. She wants to find the total number of cups she has mixed. Write a note to Berta. Explain how to solve this problem. Do not forget to explain to her how to simplify her answer.
See TWE margin.
Write each mixed number as an improper fraction. 28
31 6
1 = ______ 5__ 6
29
Write each improper fraction as a mixed number. 30
9 __ = 4
Solve. 32
1 2 ______ 4
32 3
2 = ______ 10__ 3
(Lesson 2-2, p. 41)
31
16 ___ = 3
1 5 ______ 3
(Lesson 2-2, p. 41)
FOOD Antwoine had a bag of oranges. Each orange had 8 wedges. He used 35 of the wedges for a fruit tray. Write an improper fraction and a mixed number for the amount of 35 oranges he used. 43
_
10
(Lesson 2-2, p. 41)
8
Chapter 3 Operations with Fractions
_ 8
Copyright © by The McGraw-Hill Companies, Inc.
Spiral Review
Lesson
3-2 Subtract Fractions with
3NS3.2 Add and subtract simple fractions. 5NS2.0 Students perform calculations and solve problems involving addition, subtraction, and simple multiplication and division of fractions and decimals. 6NS2.1 Solve problems involving addition, subtraction, multiplication, and division of positive fractions and explain why a particular operation was used for a given situation.
Like Denominators KEY Concept To subtract fractions with common denominators , subtract the numerators. The denominator remains the same. :PVTUBSUFE XJUIQJFDFT
:PVBUF QJFDFT
:PVIBWF QJFDFMFGUø
VOCABULARY common denominator the same denominator (bottom number) used in two or more fractions, also called like denominator 2 1 Example: __ and __ 4 4
9 9 @@
@@
@@
4
4
⎧ ⎨ ⎩
3-2 3 __ __ - 2 = _____ 4
common denominator (Lesson 3-1, p. 4)
1 = __ 4
like fractions fractions that have the same denominator 2 1 Example: __ and __ 5 5
Copyright © by The McGraw-Hill Companies, Inc.
Write the difference of like fractions in simplest form. Remember that a fraction is in simplest form or lowest terms when the numerator and the denominator have no common factor other than 1.
(Lesson 3-1, p. 4)
Example 1 Subtract
_5 - _3 using a drawing. Write the
8 8 difference in simplest form.
@@ @@
1. Draw 8 equal parts.
@@ @@
@@ @@
@@ @@
2. Color 5 parts. @@@@ @@@@
3. Cross out 3 of the 5 colored parts. 4. Count the number of shaded parts that are not crossed out. 5. Write the fraction in simplest form.
@@@@ @@@@@@@@@@@@ µ @@@@ @@@@@@@@@@@@ µ
5IF($'PG BOEJT
GO ON Lesson 3-2 Subtract Fractions with Like Denominators Stockdisc/SuperStock
11
YOUR TURN! 5 4 Subtract - using a drawing. Write the difference in 9 9 simplest form.
_ _
1. Draw
9
equal parts.
2. Color
5
parts.
3. Cross out
4
of the
9999 5
colored parts.
4. How many shaded parts are not crossed out? 5. Write the fraction in simplest form.
_1 9
Example 2 Subtract
1
YOUR TURN!
13 4 _ - _. Write the difference in
15 15 simplest form.
Subtract
11 7 _ - _. Write the difference in
16 16 simplest form.
16 .
1. The common denominator is 15.
1. The common denominator is
2. Subtract the numerators. 13 - 4 ___ 9 ______ = 15 15
2. Subtract the numerators. 11 - 7 ___ 4 _______ = 16 16 3. Write the fraction in simplest form.
_4 ÷ _4 = _1
3. Write the fraction in simplest form. 15 ÷ 3
15
16
5
4
4
Who is Correct? Subtract
9 3 _ - _. Write the difference in simplest form. 10
Gena
10
6 -3 =_ _9 - _3 = 9_ 10 10 10
10
Jolie
_ _ _ _ _
3 6 9-3 3 9 = 10 = 10 = 5 10 10
Circle correct answer(s). Cross out incorrect answer(s).
12
Chapter 3 Operations with Fractions
Theo
_3 6 _9 - _3 = _ = 5 10 10
10
Copyright © by The McGraw-Hill Companies, Inc.
9÷3 3 9 ___ = ______ = __
Guided Practice Subtract using drawings. Write each difference in simplest form.
2 6
1 3
1
3 5 __ __ - = ______ = ______
2
3 __ __ -2=
6 4
6
_1
9 99
4
4
Step by Step Practice 3
3 15 Find ___ - ___ . Write the difference in simplest form. 20 20 Step 1 Are the denominators the same? yes
12 20
12 20
12 ÷ 4 20 ÷ 4
3 5
______ = _____________ = ______
Step 3 Write the fraction in simplest form. Copyright © by The McGraw-Hill Companies, Inc.
15 - 3 20
_____________ = ______
Step 2 Find the difference of the numerators. Use the like denominator.
Subtract. Write each difference in simplest form. 4
5
5
-
2
3 9
÷
3 3
1 3
5 __ __ - 2 = _____________ = _____________ = ______ 9
9
12 ___ 16 9 - ___ 16 −−−−−
3 16
______
9
6
÷
6 __ 7 5 - __ 7 −−−−−
1 7
______
7
17 ___ 24 11 - ___
24 −−−−−
6 24
1 4
______ = ______
GO ON Lesson 3-2 Subtract Fractions with Like Denominators
13
Step by Step Problem-Solving Practice
Problem-Solving Strategies ✓ Draw a diagram.
Solve. Write in simplest form. 8
Understand
Read the problem. Write what you know. There are 8 key chains. chains are blue. Tory gave
2
4
of the key
of the blue key chains away.
Pick a strategy. One strategy is to draw a diagram. Draw eight equal parts.
Plan
Shade four to represent the 4 blue key chains. Cross out two of the shaded areas to show the 2 key chains that Tory gave away.
Solve
4 8
2 8
Count the unmarked shaded parts.
2 8
2 2
Write in simplest form.
Does the answer make sense? Look over your solution. Did you answer the question?
Check
GAMES Elan designs computer games. Of the games he designed, 3 2 are travel games and __ __ are racing games. What fraction shows 6 6 1 the number of games that are not travel or racing games?
_
Check off each step.
14
✔
Understand
✔
Plan
✔
Solve
✔
Check
Chapter 3 Operations with Fractions
6
Copyright © by The McGraw-Hill Companies, Inc.
1 4
÷ = _____________ = ______ ÷
99
2 8
______ - ______ = ______
9
Look for a pattern. Guess and check. Work backward.
HOBBIES Tory made 8 key chains. Four of the key chains were blue. She gave 2 of the blue key chains away. What fraction of the total number of key chains did she give away?
10
9 FITNESS Florence lives ___ of a mile from school. On Tuesday she 10 4 of a mile toward school. How much farther did she have jogged ___ 10 to go to get to school?
_5 = _1 of a mile 10
2
How is subtracting like fractions the same as subtracting whole numbers?
11
See TWE margin.
Skills, Concepts, and Problem Solving Subtract using drawings. Write each difference in simplest form. 2=1 6 7 5 2 12 __ - __ = 13 ___ - ___ = 8 8 11 11 8 4
_ _
9 9
99 999 Copyright © by The McGraw-Hill Companies, Inc.
14
3 5 __ __ - = 9
9
_2
15
9
1 = 7 - ___ ___ 10
10
_4 11
_6 = _3 5
10
See TWE margin for drawings. Subtract. Write each difference in simplest form. 2 4 2 7 2 16 __ - __ = 17 ___ - ___ = 5 5 13 13 5
_
18
20
5 7 - ___ ___ = 12
12
11 ___ 16 10 - ___ 16 −−−−−
_1 16
_1 6
19
21
9 4 = ___ - ___ 10
10
_5 13
_1 2
19 ___ 36 7 - ___ 36 −−−−−
_1 3
GO ON
Lesson 3-2 Subtract Fractions with Like Denominators
15
Solve. Write in simplest form. 22
3 FOOD At a Thai restaurant, __ of the meals are very spicy, 6 2 __ and of the meals are somewhat spicy. The rest of the meals 6 1 served are mild. What fraction of the meals is mild?
_ 6
23
4 of the game. Edna pitches __ 2 of the SOFTBALL Pilar pitches __ 9 9 3=1 game. What fraction of the game is left?
_ _ 9
24
3
2 bass MUSIC The school band’s percussion section has ___ 12 5 1 cymbals. What fraction of the drums, ___ snare drums, and ___ 12 12 percussion section plays an instrument other than these three instruments?
_1 3
@@
@@
@@
@@
Vocabulary Check Write the vocabulary word that completes each sentence. 25
27
simplest form A fraction is in when its numerator and denominator have no common factor other than 1. Writing in Math Miriam baked a pie that she cut into 8 equal 1 of the pie was gone. Make a drawing slices. The next morning, __ 4 to show how much pie is left. Then write an equation to represent this situation.
_ _
Answers may vary. Sample answer: The entire pie was 8 or 4 . 8 4 8 - 2 = 6 which in simplest form is 3 . 4 - 1 = 3 . 4 4 4 4 8 8 8
_ _ _
16
Chapter 3 Operations with Fractions
_ _ _ _
Copyright © by The McGraw-Hill Companies, Inc.
26
When you subtract like fractions, you find the difference of the numerators and write it over the common denominator .
Spiral Review Add. Write each sum in simplest form. 4 ___ 28 15 ___ + 1 15 −−−−−
5 15
(Lesson 3-1, p. 4)
7 −−−−−
1 3
7 7
______ =
3 __
31
8 1 + __ 8 −−−−−
4 8
1 2
1
5 __ 9 2 + __ 9 −−−−−
7 9
______ = ______
______
Complete to name an equivalent fraction.
Copyright © by The McGraw-Hill Companies, Inc.
7 2 + __
______ = ______
30
5 __
29
3
(Lesson 2-1, p. 34)
32
12 = ______ ___
33
15 1 = ______ __
34
14 2 = ______ __
35
18 = ______ ___
36
9 ___ = ______
37
6 24 = ______ ___
38
20
7
21
5
49
3 7
2
30
44
30
3 5
11
BASEBALL Dan got a hit 9 of the 11 times he batted in the softball tournament. Casey got a hit 18 of the 22 times he batted in the tournament. Did the boys have the same batting records? Explain.
_
Yes. Dan’s batting record was 9 . Casey’s batting record 11 18 9 was , which is in simplest form. 11 22
_
_
Lesson 3-2 Subtract Fractions with Like Denominators
17
Chapter
Progress Check 1
3
(Lessons 3-1 and 3-2)
Add. Write each sum in simplest form. 3NS3.2 7 3 4 1 4 1 __ + __ = 2 __ + __ = 5 5 5 3 3
_
4
10 = 2 _
6 4 + __ __ = 5
5
5
5
1= 7 + __ __ 9
9
_5
3
8 6 __ __ + =
_8
6
5 2 + ___ ___ =
_
9
6 __ __ -2=
_4 = _1
12
12 - ___ 7 = ___
3
9
Subtract. Write each difference in simplest form. 3NS3.2 5 2 3 8 2 4 7 ___ - ___ = 8 ___ - ___ = 11 11 15 15 11 15
_
10
5 1 = ___ - ___ 12
12
_4 = _1 12
3
11
5 9 ___ - ___ = 20
20
20
5
7
11
8
15
7
14 = 2 _ 7
_7 11
11
8
15
_4 = _1 8
2
_5 = _1 15
3
Solve. 5NS2.0, 6NS2.1 13
HOBBIES Last week Ken added 3 stamps to his collection. This week he added 5 stamps to his collection. Now Ken has 25 collector stamps. What fraction of the stamps did he collect in the past two weeks?
25
14
ART Amado is creating a painting. He has the tubes of paint shown at the right. How much more red paint does he have than purple paint?
_1 of a tube 4
15
SCHOOL Marika has 20 bonus points in math class. Yesterday she earned 2 of those points. Today she earned 5 bonus points. What fraction of her bonus points did she earn in the last two days?
_7 20
16
WORK I have 10 projects that I must complete this week. If I complete 4 projects by Wednesday, what fraction must I do before the end of the week?
_6 = _3 10
18
5
Chapter 3 Operations with Fractions
GVMM
GVMM
Copyright © by The McGraw-Hill Companies, Inc.
_8
Lesson
3-3 Multiply Fractions
5NS2.0 Students perform calculations and solve problems involving addition, subtraction, and simple multiplication and division of fractions and decimals. 5NS2.5 Compute and perform simple multiplication and division of fractions and apply these procedures to solving problems. 6NS2.1 Solve problems involving addition, subtraction, multiplication and division of positive fractions and explain why a particular operation was used for a given situation.
KEY Concept 1 of a pie. You eat half of that __ 1 slice. You have __ 4 4
VOCABULARY 1 of __ 1 of the original 1 of one pie is the same as eating __ Eating __ 2 8 4 1 · __ 1 = __ 1. pie. So, __ 2 4 8 To multiply two fractions, multiply the numerators and multiply the denominators.
factor a number that divides into a whole number evenly; also a number that is multiplied by another number Example: 3 and 5 are factors of 15.
The word of means to multiply in these problems. Notice 1 of __ 1 is the same as multiplying __ 1 and __ 1. that __ 2 4 2 4
Copyright © by The McGraw-Hill Companies, Inc.
Example 1 Find
_1 · _1 using a drawing.
4 3 1. Divide a figure into thirds. 1. Shade __ 3 2. Now cut each third into fourths. Each third is cut into fourths. 3. Shade in one of the fourths. 1 of __ 1 = ___ 1 __ 4
3
12
4. Name the fraction that represents the area 1 1 and the __ 1 models. ___ common to both the __ 3 4 12
YOUR TURN! Find
_1 · _1 using a drawing.
2 5 1. Divide a figure into halves . 1 . Shade
_ 2
2. Now cut each fifths half into 3. Shade in
1
.
of the fifths.
4. Name the shaded fraction.
_1 10
GO ON
Lesson 3-3 Multiply Fractions iStockphoto/Christine Balderas
19
Example 2 Find
YOUR TURN!
_2 · _9. Write the product in
3 8 simplest form.
_ _ _ _
1. Multiply the numerators. 2 · 9 = 2 · 9 3 8 3·8 Multiply the = 18 denominators. 24 2. Write the fraction in simplest form. ÷6 3 18 18 ___ = ______ = __ 24 24 ÷ 6 4
The GCF of 18 and 24 is 6.
2 _3 · _ . Write the product in
4 15 simplest form.
1. Multiply the numerators. Multiply the denominators.
6 3·2 =_ _ 4 · 15 = 60 2. Write the product. 3. Write the fraction in simplest form.
Example 3 Find
Find
_6
60 .
_1 10
YOUR TURN!
_2 · _9. Factor first. Write the product
Find
10 _3 · _ . Factor first. Write the
3 8 in simplest form.
5 18 product in simplest form.
1. Factor the numerators and denominators. 2 ·9 = 2 · 3·3 3 2·2·2 3 8
1. Factor the numerators and denominators.
_ _ _ _
2. Cross out all equivalents of one. / 2 · 3 · 3/ __
=
3 _ _3 =
2 / · 2 · 2 · 3/
2·2 4 3. Multiply. Write in simplest form.
_3
2. Cross out all equivalents of one. 3. Multiply. Write in simplest form.
_1
4
3
Who is Correct? Find
_2 · _3. Write the product in simplest form. 5
5
Liang 6 _2 · _3 = _ 25 5
5
Romy _2 · _3 = _65 = 1_15 5
5
Blythe 2 6 _2 · _3 = _ =_ 10 5 5
Circle correct answer(s). Cross out incorrect answer(s). 20
Chapter 3 Operations with Fractions
5
Copyright © by The McGraw-Hill Companies, Inc.
=
2 · 5 _3 · _________________ 5 3 · 2 · 3 3 · 2 · 5 = _______________________ 5 · 3 · 2 · 3
Guided Practice Multiply using drawings. Write each product in simplest form. 1
2
1 2 2 4 1 1 = ______ 1 of __ __ 2 3 6 1 = ______ 1 · __ __
Step by Step Practice 3
3 6 Find ___ · __. Write the product in simplest form. 10 7 Step 1: Factor the numerators. Factor the denominators.
3
2
· 3 __ 6 3 ___ · = ____________ · ____________
5
10 7
·
3
2
7
2
3· · = ______________ · ·7
Copyright © by The McGraw-Hill Companies, Inc.
5
2
2 2
Step 2: Cross out all equivalent forms of one.
·3·3 = ____________ ·5·7
Step 3: Multiply the remaining numerators and denominators. Write the fraction in simplest form.
=
9 _ 35
Multiply. Write each product in simplest form. 4
3 2
·
2 2
·
2 2
·
2 3
1 1
3 __ 8 __ · = ______________________ = ______ = 8 3
·
·
2 18
1 9
6
5 ______ ______ 4 · __ __ = =
6 24
1 4
8
3 ______ ______ 1 · __ __ = =
5
2 = ______ = ______ 1 · __ __
7
3 __ __ · 2 = ______ = ______
3 6
8 3
·
1
5 6
9 4
20 30
2 3
3 36
1 12
GO ON
Lesson 3-3 Multiply Fractions
21
Step by Step Problem-Solving Practice
Problem-Solving Strategies ✓ Draw a diagram.
Solve. 9
Use logical reasoning. Solve a simpler problem. Work backward.
HOBBIES Eleanor bought a sheet of poster board. She 2 of the poster board for a project. needs one-half of __ 3 How much will she have left? Understand
_1
_2
Read the problem. Write what you know. She will use 2 of 3 of the poster board. Subtract what she uses from 1 to find the amount that will be left.
Plan
Pick a strategy. One strategy is to draw a diagram. You need to find how much she will use in order to figure out what will be left.
Solve
Draw a picture and divide it into thirds. Then divide the thirds in half and mark out one-half of the shaded section.
2 3 6 2 ÷ 2 1 = _____________ = ______ 6 ÷ 2 3
1 · __ 2 = ______ __ 2
How much poster board did Eleanor use? How much will she have left?
_
_1 3
_
Eleanor will use 1 of her poster board and have 2 left. 3 3 ⎛3 ⎞ 1 2 ⎪ - = ⎥ 3 3⎠ ⎝3
_ _ _
Check
22
Does the answer make sense? Look over your solution. Did you answer the question?
Chapter 3 Operations with Fractions
Copyright © by The McGraw-Hill Companies, Inc.
One of those halves 1 2 represents __ of __. 2 3
10
SCHOOL Mr. Ortega had 27 students in his algebra class. On the 8 last exam, __ of his students received a passing grade. How many 9 students passed the exam? 24 How many students did not pass the exam?
11
3
✔
Understand
✔
Plan
✔
Solve
✔
Check
Check off each step.
2 of an hour. MUSIC Keith practices the saxophone every day for __ 3 How many hours does he practice each week? 4 2 hours
_ 3
12
Suppose you factor the numerators and denominators first and then cancel equivalents of one. What step should you not have to do after you multiply?
See TWE margin.
Skills, Concepts, and Problem Solving Copyright © by The McGraw-Hill Companies, Inc.
Multiply using drawings. Write each product in simplest form.
6 12
1 2
13
3 ______ ______ 2 · __ __ = =
14
3 ______ 1 · __ __ =
3 4
8 4
3 32
Multiply. Write each product in simplest form. 15
9 2 · ___ __ =
18
3 1 · __ 2 · __ __ =
3 20
2 3 4
_3 10
_1 4
16
2 · ___ 21 = __
_1
17
3 2 · __ 2 · __ __ =
_1
19
2 · ___ 12 = ___
_3
20
8 __ 3 1 __ · · __ =
1
7 18
16 10
3
20
3 4 5
3 4 2
5
GO ON Lesson 3-3 Multiply Fractions
23
Solve. Write your answer in simplest form. 21
BAKING Hilda baked a cake. She served half of it and left the rest on the table. Lorenzo ate half of the cake on the table. Then Savannah ate half of the cake she found on the table. What fraction 1 of the whole cake was left after Savannah ate her portion?
_
22
8 2 of the bag Careta bought a bag of oranges. She kept __ for 3 herself. She gave the rest of the oranges to 3 friends. If each of her friends received 2 oranges, how many oranges did Careta keep?
RETAIL
12
23
A pizza parlor has an all-you-can-eat lunch buffet. 5 7 of a mushroom pizza when There were __ of a sausage pizza and __ 8 8 2 of each amount Rogé and Cedrick arrived. Each of them ate __ 3 left. What fraction of a pizza did the boys eat? FOOD
12 or 1 pizza _ 12
Vocabulary Check Write the vocabulary word that completes each sentence.
factor
24
A(n)
25
A number that is multiplied by another number is called a(n) factor .
26
of a number divides that number evenly.
3 by the Commutative 3 of ___ 21 × ___ 21 is the same as ___ The first friend is correct. ___ 14 42 42 14 Property of Multiplication.
Spiral Review
_
_
Subtract. Write each difference in simplest form. (Lesson 3-2, p. 11) 1 1 8 3 2 1 27 ___ - ___ = 28 __ - __ = 7 7 7 14 14 2 Use < , =, or > to compare the fractions. 30
1 __
24
Chapter 3 Operations with Fractions
3
> __2 7
31
4 __ 9
29
4 __ __ -2=
32
6 ___
5
5
(Lesson 2-4, p. 59)
< __1 2
11
< __2 3
_2 5
Copyright © by The McGraw-Hill Companies, Inc.
3 21 . One friend says to Writing in Math You want to find ___ of ___ 14 42 3 3 3 21 × ___ multiply ___ . Another friend says to multiply ___ × ___. 42 14 14 14 A third friend says either equation will work. Which friend is correct? Explain why.
Lesson
3-4 Divide Fractions
5NS2.5 Compute and perform simple multiplication and division of fractions and apply these procedures to solving problems. 6NS2.1 Solve problems involving addition, subtraction, multiplication, and division of positive fractions and explain why a particular operation was used for a given situation.
KEY Concept The product of a fraction and its reciprocal is 1. 3 8 3 8 For example, the reciprocal of __ is __ , because __ · __ = 1. 8 3 8 3 3 __ 8 3____ · 8 ___ 24 __ · = = =1 8 3 8 · 3 24 Notice that the reciprocal of a fraction is the fraction obtained by switching the numerator and denominator of the original fraction. A diagram can help you to 3 1 determine the quotient of __ ÷ __ . 4 8 To divide by a fraction, multiply by the reciprocal of the divisor. Notice that the result is the same as that in the diagram above. EJWJEFOE
@@
@@
VOCABULARY reciprocals two numbers whose product is 1 Example: The reciprocal 3 5 of __ is __ because 5 3 3 __ 5 __ · = 1. 5 3
5IFSFBSF @@ FJHIUITJO µ @@ 4P @@
.VMUJQMZCZ UIFSFDJQSPDBMPG
Copyright © by The McGraw-Hill Companies, Inc.
EJWJTPS
To divide fractions, change the operation to multiplication, and use the reciprocal of the divisor.
Find the reciprocal of
YOUR TURN!
_2. 3
1. Write the reciprocal. 3 2 2. Check your answer by multiplying the two fractions. 2·3=2·3 =1 3·2 3 2 3 2 is __ The reciprocal of __ . 3 2
_
5 Find the reciprocal of . 7
7 5
1. Write the reciprocal. ______
_
_ _ _
dividend a number that is being divided divisor the number by which another number is divided divisor ↑ Example: 6 ÷ 3 = 2 ↓ ↓ dividend quotient
µ u
Example 1
quotient the answer to a division problem
2. Check your answer by multiplying the two
5 7
7 5
5 7
7 5
· fractions. ______ · ______ = ____________ = 1 ·
7 5
5 3. The reciprocal of __ is ______ . 7
GO ON
Lesson 3-4 Divide Fractions
25
Example 2 1 1 Find _ ÷ _ using diagrams. 2
6 1. 1. Model __ 2 2. Look at the denominator of the divisor. You need to find how many sixths are 1 . Change the model to have 6 sections. in __ 2
YOUR TURN! 3 1 using diagrams. Find ÷ 4 12 3 1. Model __. 4
_ _
2. Look at the denominator of the divisor. You need to find how many twelfths are 3 in __. Change the model to have that 4 many sections.
