California Math Triumphs Volume 3 Ratios Rates and Percents Teacher Wraparound Edition

Authors Basich Whitney • Brown • Dawson • Gonsalves • Silbey • Vielhaber

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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: 978-0-07-878207-7 MHID: 0-07-878207-4 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 3A

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

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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

Instructional Planning and Support

ELL Support and Vocabulary

Beatrice Luchin

ReLeah Cossett Lent

Mathematics Consultant League City, Texas

Author/Educational Consultant Alford, Florida

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CONTRIBUTING AUTHORS

California Advisory Board CALIFORNIA ADVISORY BOARD

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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

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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 3A Ratios, Rates, and Percents Chapter

Ratios and Rates

1

1-1 Ratios ..................................................................................4. 6NS1.2

1-2 Rates and Unit Costs ......................................................11 3NS2.7, 6AF2.2

Progress Check 1 .............................................................18 1-3 Probability as a Ratio......................................................19 6SDAP3.3

Assessment Study Guide .....................................................................26 Chapter Test .....................................................................28 Standards Practice ...................................................30

Standards Addressed in This Chapter 3NS2.7 Determine the unit cost when given the total cost and number of units. 6NS1.2 Interpret and use ratios in different contexts (e.g., batting averages, miles per hour) to show the relative size of two quantities, using appropriate notations (a/b, a to b, a:b). 6AF2.2 Demonstrate an understanding that rate is a measure of one quantity per unit value of another quantity. 6SDAP3.3 Represent probabilities as ratios, proportions, decimals between 0 and 1, and percentages between 0 and 100 and verify that the probabilities computed are reasonable; know that if P is the probability of an event, 1 - P is the probability of an event not occurring.

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Joshua Tree National Park

Chapters 1 and 2 are contained in Volume 3A. Chapters 3 and 4 are contained in Volume 3B.

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Contents Chapter

Percents, Fractions, and Decimals

2

Standards Addressed in This Chapter 2-1 Introduction to Percents ................................................34 5NS1.2

2-2 Percents, Fractions, and Decimals ................................41 5NS1.2

Progress Check 1 .............................................................48 2-3 Compare Data Sets of Different Sizes...........................49 5SDAP1.3, 6NS1.2

Assessment Study Guide .....................................................................56 Chapter Test .....................................................................58

5NS1.2 Interpret percents as a part of a hundred; find decimal and percent equivalents for common fractions and explain why they represent the same value; compute a given percent of a whole number. 5SDAP1.3 Use fractions and percentages to compare data sets of different sizes. 6NS1.2 Interpret and use ratios in different contexts (e.g., batting averages, miles per hour) to show the relative size of two quantities, using appropriate notations (a/b, a to b, a:b).

Standards Practice ...................................................60 Merced River near Yosemite National Park

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viii CORBIS

Contents Chapter

Using Percents

3

3-1 Calculate Percents ............................................................4 5NS1.2, 6NS1.4

3-2 Solve Percent Problems ..................................................11 6NS1.3, 6NS1.4, 7NS1.7

Progress Check 1 .............................................................20 3-3 Interest Problems.............................................................21 6NS1.4, 7NS1.7

3-4 Percent of Change .......................................................... 29 7NS1.6, 7NS1.7

Progress Check 2 .............................................................36 Assessment Study Guide .....................................................................37 Chapter Test .....................................................................40 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Standards Practice ...................................................42 Manhattan Beach Pier

Chapters 1 and 2 are contained in Volume 3A. Chapters 3 and 4 are contained in Volume 3B.

Standards Addressed in This Chapter 5NS1.2 Interpret percents as a part of a hundred; find decimal and percent equivalents for common fractions and explain why they represent the same value; compute a given percent of a whole number. 6NS1.3 Use proportions to solve problems (e.g., determine the value of N 4 N if __ = ___, find the length of a side of 7 21 a polygon simiular to a known polygon). Use cross-multiplication as a method for solving such problems, understanding it as the multiplication of both sides of an equation by a multiplicative inverse. 6NS1.4 Calculate given percentages of quantities and solve problems involving discounts at sales, interest earned, and tips. 7NS1.6 Calculate the percentage of increases and decreases of a quantity. 7NS1.7 Solve problems that involve discounts, markups, commissions, and profit and compute simple and compound interest.

ix Larry Brownstein/Getty Images

Contents Chapter

Rates and Proportional Reasoning

4

Standards Addressed in This Chapter 4-1 Proportions ......................................................................46 6NS1.3

4-2 Unit Conversions ............................................................53 3AF1.4, 3MG1.4, 6AF2.1

Progress Check 1.............................................................60 4-3 Solve Rate Problems .......................................................61 3AF2.1, 3AF2.2, 6AF2.3

4-4 Solve Problems Using Proportions ............................. 69 3AF2.1, 6NS1.3, 7AF4.2

Progress Check 2.............................................................76 Assessment Study Guide ....................................................................77 Chapter Test ....................................................................80 Standards Practice...................................................82

3AF2.1 Solve simple problems involving a functional relationship between two quantities (e.g., find the total cost of multiple items given the cost per unit). 3AF2.2 Extend and recognize a linear pattern by its rules (e.g., the number of legs on a given number of horses may be calculated by counting by 4s or by multiplying the number of horses by 4). 3MG1.4 Carry out simple unit conversions within a system of measurement (e.g., centimeters and meters, hours and minutes). 6NS1.3 Use proportions to solve problems (e.g., determine the value of N 4 N if __ = ___, find the length of a side of 7 21 a polygon simiular to a known polygon). Use cross-multiplication as a method for solving such problems, understanding it as the multiplication of both sides of an equation by a multiplicative inverse. 6AF2.1 Convert one unit of measurement to another (e.g., from feet to miles, from centimeters to inches). 6AF2.3 Solve problems involving rates, average speed, distance, and time. 7AF4.2 Solve multistep problems involving rate, average speed, distance, and time or a direct variation.

x CORBIS

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Reconstructed house in a restored Hoopa Valley Tribe village, Humboldt County

3AF1.4 Express simple unit conversions in symbolic form (e.g., ___ inches = ___ feet × 12).

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.

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

1 What is the title of Chapter 1? 2

What is the Key Concept of Lesson 1-2?

3

How do you know where the practice begins?

4

What are the vocabulary words for Lesson 2-1?

5

How many Examples are presented in Lesson 1-3?

6

What California Standards are covered in Lesson 2-3?

7

How does the Step-by-Step Practice on page 7 help you?

8

What do you think is the purpose of the Spiral Review on p. 25?

9

On what pages will you find the Study Guide for Chapter 2?

10

In Chapter 1, find the logo and Internet address that tells you where you can take the Online Readiness Quiz.

1

Chapter

1

Ratios and Rates Do you know what a batting average is? A batting average is a comparison of two numbers. It is the ratio of the number of hits to the total number of at bats. The players who have higher batting averages have more hits when they are up to bat.

Copyright © by The McGraw-Hill Companies, Inc.

2

Chapter 1 Ratios and Rates

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STEP

1 Quiz

2 Preview

STEP

Are you ready for Chapter 1? Take the Online Readiness Quiz at ca.mathtriumphs.com to find out. Get ready for Chapter 1. Review these skills and compare them with what you’ll learn in this chapter.

What You Know

What You Will Learn

You know how to write fractions to represent parts of a group.

Lesson 1-1

3 __ of the birds are bluebirds. 5

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You know how to simplify fractions. 150 ÷ 30 5 Examples: ________ = __ = 5 30 ÷ 30 1

Ratios are a way to compare numbers. A common way to write a ratio is as a fraction in simplest form. There are 3 bluebirds for every 5 birds. 3 The ratio of bluebirds to birds is __. 5 3 The ratio of bluebirds to red birds is __. 2 Lesson 1-2

TRY IT

A rate is a ratio that compares different units. When a rate has a denominator of 1, it is a unit rate .

Simplify each fraction. 36 1 ___ = 12

A teacher hands out 150 pencils to a class of 30 students. Each student gets the same number of pencils.

2

117 = ____

3

90 = ___

13 6

You know that fractions represent parts of a whole or parts of a set.

The fraction that represents the 2. number of pennies in the set is __ 7

How many pencils for each student? Each 5 pencils 150 pencils _________ ___________ = → student gets 30 students 1 student 5 pencils. Lesson 1-3 Suppose you have the coins shown at left in your pocket. You pull a coin out of your pocket without looking. The chances are that it would be a penny 2. are __ 7 The chances of something happening is the probability that it will happen.

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Lesson

1-1 Ratios KEY Concept

_

Ratios are a way to compare numbers. A ratio is a comparison of two quantities by division. Ratios can compare a part to a part, a part to a whole, or a whole to a part. There were exactly 3 boys for every 5 students. 3 The ratio of boys to students is __. 5 Other ways to write the ratio of boys to students are: 3 to 5

6NS1.2 Interpret and use ratios in different contexts to show the relative sizes of two quantities using appropriate a notations ( , a to b, a:b). b

3 out of 5

3:5

A rate is a ratio of two measurements that have different units. 300 miles in 5 days

4 pounds of turkey for 16 people

Miles and days are different kinds of units.

Pounds and people are different kinds of units.

VOCABULARY ratio a comparison of two numbers by division; the ratio of 2 to 3 can be stated as 2 out of 3, 2 2 to 3, 2:3, or __ 3 rate a ratio of two measurements or amounts made with different units Example: 300 feet per 15 seconds

Ratios can be written in simplest form.

Write the ratio that compares the number of circles to the number of triangles. Explain the meaning of the ratio. 1. Write the ratio with the number of circles in the numerator and the number of triangles in the denominator. circles 4 __ triangles 5 2. The only common factor of 4 and 5 is 1. The ratio is in simplest form. 3. The ratio of the number of circles to the number of triangles 4 , 4 to 5, or 4:5. is written as __ 5 4. The ratio means for every 4 circles, there are 5 triangles. 4

Chapter 1 Ratios and Rates

Copyright © by The McGraw-Hill Companies, Inc.

Example 1

YOUR TURN! Write the ratio that compares the number of circles to the total number of figures. Explain the meaning of the ratio. 1. Write the ratio. ______

circles total figures

2. The numerator and denominator have a common factor of . Write the fraction in simplest form.

÷ ______ = ______________ = ______ ÷

3. Write the ratio of the number of circles to the number of figures.

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4. What does the ratio mean?