@@
3. There are 3 sixths in _1_.
@@
_3_ = _1_
2 6 4. Write the division sentence. _1_ ÷ _1_ = 3 2 6
2
Example 3 Find
_1 ÷ _1.
_
_1 ÷ _1 3
2
_1 · _2
3 1 3. Multiply. Write in simplest form. 1·2 =2 3 3·1
_ _
26
Chapter 3 Operations with Fractions
9
YOUR TURN! Find
3 _2 ÷ _ .
10 3
5 10 1. Write the reciprocal of the divisor. ______
2. Multiply by the reciprocal of the divisor.
2 5
10 3
______ · ______
3. Multiply. Write in simplest form.
2 · 10 5 · 3 20 4 = ______ = ______ 15 3 ____________
Copyright © by The McGraw-Hill Companies, Inc.
3 2 1. 1. Write the reciprocal of the divisor __ 2 2 1 2. Multiply by the reciprocal of the divisor.
3 3. How many twelfths are in __? 4 4. Write the division sentence. 3 ___ __ ÷ 1 = 9 4 12
Who is Correct? Find
_3 ÷ _1. 8
4
Kazuo
Paco
_3 ÷ _1 = _38 · _41 4 8 ÷4 _ 12 = 12 _ = 4 ÷ 8 8 _3 =
_3 ÷ _1 = _38 · _14 4 8 _3 =
Ted
_3 ÷ _1 = _38 · _41 = 4 3 · 2⁄ · 2⁄ 2 · 2⁄ · 2⁄ 3 = 2
8
_
2
_
2
Circle correct answer(s). Cross out incorrect answer(s).
Guided Practice Divide using diagrams. 2 1 4 1 __ ÷ __ = 3 6
Copyright © by The McGraw-Hill Companies, Inc.
2
1= 2 ÷ __ __ 3
9
6
Find the reciprocal of each number. 3
1 __
6
4 __
2
7
_2 1
4
3 __
_7 4
7
1 __
_4
5
5 __
_8 = 8
8
9 ___
4 8 1
3
6
12
_6 5
12 _ 9
GO ON Lesson 3-4 Divide Fractions
27
Step by Step Practice 9
_ _
Find 2 ÷ 2 . Write the quotient in simplest form. 5 3 Step 1 Write the reciprocal of the divisor.
3 2
2 . Its reciprocal is ______. The divisor is __ 3 2 2 ÷ __ __
Step 2 Multiply by the reciprocal of the divisor.
5
3
3 2
2 · ______ = __ 5
Step 3 Multiply the fractions.
3 2
2· = ________ 5·
6 10
3 5
Step 4 Write your answer in simplest form. = ______ = ______
6 7
2 1
11
1 3 ÷ __ __ =
12 _
_3
7
4
7
12 7
· = ____________ = ______ ·
12
5 1 ÷ __ __ =
_3
13
8 2 ÷ __ __ =
14
1 ÷ __ 7= __
_2
15
6 2 ÷ ___ __ =
28
Chapter 3 Operations with Fractions
2
4
6
8
5
7
3
5
9
10
4
_2 3
Copyright © by The McGraw-Hill Companies, Inc.
Divide. Write each quotient in simplest form. 6 __ __ 10 ÷1 7 2
Step by Step Problem-Solving Practice
Problem-Solving Strategies ✓ Draw a diagram.
Solve. 16
Use logical reasoning. Solve a simpler problem. Work backward. Act it out.
INDUSTRY Rolls of metal are cut into sheets. If a 5 factory worker needs __ of a roll to cut 6 equal sheets, 8 what part of the roll will each sheet be? Understand
Read the problem. Write what you know.
Six
sheets are cut from
_5 8
of a roll.
Pick a strategy. One strategy is to draw a diagram. 5 To divide __ into 6 equal parts, draw a rectangle. 8 Divide it vertically into 8 equal parts. Shade 5 parts. There are 6 equal Then divide it horizontally into 6 equal parts.
Plan
parts that make up each eighth. Each one 1 is equal to ___. 48
Count the shaded parts that make up each sixth.
Solve
5 6 8 1 5 1 5 = ______ · ______ = ______ 8 6 48
5 __ ÷ 6 = ______ ÷ ______
Copyright © by The McGraw-Hill Companies, Inc.
8
The answer has to be less than the dividend. Is your 5 answer less than __? Did you answer the question? 8
Check
17
5 48
Each sixth is ______.
_
CELEBRATIONS For Molly’s birthday party, she has 4 cups of potato 3 62 chips. If one serving is __ of a cup, how many servings are there? 5 3 Check off each step.
✔
Understand
✔
Plan
✔
Solve
✔
Check GO ON Lesson 3-4 Divide Fractions
29
18
ART Saheel and Fidel attended an art exhibit. They saw a painting that was created from a series of dots. Together they 1 of the painting. counted 240 dots in an area that was ___ 36
8,640 dots
About how many dots cover the entire painting?
Write a note to an absent student explaining how to divide fractions. Be sure to use the word reciprocal.
19
See TWE margin.
Skills, Concepts, and Problem Solving Divide using diagrams. Write each quotient in simplest form. 20
1= 1 ÷ __ __ 3
9
_9 = 3 3
_4 = 2
21
1= 1 ÷ __ __
23
5 ______ 4 ÷ __ __ =
2
4
2
Divide. Write each quotient in simplest form. 22
26
25
27
24 5 6 25 30 = ______ 5 12 = ______ 2 ÷ ___ __ 3 15 36 6 3 1 3 __ ÷ 15 = ______ = ______ 5 75 25
Solve. Write the answer in simplest form. 28
2 of a foot long. He has CARPENTRY Aaron has a board that is __ 3 to cut 4 equal pieces of wood from this piece. What will be the length of each piece?
29
30
_1 of a foot long 6
1 of a FOOD The cafeteria baked 23 pies. A serving is __ 8 184 pie. How many servings can they get from 23 pies? Chapter 3 Operations with Fractions
Copyright © by The McGraw-Hill Companies, Inc.
24
18 7 9 35 24 8 9 2 ÷ ___ __ = ______ = ______ 3 12 27 9 3 1 3 __ ÷ 12 = ______ = ______ 4 48 16 5 ______ 2 ÷ __ __ =
30
3 DINING You and a friend are trying to split up __ of a pizza evenly. 4 How much of the pizza do you each get? Use a diagram to justify your answer.
_3 8
Vocabulary Check Write the vocabulary word that completes each sentence.
divisor
31
In the division problem 24 ÷ 8 = 3, the
32
A fraction made from another fraction by switching the numerator reciprocal and denominator is the of that fraction.
33
The
34
1. Writing in Math Explain how you would divide 5 by __ 4
quotient
is 8.
is the answer to a division problem.
_
__ _
Sample answer: To divide 5 by 1 , multiply 5 by the reciprocal of 1 . 5 × 4 = 20. 4 4 1 1
Spiral Review Multiply. Write each product in simplest form.
Copyright © by The McGraw-Hill Companies, Inc.
35
38
3 3 20 10 1 3 ______ 1 · __ 2 · __ __ = 2 3 4 4 9 2 · ___ __ = ______
36
39
(Lesson 3-3, p. 19)
1 7 18 3 3 12 = ______ 2 · ___ ___ 16 10 20 21 = ______ 2 · ___ __
Subtract. Write each difference in simplest form. 41
4 - __ 2= __ 5
Solve. 43
5
_2
40
(Lesson 3-2, p. 11)
42
5
37
7 - ___ 2 = ___ 13
13
1 3 4 5 5 3 3 1 ______ 8 __ ___ · · __ = 11 4 2 11 3 ______ 2 · __ 2 · __ __ =
_5 13
(Lesson 2-4, p. 59)
ENTERTAINMENT Anita and Fina are sorting cable wires for their 5 stereo devices. They have three packages of cable with __ of a yard, 6 11 of a yard, and __ 5 of a yard of cable. Which package has the ___ 8 12 longest piece of cable? Which has the shortest piece?
5 is the shortest. 11 of a yard is the longest piece. _ _ 12
8
Lesson 3-4 Divide Fractions
31
Chapter
Progress Check 2
3
(Lessons 3-3 and 3-4)
Multiply. Write each product in simplest form. 5NS2.0 1
5 1 · __ __ =
3
9 ___ ___ · 4 =
5
8 __ 1= __ · 7 · __
3 8
11 18
_5 24
_2 11
_4 27
9 6 7
_6
2
3 2 · __ __ =
4
6 8 ___ ___ · =
6
13 ___ 3 ___ __ · · 1 =
35
5 7
15 16
4
6
_1 5
_1 8
13
Divide. Write each quotient in simplest form. 5NS2.0 7
5 3 __ __ ÷ =
9
10 5 ___ ÷ ___ =
11
2÷8= __
8
4
_6 5
_1 8
3
12
5
_1 20
8
2 ÷ __ 1= __
10
14 = 7 ÷ ___ ___
12
5 __ ÷ 20 =
3
18
9
6
4
_1 12
3
_1 36
Donna cut?
_3 10
14
1 of the recipe shown. How BAKING Alonso wants to make only __ 2 much sugar will he need?
_1 cup 16
32
Chapter 3 Operations with Fractions
CUPOFMILK CUPOFSUGAR STICKOFBUTTER .IXALLINGREDIENTS
Copyright © by The McGraw-Hill Companies, Inc.
Solve. Write the answer in simplest form. 5NS2.5, 6NS2.1 3 1 13 NATURE In a rose garden, __ of the roses are red. Donna cut __ of 5 2 them for a flower arrangement. What fraction of the red roses did
Lesson
3-5 Add Fractions with Unlike Denominators KEY Concept To add unlike fractions, you have to rename the fractions so that they have like denominators. Use equivalent forms of 1 to write the fractions so that each fraction has the same least common denominator (LCD). Ways to Find the LCD for 4 + 1 6 8 1. Find a common multiple of both denominators.
_ _
×3 × 4 + 1_____ 4 + __ 1 = 4_____ __ 6
6×4
8
8×3
16 3 = ___ + ___ 24 24 19 = ___ 24
A common multiple of 6 and 8 is 24.
Copyright © by The McGraw-Hill Companies, Inc.
2. Use the prime factorization of each denominator. 1 = _____ 1 4 + __ 4 + _________ __ 6 8 2×3 2×2×2 1×3 4 × 2 × 2 + ____________ = ____________ 2×2×2×3 2×2×2×3 3 19 16 × ___ = ___ = ___ 24 24 24 Sometimes you must rename both fractions in a problem. Only rename one fraction if one of the denominators is the LCD.
Example 1 Find
3 6 a model.
_ _ _
Find
unlike fractions fractions with different denominators 2 1 Example: __ and __ 5 3 least common denominator (LCD) the least common multiple of the denominators of two or more fractions, used as a denominator Example: The LCD of 1 1 __ and __ is 6. 2 3 prime factorization a way of expressing a composite number as a product of its prime factors
@@
@@
@@
_1 + _2 using
5 2 a model.
2. Divide the circle with the thirds into sixths to create like denominators. 3. Add the fractions. 5 4 1 + = 6 6 6
VOCABULARY
YOUR TURN!
_2 + _1 using
1. Model each fraction.
3NS3.2 Add and subtract simple fractions. 5NS2.0 Students perform calculations and solve problems involving addition, subtraction, and simple multiplication and division of fractions and decimals. 6NS2.1 Solve problems involving addition, subtraction, multiplication, and division of positive fractions and explain why a particular operation was used for a given situation.
@@
1. Model each fraction. 2. Divide each circle into tenths to create like denominators. 3. Add the fractions. 5 9 4 + = 10 10 10
_ _ _
@@
@@
@@
@@
GO ON
Lesson 3-5 Add Fractions with Unlike Denominators
33
Example 2 Find
YOUR TURN!
_1 + _1. Use prime factorization to find
6 8 the LCD.
1. Write the prime factorization of each denominator. 6 = 2 · 3 8=2·2·2 2. The LCM of the numbers contains only the factors in each number. If there are common factors, include them only one time. The denominators 6 and 8 have one factor in common, 2.
Find
7 6 the LCD.
1. Write the prime factorization of each denominator. 7·1 2·3 7= 6= 2. The LCD for these fractions is
_2 + _1
3. Rename each fraction using 24 as the denominator.
=
·
⎛ ⎪ ⎝
8·
_ _ _ _
⎞ ⎥ 3⎠
4. Add the like fractions.
19 _
3 4 _ _ + 24
6
= 12 + 7 42 42
1 ·3 ·4 _ _ + ⎞ ⎥ 4⎠
7
=2·6 +1·7 7·6 6·7
-$% uuu
⎛1 ⎪ ⎝6
42
24
_ _ _
Who is Correct?
Ernesto 12 = 2 × 2 × 3 16 = 2 × 2 × 2 × 2 2=2×1 The LCD is 2.
5 _ 3 1 _ , , and _. 12 16
2
Taina , 36, 48 Multiples of 12: 12, 24 , 48 Multiples of 16: 16, 32 Multiples of 2: 2, 4, 6, 8, 10, 12, . . . 48 The LCD is 48.
Circle correct answer(s). Cross out incorrect answer(s). 34
Chapter 3 Operations with Fractions
Joy 12 × 16 × 2 = 384 The LCD is 384.
Copyright © by The McGraw-Hill Companies, Inc.
4. Add the like fractions. 4 + 3 = 7 24 24 24
Find the LCD for the fractions
42 .
3. Rename each fraction so they have the LCD as their denominator.
"MMUIFPUIFSGBDUPST PGBOE
5IFDPNNPO GBDUPS
_2 + _1. Use prime factorization to find
Guided Practice Add using models. 5 1 1 1 __ + __ = 2 8 8
_
2
@@
20 _
3 ___ __ + 11 = 4
@@
@@
@@
12
12
@@
@@
@@
Name the LCD for each pair of fractions. 4 1 1 2 3 __, __ 15 4 __, __ 14 5 3 2 7
@@
18
5 2 , ___ __
5
3 18
6
3 5 __ __ , 6 8
24
Step by Step Practice 7
1 . Write the sum in simplest form. 2 + __ Find __ 3 4
12 .
Copyright © by The McGraw-Hill Companies, Inc.
Step 1 Determine the LCD. The LCD of 3 and 4 is
3=3·1
4=2·2
Step 2 Rename each fraction as an equivalent fraction. 1 2 __ __ + 3 4 ⎛
=
⎪3 ⎝
4 ⎞ + ⎛ __1 · ______ 3 ⎞ = ______ 8 + ______ 3 ⎥ ⎪4 ⎥ 4 ⎠ ⎝ 3 ⎠ 12 12
2 · ______ __
8 12
3 12
11 12
= ______ + ______ = ______
Step 3 Add the fractions with common denominators.
Add. Write each sum in simplest form. 8
9
2 __ 6
+
1 __ 3
3 1 + __ __ 2
4
⎛
=
1 ⎞ ⎥ 1 ⎠
2 · ______ __
⎪6 ⎝
_5 4
⎛
+
⎪3 ⎝
10
2 ⎞ ⎥ 2 ⎠
1 · ______ __
3 ___ __ + 1 5
10
2
2
4 6
2 3
+ = ____________ = ______ = ______
_7 10
6
11
5 2 + __ __ 4
6
16 _ 12
GO ON
Lesson 3-5 Add Fractions with Unlike Denominators
35
Step by Step Problem-Solving Practice
Problem-Solving Strategies Draw a diagram.
Solve. Write your answer in simplest form. 12
✓ Look for a pattern.
SPORTS Alano increases his soccer training sessions by the 1 hour. same amount every 3 days. His first training lasts __ 2 5 1 hours He trains __ of an hour the second day and 1__ the 6 6 third day. How much time will he spend training during the fourth session? Understand
_1
Read the problem. Write what you know. Alano began training
_
2
Later training sessions lasted 1 1 hours. Plan
Solve
Guess and check. Work backward. Solve a simpler problem.
_5
hour each day.
6
of an hour and
6
Since he will increase his sessions by the same amount, one strategy is to look for a pattern. Find out how much time he increases each training session. Then add that amount to his last time. 5 1 will give you __ ? What number added to __ 2 6 5 1 . Find the LCD of __ 1 and __ Rename __ . 2 2 6 Multiples of 2 = 1 ______ __ Multiples of 6 = = 2 LCD = 6
2, 4, 6 6 , 12, 18
3 6
6
7 6
1 = ______ 1__ 6
1 as an improper fraction. Write 1__ 6 Add the fractions.
third training + time increase = fourth session
7 6
______
Write in simplest form. Check
36
+
2 6
______
9 6
÷
3 3
÷
9 6
______
=
3 2
___________ = ____
Because you are adding positive numbers, the sum needs to be greater than each addend.
Chapter 3 Operations with Fractions
2 6
3 6
5 ______ ______ __ = -
Copyright © by The McGraw-Hill Companies, Inc.
Subtract to find the time increase from the first to the second trainings.
13
GARDENS The Joneses and the Smiths plant same-sized gardens. 1 of their garden in corn. The Smith family The Jones family plants __ 5 3 plants ___ of their garden in corn. Write a fraction that shows how 10 1 much of the total garden space is planted in corn.
_ 2
Check off each step.
14
✔
Understand
✔
Plan
✔
Solve
✔
Check
4 of a mile to school every morning. FITNESS Sumatra walks east __ 9 1 of a mile On Saturday she walked to a friend’s house, which is __ 3 farther east from her school. How far did Sumatra walk to and
14 mi _
from her friend’s house on Saturday?
9
How is adding unlike fractions the same as adding like fractions?
15
See TWE margin.
Copyright © by The McGraw-Hill Companies, Inc.
Skills, Concepts, and Problem Solving Name the LCD for each pair of fractions. 2 1 1 3 16 ___, __ 22 17 __, ___ 16 11 2 4 16
18
40
3 , __ 1 ___ 10 8
Add using models. Write each sum in simplest form. 19
_7
2= 1 + __ __ 2
20
5 3 + ___ __
21
2 3 + __ __
4
8
6
3
12
4
_7 6
_7
8
GO ON Lesson 3-5 Add Fractions with Unlike Denominators
37
_
Add. Write each sum in simplest form. 17 9 2 2 12 22 __ + ___ = 23 __ + ___ = 3 12 3 15 12 25
25 _
8 3 ___ + __ = 12
26
24
8
3 1 + __ __ = 2
5
22 _ 5
11 _ 10
24
3 9 ___ + ____ =
27
7 + __ 2= ___
10 11
100
3
39 _
100 43 33
_
Solve. Write the answer in simplest form. 28
PAINTING Lina and Marina were painting a house. On Monday 3 1 they painted __ of the house. On Tuesday they painted another __ 8 4 of the house. What fraction of the house did they have left to paint?
_3 8
TRAVEL At a gas station Kurt asked for directions to the nearest 5 town. The attendant told him to go __ of a mile south and then 8 1 of a mile east. How far does Kurt have to drive to get to ___ 16 11 of a mile the town? 16 3 7 30 MUSIC Fiona practiced the piano ___ of an hour Monday, ___ of an 15 10 3 hour Wednesday, and __ of an hour Friday. How many hours did 5 3 Fiona practice this week? 29
_
_ 2
Vocabulary Check Write the vocabulary word that completes each sentence. Unlike fractions have different
32
The least common multiple of the denominators of two or more fractions is the least common denominator .
33
Copyright © by The McGraw-Hill Companies, Inc.
denominators
31
.
3 5 3 2 = ___ 2 = ___ + ___ Writing in Math Correct Sasha’s mistake. ___ + __ 12 8 12 12 12 Explain how to find the correct answer.
See TWE margin.
Spiral Review Divide. Write each quotient in simplest form. (Lesson 3-4, p. 25) 5 2 5 3 2 5 34 __ ÷ __ = 35 ___ ÷ __ = 9 3 12 4 15 9
_
38
Chapter 3 Operations with Fractions
_
36
3 3 ÷ __ = 4
4
Lesson
3-6 Subtract Fractions with Unlike Denominators KEY Concept When you subtract fractions that are unlike fractions , you have to rename the fractions so that they have like denominators. Use equivalent forms of 1 to write the fractions so that each fraction has the least common denominator (LCD). Ways to Find the LCD for 7 - 3 6 15 1. Find a common multiple of both denominators. Multiples of 6: 6, 12, 18, 24, 30 Multiples of 15: 15, 30 , 45
_ _
7×5 3 3×2 7 - ___ __ = _____ - ______ 6
15
6×5
Copyright © by The McGraw-Hill Companies, Inc.
6 35 = ___ - ___ 30 30 29 = ___ 30
15 × 2 A common multiple of 6 and 15 is 30.
2. Use the prime factorization of each denominator. 3 3 7 - _____ 7 - ___ __ = _____ 6 15 2 × 3 3 × 5 7×5 3×2 = _________ - _________ 2×3×5 2×3×5 35 6 = ___ - ___ 30 30 29 = ___ 30
3NS3.2 Add and subtract simple fractions. 5NS2.0 Students perform calculations and solve problems involving addition, subtraction, and simple multiplication and division of fractions and decimals. 6NS2.1 Solve problems involving addition, subtraction, multiplication, and division of positive fractions and explain why a particular operation was used for a given situation.
VOCABULARY unlike fractions (Lesson 3-5, p. 33)
fractions with different denominators 1 2 Example: __ and __ 5 3 least common denominator (LCD) (Lesson 3-5, p. 33) the least common multiple of the denominators of two or more fractions, used as a denominator 1 1 1 Example: ___, __, __; 12 6 8 LCD is 24 prime factorization (Lesson 3-5, p. 33)
a way of expressing a composite number as a product of its prime factors
In some problems, both fractions have to be renamed. In other problems, only one fraction has to be renamed because one of the denominators is the least common denominator.
GO ON Lesson 3-6 Subtract Fractions with Unlike Denominators
39
Example 1 Find
_1 - _1 using a model.
3 6 1. Divide the circle with the thirds into sixths to create like denominators.
2. Subtract the fractions. 1-1=2-1=1 3 6 6 6 6
@@
_ _ _ _ _
@@
YOUR TURN! 1 2 Find - using a model. 3 9
_ _
@@
@@
1. Model each fraction as a circle. @@
2. Divide each circle into ninths to create like denominators.
@@
@@ @@
@@
3 - __ 2 = __ 1 __
3. Subtract the fractions.
9
9
9
Example 2 Find
1 1 _ - _. Use prime factorization to find the LCD. 12
15
1. Write the prime factorization of each denominator.
3. Rename each fraction using 60 as the denominator. 4. Subtract the fractions.
LCD = 2 · 2 · 3 · 5 = 60 5 1·4 4 _ _ 1·5 - _ _ = 60 60 15 · 4 12 · 5 4 1 _ 5 -_ _ = 60
60
60
YOUR TURN! 5 7 Find . Use prime factorization to find the LCD. 8 12
_ _
1. Write the prime factorization of each denominator. 2. The LCD for these fractions is
24 .
8= 2·2·2 12 = 2 · 2 · 3
3. Rename each fraction so they have common denominators. 4. Subtract the fractions.
40
11 _ 24
Chapter 3 Operations with Fractions
·3 -_ 5·2 =_ 10 21 - _ _7 - _5 = 7_ 8
12
8·3
12 · 2
24
24
Copyright © by The McGraw-Hill Companies, Inc.
2. The LCM of the numbers contains only the factors in each number. If there are common factors, include them only one time. The denominators 12 and 15 have one factor in common, 3.
12 = 2 · 2 · 3 15 = 3 · 5
Who is Correct? Find
5 2 _ - _. 11
5
Wendy _5 - _2 = _36 = _12
Ruben _ 3 25 - 22 = _ _5 - _2 = _ 55 55 55 5
11
11
Rupert _ 16 25 - 9 = _ _5 - _2 = _ 55 55 55
5
11
5
Circle correct answer(s). Cross out incorrect answer(s).