Example 2

YOUR TURN!

Write the ratio as a fraction in simplest form.

Write the ratio as a fraction in simplest form.

4 red hats out of 10 total hats

15 puppies to 18 kittens

1. Write the ratio with the number of red hats in the numerator and the total number of hats in the denominator. 4 ___ 10

1. Write the ratio.

2. The numerator and denominator have a common factor of 2. Divide each by 2 to write the fraction in simplest form. 4 ÷ 2 __ 4 = _______ ___ =2 10

10 ÷ 2

5

______

2. The numerator and denominator have a common factor of . Write the fraction in simplest form. ÷ ______ = ______________ = ______ ÷ GO ON Lesson 1-1 Ratios

5

Example 3

YOUR TURN!

Write the ratio of the width to the length in the rectangle as a fraction in simplest form.

Write the ratio of the length to the width in the rectangle as a fraction in simplest form. 6 cm

3 cm 16 cm

12 cm

1. Write the ratio as a fraction with the width over the length. 3 ___ 12 2. The numerator and denominator have a common factor of 3. Divide each by 3 to write the fraction in simplest form. 3 ÷_____ 3 = _____ 3 = __ 1 ___ 12 ÷ 3

12

______

1. Write the ratio.

2. The numerator and denominator have a . Write the common factor of fraction in simplest form. ÷ ______ = ______________ = ______

4

÷

Who is Correct? Write the ratio as a fraction in simplest form.

Lacey 12

6

Salvatore

_8 = _2 12

3

Ivan

_8 = 2

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_8 = _4

8 pens to 4 pencils

4

Circle correct answer(s). Cross out incorrect answer(s).

Guided Practice Use the diagram to write each ratio as a fraction in simplest form.

1

The number of red counters to the number of blue counters is

.

2

The number of red counters to the total number of counters is

.

3

The number of blue counters to the total number of counters is

6

Chapter 1 Ratios and Rates

.

Step by Step Practice 4

An aquarium has 7 guppies, 3 angelfish, 5 mollies, and 6 danios. Write the ratio of each type of fish to the total number of fish in the aquarium. Write each as a fraction in simplest form. Step 1 The total number of fish is This will be the

. in the fraction.

Step 2 Write a ratio for the number of guppies to the total number of fish. What is the common factor of the numerator and denominator? Write the fraction in simplest form.

÷

______ = ______________ = ______

÷ Step 3 Write a ratio for the number of angelfish to the total number of fish. What is the common factor of the numerator and denominator? Write the fraction in simplest form.

÷

______ = ______________ = ______

÷

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Step 4 Write a ratio for the number of mollies to the total number of fish. What is the common factor of the numerator and denominator? Write the fraction in simplest form. Step 5 Write a ratio for the number of danios to the total number of fish. What is the common factor of the numerator and denominator? ÷ Write the fraction in simplest form. ______________ = ______ ÷

Write each ratio as a fraction in simplest form. 5

In a box of granola bars, there are 6 cinnamon bars and 3 almond bars. Write the ratio of almond bars to cinnamon bars. almond granola bars

→

cinnamon granola bars

→

÷

______ = ______________ = ______

÷

GO ON Lesson 1-1 Ratios

7

6

In a sports equipment closet, there are 10 softballs, 4 basketballs, and 3 soccer balls. Write the ratio of soccer balls to the total number of balls.

7

In a bag of 18 marbles, there are 16 that are not white. Write the ratio of white marbles to nonwhite marbles.

8

In a classroom, there are 24 students and 5 computers. Write the ratio of students to computers.

Step by Step Problem-Solving Practice

Problem-Solving Strategies

Solve. 9

AGES Clarence is 16 years old, and his sister Tahnya is 10 years old. In two years, what will be the ratio of Clarence’s age to Tahnya’s age? Understand

Read the problem. Write what you know. Clarence is years old. Tahnya is years old. In 2 years, Clarence will be years old, and Tahnya will be years old.

Plan

Pick a strategy. One strategy is to solve a simpler problem.

Solve

First, write the ratio of their ages in two years.

Is there still a common factor? Divide the numerator and denominator by 3.

÷

______ = ______________ = ______

÷

Is there still a common factor? Write the ratio in simplest form. Check

8

Does the answer make sense? Look over your solution. Did you answer the question?

Chapter 1 Ratios and Rates

÷

______ = ______________ = ______

÷

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To write the ratio in simplest form, divide the numerator and denominator by a common factor. Divide the numerator and denominator by 2.

Look for a pattern. Guess and check. Act it out. ✓ Solve a simpler problem. Work backward.

10

FOOTBALL In the NFL 2005 playoffs, the Pittsburgh Steelers played the Seattle Seahawks. The Steelers’ season record was 11 wins and 5 losses. The Seahawks’ season record was 13 wins and 3 losses. What was the ratio of wins for the Steelers to wins for the Seahawks? Check off each step. Understand Plan Solve Check

11

TENNIS Jena and Niles played 20 sets of tennis. Jena won 12 of them. Write a ratio of Jena’s wins to the total number of sets in simplest form. What is a ratio? Explain using examples.