Guided Practice
_
Subtract using models. 7 3 5 1 __ - ___ = 6 12 12 2
@@
_1
1= 1 - __ __ 4
@@
@@
@@
12
6
@@
@@
@@
@@
Step by Step Practice
Copyright © by The McGraw-Hill Companies, Inc.
3
2 . Write the difference in simplest form. 1 - __ Find __ 2 5 Step 1 Determine the LCD.
2=2·1 5=5·1 The LCD of 2 and 5 is
10 .
Step 2 Rename each fraction as an equivalent fraction with 2 1 - __ __ a denominator of 10 . 2 5 ⎞ ⎛ ⎞ ⎛ 2 1 = __ · ______ - __ · ______ = ______ - ______ 5 2 ⎝ ⎠ ⎝ ⎠ Step 3 Subtract the fractions with common denominators.
⎪
5 ⎥ ⎪ 5
2 ⎥ 2
5 4 10 10 5 - 4 1 = _____________ = ______ 10 10
5 10
4 10
= ______ - ______
GO ON
Lesson 3-6 Subtract Fractions with Unlike Denominators
41
1 ⎥ ⎪ 1
2 ⎥ 2
6
4
2 14
1 7
4
⎛ ⎞ ⎛ ⎞ 6 6 2 2 ___ - __ = ___ · ______ - __ · ______ = _____________ = ______ = ______ 7 14 7 14 ⎝ ⎠ ⎝ ⎠
5
3 11 - __ ___ =
8
5 __ __ -1=
⎪
5
12
8
3
19 _ 60
_7 24
14
6
8 1= ___ - __
9
3 27 = ___ - ____
15
10
_5 = _1
7
3 __ __ -2=
3 _
10
10 __ ___ -1=
15
5
3
100
100
Step by Step Problem-Solving Practice
13 _
_9 22
Read the problem. Write what you know. The girls collected
50
_1
of a ton of canned goods.
Their goal was to collect Plan
2
36
Draw a diagram. Look for a pattern. ✓ Use logical reasoning. Guess and check. Work backward.
COMMUNITY SERVICE Kathy and Paloma collected 13 ___ of a ton of canned goods for a food bank. Their goal 50 1 of a ton. Write a fraction that shows the was to collect ___ 10 difference between their goal and their collection. Understand
11
9
Problem-Solving Strategies
Solve. Write the answer in simplest form. 11
4
19 _
10
of a ton.
Pick a strategy. One strategy is to use logical reasoning.
Solve
10 ×
50 is a multiple of 10. 1 as a fraction with Write ___ 10 a denominator of 50. Subtract the like fractions.
5
= 50
1 10
1 · 5 10 · 5
5 50
______ = ____________ = ______
_
So, the difference between their goal and the 4 of a ton. amount they collected is Check
42
25
Because you are subtracting from 13 ___ , the difference needs to be less 50 13 13 than ___. Is your answer less than ___? 50 50
Chapter 3 Operations with Fractions
13 5 50
8 50
4 25
_____________ = ______ = ______
Copyright © by The McGraw-Hill Companies, Inc.
Compare the denominators of the two fractions. What relationship do you see?
12
CLEANING Pooja and Alise were washing the walls of a 7 of one wall, while room. At the end of an hour, Pooja had washed __ 8 2 of a different wall of the same size. Write a Alise had washed __ 3 fraction that shows how much more Pooja had washed than Alisa. Check off each step.
13
✔
Understand
✔
Plan
✔
Solve
✔
Check
_5 24
2, 1 , froze another __ FOOD Kevin baked a meatloaf. He ate __ 5 5 2 and left the remaining meatloaf in the refrigerator. His sister ate ___ 15 of the meatloaf in the refrigerator. What fraction of the meatloaf in the refrigerator was left?
_4 15
Explain the first step to subtracting unlike fractions.
14
See TWE margin.
Copyright © by The McGraw-Hill Companies, Inc.
Skills, Concepts, and Problem Solving Subtract using drawings. Write each difference in simplest form. 1 1= 1 1 11 - __ ___ 15 __ - __ = 16 2 3 12 4 6
_
_8 = _2 12
3
@@
@@
@@
@@
@@
_
_
Subtract. Write each difference in simplest form. 5 21 3 9 2 1 17 __ - __ = 18 ___ - ____ = 12 100 3 4 10 100 20
2 __ 3 1 - ___
12 −−−−−
_7 12
21
@@
2 __
19 22
3 7 ___
4 - __ 1= __ 7
3
_5
@@
21
9 ___ 10 29 - ____
15 −−−−−
_3 = _1 15
@@
5
100
61 _ 100
GO ON
Lesson 3-6 Subtract Fractions with Unlike Denominators
43
Solve. Write the answer in simplest form. 23
15 NATURE On Monday, Juanita’s sunflower was ___ inches tall. It 16 3 __ had grown of an inch since Friday. How tall was Juanita’s 8 9 sunflower on Friday?
_ 16
24
1 of it HOBBIES Brian has a project to complete. He decides to do __ 8 3 on Monday, ___ of it on Tuesday, and to finish the project on Friday. 16 Write a fraction to represent how much of the project Brian will do
11 _
on Friday.
16
Vocabulary Check Write the vocabulary word that completes each sentence. 25 26
27
28
Unlike
fractions have different denominators.
The least common multiple of the denominators of two or more fractions is the least common denominator (LCD) .
unlike fractions To subtract first change them into like fractions. Writing in Math
, you have to @@
1 of the circle is yellow. 2 of the circle is blue. __ __
3 9 What fraction of the circle is red? Explain how to solve this problem. Set up an equation that shows your work.
@@ @@
Spiral Review
_
Add. Write each sum in simplest form. (Lesson 3-5, p. 33) 19 16 5 1 1 3 29 __ + __ = 30 __ + __ = 3 9 4 7 28 9 Multiply. Write 4 20 32 __ · ___ = 5 36 35
_
each product in simplest form. (Lesson 3-3, p. 19) 4 __ 2 3 1 33 __ · __ = 3 2 9
8
44
3 1 + __ __ =
34
8 3 __ ___ · 1 · __ =
3 2 of the singers are sopranos, __ are SCHOOL In the school chorus, __ 8 8 1 are tenors. The others are baritone or bass. What altos, and __ 8 2=1 fraction of the chorus are baritone or bass?
_ _
Chapter 3 Operations with Fractions
4
25 _
31
6
7
16 4 9
42
_1 24
Copyright © by The McGraw-Hill Companies, Inc.
See TWE margin.
Chapter
Progress Check 3
3
(Lessons 3-5 and 3-6)
Name the LCD for each pair of fractions. 3NS3.2 1 5 1 __, ___ 27 2 9 27 3
45
2 , __ 4 __ 5 9
4
32
1 , ___ 7 __ 8 32
56
5 __ __ ,1 7 8
Add. Write each sum in simplest form. 3NS3.2, 5NS2.0 5
3 1 + ___ __ =
7
3 __ __ +1=
9
1 + __ 4= __
2 4 3
13 6 9
19 _
6
5 ___ 5 __ + =
95 _
11 _ 12
8
3 1= ___ + __
_8 = _4
_7 9
10
9 3 ___ + __ =
21 _
26
8
11
2
10 16
88
10
Copyright © by The McGraw-Hill Companies, Inc.
11
10 __ ___ -1=
13
3 __ __ -1=
3
15 5
4
_1 3
12
7 - __ 1= ___
_7 20
14
7 - __ 1= __
8
40 9
16
4
Subtract. Write each difference in simplest form. 3NS3.2, 5NS2.0
7
5
_1 20
40 _ 63
Solve. 3NS3.2, 5NS2.0 15
FITNESS Elias rode his bike to his grandmother’s house, which 2 mile west from his house. He then rode to the park, was __ 5 3 which was __ mile farther west from his grandmother’s house. 8 How far did Elias ride if he rode from his house to
_
the park and back home again?
1 11 miles 20
16
NATURE Jasmine measured her pet snake 1 of an today (shown below). He had grown __ 3 inch since the last measurement. How long was Jasmine’s snake when she measured him last?
_
4 5 inches 6
JO
Lesson 3-6 Subtract Fractions with Unlike Denominators Paul Whitten/Photo Researchers, Inc.
45
Chapter
3
Study Guide
Vocabulary and Concept Check common denominator, p. 4 dividend, p. 25 divisor, p. 25
Write the vocabulary word that completes each sentence. Some words may be used more than once. 1
Fractions that have the same denominator are called
factor, p. 19 least common denominator (LCD), p. 33
like fractions 2
Two numbers whose product is 1 are
like fractions, p. 4
reciprocals
mixed number, p. 4 prime factorization, p. 33
simplest form, p. 4 unlike fractions, p. 33
.
3
factor A(n) of a number is a number that divides evenly into that number.
4
2 are examples of 1 and __ __
5
Like denominators are also called
quotient, p. 25 reciprocal, p. 25
.
6
7
common denominators 6
2 · 3 · 3 is the
unlike fractions
.
prime factorization
8
46
reciprocals
least common denominators (LCD) or like denominators
Chapter 3 Study Guide
of 18.
Copyright © by The McGraw-Hill Companies, Inc.
Label each diagram below. Write the correct vocabulary term in each blank. 7
.
Lesson Review
3-1
Add Fractions with Like Denominators
Add. Write each sum in simplest form. 9
3 2 + __ __ =
_5
10
5 __ 3 __ + =
_8
11
3 __ __ +1=
_4 = 1
7 9
9
4
4
Example 1 Find
_1 + _2. Write the sum in simplest form.
6 6 1 __ 1+2 __ + 2 = _____ 6 6 6 3 3÷3 1 = __ = _____ = __ 6 6÷3 2
7
7
(pp. 4–10)
9
4
1. The common denominator is 6. 2. Add the numerators. 3. Write in simplest form.
3-2
Subtract Fractions with Like Denominators
Copyright © by The McGraw-Hill Companies, Inc.
Subtract. Write each difference in simplest form. 12
8 4 = ___ - ___
_4 = _1
13
9 3 ___ - ___ =
_6 = _2
14
7 - __ 4 = __
12
12
15 9
3-3
15
9
Find
6 4 _ - _. Write the difference in
3
10 10 simplest form.
15
5
1. The common denominator is 10. 6-4 2 2. Subtract the numerators. _____ = ___ 10 10 2÷2 2 = ______ 1 = __ 3. Simplify. ___ 10 10 ÷ 2 5
_3 = _1 9
3
(pp. 19–24)
Multiply. Write each product in simplest form. 3 2 9 15 __ · ___ = 3 16 8
_
16
5 ___ 3 ___ · =
_1
17
3 ___ 15 __ · =
_1
5 18
Example 2
12
Multiply Fractions
12 10
(pp. 11–17)
8
2
Example 3 Find
8 _3 · _ . Write the product in simplest form. 4
12
Multiply the numerators and multiply the denominators. Write the fraction in simplest form. 5IF($'PGBOEJT
u µ u µ u Chapter 3 Study Guide
47
3-4
Divide Fractions
Example 4
(pp. 25–31)
Divide. Write each quotient in simplest form. 18
10 5 ___ __ ÷ = 6
18
2
2
1= 4 ÷ __ __
_4 = 1_1
20
1= 2 ÷ __ __
_4 = 1_1
21
3 5 __ __ ÷ =
_5
3
2
8
3-5 22
3
4
3
3
3
3
_1 ÷ _1. Write the quotient in
5 9 simplest form.
_3 = 1_1
19
9
Find
Write the reciprocal of the divisor. 5 1 __ __ 5 1 1 1 ÷ __ __ Change division to 9 5 multiplication. Multiply by the reciprocal of the 5 1 · __ __ 9 1 divisor.
6
Add Fractions with Unlike Denominators
1 using diagrams. 1 + __ Find __ 2 4
1____ · 5 __ 5 = 9·1 9
Multiply the fractions.
_3 4
(pp. 33–38)
Example 5 Find
_3 + _1 using diagrams. 4
8
@@
@@
Divide the circle with the fourths into eighths to create like denominators. Add. Write each sum in simplest form. 1 1 23 ___ + __ 12 8
_2 + _3 = _5 24
24
24
2 + __ 1 ___ 15
9
11 _6 + _5 = _ 45
48
24
45
45
Chapter 3 Study Guide
@@
Add the fractions. 6 __ 7 __ + 1 = __ 8 8 8
@@
Copyright © by The McGraw-Hill Companies, Inc.
3-6 25
Subtract Fractions with Unlike Denominators (pp. 39-44) 2 1 using diagrams. 1 - __ Find __ 3 9 9 Example 6
_
Find
4 8 diagrams.
@@
@@
@@
Divide the circle with the fourths into eighths to create like denominators.
@@
_1 - _1 using
@@
@@
@@
@@
Subtract the fractions. 1 - __ 1 = __ 2 - __ 1 = __ 1 __ 4
Subtract. Write each difference in simplest form. 3 1 1 26 __ – ___ = 7 28 28 Copyright © by The McGraw-Hill Companies, Inc.
27
28
28
Find
28
10
10
10
LCD = 2 · 2 · 2 · 2 · 3 · 3 = 144
_1 4 12 3 _3 - _5 = _9 - _5 = _4 = _1 12
12
12
12
19 _ 27 19 21 - _ 2 =_ _7 - _2 = _
7 2 29 __ - ___ = 9 27
9
27
27
27
27
1 _1 – _ . Use prime factorization to
9=3·3 16 = 2 · 2 · 2 · 2
5 3 ___ __ =
4
8
Write the prime factorization of each denominator.
_3 5 10 10 _2 - _1 = _4 - _1 = _3 10
8
9 16 find the LCD.
1 = 2 - ___ __
5
28
28
8
Example 7
_ _1 - _1 = _4 - _1 = _3 7
8
Rename each fraction using 144 as the denominator. ⎛ 1 · 16 ⎞ ⎪ _____ ⎥ ⎝ 9 · 16 ⎠
3
⎛ 1 · 9 ⎞ - ⎪ _____ ⎥ ⎝ 16 · 9 ⎠
9 16 = ____ - ____ 144 144 Subtract the fractions. 16 9 7 ____ - ____ = ____ 144
144
144
Chapter 3 Study Guide
49
Chapter
Chapter Test
3
Add. Write each sum in simplest form. 3NS3.25, NS2.0 1 1 3 1 1 2 1 __ + __ = 2 ___ + ___ = 4 8 8 12 12 2
_
_
3
_5
1 + __ 2= __ 6
12
8
Subtract. Write each difference in simplest form. 3NS3.2, 5NS2.0 2 1 5 1 1 4 4 __ - __ = 5 __ - __ = 6 6 6 2 9 3 18
3 __ __ -1=
_4
Multiply. Write each product in simplest form. 5NS2.0, 5NS2.5 3 1 3 14 2 9 7 __ · ___ = 8 __ · ___ = 5 4 3 10 7 24
9
7 · __ 4= __
_7
12
8 2 ÷ ___ __ =
_5 = 1_1
_
_
_ _
Divide. Write each quotient in simplest form. 5NS2.0, 5NS2.5 3 3 1 4 1 11 10 __ ÷ __ = 11 ___ ÷ __ = 4 2 6 10 5 2
_
_
5
15
3
18
8 9
3
15
4
4
13
1 cup BAKING Marta has a cookie recipe that calls for __ 4 2 cup sugar. sugar, and Amos has a cookie recipe that calls for __ 4 How much sugar is needed altogether for their recipes?
_3 cup
4DIPPM
4
14
Copyright © by The McGraw-Hill Companies, Inc.
Solve. 5NS2.0, 6NS2.1
TRAVEL Tavio rode his bike to Dan’s house from his house. Then he rode his bike to school. How many miles did he ride in all? Use the map shown at the right.
_
1 1 mile 4
NJMF @@ %BOT IPVTF NJMF @@
5BWJPT IPVTF
GO ON 50
Chapter 3 Test
15
FOOTBALL Coach Ring had football practice one day when it was 2 of snowing. He had 30 members on his football team, but only __ 3 them showed up. How many members of the football team were at 20 the practice?
16
CRAFTS Kelsey has 3 yards of fabric for crafts she is 1 yard of material, making. If each craft she makes requires __ 4 12 crafts how many crafts can she make with the 3 yards?
17
1 of the chicken FOOD At the school cafeteria, the fifth graders ate __ 9 3 nuggets, and the sixth graders ate __ of the chicken nuggets. What 8 fraction of the chicken nuggets was eaten by the fifth and sixth graders?
18
35 _ 72
LANDSCAPING The Green Meadows landscaping company had a project at the swim club. Their goal was to work Monday, Tuesday, and Wednesday, and finish on Thursday. On Monday 1 of the job. On Tuesday they completed they completed __ 4 3 of the job. How 1 of the job. On Wednesday they completed __ __ 8 5 7 much was left for them to finish on Thursday?
_
Copyright © by The McGraw-Hill Companies, Inc.
40
Correct the mistakes. 6NS2.1 3 1 19 FITNESS Fabio jogged ___ mile before school and ___ mile after 10 10 school. He proudly said to his friend, “I jogged a half mile today.” What is wrong with Fabio’s statement?
See TWE margin.
20
TRIPS A group of students went on a summer digging trip. While digging they found two ancient bone fragments. The first fragment 5 1 inch. One of measured ___ inch. The second fragment measured __ 8 12 the students reported to their professor that the first fragment 1 inch longer than the second. By what amount was the was __ 4 student incorrect?
See TWE margin. Chapter 3 Test
51
Chapter
Standards Practice
3
Choose the best answer and fill in the corresponding circle on the sheet at right. 1
1 gallon of milk for one recipe Zita used __ 8 4 gallon of milk for another recipe. and __ 8 How much milk did she use for both recipes? 5NS2.0
_
2
A 5 gallon 16
1 gallon C __ 2
3 gallon B __ 8
5 gallon D __ 8
4
9
5
6
_
Seth
_
12 or 1 whole A ___ 12
1 4 or __ C ___ 3 12
5 B ___ 12
2 or __ 1 D ___ 6 12
11 foot C ___ 15
1 foot B __ 5
12 foot D ___ 5
Which symbol makes this sentence true? 5NS2.5
_1 × _3 3
12 4 12
4 foot A ___ 15
7
5
_2 ÷ _5 8
4
F >
H <
G =
J +
_ _
Write the sum in simplest form: 1 + 2 . 12 3 3 3 5NS2.0 A ___ C __ 15 4 1 B __ 4
9 D ___ 10
GO ON 52
Chapter 3 Standards Practice
Copyright © by The McGraw-Hill Companies, Inc.
Three friends split a large pizza. The table shows how much of the pizza each friend ate. How much more pizza did Jaden eat than Seth? 5NS2.0
Jaden
6 J __ 9
Gracie needs to cut a rope into 3 equal pieces. The rope is 4 foot long. 5 How long will each piece be? 5NS2.0,
7 J __ 6
_2 12 _6
2 H __ 9
5NS2.5
_ _
Dominic
5NS2.5
3 2 F ___ 27 2 G ___ 12
Which fraction has a value equal to 5 - 2 ? 3NS3.2 6 6 3 7 H ___ F __ 0 13 1 G __ 2
3
_2 × _1 =
8
Miguel is hiking through a state park. He walks 1 mile on Pike’s Lake Trail. 2 How much more does he need to hike to complete this trail? 5NS2.0, 5NS2.5
_
Trail Eagle’s Bluff Pike’s Lake Riverside
Copyright © by The McGraw-Hill Companies, Inc.
9
12
10
1 mile H __ 5
3 mile G __ 8
6 mile J __ 8
Which symbol makes this sentence true? 3NS3.1 10
10
_5 mile 6 _7 mile 8 7 _ mile
_4 5
Order these fractions from least to greatest: 5 , 3 , 7 . 3NS3.1 6 4 12 5 ___ 3 ___ 5 , __ 3 , __ , 7 C __ , 7 A __ 6 4 12 4 6 12
___
3 __ 5 7 , __ B ___ , 12 4 6
Length
2 mile F __ 6
_7
11
5 3 , ___ 7 , __ D __ 4 12 6
_
Write 6 in simplest form. 4NS1.5 8 3 2 H __ F __ 3 4 3 J __ 8
1 G __ 2 1
A
B
C
D
2
F
G
H
J
3
A
B
C
D
4
F
G
H
J
5
A
B
C
D
6
F
G
H
J
A >
C <
7
A
B
C
D
B =
D +
8
F
G
H
J
9
A
B
C
D
10
F
G
H
J
11
A
B
C
D
12
F
G
H
J
Which mixed number does the model represent? 4NS1.5, 4NS1.7
Success Strategy 4 F 1___ 10
5 H 1__ 9
1 G 1__ 2
3 J 1__ 5
After answering all the questions, go back and check signs and operations, making sure you worked each problem correctly. Chapter 3 Standards Practice
53
Chapter
4
Positive and Negative Fractions and Decimals We use money every day. Before lunch you count your money. You may borrow a few cents from a friend. Our money system is based on decimals.
Copyright © by The McGraw-Hill Companies, Inc.
54
Chapter 4 Positive and Negative Fractions and Decimals
Michael Newman/PhotoEdit
STEP
STEP
1 Quiz
Are you ready for Chapter 4? Take the Online Readiness Quiz at ca.mathtriumphs.com to find out.
2 Preview
Get ready for Chapter 4. Review these skills and compare them with what you will learn in this chapter.
What You Know
What You Will Learn
You know how to read price tags.
Lessons 4-1 and 4-2 Our money system is based on decimals. The price is a decimal.
$12.50 These items cost 12 dollars and 50 cents. You know how to add and subtract whole numbers. Examples:
45 + 34
809 - 46
79
763
TRY IT! 234 1 + 12
2
Copyright © by The McGraw-Hill Companies, Inc.
246 3
412 + 39
4
284
×
11 7 77
44 6 264
TRY IT!
50 6
252 7
After you line up the decimal points, you can add and subtract decimals like whole numbers. 4.5 + 3.4 −−−−− 7.9
8.09 -0.46 −−−−− 7.63
349
Examples:
25 × 5
Lessons 4-5 and 4-6
425 - 76
You know how to multiply and divide whole numbers.
5
decimal point
This is the cents, or part of the dollar.
451
334 - 50
42 × 6
This is the dollar amount.
2 100
Lessons 4-7 and 4-8 Multiply and divide decimals as if the numbers were whole numbers. Then place the decimal point in the answer. 4.4 × 6 −−− −−−− 26.4
1.1 7 7.7
11 8
3 33
125 55
Lesson
4-1 Introduction to Decimals KEY Concept Decimals and fractions represent both whole numbers and numbers between consecutive whole numbers. A decimal is another way to show a fraction that has a denominator of 10, 100, and so on.
VOCABULARY
In a decimal, the digit(s) to the left of the decimal point represent a whole number. The digit(s) to the right of a decimal point represent a fraction. A digit one place to the right of the decimal point is a tenth .
A digit two places to the right of the decimal point is a hundredth .
XIPMFOVNCFS
UFOUI
IVOESFEUI
hundredths the second decimal place to the right of the decimal point; one of one hundred equal parts Example: In 4.57, 7 is in the hundredths place.
Read a decimal from left to right. Place-Value Chart Value Place Value
1,000
100
thousands hundreds
Example
1
2
Words
one thousand
two hundred
10
1
tens
ones
3
4
thirty-four
0.1
0.01
tenths hundredths · and
5
6
fifty-six hundredths
Place values are based on powers of ten. From left to right, the value of each place is divided by 10.
56
Chapter 4 Positive and Negative Fractions and Decimals
Copyright © by The McGraw-Hill Companies, Inc.
decimal a number that can represent a whole number and a fraction; a decimal point separates the whole number from the fraction tenths the first decimal place to the right of the decimal point; one of ten equal 1 parts or ___ 10 Example: In 3.6, 6 is in the tenths place.