12

Skills, Concepts, and Problem Solving

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Use the diagram to write each ratio as a fraction in simplest form. 13

apples and bananas to plums and pears

14

fruit that is not pears to total pieces of fruit

15

apples to bananas and plums

Write each ratio as a fraction in simplest form. 16

Raymond had 6 hits out of 10 at bats.

17

The grapes were $6 for 3 pounds.

18

Samantha jogged 10 miles in 100 minutes.

19

There are 12 puppies to 15 kittens at the pet store. 100 in.

Write the ratio of width to length in each rectangle as a fraction in simplest form. 4m

20 16 m

220 in.

21

GO ON Lesson 1-1 Ratios

9

Write the ratio of length to width in each rectangle as a fraction in simplest form. 12 m

22

7 mm

23

50 m 11 mm

SPORTS The batting average is the ratio of the number of hits to the total number of at bats. Refer to the table to answer Exercises 24–26. 24

Which players had the same batting average? What is that batting average?

25

26

Did the player with the most hits have the highest batting average? Explain.

Player

Hits

At Bats

Mary

12

36

Dwaine

24

56

Juan

27

81

Ramon

20

45

Bob

15

45

Explain the meaning of Juan’s batting average.

Write the vocabulary word that completes each

27

A(n)

compares two quantities.

28

A(n) also compares two quantities, but it compares two quantities with different units.

29

Writing in Math different ways.

Write the ratio of 2 pens out of a total of 3 pens four

Solve. 30

FITNESS Monique and Emil played 8 sets of racquetball. Monique won 6 of them. Write a ratio of Monique’s wins to the total number of sets in simplest form.

10

Chapter 1 Ratios and Rates

CORBIS

Copyright © by The McGraw-Hill Companies, Inc.

Vocabulary Check sentence.

Lesson

1-2 Rates and Unit Costs 3NS2.7 Determine the unit cost when given the total cost and number of units. 6AF2.2 Demonstrate an understanding that rate is a measure of one quantity per unit value of another quantity.

KEY Concept A rate is a ratio of two measurements having different units. 300 miles 300 miles in 5 days  _________ 5 days

VOCABULARY

The units miles and days are different.

rate a ratio of two measurements or amounts made with different units Example: 300 feet per 15 seconds (Lesson 1-1, p. 4)

When a rate is simplified so that it has a denominator of 1 unit, it is called a unit rate . 50 miles 50 miles per hour  ________ 1 hour

The denominator is 1 unit.

ratio a comparison of two numbers by division

Unit cost is the cost of a single item or unit of measurement.

(Lesson 1-1, p. 4)

The cost of a 12-ounce jar of jam is $2.49. 2.49 ____

21 cents 12  2.49  about 0.21  ________ 12  1 ounce

unit rate a rate that describes how many units of the first type of quantity are equal to 1 unit of the other type of quantity Example: 50 miles per hour

The unit cost is 21 cents per ounce.

Copyright © by The McGraw-Hill Companies, Inc.

Rates are often written using abbreviations, such as 300 mi/5 days, 60 mi/h, or $0.21/oz.

Example 1

unit cost the cost of a single item

YOUR TURN!

Write the rate 50 claps in 5 seconds as a fraction. Find the unit rate. 50 claps 1. Write the rate as a fraction. _________ 5 seconds

2. Find an equivalent rate with a denominator of 1. The numerator and denominator have a common factor of 5. Divide each by 5. 10 claps 50 claps ÷ __ 35 ________________ = _________ 4 5 seconds ÷ 5 1 second 3. Name the unit rate. 10 claps per second or 10 claps/s

Write the rate 90 miles in 2 hours as a fraction. Find the unit rate. 1. Write the rate as a fraction.

miles ____________ hours

2. The numerator and denominator have . a common factor of miles miles ÷ ___________________ = ____________ hours ÷

hour

3. Name the unit rate. GO ON Lesson 1-2 Rates and Unit Costs

11

Example 2 Find the unit rate for selling 300 tickets in 6 days. Use the unit rate to find the number of tickets sold in 5 days. 1. Write the rate as a fraction. 300 tickets __________ 6 days 2. Find an equivalent rate with a denominator of 1. Divide the numerator and denominator by 6. 300 ÷6 50 ________ = ___ 6÷6 1 3. The unit rate is 50 tickets/day. 4. To find how many tickets will be sold at this rate in 5 days, multiply the numerator and denominator by 5. 250 tickets 50 tickets × 5 _____________ = __________ 5 days 1 day × 5 At this rate, 250 tickets will be sold in 5 days.

YOUR TURN! Find the unit rate for traveling 165 feet in 15 seconds. Use the unit rate to find the number of feet traveled in 120 seconds. 1. Write the rate as a fraction. ft __________ s 2. Divide the numerator and denominator . by ft ft ÷ _________________ = ________ s

s ÷ 3. The unit rate is

/

.

4. Multiply the numerator and denominator by . ft × ft ___________________ = ____________ s ×

s

feet will be At this rate, traveled in 120 seconds.

Ms. Tuttle bought a box of greeting cards for $5.75. The box contains 12 cards. Find the unit cost to the nearest cent. 1. Write the rate as a fraction. $5.75 ________ 12 cards 2. Divide the numerator by the denominator. 11 rounded to the 3. 0.47___ 12 nearest cent is $0.48

0.47 5.75 12  48 95 84 11 Each card costs about $0.48. 12

Chapter 1 Ratios and Rates

YOUR TURN! Mr. Jonas bought a box of oranges for $12.50. The box contains 15 oranges. Find the unit cost to the nearest cent. 1. Write the rate as a fraction.