IPMETUIFQMBDFXIFO UIFSFJTOPXIPMFOVNCFS UFOUI BOETPPO
3NS3.4 Know and understand that fractions and decimals are two different representations of the same concept. 4NS1.6 Write tenths and hundredths in decimal and fraction notations and know the fraction and decimal equivalents for halves and fourths. 4NS1.7 Write the fraction represented by a drawing of parts of a figure; represent a given fraction by using drawings; and relate a fraction to a simple decimal on a number line.
Example 1
YOUR TURN!
Write two and sixty-four hundredths as a decimal and as a fraction in simplest form.
Each square is divided into 100 equal parts. 1 of the square. Each part is ____ 100
Copyright © by The McGraw-Hill Companies, Inc.
100 1. ____ of the first square is shaded. 100 100 ____ of the second square is shaded. 100 This equals 2 wholes or 2. 64 2. ____ of the third square is shaded. This 100 equals sixty-four hundredths or 0.64. 64 64 3. Add the parts. 2 + ____ = 2____ 100 100 4. Write the number as a decimal. 2.64
Write one and eighty-two hundredths as a decimal and as a fraction in simplest form.
100
1. ____ of the first square is shaded. 100
82
2. ____ of the second square is shaded. 100
_
_
1 + 82 = 1 82 100 100 4. Write the number as a decimal. 1.82 3. Add the parts.
5. Write the number as a fraction in simplest form. Divide the numerator and denominator by an equivalent form of one.
_
1 41 50
5. Write the number as a fraction in simplest form. Divide the numerator and denominator by an equivalent form of one. 64 64 ÷ 4 16 _ _ _ = = 100
100 ÷ 4
25
64 16 _ _ =2
2
100
25
Example 2 Write 4.38 in words. 1. Name the decimal by the place value of the last digit to the right of the decimal point. 4.38 hundredths 2. There is a 4 to the left of the decimal point. At the decimal point, write “and.” So, 4.38 is four and thirty-eight hundredths.
YOUR TURN! Write 6.4 in words. 1. What place value is the last digit to the right of the decimal point? tenths . 2. There is a 6 to the left of the decimal point. At the decimal point, write “and.” So, 6.4 is
six
and
four tenths . GO ON
Lesson 4-1 Introduction to Decimals
57
Example 3
YOUR TURN!
Write the value of the red digit in 0.34 as a fraction. 1. The place value one place to the right of the decimal point is tenths.
Write the value of the red digit in 6.75 as a fraction. 1. The place value two places to the right of hundredths . the decimal point is
5
2. Multiply
2. Multiply the digit in the tenths place 1. by ___ 10 3 1 3× = 10 10 3 The value is ___. 10
1 _ .
by
100
5 1 =_ 5×_ 100
_ _
100
The value is
5 _ 100
.
Who is Correct? Write 12.07 in words.
Cybil Twelve and seven tenths
Nuna
Nam
twelve point seven
twelve and seven hundredths
Guided Practice Write a decimal and a fraction in simplest form for each model. 1
2
_
1.74 = 1 74 100
58
Chapter 4 Positive and Negative Fractions and Decimals
_ _
0.25 = 25 = 1 100 4
Copyright © by The McGraw-Hill Companies, Inc.
Circle correct answer(s). Cross out incorrect answer(s).
Step by Step Practice 3
Write four and twenty-six hundredths as a decimal. Step 1 Identify the whole number. Write it first. four and twenty-six hundredths Step 2 Place the decimal point. Step 3 Identify the digits for the decimal part.
4 4 4
· ·
2 6
hundredths
four and twenty-six hundredths
Write each number as a decimal. 4
five and eighty-four hundredths
5
·
8 4
whole number decimal point 6
5
11.7
hundredths
tenths
sixteen and thirty hundredths
7
16.30 Copyright © by The McGraw-Hill Companies, Inc.
eleven and seven tenths
four tenths
0.4
Write each number in words. 8
11.67
9
eleven and sixty-seven hundredths 10
89.4
8.03
eight and three hundredths 11
eighty-nine and four tenths
55.16
fifty-five and sixteen hundredths
Write the value of each red digit as a fraction. 12
42.8
14
6.97
_8 10
7 _ 100
13
91.63
15
34.62
_6 10 2 _ 100
GO ON
Lesson 4-1 Introduction to Decimals
59
Step by Step Problem-Solving Practice
Problem-Solving Strategies Draw a diagram.
Solve. 16
✓ Use a model.
BAKING Jessica’s brother ate 0.25 of the cake she baked. Write a fraction to show the amount of cake left. Understand
Solve a simpler problem. Act it out. Work backward.
Read the problem. Write what you know. Jessica’s brother ate
0.25
of the cake.
Plan
Pick a strategy. One strategy is to use a model.
Solve
Draw a model to represent the whole cake. Shade the amount eaten. The place value of the last digit to the right of the decimal place is hundredths . Use a grid of 100 squares to represent the entire cake. Shade 25 squares to represent 0.25 of the cake. How many squares are not shaded?
75 squares .
This represents the amount of cake not eaten. What fraction of the cake is not eaten? Write in simplest form. So,
_3
100
25 ÷ ___ = 25 100
_3 4
of the cake is left.
Write 0.25 as a fraction. Subtract from 1.
Check
.
whole cake 100 ____ 100
17
brother ate
-
25 _ 100
cake left
=
75 _ 100
PLANTS Refer to the photo caption at the right. Write a decimal to show how much water Dwaine uses to water his plant 1.5 gallons each day. Check off each step.
60
✔
Understand
✔
Plan
✔
Solve
PLANTS Dwaine waters
✔
Check
5 his plant with 1 gallons 10 of water every day.
Chapter 4 Positive and Negative Fractions and Decimals
David Young-Wolff/PhotoEdit
_
Copyright © by The McGraw-Hill Companies, Inc.
4
75 _
75 _
18
7 of a mile to school. Her friend Julie Solidad rode her bike ___ 10 walked 0.07 of a mile to school. Did the two girls travel the same distance? Explain.
_
__ _
No; Solidad traveled 7 , and Julie traveled 7 . 7 ≠ 7 . 10 100 10 100 3 Annie says that __ is equal to 0.75. Is Annie correct? Show 4 work to explain your answer.
19
_ _
_
Annie is correct. 3 · 25 = 75 and 75 = 0.75 4 · 25 100 100
Skills, Concepts, and Problem Solving Write a decimal and a fraction in simplest form for each model. 20
21
_ _
_ _
0.5 = 50 = 1 100 2
0.34 = 34 = 17 100 50
Write each number as a decimal.
Copyright © by The McGraw-Hill Companies, Inc.
22
forty-forty and nineteen hundredths
23
44.19 24
ten and six hundredths
10.06
eight and seven tenths
25
8.7
nineteen and thirteen hundredths
19.13
Write each number in words. 26
0.9
27
nine tenths 28
18.03
eighteen and three hundredths
7.17
29
seven and seventeen hundredths
3.78
three and seventy-eight hundredths
Write the value of each red digit as a fraction. 30
46.7
_7 10
31
14.23
_2 10
32
86.61
1 _ 100
33
42.95
5 _
GO ON
100 Lesson 4-1 Introduction to Decimals
61
Solve. TRAVEL Mikal asked for directions to a local park. He was told to 1 miles south and then 2__ 1 miles east. Write both numbers go 3__ 2 4 3.25, 2.5 as decimals. 5 35 MONEY Paulina promises to give her little brother __ of a dollar 4 each week for cleaning her room. Write the amount as a decimal 34
1.25 Vocabulary Check Write the vocabulary word that completes each sentence. 36
The first digit to the right of the decimal point is the tenths place. The second digit to the right is the
hundredths 37
38
place.
decimal A(n) number and a fraction.
is a number that can represent a whole
Writing in Math Write the amount of the fraction shown in the figures at the right in decimal and fraction form. 0.25 = __ 1
4
Write different fractions for the fraction shown.
_
___
Sample answer: 1 can be written as 25 , 4 , 3 . 4 100 16 12 Sample answer: a quarter of a pie or pizza, 1 of 4 legs on a pet, 1 page of 4 pages, 1 penny of 4 pennies
Spiral Review Solve. 39
NATURE There are two types of flowers in a field. The yellow flower grows 4 decimeters every two days. The blue flower grows 3 decimeters every day. Which flower grows faster? How much faster does it grow? (Lesson 3-6, p. 34)
The blue flower grows 1 dm per day faster than the yellow flower. Use < , =, or > to compare the fractions. (Lesson 2-4, p. 59) 4 2 1 7 40 ___ < __ 41 __ > ___ 36 9 2 15 62
Chapter 4 Positive and Negative Fractions and Decimals
42
8 ___ 24
= __1 3
Copyright © by The McGraw-Hill Companies, Inc.
Describe different ways to represent this amount with real-life objects.
Lesson
4-2 Decimals and Money KEY Concept Decimals form the basis of our money system. Problems about money are solved like other problems with decimals . Commonly used denominations in U.S. currency are shown in the table. Currency
Numbers
Words
$100
hundreds
$10
tens
$1
ones
$0.10 or 10¢
tenths
Copyright © by The McGraw-Hill Companies, Inc.
$0.01 or 1¢
hundredths
Different combinations of money can have the same value. EPMMBSCJMM
IBMGEPMMBST
2NS5.1 Solve problems using combinations of coins and bills. 2NS5.2 Know and use the decimal notation and the dollar and cent symbols for money.
VOCABULARY denomination a category or type in the measurement of currency, or money There are denominations of bills, such as $1 bills, $5 bills, and $10 bills. There are denominations of coins, such as pennies, nickels, dimes, quarters, and half-dollars. decimal a number that can represent a whole number and a fraction; a decimal point separates the whole number from the fraction (Lesson 4-1, p. 56)
hundredths the second decimal place to the right of the decimal point; one of 100 equal parts Example: In 4.57, 7 is in the hundredths place. (Lesson 4-1, p. 56)
RVBSUFST
RVBSUFS PSa
IBMGEPMMBS PSa EJNFT
EJNF PSa QFOOJFT
OJDLFMT
OJDLFM PSa
QFOOZ PSa
GO ON Lesson 4-2 Decimals and Money
coins: United States Mint, bills: Michael Houghton/StudiOhio
63
The whole number in a decimal expression represents the number of dollars. The coins that total less than a dollar are KEY Concept the fraction of the dollar. They are written to the right of the decimal point. For example, $4.23 is equal to four 1-dollar bills and 23 cents. Use the dollar sign, $, with dollars. Use the dollar or cent sign, ¢, to represent an amount that is less than $1.00.
You should know which coins can be exchanged for a coin with a greater denomination.
Example 1 Write the amount shown.
$3.00
+
$0.50
+
$0.05 +
$0.06
2. Use skip-counting to find the total amount. +0.25 +0.25 +0.05 +0.01 +0.01 +0.01 +0.01 +0.01 +0.01 $3.00 $3.25 $3.50 $3.55 $3.56 $3.57 $3.58 $3.59 $3.60 $3.61
64
Chapter 4 Positive and Negative Fractions and Decimals
coins: United States Mint, bills: Michael Houghton/StudiOhio
Copyright © by The McGraw-Hill Companies, Inc.
1. Find the value of each bill and coin.
YOUR TURN! Write the amount shown.
1. Find the value of each bill and coin.
$5.00
$0.25
$0.20
$0.03
2. Use skip-scounting to find the total amount. +0.25
$5.00
+0.10
$5.25
+0.10
$5.35
+0.01
$5.45
+0.01
$5.46
+0.01
$5.47
$5.48
Example 2 Represent 87¢ with the least number of coins possible.
Copyright © by The McGraw-Hill Companies, Inc.
The amount is less than $1.00. Begin with the coin of greatest value, a half-dollar. Use as many of that coin as possible. Then continue with the coin with the next largest value. IBMGEPMMBS
a
RVBSUFS
a EJNF
a
a
a
QFOOZ
a
QFOOZ
a
a
87¢ =
GO ON
Lesson 4-2 Decimals and Money coins: United States Mint, bills: Michael Houghton/StudiOhio
65
YOUR TURN! Represent 63¢ with the least number of coins possible. The amount is less than $1.00. Begin with the coin of greatest value, a half-dollar. One half-dollar =
50¢
What can you use next?
.
dime
50¢ + 10¢ 60¢ 1 half-dollar, 1 dime, 3 pennies
Add the amounts.
So, 63¢ =
.
Who is Correct? Write the amount shown.
Yang
Monica
25¢ 10¢ + 1¢ −−−− 36¢
25¢ 5¢ + 1¢ −−−− 31¢
50¢ 5¢ + 1¢ −−−− 56¢
Circle correct answer(s). Cross out incorrect answer(s).
Guided Practice Write the amount shown. 1
$0.61
2
$7.40
66
Chapter 4 Positive and Negative Fractions and Decimals
coins: United States Mint, bills: Michael Houghton/StudiOhio
Copyright © by The McGraw-Hill Companies, Inc.
Wade
Step by Step Practice 3
Represent 46¢ with the least number of coins possible. Step 1 The coin of greatest value that is less than 46 cents is a 25¢ . quarter. One quarter is Step 2
25¢
dimes
is
45¢
Step 3
45¢
plus one penny The combination of coins is
is
46¢
plus two
. .
one quarter, two dimes, and one penny
.
Represent each amount using the least number of bills and coins possible. 4
$3.13
3
How many 1-dollar bills? How many dimes? How many pennies?
1 3
Copyright © by The McGraw-Hill Companies, Inc.
The combination of currency is three 1-dollar bills , 3 pennies . 5
1 dime
, and
$23.98
one 20-dollar bill, three 1-dollar bills, 1 half-dollar, 1 quarter, 2 dimes, 3 pennies. 6
$14.34
one 10-dollar bill, four 1-dollar bills, 1 quarter, 1 nickel, 4 pennies 7
$2.09
two 1-dollar bills, 1 nickel, 4 pennies 8
$6.15
one 5-dollar bill, one 1-dollar bill, 1 dime, 1 nickel GO ON Lesson 4-2 Decimals and Money
67
Step by Step Problem-Solving Practice
Problem-Solving Strategies Draw a diagram.
Solve. 9
✓ Use logical reasoning.
MONEY Alan and Jay each ordered the same meal at a restaurant. Each boy paid $7.82 but used a different combination of bills and coins. Represent $7.82 in two ways.
Solve a simpler problem. Work backward. Guess and check.
Understand
Read the problem. Write what you know. $7.82 . Each boy paid a total of
Plan
Pick a strategy. One strategy is to use logical reasoning. First represent $7.82 using the fewest coins. Then find a combination using different coins and bills.
Solve
The least number of bills that can make $7.82 one two is 5-dollar bill and 1-dollar bills. 82¢ What amount is left? half-dollar Which coin represents the greatest amount? 50¢ + 25¢ = 75¢ 75¢ +
5¢
= 80¢
1¢
80¢ +
+
1¢
= 82¢
So, one combination for $7.82 is
one 5-dollar bill, two 1-dollar bills, one half-dollar, 1 quarter,
.
Now represent $7.82 using a different combination of bills and coins. $7 = seven 1-dollar bills +25¢ +10¢
+10¢ +10¢
50¢
70¢
82¢ = 25¢
60¢
+1¢
80¢
+1¢
81¢
82¢
A second combination of bills and coins is
seven 1-dollar bills, 2 quarters, 3 dimes, and 2 pennies Check
68
Add each amount. The total of each amount should be $7.82.
Chapter 4 Positive and Negative Fractions and Decimals
.
Copyright © by The McGraw-Hill Companies, Inc.
1 nickel, and 2 pennies
10
FASHION Lauren went to a clothing store with $40. She bought a shirt for $11.99 and paid with a 10-dollar bill and coins. What coins did she use if she did not receive any change?
Answers will vary. Sample answer: 3 half-dollars, 1 quarter, 2 dimes, 4 pennies Check off each step.
11
✔
Understand
✔
Plan
✔
Solve
✔
Check
BUSINESS Noor has 5 dimes and 1 nickel. A permanent marker costs 65¢. A washable marker costs 52¢. Which marker can Noor buy?
the washable marker 12
Vera bought a toy for her little brother and paid the cashier with a 20-dollar bill. The cashier gave Vera 32¢ in change. If one of the coins that Vera got back is a quarter, what are the other coins? How much did the toy cost?
The other coins are either 1 nickel and 2 pennies or 7 pennies.
Copyright © by The McGraw-Hill Companies, Inc.
The toy cost $19.68.
Skills, Concepts, and Problem Solving Write the amount shown. 13
14
$1.75
$7.30
GO ON Lesson 4-2 Decimals and Money coins: United States Mint, bills: Michael Houghton/StudiOhio
69
Represent each amount with the least number of bills and coins possible. 15
$17.32
one 10-dollar bill, one 5-dollar bill, two 1-dollar bills, 1 quarter, 1 nickel, 2 pennies. 16
$44.10
two 20-dollar bills, four 1-dollar bills, 1 dime 17
$10.97
one 10-dollar bill, 1 half-dollar, 1 quarter, 2 dimes, 2 pennies 18
$35.36
one 20-dollar bill, one 10-dollar bill, one 5-dollar bill, 1 quarter, 1 dime, 1 penny
Represent each amount with the greatest number of bills and coins possible. 19
$0.99
20
75¢
99 pennies 75 pennies
Solve. 21
2 half-dollars, 1 quarter, 2 dimes, 4 pennies 22
BUSINESS Ginese earned the amount shown for delivering newspapers last week. Write a decimal for the amount shown.
$12.94
Vocabulary Check Write the vocabulary word that completes each sentence. 23
denomination The lowest money system is the penny.
24
When rounding to the nearest cent, you are rounding to the
hundredths 70
of coins in our
place.
Chapter 4 Positive and Negative Fractions and Decimals
coins: United States Mint, bills: Michael Houghton/StudiOhio
Copyright © by The McGraw-Hill Companies, Inc.
MONEY Asia buys gum that costs $1.49, including tax. She pays with 9 coins. What coin combination did she use if she used the least number of coins?
25
Writing in Math Corinne and Claudio each have 3 quarters. Corinne says she has $0.75. Claudio gave an answer of $0.45. Who is correct? Explain the error of the other student.
Corinne is correct. Claudio did not count correctly.
Spiral Review Solve. 26
(Lesson 3-4, p. 21)
2 of a sheet of drywall to repair CONSTRUCTION Janet uses __ 3 3 holes of equal size. What fraction of a drywall sheet did she use to repair each hole?
27
_2 9
COOKING Domingo is baking a 2-tier cake for his mom’s birthday. 1 of a bag of sugar for both tiers. How much sugar did he He uses __ 3
_1 of a bag
use for one tier of cake?
Write each fraction in simplest form. (Lesson 2-5, p. 69) 1 3 12 2 28 ___ = 29 ___ = 15 5 18 3
_
Copyright © by The McGraw-Hill Companies, Inc.
6
_
48 ___ =
30
80
Write the unit fraction that represents the shaded region. 31
_3 5
(Lesson 1-2 p. 11)
32
_1
_1
8
9
33
34
_1 5
_1 10
Lesson 4-2 Decimals and Money
71
Chapter
Progress Check 1
4
(Lessons 4-1 and 4-2)
Write a decimal and equivalent fraction in simplest form. 3NS3.4, 4NS1.6 1
2
_
_
0.75 = 3 4
0.6 = 3 5 3
forty and ninety-one hundredths
_
4
two tenths
40.91; 40 91 100
_
0.2; 1 5
Write the amount shown. 2NS5.2 5
6
$0.62
$11.53
Represent each amount using the least number of bills and coins possible. 2NS5.2 7
$16.34
8
$22.88
one 20-dollar bill, two 1-dollar bills, 1 half-dollar, 1 quarter, 1 dime, 3 pennies Solve. 3NS3.4, 2NS5.1 9
MONEY Brenda paid $36.25 for her groceries with 9 bills and 5 coins. What possible combinations of bills and coins could she have used?
Sample answer: three 10-dollar bills, six 1-dollar bills, 5 nickels 10
FOOD At a party, 26 out of 50 deviled eggs and 13 out of 25 ham sandwiches had been eaten. Write each amount as a decimal.
eggs, 0.52; ham sandwiches, 0.52 72
Chapter 4 Positive and Negative Fractions and Decimals
coins: United States Mint, bills: Michael Houghton/StudiOhio
Copyright © by The McGraw-Hill Companies, Inc.
one 10-dollar bill, one 5-dollar bill, one 1-dollar bill, 1 quarter, 1 nickel, 4 pennies
Lesson
4-3 Compare and Order Decimals KEY Concept To compare and order decimals , you can use a number line, decimal models, or place value.
5NS1.5 Identify and represent on a number line decimals, fractions, mixed numbers, and positive and negative integers. 6NS1.1 Compare and order positive and negative fractions, decimals, and mixed numbers and place them on a number line.
VOCABULARY
Compare 1.19 and 1.1 using a number line. Numbers to the right are greater than numbers to the left. 1.19 is to the right of 1.1, so 1.19 > 1.1.
decimal a number that can represent a whole number and a fraction; a decimal point separates the whole number from the fraction (Lesson 4-1, p. 56)
;FSPTDBOCFBEEFE UPUIFSJHIUPGUIF MBTUEJHJUXJUIPVU DIBOHJOHUIFWBMVF
Compare 0.3 and 0.14 using models.
More of the first grid is shaded, so 0.3 > 0.14. Compare 2.73 and 2.37 using place value.
Copyright © by The McGraw-Hill Companies, Inc.
Line up the decimal points. Compare from left to right. So, 2.73 > 2.37.
2.73 2.37
same
7 tenths is greater than 3 tenths.
Practice all three methods for comparing decimals.
Example 1 Write 1.11, 1.21, and 0.13 in order from least to greatest. Check your answer by graphing the decimals on a number line. 1. Compare the numbers two at a time. 1.11 and 1.21: The digits in the tenths places are different. 1.21 and 0.13: The digits in the ones places are different. 0.13 and 1.11: The digits in the ones places are different. 2. Write the numbers from least to greatest.
1.11 < 1.21 1.21 > 0.13 0.13 < 1.11 0.13, 1.11, 1.21
3. Check by graphing the decimals on a number line.
GO ON Lesson 4-3 Compare and Order Decimals
73
YOUR TURN! Write 4.55, 4.15, and 5.14 in order from least to greatest. Check your answer by graphing the decimals on a number line. 1. Compare the numbers two at a time.
4.55 and 4.15 : The digits in the 4.55 > 4.15
tenths
places are different.
4.15 and 5.14 : The digits in the 4.15 < 5.14
ones
places are different.
5.14 and 4.55 : The digits in the 5.14 > 4.55
ones
places are different.
2. Write the numbers from least to greatest.
4.15, 4.55, 5.14 3. Check by graphing the decimals on a number line.
Example 2
YOUR TURN!
1. Model each decimal.
Compare 1.5 and 1.45 using decimal models. Write a statement using the symbols <, =, or >. 1. Model each decimal.
2. Compare the shaded regions.
2. Compare the shaded regions.
1.12 < 1.2 74
Chapter 4 Positive and Negative Fractions and Decimals
1.5 > 1.45
Copyright © by The McGraw-Hill Companies, Inc.
Compare 1.12 and 1.2 using decimal models. Write a statement using the symbols <, =, or >.
Example 3
YOUR TURN! Compare 26.85 and 26.78 using place value. Write a statement using the symbols <, =, or >.
Compare 18.45 and 18.46 using place value. Write a statement using the symbols <, =, or >. 1. Line up the decimal points. Start at the left. Compare digit to digit until you find the place where the numbers are different. 2. 18.45 18.46
5<6
1. Line up the decimal points. Start at the left. Compare digit to digit until you find the place where the numbers are different. 2. Tens place: 2 = 2 Ones place: 6 = 6 Tenths place: 8 > 7
5 hundredths is less than 6 hundredths.
3. Write a statement using an inequality symbol. 18.45 < 18.46
3. Write a statement using an inequality symbol.
26.85 > 26.78
Who is Correct?
Copyright © by The McGraw-Hill Companies, Inc.
Write 6.04, 1.6, and 6.4 in order from greatest to least.