2. Divide the numerator by the denominator. 3. The unit cost rounded to the nearest cent is . Each orange costs about

.

Copyright © by The McGraw-Hill Companies, Inc.

Example 3

Who is Correct? Georgie can drive 220 miles on 8 gallons of gas. Find the unit rate. Use the unit rate to find the number of miles Georgie can drive on 64 gallons of gas.

Rasha

Jeff

Janine

27 8 ÷ __ 22 ___0___ = ___ 1 8÷8

.5 27__ 8 ÷ __ 22 ___0___ = __1 8÷8

Unit rate = 27 mi/gal; 27 × 64 = 1,728 mi

Unit rate = 27.5 mi/gal; 27.5 × 64 = 1,760 mi

27.5  0.0 22 8  16 60 56 40 40 .5 mi/gal 27 is Unit rate 64 = × l /ga 27.5 mi gal 64 on 1,760 mi

Circle correct answer(s). Cross out incorrect answer(s).

Guided Practice Write each rate as a fraction. Find each unit rate.

Copyright © by The McGraw-Hill Companies, Inc.

1

2

140 words in 4 minutes

4

168 miles in 3 hours

$2.00

3

12 books in 5 days

Find each unit rate. Use the unit rate to find the unknown amount. 5

5 pounds for 8 people; □ pounds for 20 people

6

12 hours for 5 classes; □ hours for 4 classes

7

150 feet in 8 seconds; □ feet in 14 seconds

8

$5 for 4 books; □ dollars for 15 books

GO ON Lesson 1-2 Rates and Unit Costs

13

Step by Step Practice 9

Use the table to find which box of macaroni has the lowest unit cost. Round to the nearest cent. Step 1 Find the unit cost of a 12-oz package. 0.90 ____

12  0.90  about $ 12 

/oz

Step 2 Find the unit cost of a 16-oz package. __________



  about $

Box Size

Price

12 oz

$0.90

16 oz

$1.12

32 oz

$1.95

/oz

Step 3 Find the unit cost of a 32-oz package. __________



  about $

/oz

Round to the nearest cent means the nearest hundredth.

Step 4 Which package costs the least per ounce?

Which product has the lowest unit cost? Round to the nearest cent. 10

a 12-oz juice bottle for $0.75 or a 24-oz juice bottle for $1.95

 about $

/oz

24-oz bottle: __________ 

 about $

/oz

The

juice bottle costs less per ounce.

11

50-count vitamins for $5.49, 100-count vitamins for $8.29, or 150-count vitamins for $12.75

12

a 16-oz bag of apples for $2.99, a 32-oz bag of apples for $3.99, or a 48-oz bag of apples for $5.49

13

a 6-pack of yogurt for $1.99 or a 12-pack of yogurt for $3.50

14

4 shirts for $24.85 or 7 shirts for $49.49

14

Chapter 1 Ratios and Rates

Copyright © by The McGraw-Hill Companies, Inc.

12-oz bottle: __________ 

Step by Step Problem-Solving Practice

Problem-Solving Strategies Draw a diagram. Look for a pattern. Guess and check. Make a table. ✓ Solve a simpler problem.

Solve. 15

NATURE The American robin can travel 32 miles in a 20-hour flight. The grey-cheeked thrush can travel 33 miles in 6 hours. Which bird flies at a faster rate? Understand

Read the problem. Write what you know. The American robin can travel miles in a -hour flight. The grey-cheeked thrush can travel miles in a -hour flight.

Plan

Pick a strategy. One strategy is to solve a simpler problem. Find each unit rate.

Solve

Write each rate as a fraction. Find an equivalent rate with a denominator of 1. American Robin

Unit Rate of the American Robin miles 32 miles ÷ _____________ = _________ 20 hours ÷

1 hour

Unit Rate of the Grey-Cheeked Thrush

Copyright © by The McGraw-Hill Companies, Inc.

miles 33 miles ÷ ___________ = _________ 1 hour 6 hours ÷

Compare the unit rates for each bird. miles/hour < The faster rate.

miles/per hour flies at a

Does the answer make sense? Did you answer the question?

Check

16

Grey-Cheeked Thrush

BUSINESS While working at a gardening center for the summer, Elio earned $780 in 12 weeks. Find a unit rate to describe his weekly wages. Check off each step. Understand Plan Solve Check

GO ON Lesson 1-2 Rates and Unit Costs

(t)Millard H. Sharp/Photo Researchers, Inc., (b)Steve Maslowski/Visuals Unlimited

15

17

POPULATION The population of California is about 36.1 million people. Its land area is approximately 156,300 square miles. Find the population per square mile. Explain the difference between a rate and ratio. What is the difference between unit rate and unit cost?

18

Skills, Concepts, and Problem Solving Write each rate as a fraction. Find each unit rate. 19

6 pancakes in 4 minutes

20

9 feet in 12 years

21

9 feet every 10 seconds

22

21 hits out of 40 at bats

Find each unit rate. Use the unit rate to find the unknown amount. $30 for 16 ounces; □ dollars for 6 ounces

24

50 meters every 8 seconds; □ meters for 20 seconds

25

150 feet in 8 seconds; □ feet in 14 seconds

26

9 yards in 3 plays; □ yards for 4 plays

Which product has the lower unit cost? Round to the nearest cent. 27

12 golf balls for $9 or 10 golf balls for $8.50

28

32-oz shampoo bottle for $6 or 8-oz shampoo bottle for $1.75

29

40-oz can of soup for $4.49 or 25-oz can of soup for $2

30

4-pack of tissues for $3.39 or 16-pack of tissues for $14.75

Solve. 31

FUND-RAISER Liza sold 225 raffle tickets in 6 days, while Brian sold 181 tickets in 4 days. Who sold raffle tickets at a faster rate? Explain.

16

Chapter 1 Ratios and Rates

Copyright © by The McGraw-Hill Companies, Inc.

23

32

LIFE SCIENCE The heart of a rat beats about 840 times in 2 minutes, while the heart of a guinea pig beats about 1,200 times in 4 minutes. The heartbeat of a rabbit is about 1,025 beats in 5 minutes. Which animal’s heart beats the most times in one hour? Explain.

33

POPULATION Which country has the lower population per square mile? Explain. Country

Population

Area in sq miles

Japan

128,000,000

377,900

United Kingdom

59,700,000

243,000

Copyright © by The McGraw-Hill Companies, Inc.

Vocabulary Check sentence.

Write the vocabulary word that completes each

34

A ratio of two measurements or amounts of different units, where the second amount is 1 is a(n) .

35

The cost of a single item or unit is the

36

Writing in Math Which of the following statements are sometimes, always, or never true? Give an example or counterexample to illustrate. A ratio is a rate.

.

A rate is a ratio.

Spiral Review Use the diagram shown at the right to write each ratio as a fraction in simplest form. (Lesson 1-1, p. 4) 37

The number of red counters to the number of blue counters is .

38

The number of red counters to the total number of counters is

39

The number of blue counters to the total number of counters is

. .

Lesson 1-2 Rates and Unit Costs Jules Frazier/CORBIS

17

Chapter

Progress Check 1

1

(Lessons 1-1 and 1-2)

Use the diagrams to write each ratio as a fraction in simplest form. 1

2

shaded to unshaded squares

unshaded parts to total parts

Write the ratio of width to length in each rectangle as a fraction in simplest form. 3

4 4 km

10 in.

5 km 2 in.

Write each rate as a fraction. Find each unit rate. 5

45 miles in 9 minutes

6

3 tons in 75 years

8

5 long-haired cats out of 12 cats

Write each ratio as a fraction in simplest form. 19 out of 133 girls had green eyes

Which product has the lowest unit cost? Round to the nearest cent. 9

12-oz can for $1.99, a 16-oz can for $2.50, or a 32-oz can for $3.79

10

9 kiwis for $1.35, 14 kiwis for $2.25, or 20 kiwis for $3.80

Solve. 11

SPELLING Write a fraction in simplest form for the ratio of the number of vowels in California to the total number of letters.

12

SPORTS

18

Chapter 1 Ratios and Rates

Nina swam 200 feet in 44 seconds. What is her unit rate?

Copyright © by The McGraw-Hill Companies, Inc.

7

Lesson

1-3 Probability as a Ratio KEY Concept Probability is a number that measures the chance of an event happening. The probability of an event is a ratio that compares the number of favorable outcomes to the number of possible outcomes . The probability of an event is written as P(event). Suppose you roll a number cube. number of favorable outcomes P(even number) = ____________________________ total number of outcomes The even numbers are 2, 4, and 6.

3 1. The ratio of the even numbers to the total numbers is __ or __ 6 2 Notice the ratio and the probability are the same.

Copyright © by The McGraw-Hill Companies, Inc.

probability a number between 0 and 1 that measures the likelihood of an event happening

(Lesson 1-1, p. 4)

outcomes the possible results of a probability event Example: 4 is an outcome when a number cube is rolled.

Probability can also be written as a decimal or as a percent. 1 __ 0.50 50% 2 1 , means that you can expect to roll an even The probability, __ 2 number 1 out of every two times or 50% of the time.

event a set of outcomes

The probability of an event can be 0, 1, or any number between 0 and 1. equally likely to occur

VOCABULARY

ratio a comparison of two numbers by division; the ratio of 2 to 3 can be stated as 2 out of 3, 2 2 to 3, 2:3, or __ 3

3 1 number of even numbers = _______________________ = __ = __ total number of outcomes 6 2

impossible to occur

6SDAP3.3 Represent probabilities as ratios, proportions, decimals between 0 and 1, and percentages between 0 and 100 and verify that the probabilities computed are reasonable; know that if P is the probability of an event, 1 – P is the probability of an event not occurring.

certain to occur

0

1 or 0.25 4

1 or 0.5 2

3 or 0.75 4

1

0

25%

50%

75%

100%

When probability equals 0, the event is impossible. For example, the probability of rolling a 7 on a number cube is 0. When probability equals 1, the event is certain. For example, the probability of rolling a natural number that is 6 or less is 1. The probability that one event does not occur is equal to 1 - P(event does occur).