Mason
Mateo
Lawana
1.6, 6.04, 6.4
6.04, 6.4, 1.6
6.4, 6.04, 1.6
Circle correct answer(s). Cross out incorrect answer(s).
Guided Practice Write the numbers in order from least to greatest. Check your answer by graphing the decimals on a number line. 1
5.06, 5.6, 6.5
5.6, 6.5, 5.06
2
0.6, 1.6, 0.60
0.06, 0.6, 1.6
Lesson 4-3 Compare and Order Decimals
75
Compare each pair of decimals using models. Write <, =, or > in each circle to make a true statement. 3
0.24 and 0.42
4
0.33 and 0.3
0.24 < 0.42
0.33 > 0.3
Step by Step Practice 5
Compare 34.430 and 34.45. Write <, =, or > in the circle to make a true statement.
Write <, =, or > in each circle to make a true statement. 6
34.16 and 35.15 Which place is different?
ones place
34.16 < 35.15 7
1.03 < 1.13
8
0.55 > 0.5
9
72.340 < 72.430
10
17.80 > 17.180
76
Chapter 4 Positive and Negative Fractions and Decimals
Copyright © by The McGraw-Hill Companies, Inc.
Step 1 Line up the decimal points. Start at the left. Compare digit to digit until you find the place where the numbers are different. 3=3 Tens place: 34.430 4=4 Ones place: 34.45 4=4 Tenths place: Hundredths place: 3 < 5 Step 2 Which place is different? hundredths place Step 3 Write the inequality. 34.430 < 34.45
Step by Step Problem-Solving Practice
Problem-Solving Strategies ✓ Use a graph.
Solve. 11
SCIENCE In science class, students placed a seed in a dish in a dark closet. They placed another seed the same size in a dish on a window sill.
Use logical reasoning. Guess and check. Solve a simpler problem. Work backward.
After 5 days, the seed in the light became a plant that was 1.73 centimeters tall. The seed in the dark became a plant that was 0.91 centimeter tall. Which seed became the taller plant? Understand
Read the problem. Write what you know. After 5 days, the seed in the light became a plant that was 1.73 centimeters tall. The seed growing in the dark became a plant that was 0.91 centimeter tall.
Plan
Pick a strategy. One strategy is to use a graph. Use a number line from 0 to 2. Divide the number line from 0 to 1 into 10 equal parts. Do the same between 1 and 2.
Plot 1.73 and 0.91 on the number line. 0.91 is closer to zero than 1.73 . 1.73 > 0.91.
Copyright © by The McGraw-Hill Companies, Inc.
Solve
The seed growing in the
became the taller plant.
You can use base-ten blocks to check your answer.
Check
12
light
TRAVEL Marcus can walk two different ways to get to his aunt’s house. One way is 1.42 miles. The other route is 1.24 miles. Which 1.24-mile route route is the shortest? Check off each step.
✔
Understand
✔
Plan
✔
Solve
✔
Check GO ON Lesson 4-3 Compare and Order Decimals
77
13
TUTORING Esteban was tutoring Tammy. He made a decimal model by drawing a square divided into ten strips. He shaded 3 of the strips. What decimal did he model? 0.3 Explain how he could model 0.25.
Sample answer: Esteban would need a decimal model with 100 squares. He would shade 25 of the 100 squares. 14
You have used place value, base-ten blocks, and number lines to compare and order decimals. How can you use what you have learned to compare and order decimals with money (dollars and cents)?
Money is represented with decimals. Dollars are the whole numbers, and the cents amount is the fraction of the dollar or the numbers to the right of the decimal point.
Skills, Concepts, and Problem Solving Write <, =, or > in each circle to make a true statement. Check your answer by graphing the decimals on a number line. 15
22.06 < 22.16
Copyright © by The McGraw-Hill Companies, Inc.
16
0.17 > 0.130
Write the numbers in order from least to greatest. Check your answers by graphing the decimals on a number line. 17
75.02, 75.20, 74.20
78
18
98.99, 100, 98.10
Chapter 4 Positive and Negative Fractions and Decimals
Compare each set of decimals using models. Write <, =, or > in each circle to make a true statement. 19
0.52 < 0.56
20
0.07 < 0.45
Write <, =, or > in each circle to make a true statement. 21
45.38 < 45.89
22
5.04 < 5.1
23
0.62 > 0.26
24
65.13 > 6.51
25
234.95 < 236
26
890.74 > 890
27
3.28 < 3.82
28
0.91 < 0.99
Solve.
Copyright © by The McGraw-Hill Companies, Inc.
29
FITNESS Jamie started on a 3-times-a-week running program. On Monday she ran 0.75 mile. On Tuesday she was able to run 0.72 mile. Thursday she ran 1.04 miles. Order the days from the day Jamie ran the longest distance to the day she ran the shortest distance.
Thursday, Monday, Tuesday 30
HOMEWORK Harrison studies 0.85 hour after school, 0.35 hour after supper, and 0.5 hour after his favorite TV show. Order the times Harrison spends studying from least to greatest.
0.35, 0.5, 0.85 31
TRANSPORTATION The chart shown at the right shows the costs of subway, bus, and taxi trips from Ramona’s apartment to the county library. Order the costs from least to greatest.
$1.25, $1.95, $6.20
Transportion
Cost
Subway
$1.95
Bus
$1.25
Taxi
$6.20
Which choice is the least expensive?
bus GO ON Lesson 4-3 Compare and Order Decimals
79
Vocabulary Check Write the vocabulary word that completes each sentence. 32
33
34
The first digit to the right of a decimal point is the tenths place. The second digit to the right is the hundredths place.
decimal point A(n) fraction.
separates the whole number from the
Writing in Math Explain how you would order 2.32, 2.23, and 2.22 on a number line. What process would you use to make the number line? How would you label it? Now follow the procedures you have written. Describe how to plot the points.
Descriptions will vary. Sample answer: Draw a number line from 2 to 3 and plot the decimals. Since 2.2 is farthest to the left with 2.23 next, 2.32 > 2.23 > 2.22.
Spiral Review Solve. Write in simplest form. 35
(Lesson 3-6, p. 34)
Copyright © by The McGraw-Hill Companies, Inc.
EARTH SCIENCE Bobbi was growing fungus in a Petri dish. On 1 of its adult length. On Tuesday, it Monday, the fungus grew __ 3 2 of its adult length. On Wednesday, the fungus reached its grew __ 9 adult length. How much did the fungus grow on Wednesday?
_4 of its adult length 9
36
COMMUNITY SERVICE Terrance and Juan collected money to 15 donate to a charity. In the first week, they collected ___ of their goal. 24 What fraction represents how much more they want to collect?
_9 = _3 24
8
Write each mixed number as an improper fraction. (Lesson 2-2, p. 44) 35 20 3 2 37 4 __ = 38 6 __ = 8 3 8 3
_
80
_
Chapter 4 Positive and Negative Fractions and Decimals
39
4= 8 __ 5
44 _ 5
Lesson
4-4 Compare and Order Fractions and Decimals KEY Concept Fractions and decimals can be compared and ordered on a number line. On the number line, the values farthest to the left are the lesser numbers. The values farthest to the right are the greater numbers.
VOCABULARY fraction a number that represents part of a whole or part of a set 1 1 1 3 Examples: __, __, __, __ 2 3 4 4
5NS1.5 Identify and represent on a number line decimals, fractions, mixed numbers, and positive and negative integers. 6NS1.1 Compare and order positive and negative fractions, decimals, and mixed numbers and place them on a number line. 4NS1.7 Write the fraction represented by a drawing of parts of a figure; represent a given fraction by using drawings; and relate a fraction to a simple decimal on a number line.
1 < 0.75 < 1 0 < 0.33 < __ 2 Fractions and decimals are easier to compare when you can change the fractions into decimals, or the decimals into fractions with common denominators.
(Lesson 1-1, p. 4)
decimal a number that can represent a whole number and a fraction; a decimal point separates the whole number from the fraction
Memorize the equivalent forms of common fractions and decimals. This will make it easier to compare them.
(Lesson 4-1, p. 56)
Example 1 Copyright © by The McGraw-Hill Companies, Inc.
Order
25 _3, -0.6, _ , and 0.34 from least to greatest. Graph the
4 50 numbers on a number line.
_
-0.6 = -6 10
1. Change each decimal to a fraction.
0.34 =
34 _ 100
2. Rewrite the fractions to have common denominators. The LCD of 4, 10, 50, and 100 is 100. × 10 -60 _ _ -6 = -6 _ = 10
10 × 10
× 25 75 _3 = 3_ _ =
100
4
4 × 25
100
25 ×2 25 50 _ _ _ = = 50
50 × 2
100
3. Use the numerators to order the numbers. The fractions in order from least to greatest are 34 _ 50 _ 75 -60 , _ _ , , . 100 100 100 100
4. Graph the numbers on a number line.
o o
o
GO ON
Lesson 4-4 Compare and Order Fractions and Decimals
81
YOUR TURN! Order
_2, -0.9, 1_1, and -1.82 from least to greatest. Graph the
5 4 numbers on a number line.
1. Change each decimal to a fraction. -0.9 =
-9 _
-1.82 =
10
_
-1 82 100
2. Rewrite the fractions to have common denominators. The LCD is 100 .
-90 -9 -9 × 10 = _____ ___ = ________
40 × 20 = ____ 2 = 2_______ __ 5
5 × 20
10
100
10 × 10
25
1 = 1____ 1__ 4 100
100
82 ____
-1.82 = -1 100
3. The fractions in order from least to greatest are
-90 _
_
-1 82 , 100
100
40 _
,
_
1 25 100
,
100
.
4. Graph the numbers on a number line. o
o o
Example 2
_
2. With all the numbers in decimal form, line up the decimal points. 3. Compare the digits in the ones places of the positive numbers. 4 is the greatest number.
.666 _2 = 3 2.0 3
4.0 -3.62 3.67 -3.14 3.14
-18 20 -18 20 -18 2
Notice that the decimal does not end, or terminate. In this case, round to the nearest hundredth. 2 So, __ ≈ 0.67 in 3 decimal form.
4. Compare the digits in the tenths places of the remaining positive numbers. 3.67 > 3.14 5. Compare the digits in the ones places of the negative numbers. They are the same. 6. Compare the digits in the tenths places of the negative numbers. -3.62 < -3.14 7. Write all of the numbers from least to greatest. 2 -3.62, -3.14, 3.14, 3.67, 4 or -3.62, -3.14, 3.14, 3__, 4 3 82
Chapter 4 Positive and Negative Fractions and Decimals
Copyright © by The McGraw-Hill Companies, Inc.
2 Order 4, -3.62, 3 , 3.14 and -3.14 from 3 least to greatest. 2 as a decimal. One way to find the 1. Write 3__ 3 2 is to divide the numerator decimal for __ 3 by the denominator.
YOUR TURN!
_
_ _ _
1 -2 ≈ 3 3 -2 = 4 3 2 = 4
_ _
3 3 1 Order -2 , 2.78, -2 , 2 , and 3.05 from least to greatest. 4 4 3 1. Write
1 -2_ , 3
3 _3 -2_ , and 2 4
4
as decimals.
2. With all of the numbers in decimal form, line up the decimal points. Then compare them using place value.
-2.33 -2.75 2.75
-2.33
3. Compare the digits in the ones places of the positive numbers. 3.05 is the greatest number.
2.78
-2.75
4. Compare the digits in the tenths places of the remaining positive numbers. They are the same.
2.75 3.05
5. Compare the digits in the hundredths places of the remaining positive numbers. 2.78 > 2.75 6. Compare the digits in the ones places of the negative numbers. They are the same. 7. Compare the digits in the tenths places of the negative numbers. -2.75 < -2.33
Copyright © by The McGraw-Hill Companies, Inc.
_
_ _
-2 3 , -2 1 , 2 3 , 2.78, 3.05 4 3 4
8. Write all of the numbers from least to greatest.
Who is Correct?
_1 in order from least to greatest.
Write -4, 4.6, and -4
5
Opal
Prem
_1 -4 , -4, 4.6
-4.2, -4, 4.6
5
Gigi 60 20 , -4, 4_ _ -4 100 100
Circle correct answer(s). Cross out incorrect answer(s).
Guided Practice Order the numbers by graphing them on a number line. 1
0.33, 0.25, 1.50, -0.10 ĕ ĕ
2
0.01, 0.7, 0.875, 1.20
Lesson 4-4 Compare and Order Fractions and Decimals
83
Step by Step Practice 3
5 4 , and 0.47 from least to greatest. Order __, -0.75, __ 5 6 Step 1 Change each decimal into a fraction with a denominator of 100.
47
75
0.47 = ______ 100
-0.75 = - ______ 100
2 . The decimal form of __ 2 , to the nearest 4 = __ Step 2 Simplify. __ 6 3 3 0.67 hundredth, is . Written as a fraction with a
67
denominator of 100, it is _____. 100 100 5 Step 3 __ = 1 or ______. Compare these fractions with a 5 100 denominator of 100 by comparing their numerators.
-75 <
47
67
<
100
<
Step 4 Write the numbers from least to greatest.
_4
-0.75 , 0.47 ,
6
_5
,
.
5
o
Step 5 Graph the numbers on a number line to verify their order.
/PUJDFUIBUoJTUIFGBSUIFTUPO UIFMFGU TPJUJTUIFMFBTUWBMVF
1 , 0.07 from least to greatest. Graph the numbers on a 1 , 1__ Order - __ 3 2 number line.
7
33
1 = - ______ -__ 3 100
_
150
- 1 < 0.07 < 3
1 = ________ 1__ 2 100
0.07 = ______ 100 o
o
Change all decimals to fractions. Then order the numbers from least to greatest. Graph the numbers on a number line. 5
6
84
_
-0.75, -0.67, - 1 2 2
1 -0.67, -0.75, -__
8 4 , 0.25, ___ __ 2
10
__
0.25, 8 , 4 10 2
o
o
o
o
Chapter 4 Positive and Negative Fractions and Decimals
_
11 2
Copyright © by The McGraw-Hill Companies, Inc.
4
o
7
1 , -__ 1 0.20, __ 8 4
__
_ _ _
- 25 < 125 < 20 or - 1 , 1 , 0.20 4 8 100 1000 100 o
Order the numbers from least to greatest. 8 8 8 0.82, -___, 1.89 , 0.82 , 1.89 10 10
_
10
3 __ , -0.72, 0.79 4
-0.72 ,
_3 4
, 0.79
_1
9
1 , 2.02 -2 , -2.1 , 2.02 -2.1, -2__ 5 5
11
5 1 , 1 -1 1 , 1__, -1__ 8 8 8
_
Step by Step Problem-Solving Practice
Draw a diagram.
FOOD The cafeteria served pizzas of equal size at lunch. 2 of another, and Nell ate 2 of one pizza. Flor ate __ Jorge ate __ 7 8 0.50 of another. Who ate the most? Who ate the least? Understand
✓ Use logical reasoning. Solve a simpler problem. Work backward. Act it out.
Read the problem. Write what you know.
_2
_2
Jorge ate , Flor ate 7 8 0.50 of equal-sized pizzas.
Copyright © by The McGraw-Hill Companies, Inc.
_
15 8
,
Problem-Solving Strategies
Solve. 12
1
, and Nell ate
Plan
Pick a strategy. One strategy is to use logical reasoning.
Solve
0.50 is ______ of a pizza. That is half. 10
5
1 4
2 = ______. __ 8
This is
less
than half.
2 2 have the same __ and __
numerator . 7 8 It represents the same number of pieces. If a pizza is cut in 8 pieces, then the pieces ______ < ______ < 0.50 smaller will be than a pizza cut into 7 pieces. What conclusions can you make? Nell So ate the most pizza and Flor ate the least.
2 8
Check
Does the answer make sense? Look over your solution. Did you answer the question?
2 7
GO ON
Lesson 4-4 Compare and Order Fractions and Decimals
85
13
ART There are three paintings at the museum. The first painting 3 is 0.78 yard long. The second painting is __ yard long. The third 4 7 yard long. Order the lengths of the paintings from painting is __ 8 7 yd, 0.78 yd, 3 yd greatest to least.
_
_ 4
8
Check off each step.
14
✔
Understand
✔
Plan
✔
Solve
✔
Check
3, 2 , -__ NUMBER SENSE Jeanine is putting the numbers -0.67, ____ 100 4 -3, 0.625 in order on a number line. From left to right, how will the numbers be ordered?
_
-3, - 3 , -0.67, 0.625, 0.02 4 15
Sometimes you have to put numbers in the same form in order to compare them. At other times you do not need to place them in the same form. Give reasons for each.
See TWE margin.
Order the numbers from least to greatest on a number line. 9 3 2 1 1 16 -___, -1.33, -__, __ 17 2.01, -___, __, 0.08 3 5 10 10 4 o
o
o
o
o
o
Order the numbers from least to greatest.
_ _ _ _ _ _ _ _
2 -1, 0.25, -0.06, __ 5
19
3 3 __ , 2, 0.67, __
86
Chapter 4 Positive and Negative Fractions and Decimals
4
_
- 100 < - 6 < 25 < 40 or -1, -0.06, 0.25, 2 5 100 100 100 100 67 < 75 < 150 < 200 or 0.67, 3 , 3 , 2 4 2 100 100 100 100
18
2
__
Copyright © by The McGraw-Hill Companies, Inc.
Skills, Concepts, and Problem Solving
Order the numbers from greatest to least. 20
4 , 1, __ 2 , 0.20 __
_4, 1, _2, 0.20
21
1 1 , -0.50, -0.05, -__ -__ 6 5
-0.05, - 1 , - 1 , -0.50 6 5
3
3
6
6
_ _
Solve. 22
3 GEOGRAPHY On a map, Conntown is __ inch from Maple City. 4 Bonny Village is 0.80 inch from Maple City. Quintin is 0.67 inches from Maple City. Which town is farthest from Maple City? Which is closest?
Bonny Village is farthest, and Quintin is closest. Vocabulary Check Write the vocabulary word that completes each sentence. 23
A number that can represent a whole number and a fraction is a(n) decimal .
24
A(n) a set.
25
Writing in Math Explain how to order negative numbers.
fraction
represents part of a whole or part of
Copyright © by The McGraw-Hill Companies, Inc.
See TWE margin.
Spiral Review Write a decimal for the amount shown. 26
(Lesson 4-2. p. 63)
27
$1.40 $14.60 28
4 of its yearly BUSINESS In December, a local gift shop has ___ 10 1 of its sales. How much sales. In May, the same gift shop has __ 5 more of the shop’s sales are done during December than May? (Lesson 3-6. p. 34)
_1 5
Lesson 4-4 Compare and Order Fractions and Decimals coins: United States Mint, bills: Michael Houghton/StudiOhio
87
Chapter
Progress Check 2
4
(Lessons 4-3 and 4-4)
Compare each pair of decimals using models. Write <, =, or > in each circle to make a true statement. 5NS1.5, 6NS1.1 1
0.23 < 0.32
2
1.01 < 1.10
Compare each pair of decimals using place value. Write <, =, or > in each circle to make a true statement. 5NS1.5, 6NS1.1 3
41.5 > 41.05
3.33 < 3.67
4
Write <, =, or > in each circle to make a true statement. Check your answer by placing the decimals on a number line. 5NS1.5, 6NS1.1 5
0.73 > 0.37
6
11.13 > 11.03
8
8 1 –0.125, 2.06, __, – __ 7 3
_ _ _ 8 , 2.06 1 , - 0.125, _ -_ 3
7
Solve. 5NS1.5, 6NS1.1 9
FITNESS Before lunch Nicki walks 0.33 mile. After lunch she 1 mile. Did she walk farther before or after lunch? walks another __ 5
before lunch
10
3 BUSINESS A gift shop sold __ of their picture frames last 4 month. If their goal was to sell at least 75% of their picture frames, did they meet their goal? Explain.
_
Yes; when you change 3 to a decimal, it is 0.75, which is 75%. 4 88
Chapter 4 Positive and Negative Fractions and Decimals
Copyright © by The McGraw-Hill Companies, Inc.
Order the numbers from least to greatest. 5NS1.5, 6NS1.1 3 7 3 - 3 , - 7 , - 3 , - 0.25 7 -__, -0.25, -__, -__ 5 5 8 3 3 8
Lesson
4-5 Add Decimals KEY Concept You can use models to add Each row or column represents one tenth or 0.1.
4NS2.0 Students extend their use and understanding of whole numbers to the addition and subtraction of simple decimals. 5NS2.0 Students perform calculations and solve problems involving addition, subtraction, and simple multiplication and division of fractions and decimals. 5NS2.1 Add, subtract, multiply, and divide with decimals; add with negative integers; subtract positive integers from negative integers; and verify the decimals. reasonableness of the results. 7NS1.2 Add, subtract, multiply, and Each square represents one divide rational numbers and take positive rational numbers to wholehundredth or 0.01. number powers.
VOCABULARY
10 tenths can be regrouped for one whole.
10 hundredths can be regrouped for one tenth.
tenths the first decimal place to the right of the decimal point; one of ten equal 1 parts or ___ 10 Example: In 3.6, 6 is in the tenths place. (Lesson 4-1, p. 56)
hundredths the second decimal place to the right of the decimal point; one of one hundred equal parts Example: In 4.57, 7 is in the hundredths place. (Lesson 4-1, p. 56)
Copyright © by The McGraw-Hill Companies, Inc.
One whole 10-by-10 grid represents 1 or 1.0.
Another way to add decimals is to write the numbers vertically. Line up the decimal points. Then add as with whole numbers. Bring the decimal point directly down to the sum.
Example 1
YOUR TURN! Add 0.4 + 0.3.
Add 0.2 + 0.7. 1. Write the addends in vertical format. Line up the decimal points.
0.2 + 0.7 −−−−−− 0.9
2. Add each column. 3. Write a decimal point in the sum.
1. Write the addends in vertical format.
0.4 + 0.3 −−−− 0.7
2. Add each column. 3. Write a decimal point in the sum. GO ON
Lesson 4-5 Add Decimals
89
Example 2 Find 2.53 + 0.07 using decimal models. Circle any regrouping. 1. Model each number.
2. Combine the models. Regroup 10 hundredths as 1 tenth.
IVOESFEUIT
3. Write the sum as a decimal. ⎫ ⎬ ⎭
2.53 + 0.07
= 2.6
YOUR TURN! Find 1.62 + 0.39 using decimal models. Circle any regrouping. 1. Model each number.
1.62
Copyright © by The McGraw-Hill Companies, Inc.
0.39
2. Combine the models. Regroup 10 hundredths as 1 tenth.
3. Regroup 10 tenths as 1.
90
Chapter 4 Positive and Negative Fractions and Decimals
⎫ ⎬ ⎭
⎫ ⎬ ⎭
1.62 + 0.39
= 2.01
Example 3 Find the sum of 14.17 and 8.3. 1. Write the addends vertically. Line up the decimal points. 1
3FHSPVQUFOT BTPOF
1 4. 1 8. 3 + 2 2. 4
Add a zero here so the place values line up.
7 0 7
2. Bring down the decimal point to the same place in the sum. 3. Add columns from right to left. 4. The sum of 14.17 and 8.3 is 22.47. YOUR TURN! Find the sum of 36.8 and 9.52. 1. Write the addends vertically. Line up the decimal points.
3 6. 8 0 9. 5 2 + 4 6. 3 2 2. Bring down the decimal point to the same place in the sum. Copyright © by The McGraw-Hill Companies, Inc.
3. Add from right to left. The sum of 36.83 and 9.52 is
46.32 .
Who is Correct? Find the sum of 54.68 and 13.7.
Lok
Monsa
54.68 + 13.7 560.5
54.68 × 13.7 749.116
Lawson 54.68 + 13.7 68.38
Circle correct answer(s). Cross out incorrect answer(s). GO ON Lesson 4-5 Add Decimals
91
Guided Practice Add using decimal models. 1
1.45 + 0.75 =
2.20
2
0.54
0.03 + 0.51 =
Step by Step Practice 3
7 3 3 2
Find the sum of 73.66 and 32.05. Step 1 Write the addends vertically. Line up the decimal points. Step 2 Bring down the decimal point to the same place in the sum.