GO ON Lesson 1-3 Probability as a Ratio

19

Example 1

YOUR TURN! Use the spinner in Example 1 to find the probability of spinning an odd number. Write the probability as a fraction in simplest form. Explain the probability.

Use the spinner to find the probability of spinning 6. Write the probability as a fraction in simplest form. Explain the probability. 1. Count the number of sections labeled 6. Write this number in the numerator.

8

1. What number of sections are labeled with an odd number?

1

7

2

6

3 5

2. What is the total number of sections?

4

2. Count the total number of sections. Write this number in the denominator. The spinner has one 6. 1 P(6) = __ The spinner has 8 sections. 8 1 . It is already in simplest 3. The ratio is __ 8 form. 1 means that 1 out of 4. The probability of __ 8 every 8 spins should be a 6.

Example 2

3. Write a ratio for the P(black or blue).

6 9 6 __ 9

4. Write the fraction in simplest form. 6 ÷ 3 __ ______ =2 9÷3 3 5. When you take a pair of socks without looking, the probability the socks will be blue or black is two-thirds. 20

Chapter 1 Ratios and Rates

÷

______________ = ______

÷ 5. The probability of out of every be an odd number.

means that spins should

A bowl of fruit has 4 peaches, 5 plums, 5 apples, and 3 oranges. What is the probability that a peach or an orange is selected if you choose a fruit without looking? 1. The number of peaches and oranges is . 2. The number of pieces of fruit in the basket is . 3. P(peach or orange) = ______ 4. The ratio is already simplified. 5.

out of every times you take a piece of fruit from the basket without looking it will be .

Copyright © by The McGraw-Hill Companies, Inc.

2. How many pairs of socks are in the drawer in all?

4. The ratio can be simplified.

YOUR TURN!

A drawer of socks contains 3 pairs of white socks, 3 pairs of blue socks, and 3 pairs of black socks. What is the probability of choosing a pair of black or blue socks if you take 1 pair from the drawer without looking? 1. How many pairs of socks are blue or black?

3. P(odd number) = ______

Who is Correct? The ratio of green marbles to the total number of marbles in a bag is 3 . What is the probability that when you pick a marble without 10 looking it will not be green?

_

Luis 3 ___

10

Nora 3 ___

Ajay

13

7 ___ 10

Circle correct answer(s). Cross out incorrect answer(s).

Guided Practice Use the spinner to find each probability. Write the probability as a fraction in simplest form. 1 2

9

10 1

2 3

8

P(multiple of 3)

7

6 5

4

P(not a multiple of 3)

Copyright © by The McGraw-Hill Companies, Inc.

Step by Step Practice Find the probability. Write the probability as a fraction in simplest form. 3

In a bag there are 5 green chips, 3 yellow chips, 7 blue chips, and 1 red chip. Find the probability of reaching into the bag without looking and not getting a green chip. Step 1 Count the total number of chips. This is the in the fraction. Step 2 Count the number of chips in the bag that are green. This is the in the fraction. Step 3 Write a ratio in simplest form for the P(green). ______

Step 4 Find the P(not green). 1 - ______ = ______ GO ON Lesson 1-3 Probability as a Ratio

21

Find each probability. Write the probability as a fraction in simplest form. 4

In a box of mugs, there are 6 white mugs, 4 blue mugs, and 8 beige mugs. What is the probability that without looking you would choose a mug that is not white? ÷ P(not white) = ______ = _____________ = ______ ÷

5

On a serving counter, there are 3 sausage pizzas, 8 cheese pizzas, and 6 pepperoni pizzas. Find the probability of randomly selecting a piece of sausage pizza. P(sausage) =

Find each probability using a number cube. Write the probability as a fraction in simplest form.

P(roll a 3)

7

P(roll a 1 or a 6)

8

P(roll a number less than 6)

9

P(roll a number less than 3)

10

P(roll an even number)

11

P(roll a 7)

Find the probability of each event. Write the probability as a fraction in simplest form. 12

You pick a day of the week that begins with the letter T.

13

You pick a weekend day from the days of the week.

14

You pick one of the letters A, F, or G from the alphabet.

15

You pick a month that begins with the letter J.

16

You pick the letter A from the letters in CALIFORNIA.

17

You pick a day of the week that ends in the letter Y.

22

Chapter 1 Ratios and Rates

Copyright © by The McGraw-Hill Companies, Inc.

6

Step by Step Problem-Solving Practice

Problem-Solving Strategies Draw a diagram. Look for a pattern.

Solve. 18

✓ Use logical reasoning.

GENETICS The ratio of brown-haired students to the total number of students in a fifth-grade class is 22 out of 30. What is the probability if one student is picked by the teacher without looking that the student will not have brown hair?

Act it out. Solve a simpler problem.

Understand

Read the problem. Write what you know. Out of students, have brown hair.

Plan

Pick a strategy. One strategy is to use logical reasoning.

Solve

Write the ratio of brown-haired students to total students. Think: The ratio of the total number of students to the total number of students is . The probability of choosing a student who does not have brown hair is the same as the difference of the ratio of the entire class and the ratio of brownhaired students. 30 ___ 8 ___ - 22 = ___ = ______

Copyright © by The McGraw-Hill Companies, Inc.