+
1 0 5
.
6 6 0 5
.
7 1
.
Place the decimal here.
Find each sum. 4
1.1 + 0.17 = 1.27
+
1 0 1
5
.
1 0 1 7
.
2 7
.
22.03 + 0.54 −−−−−−
Write a zero as a placeholder. 6
22.57 8
44.4 + 3.53 −−−−−−
47.93 92
3.20 + 9.11 −−−−−−
7
12.31 9
23.02 + 79.34 −−−−−−−
102.36
Chapter 4 Positive and Negative Fractions and Decimals
78.3 + 67.18 −−−−−−−
145.48 10
5.92 + 58.24 −−−−−−−
64.16
Copyright © by The McGraw-Hill Companies, Inc.
Step 3 Add.
Step by Step Problem-Solving Practice
Problem-Solving Strategies
Solve. 11
BANKING Rhyan is saving for college. She had $101.82 in her savings account. She earned $55.15 one week and $43.89 the next. Rhyan put all the money that she earned into her savings account. How much does she have now? Understand
Draw a diagram. Act it out. Guess and check. Use logical reasoning. ✓ Solve a simpler problem.
Read the problem. Write what you know. Rhyan had $101.82 in her account. She deposited $55.15 and $43.89 into the account. Pick a strategy. One strategy is to solve a simpler problem.
Plan
Add the numbers two at time. Add the amounts she earned in two weeks: $55.15 + $43.89 = $99.04
Solve
Add the amount she earned to the amount already in the bank to find the total amount.
$99.04 + $101.82 = $200.86 Did you answer the question? You can check your answer by adding the three numbers in a different order.
Copyright © by The McGraw-Hill Companies, Inc.
Check
12
BUSINESS Marcos has a business mowing lawns in the summer. He mowed four lawns last week. He made $25, $15.50, $21.75 and $18.75 for the lawns. What were his weekly total earnings?
$81
Check off each step.
✔
Understand
✔
Plan
✔
Solve
✔
Check GO ON Lesson 4-5 Add Decimals
93
13
TRAVEL Corbin rode his bike 1.17 miles to visit a friend. He rode it 0.68 mile to the school on the way home and then went another 2.23 miles to the library. The distance he rode home from the library was 0.43 mile. How far did he ride altogether?
4.51 miles How is adding decimals like adding whole numbers? How is adding decimals different from adding whole numbers?
14
Sample answer: It is the same in that after you line up the place values, you add the columns from right to left just as you would add whole numbers. The difference is that the decimal point is placed in the sum.
Skills, Concepts, and Problem Solving Add using decimal models. 15
1.2 + 0.03 =
1.23
16
3.64 + 1.07 =
17
44.18 + 89.13
18
133.31 19
10.19 55.78 + 13.44
12.21 20
79.41
94
11.92 + 0.29
Chapter 4 Positive and Negative Fractions and Decimals
92.46 + 55.14
147.60
Copyright © by The McGraw-Hill Companies, Inc.
Find each sum.
4.71
21
0.06 + 0.17 =
0.23
22
11.03 + 4.5 =
15.53
23
17.6 + 8.07 =
25.67
24
4.67 + 3.6 =
8.27
25
23.89 + 45.23 =
26
33.01 + 5.1 =
38.11
27
3.06 + 5.9 =
28
7.02 + 12.8 =
19.82
29
15.76 32.04 + 8.5
30
3.77 28.44 + 11.03
69.12 8.96
56.30 31
152.9 65.08 + 3.0
220.98
43.24 32
137.4 73.09 + 8.0
218.49
Solve.
Copyright © by The McGraw-Hill Companies, Inc.
33
PHYSICS Aida and Kraig conducted an experiment using ramps. One ramp was 16.83 centimeters long. The other ramp was 5.6 centimeters long. They decided to put the two ramps together. How long was the combined ramp?
22.43 cm 34
EARTH SCIENCE In a science experiment, the angle of the sunshine coming through the window was measured monthly. In the first month, the angle measured 38.2 degrees. The next month, the angle increased by 21.03 degrees. The month after that, the angle increased 45.78 degrees. What was the total measure of the angle after the third month?
105.01 degrees 35
¡ ¡ ¡
FAMILY Clarence receives $9.25 each day he dog-sits for his neighbor. If he dog-sits 5 days, how much will he earn?
$46.25 GO ON Lesson 4-5 Add Decimals
95
Vocabulary Check Write the vocabulary word that completes each sentence. 36
37
38
decimal A(n) numbers and fractions.
is a number that can represent whole
When you add decimals, the first thing you do is line up the decimal points . Writing in Math Ethan wrote the equation: 8.9 + 14.03 = 23.2. How would you correct his mistake?
The answer should be 22.93. Ethan did not line up the place values correctly for 14.03. He mistakenly used 14.3.
Spiral Review Write <, =, or > in each circle to make a true statement. Check your answer by placing the decimals on a number line. (Lesson 4-3, p. 73) 39
12.06 < 12.16
40
(Lesson 3-4, p. 21)
41
FOOD Pete’s Pizza Place made 7 pizzas. Each serving 1 of a pizza. How many servings were there in the is __ 8 7 pizzas? 56
42
FOOD
1 of a pizza evenly between Paquito is trying to split __ 2 himself and his parents. How much of the pizza will each person 1 get? Use a diagram to justify your answer.
_
Divide a figure into halves and shade
6
1 part. Then divide that part into thirds. Each person gets 1 . 6
_
96
Chapter 4 Positive and Negative Fractions and Decimals
Copyright © by The McGraw-Hill Companies, Inc.
Solve.
0.08 < 0.80
Lesson
4-6 Subtract Decimals
4NS2.0 Students extend their use and understanding of whole numbers to the addition and subtraction of simple decimals. 5NS2.0 Students perform calculations and solve problems involving addition, subtraction, and simple multiplication and division of fractions and decimals. 5NS2.1 Add, subtract, multiply, and divide with decimals; add with negative integers; subtract positive integers from negative integers; and verify the reasonableness of the results. 7NS1.2 Add, subtract, multiply, and divide rational numbers and take positive rational numbers to wholenumber powers.
KEY Concept Decimal models can be used to subtract decimals .
Copyright © by The McGraw-Hill Companies, Inc.
Another way to subtract decimals is to write the numbers vertically to line up the decimal points. Subtract the numbers as you would with whole numbers. Bring the decimal point directly down to the difference. You cannot subtract 6 12 7.29 - 2.65 = 6 tenths from 2 7. 2 9 tenths. Regroup 1, - 2. 6 5 which equals 10 4. 6 4 7.29 - 2.65 = 4.64 tenths. Take 1 from ones place and add 10 to the tenths place.
VOCABULARY regroup to use place value to exchange equal amounts when renaming a number Example: 12 ones = 1 ten 2 ones difference the answer or result of a subtraction problem
When subtracting decimals, remember to add zeros as needed so that both numbers have the same place value.
Example 1 Find 1.77 – 0.9 using decimal models. 1. Model 1.77. 2. You need to take away 9 tenths, but there are not 9 rods in the model. Regroup. Separate the one whole into 10 tenths.
3. Subtract 0.9 or 9 tenths. The remaining blocks represent the difference. 4. The model of the difference has 8 tenths rods and 7 hundredths blocks. The difference is 0.87. 1.77 - 0.9 = 0.87
GO ON
Lesson 4-6 Subtract Decimals
97
YOUR TURN! Find 1.43 - 0.28 using decimal models. 1. Model 1.43.
1.43 8
2. You need to take away to regroup? yes Replace
1
hundredths units. Do you need
tenths rod with
10
hundredths units.
3. Subtract 0.28 . The remaining blocks represent the difference.
4. The model of the difference has 1 whole square, rod, and 5 units. The difference is 1.15 .
1
1.43 - 0.28 = 1.15
Example 2
YOUR TURN! Find 72.40 - 59.65. 6 3
16
11
1. Write the numbers vertically. Line up the decimal points.
4 7. 1 3 - 2 9. 7 0 1 7. 4 3 2. Bring down the decimal point to the same Add a zero so place in the difference. that the number of place values is 3. Subtract the the same. hundredths column. 4. You need to regroup to subtract the tenths column. Take one 1 and replace it with 10 tenths. 5. You need to regroup to subtract 9 from 6. Take one 10 and replace it with ten 1s.
1. Write the numbers vertically. Line up the decimal points.
13
2. Bring down the decimal point to the same place in the difference. 3. Subtract the hundredths column. Do you need to regroup? yes 4. Subtract the tenths column. Do you need to regroup? yes 5. Subtract the ones column. Do you need to regroup? yes 6. Subtract the tens column.
6. Subtract the tens column. 98
11
7 2. 4 - 5 9. 6 1 2. 7
Chapter 4 Positive and Negative Fractions and Decimals
10
0 5 5
Copyright © by The McGraw-Hill Companies, Inc.
Find 47.13 - 29.7.
Who is Correct? Find 20.57 - 16.98.
Gabriel
Theresa
Dory
1 141 \ \20.57 - 16.98
1
19 4 \ \20\ 57 - 1698
1
19 41 \ \20\ .57 - 16.98 3.59
0.0359
4.59
Circle correct answer(s). Cross out incorrect answer(s).
Guided Practice
Copyright © by The McGraw-Hill Companies, Inc.
Subtract using decimal models. 1
2.03 - 0.14 =
1.89
2
1.19 - 1.05 =
0.14
3
0.67 - 0.29 =
0.38
4
0.71 - 0.56 =
0.15
GO ON Lesson 4-6 Subtract Decimals
99
Step by Step Practice 5
Find 121.09 - 85.6. Step 1 Write the numbers vertically. Line up the decimal points. Add any zeros to line up place values. Step 2 Bring down the decimal point to the same place in the difference.
1 2 1 - 8 5
. .
0 9 6 0
3 5
.
4 9
Step 3 Subtract the hundredths column. Do you need to regroup? no Step 4 Subtract the tenths column. Do you need to regroup? yes Step 5 Subtract the ones column. Do you need to regroup? yes Step 6 Subtract the tens column. Do you need to regroup? yes
Subtract. 0.33 6 0.09 −−−−−
7
-
0 0 0
9
2.06 .45 −−−−−
1.61
100
8
16.12
.
3 3 0 9
.
2 4
.
10
$42.75 - $18.98
$23.77
Chapter 4 Positive and Negative Fractions and Decimals
14 3.78 −−−−−
10.22
11
$13.25 - $11.26
$1.99
Copyright © by The McGraw-Hill Companies, Inc.
0.24
19.77 -3.65 −−−−−
Step by Step Problem-Solving Practice
Problem-Solving Strategies Draw a diagram. Use logical reasoning. Act it out. Solve a simpler problem. ✓ Work backward.
Solve. 12
GIFTS Ivan received $20 from his grandmother for his birthday. He spent $5.13 on trading cards and $2.98 on a book. He then gave $5 to his brother for helping him clean the garage and put the rest of the money into his savings account. How much of the $20 did Ivan save? Understand
Read the problem. Write what you know.
$20 . He spent $ 5.13 , $ Ivan has 5.00 and $ , and then saved the rest. Pick a strategy. One strategy is to work backward.
Plan
You can begin with $20. Subtract each of the three amounts he spent. The final difference is the amount he saves.
Ivan saved
Copyright © by The McGraw-Hill Companies, Inc.
$14.87 -
$5.13 trading cards
$2.98
money left
$11.89 money left
book
$5 money to brother
=
$14.87
=
$11.89
=
$6.89
money left
money left
savings
$6.89
Check your answer by adding the amount he spent to the amount saved. The total should be $20.
Check
14
-
,
Fill in the blanks to find the amount he saved.
Solve
13
$20 birthday money
2.98
GARDENING Samili grows tomatoes. She cans them in the summer to use during the winter. She started with 10.25 pounds of tomatoes. She canned 1.75 pounds, 2.13 pounds, and 1.98 pounds. How many pounds of tomatoes did she not can? 4.39 lb Check off each step.
✔
Understand
✔
Plan
✔
Solve
✔
Check
COMMUNITY SERVICE Ian and Tito participated in a walk-a-thon to raise money for charity. Together, they walked a total of 33.08 miles. Their goal was 20.5 miles. How much farther than their 12.58 miles goal did they walk?
GO ON
Lesson 4-6 Subtract Decimals
101
Write a few sentences for a student who was absent from class explaining how to decide whether you have to regroup when subtracting decimals.
15
Sample answer: After the numbers are written vertically, look at each column of numbers. If the second number in the column is greater than the first number, then you have to regroup.
Skills, Concepts, and Problem Solving Subtract using decimal models. 16
1.11 - .99 =
.12
17
2.13 - 0.03 =
2.10
Subtract. 18
1.0 0.73 −−−−−
19
0.27
14.89 0.7 −−−−−
20
14.19
88.06 1.75 −−−−−
21
86.31
9.75
0.17 - 0.06 =
0.11
23
11.03 - 4.5 =
6.53
24
17.6 - 8.07 =
9.53
25
25.4 - 7.98 =
17.42
26
CIVICS In a local election, one candidate received 24.85 percent of the vote. Another candidate received 33.13 percent. What percent more votes did the second 8.28 candidate receive than the first?
27
FOOD The cafeteria had 36 cups of flour at the beginning of the week. One day 10.33 cups of flour were used, and another day 5.25 cups were used. On Friday, the chef wanted to make a recipe yes that needed 16.75 cups of flour. Did she have enough? If so, how much flour was left at the end of the day on Friday?
3.67 cups
102
Chapter 4 Positive and Negative Fractions and Decimals
Copyright © by The McGraw-Hill Companies, Inc.
22
Solve.
20.96 11.21 −−−−−−
28
ENGINEERING An engineer designed a machine that weighed 2.75 tons. The owner of the factory that will use the machine wanted it to weigh 0.06 ton less. How many tons did the owner want the machine to weigh? 2.69 tons
Vocabulary Check Write the vocabulary word that completes each sentence. 29
regroup When you rename a number.
30
The answer to a subtraction problem is called the
31
Writing in Math Write about a real-life situation in which you would subtract decimals.
, you exchange equal amounts to
difference
.
Answers will vary. Sample answer: making a withdrawal from a bank account
Spiral Review Solve. 32
(Lesson 4-5, p. 89)
SCIENCE A compound contains 0.08 liter of acid, 1.16 liters of water, and 0.78 liter base. How many total liters are in the compound?
Copyright © by The McGraw-Hill Companies, Inc.
2.02 L 33
ASTRONOMY A comet traveled across the sky 0.16 million miles from Earth. Five years later the same comet was 0.07 million miles farther from Earth. How far was the comet from Earth the second time?
0.23 million miles
Add. Write the sum in simplest form. (Lesson 3-5, p. 28) 23 6 10 24 4 7 =1 34 ___ + ____ = 35 ___ + ___ = 10 100 50 12 24 24
_
_
Divide. Write the quotient in simplest form. (Lesson 3-4, p. 21) 3 1 9 1 2 3 37 __ ÷ __ = 38 __ ÷ __ = 2 3 4 8 3 8
_
_
36
3 1= ___ + __
_7
39
5 2= ___ ÷ __
35 _
16
12
4
7
16
24
Lesson 4-6 Subtract Decimals
103
Chapter
4
Progress Check 3
(Lessons 4-5 and 4-6)
Add using decimal models. 4NS2.0, 5NS2.0, 5NS2.1, 7NS1.2 1
0.07 + 0.94 =
1.01
2
0.11 + 1.03 =
1.14
Add. 4NS2.0, 5NS2.0, 5NS2.1, 7NS1.2 3
4.95 + 56.03 =
60.98
4
13.07 + 0.29 =
13.36
5
6.92 + 11.08 =
18
6
24.85 + 4.76 =
29.61
Subtract using decimal models. 4NS2.0, 5NS2.0, 5NS2.1, 7NS1.2 7
0.17 - 0.10 =
0.07
8
1.03 - 0.2 =
0.83
9
48.09 - 10.7 =
37.39
10
8.11 - 0.73 =
7.38
11
78.03 - 19.5 =
58.53
12
34.13 - 6.89 =
27.24
Solve. 4NS2.0, 5NS2.0, 5NS2.1, 7NS1.2 13
ART A fifth-grade class is making a mural. Three groups of students are creating different parts of the mural. Then the three parts will be combined to make the mural. One group makes a piece 3.50 yards long. The other two groups make pieces that are each 4.22 11.94 yards yards long. How long will the mural be?
14
MONEY Sharika and her friend set up lemonade stands at opposite sides of the neighborhood. Sharika made $43.12. Her friend made $36.83. How much more did Sharika make?
$6.29
104
Chapter 4 Positive and Negative Fractions and Decimals
Copyright © by The McGraw-Hill Companies, Inc.
Subtract. 4NS2.0, 5NS2.0, 5NS2.1, 7NS1.2
Lesson
4-7 Multiply Decimals
5NS2.0 Students perform calculations and solve problems involving addition, subtraction, and simple multiplication and division of fractions and decimals. 7NS1.2 Add, subtract, multiply, and divide rational numbers (integers, fractions, and terminating decimals) and take positive rational numbers to KEY Concept whole-number powers. 5NS2.1 Add, subtract, multiply, and divide with decimals; add with To multiply decimals , multiply the numbers as if they were negative integers; subtract positive whole numbers. Then place the decimal point in the product . integers from negative integers; and verify the reasonableness of the results.
336 × 22 −−−−− 672 + 6720 −−−−− 7392
Count the number of places to the right of the decimal point in each factor. 3. 3 6 0. 2 2 2 places
VOCABULARY factor a number that divides into a whole number evenly; also a number that is multiplied by another number Example: 5 and 7 are factors of 35.
2 places
Add the number of decimal places in each factor . This is the number of decimal places that needs to be in the product. 0.7 3 9 2
Begin on the right side.
(Lesson 3-3, p. 15)
Count left 4 places. Place the decimal here.
product the answer or result of a multiplication problem
The product of 3.36 and 0.22 is 0.7392.
exponent a number that indicates how many times a number or expression is to be multiplied by itself Example: In the equation 3 2 = 9, the exponent is 2.
When a number is raised to a power, you must count the decimal places in all of the factors.
Copyright © by The McGraw-Hill Companies, Inc.
⎧ ⎨ ⎩
1.2 3 = 1.2 × 1.2 × 1.2 = 1.728
⎧ ⎨ ⎩
1.2 2 = 1.2 × 1.2 = 1.44 Count 2 decimal places.
Count 3 decimal places.
Unlike adding and subtracting decimals, you cannot place the decimal point in the answer until the last step.
Example 1 Find 7.53 × 0.7. 753 × 7 = 5271
1. Multiply the factors, ignoring the decimal points for now. 2. Begin to the right of each factor. Count the number of places from the right to the decimal point.
7.5 3
0.7
3. Add the number of decimal places for both factors. 2 + 1 = 3
2 places
1 place
4. Begin to the right of the product. Count left 3 places. Place the decimal point there.
5.2 7 1
5. The product is 5.271.
7.53 × 0.7 = 5.271
3 places
GO ON
Lesson 4-7 Multiply Decimals
105
YOUR TURN! Find 22.86 × 0.05. 1. Multiply the factors, ignoring the decimal points for now.
2286
5
×
=
11430
2. For each factor, count the number of places from the right to the decimal point. 3. Add the number of decimal places for both factors. 2 2 4 + =
4
4. Begin to the right of the product. Count left Place the decimal point there. 5. The product is
1.1430 .
22.86 × 0.05 =
places.
1.143
Example 2 Find 1,647 × 0.09. 1647 × 9 = 14823
1. Multiply the factors, ignoring the decimal points for now. 2. Begin to the right of each factor. Count the number of places from the right to the decimal point.
1647.
0.09
3. Add the number of decimal places for both factors. 0 + 2 = 2 0 places
1,647 × 0.09 = 148.23
5. The product is 148.23 YOUR TURN! Find 92 × 0.7.
1. Multiply the factors, ignoring the decimal points for now.
92
×
7
=
644
2. For each factor, count the number of places from the right to the decimal point. 3. Add the number of decimal places for both factors. 0 1 1 + =
1
4. Begin to the right of the product. Count left Place the decimal point there. 5. The product is 106
64.4
.
92 × 0.7 =
64.4
Chapter 4 Positive and Negative Fractions and Decimals
place.
2 places Copyright © by The McGraw-Hill Companies, Inc.
4. Begin to the right of the product. Count left 2 places. Place the decimal point there.
Example 3
YOUR TURN! Multiply 8 × 0.6 2.
2
Find 5 × 1.4 . 1. Simplify the factor with the exponent first.
1. Simplify the factor with the exponent first.
0.6
1.4 × 1.4 = 1.96
1.4 has 1 decimal place. 1.4 has 1 decimal place. So the product will have 2 decimal places. 2. Multiply, ignoring the decimal point for now. 5 × 196 = 980
5 has 0 decimal places. 4. Begin to the right of the product. Count left 2 places. The product is 9.8
=
0.36
2. Multiply, ignoring the decimal point for 8 36 now. × =
288
0.36 8
3.
3. 1.96 has 2 decimal places.
0.6
×
has 2 decimal places. has 0 decimal places.
4. Begin to the right of the product. Count left 2 places. The product is 2.88. 8 × 0.6 2 =
2.88
:PVDBOPNJU UIFmOBMJOUIF QSPEVDU
5 × 1.4 2 = 9.8
Who is Correct?
Copyright © by The McGraw-Hill Companies, Inc.
Multiply 3.16 × 0.02.
Abir
Gracie
3.16 2 x−−0.0 −−−− 622
3.16 2 x−−0.0 −−−− 632
0.0622
0.0632
Teo 3.16 2 x−−0.0 −−−− 632 0.632
Circle correct answer(s). Cross out incorrect answer(s).
Guided Practice Multiply. 1
0.03 × 7.0 = 0.03 × 7.0 −−−−
0.21
2 0 2
0.21 decimal places decimal places decimal places
2
4.076
10.19 × 0.4 = 10.19 × 0.4 −−−−
4.076
2 1 3
decimal places decimal places decimal places
Lesson 4-7 Multiply Decimals
107
Step by Step Practice 3
Find 223.01 × 11.9. Step 1 Multiply the factors, ignoring the decimals. Step 2 Begin at the right. Count the number of decimal places in both factors.
3
Step 3 Begin at the right of the product. Move left places.
2 2 3 . 0 1 1 1 . 9 × 2 6 5 3 223.01 × 11.9 =
.
2
decimal places
1
decimal place
8 1 9 2,653.819
Multiply. 4
10.1 × 4.0
. .
1 0
4 0
.
4 8.1267
6
43.01 × 11.1 = 477.411
8
83.2 × 5.09 = 423.488
10
9.1 × 3.5 2 = 111.475
100
12
3 2 ×4 2 =
144
7.2
14
6 × 2.1 3 =
55.566
5
1.03 × 7.89 =
7
77.55 × 0.03 =
9
2.5 × 1.3 2 =
11
22 × 52 =
13
5 × 1.2 2 =
108
2.3265 4.225
Chapter 4 Positive and Negative Fractions and Decimals
Copyright © by The McGraw-Hill Companies, Inc.
1 0 4 ×
Step by Step Problem-Solving Practice
Problem-Solving Strategies Draw a diagram.
Solve. 15
✓ Use logical reasoning.
MONEY Evita is saving change to buy bottled water after gymnastics class for herself and two of her friends. The water costs $0.75. How many quarters will Evita need? Understand
Act it out. Solve a simpler problem. Work backward.
Read the problem. Write what you know.
3 Evita needs times $0.75 water for herself and her friends.
to buy
Pick a strategy. One strategy is to use logical reasoning.
Plan
Read the problem. You can find how much money she needs by multiplying. Then figure out how many quarters make up the amount. Multiply to find how much money she needs.
Solve
3
×
0.75
=
$2.25
A quarter is $0.25. Four quarters equal $1. Eight 8 quarters equal $2. So, $2.25 will be 1 9 quarters plus more, or quarters. Does the answer make sense? Think about the problem a different way. Each bottle costs 3 quarters. Three bottles cost 9 quarters.
Copyright © by The McGraw-Hill Companies, Inc.
Check
16
BABY-SITTING Chloe charges $5.75 an hour to baby-sit. She baby-sat last night for 3.25 hours. How much did she earn? Round to the nearest cent if necessary. $18.69 Check off each step.
✔
UNDERSTAND
✔
Plan
✔
Solve
✔
Check
To round to the nearest cent, look at the place to the right of the hundredths place in the product. If the digit is 5 or above, round up. If the digit is less than 5, the digit to be rounded stays the same.
GO ON Lesson 4-7 Multiply Decimals
109
17
EARTH SCIENCE Lamar is growing a plant in botany class. He finds that when he keeps the living conditions the same, the plant grows 0.35 centimeters each week. How much will it grow in 3.5 weeks if Lamar continues to keep the living conditions the same?
1.225 cm
Explain the final step in multiplying decimals.
18
Count the number of decimal places in the factors. EARTH SCIENCE
Place that number of decimal places in the product.
Plants require certain living conditions, such as sunlight and water.
Skills, Concepts, and Problem Solving Multiply. 19
0.05 × 80 −−−−
4.00 21
4
0.05 × 80 =
2 0 2
20
19.8 × 0.6 −−−−
25
0.15 × 0.23 −−−−−
28
12.32 × 6 −−−−
73.92
110
23
10.19 × 3.3 −−−−
26
0.32 × 0.16 −−−−−
29
120.05 × 4 −−−−−
480.2
Chapter 4 Positive and Negative Fractions and Decimals
Suzahhah Skelton
0.06 × 11.3 −−−−−
36.1 × 1.9 −−−−
68.59
0.0512
31
decimal places decimal places decimal places
0.678
33.627
0.0345
30
0.009 × 4.23 −−−−−
0.03807
11.88
27
26.64
1 1 2
9.45 × 8 −−−−
75.6
32
147.02 × 5 −−−−−
735.1
Copyright © by The McGraw-Hill Companies, Inc.
22
0.01491
24
3.6 × 7.4 −−−−
decimal places decimal places decimal places
0.007 × 2.13 −−−−−
3.6 × 7.4 = 27.4456
57.3
33
10 × 5.73 =
35
17.6 × 99.78 = 1,756.128
37
1.6 × 4 3 =
39
12.8 × 2.2 2 =
102.4
61.952
$35.70
34
$11.9 × 3 =
36
13.9 × 55.14 = 766.446
38
5.2 × 2 4 =
40
24.8 × 5.1 2 = 645.048
83.2
Solve. 41
COOKING Lawrence is preparing the recipe at right. He is tripling the recipe. How much of each ingredient does he need?
He needs 3.75 cups of noodles, 6.99 lbs of beef, and 14.25 cups of broth.
Copyright © by The McGraw-Hill Companies, Inc.
42
-ARRYsS#EEFAND/OODLES CUPSDRYNOODLES LBSSHREDDEDBEEF CUPSOFBEEFBROTH
MONEY Tyrell wants to save $200 this summer to use for school clothes in the fall. He saves $17.45 a week for 10.5 weeks. How much will he have at the end of the 10.5 weeks? Will he reach his goal?
$183.23. He will not reach his goal.
Vocabulary Check Write the vocabulary word that completes each sentence. 43
44
factor A(n) another number.
is a number that is multiplied by
When you multiply two factors, the answer is the product .
GO ON Lesson 4-7 Multiply Decimals
111
45
Writing in Math How do you determine where to put the decimal point in a product?
When you are done multiplying the numbers, you count the total number of decimal places in each of the factors and put the decimal point in the product that many decimal places from the right.
Spiral Review Write a decimal and a fraction in simplest form for each number.
_
(Lesson 4-1, p. 56)
46
nineteen and seven hundredths
47
three tenths
48
ten and ten hundredths
_
0.3, 3 10
19.07, 19 7 100
_
10.10, 10 1 10
Write each fraction in simplest form. Divide by the GCF. (Lesson 2-5, p. 69) 3 1 75 16 2 30 3 4 49 ____ = 50 ___ = 51 ___ = 52 ___ = 50 5 100 4 24 3 16 4
_
Solve.
_
_
(Lesson 2-4, p. 59)
1 of her allowance on movie ENTERTAINMENT Gracia spends __ 4 1 on books. She saves the rest. On what 1 on clothes, and __ tickets, __ 5 4 does she spend the least?
Copyright © by The McGraw-Hill Companies, Inc.
53
_
books 54
GARDENING A gardener grows produce to sell each summer. 2 of her 1 of her total crop. In July, she sold __ In June, she sold __ 3 5 total crop. In August, she sold the rest. Which month did the gardener sell the least amount of her crop? The most?
112
July
Chapter 4 Positive and Negative Fractions and Decimals
Jeff Greenberg/PhotoEdit
August
. ENTERTAINMENT Gracia spends part of her allowance on books each month.
Lesson
4-8 Divide Decimals
5NS2.0 Students perform calculations and solve problems involving addition, subtraction, and simple multiplication and division of fractions and decimals. 7NS1.2 Add, subtract, multiply, and divide rational numbers (integers, fractions, and terminating decimals) and take positive rational numbers to KEY Concept whole-number powers. 5NS2.1 Add, subtract, multiply, and divide with decimals; add with You can divide decimals in different ways. One way is to negative integers; subtract positive round the divisor and dividend to numbers that will give integers from negative integers; and verify the reasonableness of the you an estimate of the quotient . Then divide just like whole results.
numbers. Your estimate will help you decide where to place the decimal. 7.98 ÷ 2.1 ≈ 8 ÷ 2 ≈ 4
Another way to divide decimals is to multiply the divisor by a power of 10 to change it to a whole number. Multiply the dividend by the same power of ten. Then place the decimal in the same place in the quotient as it is in the dividend. dividend 7.98 ×100 798
÷ ÷
divisor 2.1 ×100 210
= 3.8
VOCABULARY divisor the number by which the dividend is being divided Example: In 3 9 , 3 is the divisor (Lesson 3-4, p. 21) dividend a number that is being divided Example: In 3 9 , 9 is the dividend. (Lesson 3-4, p. 21) quotient the answer to a division problem Example: In 6 ÷ 3 = 2, 2 is the quotient. (Lesson 3-4, p. 21)
Copyright © by The McGraw-Hill Companies, Inc.
Example 1 Divide. Place the decimal point by estimation.
19 142.31
1. Round the dividend and divisor to make an estimate. 19 is close to 20.
142.31 is close to 140.
140 ÷ 20 = 7. The quotient will be about 7. 2. Divide without regard to the decimal points. 3. Place the decimal in the quotient so the answer is about 7. The quotient is 7.49. 4. You can check your answer by multiplying the quotient by the divisor. 7.49 × 19 = 142.31
7.49 19 142.31 - 133 93 - 76 171 - 171 0
GO ON Lesson 4-8 Divide Decimals
113
YOUR TURN! Divide. Place the decimal point by estimation.
28 120.96
1. Round the dividend and divisor to make an estimate. 28 is close 30 120 . to . 120.96 is close to 120 30 4 ÷ = . The quotient will be about 4 . 2. Divide without regard to the decimal points.
4
3. Place the decimal in the quotient so the answer is about 4.32 . The quotient is
.
432 120.96 28 - 112 89 - 84 56 - 56 0
4. Check your answer by multiplying the quotient by the divisor.
4.32
×
28
=
120.96
Example 2 Divide. Multiply to make the divisor a whole number.
0.5 39.55
1. To eliminate the decimal in the divisor, multiply the divisor by 10. 10 × 0.5 = 5.0
Place the decimal point in the quotient directly above where it is in the dividend.
YOUR TURN! Divide. Multiply to make the divisor a whole number.
3.2 22.72
7.1 1. To eliminate the decimal in the divisor, multiply the divisor by 10. 10 × 3.2 = 32.0 2. Multiply the dividend by 10. 10 × 22.72 = 227.2 3. Divide. 4. The quotient is 7.1 . Check your answer by multiplying. 114
Chapter 4 Positive and Negative Fractions and Decimals
32 227.2
-224 32 32 0
Copyright © by The McGraw-Hill Companies, Inc.
When you multiply by 10, the decimal moves to the right 1 place. 79.1 5 395.5 2. Multiply the dividend by the same power of 10. - 35 10 × 39.55 = 395.5 45 - 45 The decimal moves 1 place to the right. 05 3. Divide. -5 0 4. The quotient is 79.1. Check your answer by multiplying.
Who is Correct? Divide. Multiply to make the divisor a whole number. 9.44 ÷ 3.2
Darnell
Wyatt
Aneisa
9.44 ÷ 3.2 = (10)9.44 ÷ 3.2(10) = 94.4 ÷ 32 = 2.95
9.44 ÷ 3.2 = (100)9.44 ÷ 3.2(10) = 944 ÷ 32 = 29.5
9.44 ÷ 3.2 = = (100)9.44 ÷ 3.2(100) 944 ÷ 320 = 295
Circle correct answer(s). Cross out incorrect answer(s).
Guided Practice Divide. Place the decimal by estimation.
0.21 1
37 7.77
ɄNFBOT BCPVUFRVBMUP
0.75 40 37 ≈ 8 7.77 ≈ Estimate ≈
2
51 38.25
50 51 ≈ 40 38.25 ≈ 0.8 Estimate ≈
0.2
Step by Step Practice Copyright © by The McGraw-Hill Companies, Inc.
Divide. Multiply to make the divisor a whole number. 3
Step 1
0.75 3.90
Multiply the divisor by 100. The decimal point will move to the right two places. 100 × 0.75 =
75
Step 2 Multiply the dividend by the same power of 10. 100 × 3.90 =
390
Step 3 Divide. Move the decimal point directly up from its position in the dividend. Step 4 To check, multiply. 5.2 × 0.75 = 3.90
2
5.
7 5 3 9 0 3 7 5 1 5 1 5
.
Write a zero so you can keep dividing.
0
0 0
0
GO ON Lesson 4-8 Divide Decimals
115
Divide. 4
12.2 45.75
10
12.2 × 45.75 ×
10
= =
122 457.5
3 . 7 5 1 2 2 4 5 7 ·ç 5 0 0.66 5
2.4 1.584
19.6 6
90 7
0.07 6.3
28.1 8
38.20 9
1.5 57.30
5.2 101.92
32.3 907.63
61.3 10
9.8 600.74
83.2 474.24 12 5.7
4.9 13
116
1.21 5.929
7.6 14
Chapter 4 Positive and Negative Fractions and Decimals
0.42 3.192
Copyright © by The McGraw-Hill Companies, Inc.
31.4 91.06 11 2.9
Step by Step Problem-Solving Practice
Problem-Solving Strategies Draw a diagram.
Solve. 15
✓ Use logical reasoning.
GAMES Vito is purchasing 15 game tokens for a total cost of $3.75. How much does each token cost? Understand
Read the problem. Write what you know. 15 tokens cost $3.75.
Plan
Pick a strategy. One strategy is to use logical reasoning. You know how much 15 tokens cost. In order to find how much 1 costs, you need to divide by 15.
Solve a simpler problem. Work backward. Act it out.
GAMES $3.75 worth of game tokens.
15 3.75 The divisor is a whole number, so you do not need to multiply by a power of 10.
Solve
0.25 15 3.75
Where should you place the decimal point in the quotient?
directly above where it is in the dividend Each token costs
Does the answer make sense? Look over your solution. Did you answer the question?
Check
Copyright © by The McGraw-Hill Companies, Inc.
$0.25 .
Check your division by multiplying the factors. 0.25 15 3.75 × =
16
17
MONEY Quasam bags groceries at the local grocery store. He earned $81.25 one week. Quasam worked 13 hours that week. What was his hourly wage? $6.25 Check off each step.
✔
Understand
✔
Plan
✔
Solve
✔
Check
FITNESS Mary spent the afternoon biking with her friend. At the end of the day, she had traveled 50.05 miles. If she averaged the same speed per hour and she rode her bike for 3.25 hours, at what 15.4 miles per hour speed did she travel?
GO ON
Lesson 4-8 Divide Decimals Steve Cole/Getty Images
117
18
When dividing by a decimal, multiply the divisor by a power of ten. What is a power of ten? How do you know which power of ten to use?
See TWE margin.
Skills, Concepts, and Problem Solving Divide. Place the decimal using estimation.
0.32
0.93
20.48 19 64
51.15 20 55
60 64 ≈ 20 20.48 ≈ 0.33 quotient ≈
60 55 ≈ 50 51.15 ≈ 0.8 quotient ≈
Divide. Multiply to make each divisor a whole number.
2.5 21
7.1 17.75
160 22
2.13 1.491 23 0.7
33 19.8 24 0.6
115 0.05 5.75
30 26
10.3 27
3.34 34.402
3.8 28
3.1 29
5.6 17.36
1.19 35.70
73.5 279.3
9.2 30
4.8 44.16
1.22
6.13
0.1098 31 0.09
0.1839 32 0.03
6 33
5.42 32.52
4 34
3.78 15.12
Solve. 35
118
MONEY Galeno mowed his neighbor’s yard this summer and earned $192.50. If Galeno earned $17.50 each time, how many times 11 did he mow the lawn? Chapter 4 Positive and Negative Fractions and Decimals
Copyright © by The McGraw-Hill Companies, Inc.
25
0.05 8.0
36
37
3 BIOLOGY A garden snake grows 3__ inches a year. If the 4 growth of the snake shown is the same every year, 2.98 years how old is it? 1 - cup measure to measure COOKING Jin used a __ 4 2.75 cups of rice. How many times did she fill the cup?
This snake is 11.175 inches long.
11
Vocabulary Check Write the vocabulary word that completes each sentence.
quotient
38
The answer in a division problem is the
39
When you multiply the you must also multiply the power of 10.
40
Writing in Math Why would you estimate before dividing a decimal?
divisor dividend
.
by a power of 10, by the same
See TWE margin.
Spiral Review Copyright © by The McGraw-Hill Companies, Inc.
Solve.
(Lesson 4-3, p. 73)
41
RACES Annette’s and Leigh’s times in a race are shown. Annette Who had the faster time?
42
BUSINESS A bank pays interest on money in a savings account. The amount of interest that is calculated for March is $5.13. The amount of interest that is calculated for June is $5.14. For which month did the bank pay more June interest?
Add. Write the sum in simplest form. 2 1 __ __ 43 44 3 2 10 1 ___ __ + + 15 9 −−−− −−−−−
_7
_7
9
6
Annette
Leigh
(Lesson 3-5, p. 28)
45
3 __ 4 2 + __ 5 −−−−
23 _ 20
46
2 __ 3 3 + __ 7 −−−−
23 _ 21
Lesson 4-8 Divide Decimals GK Hart/Vikki Hart/Getty Images
119
Chapter
4
Progress Check 4
(Lessons 4-7 and 4-8)
Multiply. 5NS2.0, 5NS2.1, 7NS1.2 5.09 3.13 1 2 × 0.03 × 10.2 −−−−− −−−−−
51.918
3
0.0939
7.16 × 3.2 −−−−
4
22.912
11.07 × 0.1 −−−−
1.107
5
0.07 × 94 =
6.58
6
0.11 × 10 =
1.1
7
4.95 × 5.3 =
26.235
8
13.07 × 2.1 =
27.447
9
3.6 × 2.5 2 =
22.5
10
9.2 × 1.5 2 =
20.7
Divide. Place the decimal by estimation. 5NS2.0, 5NS2.1, 7NS1.2 11
10.9 ÷ 0.25 =
13
9.3 ÷ 17 =
43.6
158.1
12
8.04 ÷ 67 =
14
7.26 ÷ 1.32 =
0.12
5.5
44.3 15
1.3 57.59
1.1 16
2.2 2.42
63.2 17
7.1 448.72
Solve. 5NS2.0 19
20
120
MONEY Antwan earned $90 for washing his grandmother’s car every month. If he earned $7.50 each time he washed the car, for 12 how many months did he wash it? 2 ¢ for every envelope he addressess. If he BUSINESS Tang earns 5__ 5 addresses 275 envelopes, how much will he earn? $14.85 Chapter 4 Positive and Negative Fractions and Decimals
44.8 18
8.3 371.84
Copyright © by The McGraw-Hill Companies, Inc.
Divide. Multiply to make each divisor a whole number. 5NS2.0, 5NS2.1, 7NS1.2
Lesson
4-9 Operations with Positive and Negative Numbers KEY Concept When you perform operations with both positive and negative numbers, remember the rules: To add two numbers with the same signs: • Add their absolute values. • Use the sign of the addends in the answer. Example: -3 + (-3) = -6
4NS1.8 Use concepts of negative numbers. 5NS2.1 Add, subtract, multiply, and divide with decimals; add with negative integers; subtract positive integers from negative integers; and verify reasonableness of results. 6NS2.3 Solve addition, subtraction, multiplication, and division problems, including those arising in concrete situations, that use positive and negative integers and combinations of these operations. 7NS1.2 Add, subtract, multiply, and divide rational numbers (integers, fractions, and terminating decimals) and take positive rational numbers to whole-number powers.
To add two numbers with different signs: • Subtract their absolute values.
VOCABULARY
• Use the sign of the addend with the greater absolute value in the answer.
fraction a number that represents part of a whole or part of a set 1 1 1 3 Example: __, __, __, __ 2 3 4 4
Example: (-3) + 3 = 0 To subtract two numbers:
(Lesson 1-1, p. 4)
• Add the opposite of the second number. Example: (-3) - 3 = -6
(-3) + (-3) = -6
Copyright © by The McGraw-Hill Companies, Inc.
To multiply and divide two numbers: • The product of two numbers with the same sign is positive. The quotient of two numbers with the same sign is positive. Example: 2×3=6 6÷3=2
decimal a number that can represent whole numbers and fractions; a decimal point separates the whole number from the fraction (Lesson 4-1, p. 56)
(-2) × (-3) = 6 (-6) ÷ (-3) = 2
• The product of two numbers with different signs is negative. The quotient of two numbers with different signs is negative. Example: (-2) × 3 = -6 (-6) ÷ 3 = -2
2 × (-3) = -6 6 ÷ (-3) = -2
GO ON Lesson 4-9 Operations with Positive and Negative Numbers
121
Example 1 Find 23.1 + (–45). 20 + (-50) = 20 -50 = -30
1. Estimate.
2. The addends have different signs. To add numbers with different signs, subtract their absolute values.
45 – 23.1
3. Write the numbers vertically. Line up the decimal points. (Add a zero so that the number of place values is the same.) Bring the decimal point to the same place in the difference.
45.0 – 23.1 −−−−−− 21.9
4. Use the sign of the addend with the greater absolute value in the answer. 45 > 23.1
The sign of -45 is negative, so 23.1 + (–45) = –21.9.
5. Check your answer against your estimate for reasonableness. YOUR TURN! Find -35.4 - 17.3.
-40 - 20 = -60 subtraction . To subtract numbers with 2. The operation is 1. Estimate.
–35.4 +
different signs, add the opposite of the second number. The opposite of 17.3 is -17.3 .
-17.3
35.4 + -17.3 35.4 + 17.3 −−−−−− 52.7
To add numbers with the same signs, add their absolute values. 3. Write the numbers vertically. Line up the decimal points. Bring the decimal point to the same place in the sum.
5. Check your answer against your estimate for reasonableness.
Example 2
_1 ×
Find 2
⎛ ⎪ ⎝
_1 .
–5
⎞ ⎥ 2⎠
3 1. The operation is multiplication. 2. Change the mixed numbers to improper fractions. Multiply each whole number by the denominator of the mixed number. Add each result to the numerator. 3. Multiply. The product of two numbers with different signs is negative. 4. Write the product as a mixed number in simplest form. 122
Chapter 4 Positive and Negative Fractions and Decimals
_ _
_
7 1 2 = 3 3
_7 × – _ 11 3
⎛ ⎪ ⎝
_
1 –5 = – 11 2 2
_ _ _
7 × (-11) 3×2 2 -77 = 6 5 = -12 6 ⎞ ⎥ ⎠
=
Copyright © by The McGraw-Hill Companies, Inc.
4. Use the sign of the numbers that are added in the answer. So, -35.4 - 17.3 = - 52.7 .
YOUR TURN! ⎛
Find ⎪ ⎝
_4 ÷ - _1 . ⎞ ⎥ 5⎠
⎛ ⎪ ⎝
⎞ ⎥ 2⎠
1. The operation is
division
.
_
_
2 -1 → 1⎛ 4⎞ ⎛ 1⎞ 2 ⎪- ⎥ ÷ ⎪- ⎥ 3. Change division to multiplication. Multiply by the reciprocal of ⎝ 5⎠ ⎝ 2⎠ the divisor. ⎛ 4 ⎞⎥ × ⎛⎪- 2 ⎞⎥ ⎛ ⎞ ⎪ 8 -2 ⎛ 4⎞ ⎝ ⎪ ⎥= 5⎠ ⎝ 1⎠ ⎪- ⎥ × 4. Multiply. The product of two ⎝ 1 ⎠ 5 ⎝ 5⎠ numbers with like signs is positive. 8 3 = 1 5 5 5. Write the product as a mixed number in simplest form. 2. To divide fractions, write the reciprocal of the divisor.
_ _
_ _
_
_
_
_
_
Who is Correct? Find 20
_1 ÷ 2
_
⎛ 1⎞ ⎪- 2 ⎥ ⎝ 4⎠
.
_1 ⎛ _1 ⎞ 20 ÷ ⎪ -2 4 ⎥⎠ 2 ⎝ 4⎞⎥ 41 ÷ ⎛⎪ -_ _ = 2 ⎝ 9⎠ 2 -8 _ = 9 _1 = -9
_1 _1 20 × 2 = 4 2 5 2.2 × 20.5 = 46.125 Copyright © by The McGraw-Hill Companies, Inc.
Kent
Veta
Jora
_1 ÷ ⎛⎪-2 _14⎞⎥ ⎠ 2 ⎝ 2 _ 41 ÷ _ = 9 2 82 _ = 9 _1 =9
20
9
9
Circle correct answer(s). Cross out incorrect answer(s).
Guided Practice Estimate. Perform each indicated operation. Then compare to your estimate for reasonableness. 25 ⎞ 4 ⎛⎪ ___ __ -10 1 2 (-3.4) × (- 5.5) = × - ⎥ = 5 ⎝ 2⎠ 3
(-45.8) + 17.9 =
-27.9
4
5 (-0.5) ÷ __ = 8
18.7
-0.8 GO ON
Lesson 4-9 Operations with Positive and Negative Numbers
123
Step by Step Practice 5
Find (-2.31) ÷ 5.5. Step 1 Estimate. Step 2 The operation is division. Step 3 Divide. Place the decimal point in the quotient directly above its position in the dividend. Step 4 The quotient of two numbers with different signs is negative. Write the quotient as a negative number.
Add, subtract, multiply, or divide. ⎛ 2 ⎞⎥ × ⎛⎪-1 __ 1 ⎞⎥ = 1 ⎪- __ 6 ⎝ ⎝ 3⎠ 2⎠ 8
(-35.07) - 14.23 =
-49.3
7
1 5.6 ÷ 9 __ = 3
9
1 (-1.125) + 2 __ = 4
,PSFZ ĕ ĕ
Plan
Pick a strategy. One strategy is to solve a simpler problem.
Solve
Answer each question by using the number line. Bill 1. Who has the most money? 2. How much money do Korey and Margo have together? -2 + 5 = $3
Check
Use models or counters to represent the money each person has. Then answer the questions.
Chapter 4 Positive and Negative Fractions and Decimals
.BSHP
#JMM
Copyright © by The McGraw-Hill Companies, Inc.
124
1.125
Draw a diagram. Look for a pattern. Guess and check. ✓ Solve a simpler problem. Work backward.
MONEY The number line shows the amount of money that Marianne, Korey, Margo, and Bill have either in their wallets or owe their parents. (A negative value represents a debt, or money owed.) Who has the most money? How much money do Korey and Margo have together? Read the problem. Write what .BSJBOOF you know. Use the number line ĕ ĕ ĕ to determine how much money each person has. Marianne has $ -9 . Korey has $ -2 . Margo has $ 5 . Bill has $ 8 .
0.6
Problem-Solving Strategies
Solve.
Understand
55 23.1
(-2.31) ÷ 5.5 = -0.42
Step by Step Problem-Solving Practice 10
0.42
Solve. Name the operation you used. 11
SAFETY Brett is making a stop-sign poster for a talk about safety to a first-grade class. He will put a strip of black paper around the perimeter of the stop sign. Each side of the stop sign has a length 3 of 16__ inches. How long will the strip of paper need to be? 4
134 inches (11 feet and 2 inches); addition or multiplication Check off each step.
12
✔
Understand
✔
Plan
✔
Solve
✔
Check
SAFETY A stop sign
GARDENING Isidro is connecting 3 garden hoses to make one hose long enough to put a sprinkler 20 feet from the outside water faucet. The green hose is 6.25 feet long. The orange hose is 3 5 __ feet long, and the black hose 6.5 feet long. Find the length of 4 18.5 feet the longer hose he will make. Will it be as long as he needs? The operations used were
addition and subtraction
.
In your own words, write the rules for multiplying and dividing positive and negative numbers.
13 Copyright © by The McGraw-Hill Companies, Inc.
no
Sample answer: When the signs of the numbers are the same, the answer is a positive number. When the signs of the numbers are different, the answer is a negative number.
Skills, Concepts, and Problem Solving Add 14
⎛ 3⎞ ⎪ -2__ ⎥ ⎝ 4⎠
16
4.56 + (-0.003)
4 + 3__ 5
_
11 20
4.557
⎛
15
⎛ 5 ⎞⎥ ⎪ -__ ⎝ 7⎠
⎞
17
(-9.87) + (-23.43)
5⎥ + ⎪⎝-__ 6⎠
_
-1 23 42
-33.3 GO ON
Lesson 4-9 Operations with Positive and Negative Numbers Comstock Images/Alamy
125
Subtract. 1 1 18 5__ + 3__ 2 3 20
_
85 6
Multiply. ⎛ 3⎞ 5 22 ⎪-__⎥ × __ ⎝ 8⎠ 6 24
11.93
(–2.57) - (-14.5)
_
-5 16
1.125
(-6.25) × (-0.18)
Divide. 26 28
⎛ 4⎞ ⎪-__⎥ ⎝ 9⎠
2 ÷ __ 3
0.36 ÷ (-1.5)
_
-2 3 -0.24
_
- 13 44
19
5 3 ___ - __
21
24.88 - (-13.95)
23
3 ⎞⎥ 2 × ⎛⎪-__ 2__ 7 ⎝ 4⎠
25
15.9 × 3.48
27 29
11
4
⎛ 1⎞ ⎪-3__⎥ ⎝ 2⎠
⎛
38.83
_
-1 5 7 55.332
⎞
1⎥ ÷ ⎪-__ ⎝ 4⎠
(-75) ÷ (-1.2)
14 62.5
Solve. 30
TEMPERATURE Read the photo caption to the right. Which planet has the lower average temperature?
Saturn What is the difference in temperatures?
56ºF STOCK Your stock lost 53 points on Monday and 23 points on Tuesday, but it gained 67 points on Wednesday. What was the change in your stock for these three days?
-9 points 32
GEOGRAPHY The elevation of the Puerto Rican Trench in the Atlantic Ocean is -8,605 meters. The elevation of the Mariana Trench in the Pacific Ocean is -10,924 meters, and the elevation TEMPERATURE The of the Java Trench in the Indian Ocean is -7,125 meters. Write the average temperature on trenches in order from lowest to highest elevation. Saturn is –218°F. The
Mariana, Puerto Rican, Java
.
What is the difference in elevations between the lowest and highest trenches?
3,799 meters 126
Chapter 4 Positive and Negative Fractions and Decimals
StockTrek/Getty Images
average temperature on Jupiter is –162°F.
Copyright © by The McGraw-Hill Companies, Inc.
31
Vocabulary Check Write the vocabulary word that completes each sentence. 33
decimal A(n) is a number in which a decimal point separates the whole number from the fraction.
34
fraction A(n) whole or part of a set.
35
is a number that represents part of a
Writing in Math Name three situations in which negative integers apply to real-world situations.
Sample answer: temperatures below 0º; losing yards in a football game; owing someone money
Spiral Review Divide.
(Lesson 4-8, p. 113)
2.56
4.89
4.608 36 1.8
1.467 37 0.3
4
Copyright © by The McGraw-Hill Companies, Inc.
38
6.24 24.96
6 39
7.6 45.6
Write <, =, or > in each circle to make a true statement. (Lesson 4-3, p. 73)
40
11.8 < 18.1
41
9.9 < 9.99
42
1.45 > 1.35
43
0.89 > 0.88
Solve. 44
(Lesson 3-5, p. 33)
3 3 READING Bruce read for __ of an hour on Friday, for ___ of 5 10 1 of an hour on Sunday. Write a an hour on Saturday, and for __ 2 fraction to represent the number of hours Bruce read over those three days.
_
12 h 5
Lesson 4-9 Operations with Positive and Negative Numbers
127
Chapter
4
Study Guide
Vocabulary and Concept Check decimal, p. 56
Write the vocabulary word that completes each sentence. Some words may be used more than once.
denomination, p. 63 difference, p. 97
1
exponent, p. 105 hundredths, p. 56
2
regroup, p. 97 tenths, p. 56
3
4
5
difference A(n) problem.
is an answer to a subtraction
exponent A(n) is multiplied by itself.
is the number of times a base
regroup is to use place value to To exchange equal amounts when renaming a number. Denomination
is a kind or class in the measurement of currency, or money.
decimal is a number that can A(n) represent a whole number and a fraction.
Label each diagram below. Write the correct vocabulary term in each blank.
tenths
6
7
exponent
8
43
28.75
hundredths 3.67
Lesson Review
9
Introduction to Decimals
Write a decimal and a fraction in simplest form for the model.
(pp. 56-62)
Example 1 Write one and fifty-six hundredths as a decimal and a fraction in simplest form.
Write as a decimal. 1.56
_
3.72 as decimal; 3 18 25 as fraction 128
Chapter 4 Study Guide
Write as a fraction in simplest form. Divide the numerator and denominator by an equivalent form of one. 56 ÷ 4 56 14 = 1___ 14 1____ = 1 + _______ = 1 + ___ 25 25 100 100 ÷ 4
Copyright © by The McGraw-Hill Companies, Inc.
4-1
10
Example 2
Write 15.02 in words.
fifteen and two hundredths 11
Write 6.12 in words. Name the decimal part by the place value of the last digit to the right of the decimal point.
Write 24.8 in words.
twenty-four and eight tenths
6.12 hundredths
12
Write 105.16 in words.
one hundred five and sixteen hundredths
4-2
Decimals and Money
(pp. 63-71)
Represent each amount using the least number of bills and coins possible. 13
$5.76
one 5-dollar bill, 1 half-dollar,
Copyright © by The McGraw-Hill Companies, Inc.
1 quarter, 1 penny 14
There is a 6 to the left of the decimal point. At the decimal point, write “and.” 6.12 is six and twelve hundredths
Example 3 Represent $2.40 with the least number of bills and coins possible. So, $2.40 =
$11.42
one 10-dollar bill, one 1-dollar bill, 1 quarter,
UXPCJMMT RVBSUFS EJNF OJDLFM
1 dime, 1 nickel, 2 pennies
Chapter 4 Study Guide coins: United States Mint, bills: Michael Houghton/StudiOhio
129
4-3
Compare and Order Decimals
(pp. 73-80)
Compare each pair of decimals. Write <, =, or > in each circle to make a true statement. 15
243.69 > 243.68
16
98.47 < 98.67
17
7.35 < 8.35
18
3.50 = 3.5
Example 4 Compare 37.08 and 37.10 using place value. Write a statement using <, =, or >. Start at the left. Compare digit to digit until you find the place where the numbers are different. tens place: ones place: tenths place: 37.08
3=3 7=7 0<1 37.10
Then write a statement using an inequality symbol. 37.08 < 37.10
4-4
Compare and Order Fractions and Decimals
Order the numbers from least to greatest. 2 1 19 -2.79, 1 __, -1 __, 1.5 3 2
-2.79 , 20
_
_
1.5
-5.05 ,
,
13 _ 12
Chapter 4 Study Guide
3
-8.97 , 1.2
_
-2.024 , 11 3.3 , 3
12
,
,
-0.54 , 3 0.54 , 2
13 11 , -2.024, 3.3, ___ ___ 3
130
,
3 1 , 0.54, -0.54, __ -__ 2 2
-1 2
22
,
1 , 1.2 -8.97, -5.05, -8 __ 2
-8 1 2
21
2
2 1_
_
Example 5
_
_
_
1 1 1 Order 2 , 1.09, -3 , -2 from least 4 4 3 to greatest. Order the two negative numbers first. 1 < -2__ 1 -3 __ 3 4 One of the positive numbers is a mixed number. The other is a decimal between 1 and 2. Compare their whole number parts. 1<2 1 Order them using <. 1.09 < 2__ 4 Write all the numbers from least to greatest. 1 1 , -2__ 1 , 1.09, 2__ -3 __ 3 4 4
Copyright © by The McGraw-Hill Companies, Inc.
1 -1_
(pp. 81-87)
4-5 23
Add Decimals
(pp. 89–96)
Find 1.47 + 0.82 using decimal models. Represent each number with blocks or pictures.
Example 6 Find 1.28 + 0.12 using decimal models. Model each number.
Regroup 10 tenths as 1.
Regroup 10 hundredths as 1 tenths.
IVOESFEUIT
Copyright © by The McGraw-Hill Companies, Inc.
1.47 + 0.82 =
4-6
2.29
Subtract Decimals
Subtract. 24
45.06 - 23.19 =
21.87
Write the sum as a decimal. 1.28 + 0.12 = 1.4
(pp. 97-103)
Example 7 Find 25.3 - 19.72.
25
89.50 - 34.75 =
26
4.56 - 4.35 =
27
99.99 - 86.98 =
54.75 0.21 13.01
Write the numbers vertically. Line up the decimal points. 1 14 12 10 2 5. 3 0 - 1 9. 7 2
Add a zero so that the number of place values is the same.
5. 5 8 Bring down the decimal point to the same place in the difference. Regroup as needed.
Chapter 4 Study Guide
131
4-7
Multiply Decimals
(pp. 105-112)
Multiply. 28
1,524 × 0.8 = 1,219.2
Example 8 Find 2.38 × 1.5.
37.36
29
4 × 9.34 =
30
0.09 × 34.56 =
31
6.11 × 0.3 =
3.1104 1.833
Multiply the factors, ignoring the decimal points for now. 238 × 15 = 3570 Add the number of decimal places for both factors. 2 + 1 = 3 The product is 3. 5 7 0
Place decimal here.
4-8
Divide Decimals (pp. 113-119)
Divide. 32
Example 9
19.5 2.85 55.575
34
16.5 ÷ 5.5 =
592.12 ÷ 45.2 =
3
13.1
To eliminate the decimal in the divisor, multiply the divisor by 10. Multiply the dividend by the same power of 10. Divide.
8.0 34 272.0 272
10 × 3.4 = 34.0 10 × 27.2 = 272.0
Place the decimal point in the quotient.
0 The quotient is 8.0. Check your answer by multiplying the quotient by the divisor. 8.0 × 3.4 = 27.2
132
Chapter 4 Study Guide
Copyright © by The McGraw-Hill Companies, Inc.
33
Divide. Multiply to make the divisor a whole number. 3.4 27.2
4-9
Operations with Positive and Negative Numbers (pp. 121-127)
Solve. 35
(-34.54) + (-72.4) = -106.94
Example 10 Find 44.8 + (-21.6).
36
37
38
90.3 - (-34.7) =
(-6.45) × (-5.6) =
(-36.48) ÷ 2.4 =
125
36.12
-15.2
1. The operation is addition. The addends have different signs. To add numbers with different signs, subtract their absolute values. 2. Write the numbers vertically. Line up the decimal points.
44.8 – 21.6
44.8 21.6 −−−−−− 23.2
3. Use the sign of the addend with the greater absolute value in the answer. The sign of 44.8 is positive, so 44.8 + (-21.6) = 23.2.
Copyright © by The McGraw-Hill Companies, Inc.
44.8 > 21.6
Chapter 4 Study Guide
133
Chapter
4
Chapter Test
Write the equivalent decimal and fraction in simplest form. 3NS3.4, 4NS1.6, 4NS1.7
_
0.75 = 3 4
1
2
Thirteen and fifty-three hundredths
_
13.53; 13 53 100
Write a decimal for the amount shown. 2NS5.2 3
4
$1.14
.
$10.11 Represent $19.24 using the least number of bills and coins possible. 2NS5.2 5
$19.24
One 10-dollar bill, one 5-dollar bill, four 1-dollar bills, 2 dimes, 4 pennies Compare 0.25 and 0.17 using models. Write <, =, or > in each circle to make a true statement. 6NS1.1 0.25 > 0.17
Compare each pair of decimals using place value. Write <, =, or > in each circle to make a true statement. 6NS1.1 7
28.9 = 28.90
8
7.10 > 7.01
Write <, =, or > in each circle to make a true statement. Check your answer by placing the decimals on a number line. 5NS1.5, 6NS1.1 9
0.45 < 0.7
134
Chapter 4 Test
coins: United States Mint, bills: Michael Houghton/StudiOhio
Copyright © by The McGraw-Hill Companies, Inc.
6
Order the numbers from least to greatest. 5NS1.5, 6NS1.1, 4NS.18 4 7 2 -7 , - 4 , -0.75 , -2 10 -__, -0.75, -__, -__ 5 5 8 3 8 3
_
11
1 , -__ 1 –0.35, 0.33, __ 2 3
_
-0.35 ,
_
-1 3
_
,
0.33
_1
,
2
Add. 4NS2.0, 5NS2.0 12
7.83 + (-48.95) =
29.06
-41.12
13
28.11 + 0.95 =
76.99
15
277.18 - 69.41 =
17
1.93 × 2.4 −−−−−
Subtract. 4NS2.0, 5NS2.0 14
63.19 - (-13.8) =
207.77
Multiply. 5NS2.0, 7NS1.2 16
0.29 × 1.7 −−−−−
0.493
4.632
Divide. 5NS2.0 18
4.485 ÷ (-1.15) =
-3.9
19
28.6 7.5 214.5
Copyright © by The McGraw-Hill Companies, Inc.
Solve. 3NS3.4, 5NS1.5, 6NS1.1, 4NS2.0, 5NS2.0 20
LANDSCAPING On Monday the Windy Branch Landscaping Company had planted 3 out of the 5 trees to be planted and 7 of the 14 shrubs to be planted. Write each amount planted as a decimal.
trees, 0.6; shrubs, 0.5 21
FOOD How much more were the café’s dinner sales than its lunch sales?
/PTUBMHJB$BGÏ 5PUBM-VODI4BMFT 5PUBM%JOOFS4BMFT
$297.04 Correct the mistakes. 5NS2.0, 5NS2.1 22
Mirna says that her quotient is wrong because the check does not have a product equal to the divisor. What mistake did Mirna make?
75.0 .5 97 1.3 91 65 65 0
See TWE margin. Chapter 4 Test
135
Chapter
4
Standards Practice
Choose the best answer and fill in the corresponding circle on the sheet at right.
_
Savon has 1 3 dollars in his pocket. 4 Which point on the number line represents this amount of money?
1
"
$
#
A A
C C
B B
D D
5
5NS2.0, 7NS1.2
A 94.632
C 6,174.40
B 617.44
D 617,640
%
6
3NS3.4, 4NS1.6, 5NS1.5 2
72.64 × 8.5 =
Round 43.61 to the nearest tenth.
The 25 students in Mr. Marcero’s class raised money to go on a class trip. They collected $1,524 altogether. If each student had to raise the same amount of money, what was the cost of the trip per student? 5NS2.0, 7NS1.2
F 40
H 43.7
F $51.20
H $61.00
G 43.6
J 44
G $60.96
J $69.60
4NS1.6, 6NS1.1 3
A 218.12 miles
C 319.12 miles
B 318.22 miles
D 319.22 miles
7
Order these fractions and decimals from least to greatest: 9 , -0.35, 0.8, - 1 . 12 4 9 1 ___ __ A - , , -0.35, 0.8 6NS1.1, 4NS1.8 4 12 9 1 , -0.35 B 0.8, ___, -__ 12 4 9 1 , ___ , 0.8 C -0.35, -__ 4 12 9 , -0.35, -__ 1 , 0.8 D ___ 12 4
8
Ashlynn collected the spare change she found around her house. She found 7 quarters, 8 dimes, 13 nickels, and 28 pennies. How much money did she collect? 2NS5.1, 2NS5.2
4SN2.0, 5NS2.0, 7NS1.2 4
Emilio finished his first race in 34.07 seconds. He finished his second race in 33.89 seconds. How much faster did Emilio run the second race? F 0.18 seconds
H 0.22 seconds
G 0.42 seconds
J 1.82 seconds
4SN2.0, 5NS2.0, 7NS1.2
_
_
F $1.48
H $3.48
G $3.20
J $3.68 GO ON
136
Chapter 4 Standards Practice
Copyright © by The McGraw-Hill Companies, Inc.
Kamryn drove 186.37 miles to her grandmother’s house. Then she drove another 132.85 miles before she needed to stop for gas. How many miles did she drive before she stopped for gas?
9
Alvar is building a model plane that is 1 the size of the actual plane. What 12 is the length of the model if the actual plane length is shown below?
_
36 feet
12
Which symbol makes the sentence true? 2-1 7-1 5NS2.0, 6NS1.1 3 6 8 4 F > H <
_ _
_ _
G =
J +
ANSWER SHEET Directions: Fill in the circle of each correct answer.
1 feet A __
3 B 3 feet
C 6 feet D 36 feet
5NS2.0, 7NS1.2
Copyright © by The McGraw-Hill Companies, Inc.
10
_
Josie has 2 of a pizza she would like to 3 split among 4 friends. What part of the pizza will each person get? 8 12 F ___ H __ 2 3 1 1 G __ J ___ 6 10
5NS2.0, 7NS1.2
11
_3 + _1 = 5NS2.0 5
4 3 A ___ 20 4 B __ 9
1
A
B
C
D
2
F
G
H
J
3
A
B
C
D
4
F
G
H
J
5
A
B
C
D
6
F
G
H
J
7
A
B
C
D
8
F
G
H
J
9
A
B
C
D
10
F
G
H
J
11
A
B
C
D
12
F
G
H
J
Success Strategy 13 C ___ 20 17 D ___ 20
Use estimates to check your answers. Substitute estimate figures for the numbers in each problem. Calculate and compare with the answers you chose. If your estimated answers are very different, recheck your work.
Chapter 4 Standards Practice Getty Images
137
Index A addition decimals fractions 4–10, 33–38 Answer sheet, 53, 137 Assessment, 50–51, 133–135
C California Mathematics Content Standards, 4, 11, 19, 25, 33, 39, 56, 63, 73, 81, 89, 97, 105, 113, 121 Chapter Preview, 3, 55 Chapter Test (range), 50–51, 134–135 common denominator, 4, 11 Correct the Mistakes, 51, 135
D
denomination, 63 difference, 97
factor, 19, 105 fraction adding, 4–11, 33–38, 121–127 common denominators, 4–11, 11–17 comparing, 81–87 decimals to, 79–85 dividing, 25–31, 121 least common denominator (LCD), 33–38, 39–44 like, 4–11, 11–17, 33–38, 39–44 multiplying, 19–23, 121–127 ordering, 81–87 simplest form, 4–11, 11–17, 19–24, 25–31, 33–38, 39–44 subtracting, 11–17, 39–44, 121–127 unlike, 33–38, 39–44
H hundredths, 56, 63, 73, 89
L
dividend, 25, 113
least common denominator (LCD), 33–38, 39–44
division decimals, 113–118 fractions, 25–31
like fractions, 4–11, 11–17, 33–38, 39–44
divisor, 25, 113
E exponent, 105
M Mathematical Reasoning, see Step-by-Step Problem Solving mixed number, 4 money, 63–71 multiplication decimals, 105–112 fractions, 19–24
138
Index
N Number sense, 4, 11, 19, 25, 33, 39, 56, 63, 73, 81, 89, 97, 105, 113, 121
P prime factorization, 33, 39 Problem-Solving, see Step-byStep Problem Solving product, 105 Progress Check, 18, 32, 45, 72, 86, 104, 120
Q quotient, 25, 113
R Real-World Applications art, 18, 30, 84, 104 astronomy, 103 baby-sitting, 109 baking, 24, 32, 50, 60, 134 banking, 91 baseball, 17 biology, 119 books, 8 business, 69, 70, 87, 93, 120 carpentry, 30 celebrations, 29 civics, 102 cleaning, 43 community service, 42, 80, 101 construction, 71 cooking, 71, 85, 111 crafts, 51 dining, 31 earth science, 95, 110 engineering, 103 entertainment, 31, 112 family, 8, 95 fashion, 69
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
decimal, 56–62, 63–71, 73–80, 81–87, 89–96, 97–103, 105– 112, 113–118, 121–127
F
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
fitness, 15, 37, 45, 51, 79, 88, 117 food, 10, 16, 24, 30, 43, 51, 72, 85, 96, 102, 135 football, 51 games, 14 gardening, 101, 112 gardens, 37 geography, 87, 126 gifts, 101 hobbies, 9, 14, 18, 22, 44 homework, 79 industry, 29 landscaping, 51, 134 money, 62, 68, 70, 72, 104, 109, 111, 117, 118, 120, 125 music, 16, 23, 38 nature, 32, 44, 45, 62 number sense, 84 nutrition, 9 painting, 38 physics, 95 plants, 60 races, 119 reading, 127 retail, 24 school, 18, 23, 44, 117 science, 77, 103 softball, 16 sports, 36 stock, 126 temperature, 126 travel, 38, 50, 62, 77, 94 trips, 51 weather, 8 work, 18 reciprocals, 25–31
Reflect, 9, 15, 23, 30, 37, 43, 61, 69, 78, 86, 94, 102, 110, 118, 125
subtraction decimals, 97–103 fractions, 11–17, 39–44
regroup, 97
Success Strategy, 53, 137
S signed numbers, 119–125 simplest form, 4–11, 11–17, 19–24, 25–31, 33–38, 39–44 Spiral Review, 10, 17, 24, 31, 38, 44, 62, 71, 80, 87, 96, 103, 112, 119, 127 Standards Practice, 52–53, 136–137 Step-by-Step Practice, 6, 13, 21, 28, 35, 41, 59, 67, 76, 84, 92, 100, 108, 115, 124 Step-by-Step Problem Solving Practice, 8, 14, 22–23, 29, 36– 37, 42–43, 60, 68–69, 77, 85– 86, 93, 101, 109, 117, 125 Draw a diagram, 8, 14, 22, 29, 117 Look for a pattern, 36 Solve a simpler problem, 93, 125 Use a graph, 77 Use a model, 60 Use logical reasoning, 42, 85, 109 Work backward, 68, 101 Study Guide, 46–49, 128–132
T tenths, 56, 73, 89
U unlike fractions, 33–38, 39–44
V Vocabulary, 4, 11, 19, 25, 33, 39, 56, 63, 73, 81, 89, 97, 105, 113, 121 Vocabulary and Concept Check, 46, 128 Vocabulary Check (might be a range), 10, 16, 24, 31, 38, 44, 63, 72, 80, 87, 96, 103, 111, 119, 126
W Who is Correct?, 6, 12, 20, 27, 34, 41, 58, 66, 75, 83, 91, 99, 107, 123 Writing in Math, 10, 16, 24, 31, 38, 44, 63, 72, 80, 87, 96, 103, 111, 119, 126
Index
139