30

30

30

15

The probability of choosing a student who does not have brown hair is . Check your answer. The sum of the probability that an event occurs and the probability that the event does not occur is 1. Is the sum of your probabilities equal to one? Explain.

Check

19

FOOD The probability of buying a dozen bagels and receiving an extra bagel is 2 out of 100. Find the probability of not receiving an extra bagel. Check off each step. Understand Plan Solve Check

GO ON Lesson 1-3 Probability as a Ratio

23

20

GAMES The probability of choosing a black marble out of a bag of 3 marbles without looking is ___. What is the probability of not picking 14 a black marble? How are probability and ratios the same?

21

Skills, Concepts, and Problem Solving Use the spinner to find each probability. Write the probability as a fraction in simplest form. 22

P(white)

24

Add your answers to Exercises 22 and 23. What is their sum?

23

P(not white)

Use the basket of fruit to find each probability. Write the probability as a fraction in simplest form. Write the ratio for the number of plums and pears to the total number of fruit.

26

Write the ratio for the number of apples to the number of bananas.

27

What is the probability of choosing fruit from the basket without looking and getting an apple or a banana?

28

What is the probability of choosing fruit from the basket without looking and getting a fruit that is not an apple or banana?

Find each probability. Write the probability as a fraction in simplest form. 29

7 red hats, 9 green hats, and 4 blue hats; P(blue hat)

30

2 small popcorn bags, 5 medium popcorn bags, and 3 large popcorn bags; P(small or large popcorn bags)

31

9 fourth graders, 6 second graders, and 2 third graders; P(not a second grader)

24

Chapter 1 Ratios and Rates

Copyright © by The McGraw-Hill Companies, Inc.

25

32

LANGUAGE Suppose the letters of California are placed in a bag. A letter is pulled out without looking. What is the probability that the letter is an i ?

33

SEWING The ratio of blue buttons to the total number of buttons 4 . What is the probability if a button is chosen without in a tin is __ 9 looking that the button will not be blue?

Vocabulary Check sentence.

Write the vocabulary word that completes each

34

is a number between 0 and 1 that measures the likelihood of an event.

35

A(n)

36

Writing in Math Write an example of a situation in which the probability of an event occurring is 0.

compares two quantities.

Spiral Review Find each unit rate. Use the unit rate to find the unknown rate.

Copyright © by The McGraw-Hill Companies, Inc.

(Lesson 1-2, p. 11)

37

10 feet every 50 seconds; □ feet for 30 seconds

38

9 hits out of 36 at bats; □ hits for 44 at bats

Solve.

(Lesson 1-2, p. 11)

39

HEALTH After running in a race, Thomas’s heart rate is 111 beats per minute. After running the same race, Lena’s heart rate is 235 beats every 2 minutes. Who has a faster heart rate?

40

INSECTS

Which caterpillar travels more slowly?

This type of caterpillar travels 6 meters in 5 hours.

This type of caterpillar travels 30 meters in 20 hours. Lesson 1-3 Probability as a Ratio

Dorling Kindersley/Getty Images

25

Chapter

1

Study Guide

Vocabulary and Concept Check event, p. 19

Write the vocabulary word that completes each sentence.

outcomes, p. 19

1

is a number between 0 and 1 that measures the likelihood of an event happening.

2

A(n) is a ratio of two measurements or amounts made with different units, such as 2 miles in 5 minutes.

3

When finding the probability, put the number of possible in the denominator.

4

A(n) numbers by division.

5

The probability of a(n) written as a fraction.

probability, p. 19 rate, p. 4 ratio, p. 4 unit cost, p. 11 unit rate, p. 11

is a comparison of two can be

Write the correct vocabulary term in each blank. 6

7

$2.73 per gallon

65 miles per hour

1-1

Ratios

(pp. 4–10)

Write each ratio as a fraction in simplest form. 8

16 apples to 24 oranges

9

5 strawberries to 15 cherries

10

12 televisions to 4 radios

Example 1 Write the ratio as a fraction in simplest form. 3 white shirts out of 15 total shirts Write the ratio with the number of white shirts in the numerator and the total number of shirts in the denominator. 3 ___ 15 Write the fraction in simplest form. 3 ÷ __ 33 __ _______ =1 15 ÷ 43

26

Chapter 1 Study Guide

5

Copyright © by The McGraw-Hill Companies, Inc.

Lesson Review

1-2

Rates and Unit Costs

(pp. 11–17)

Write each rate as a fraction. Find each unit rate. 11

12

Example 2 Write the rate 80 beats per 10 seconds as a fraction. Find the unit rate.

100 miles in 4 hours Write the rate as a fraction. 80 beats __________ 10 seconds

60 gallons in 5 minutes

Find an equivalent rate with a denominator of 1. ÷ 10 80 beats 3 8 beats __ ________________ = ________ 4 10 seconds ÷ 10 1 second Name the unit rate. 8 beats per second or 8 beats/s

1-3

Copyright © by The McGraw-Hill Companies, Inc.

13

Probability as a Ratio

(pp. 19–25)

Use the spinner in Example 3 to find the probability of spinning a 3 or a 4. Write the probability as a fraction in simplest form. Explain the probability.

Example 3 Use the spinner to find the probability of spinning an even number. Write the probability as a fraction in simplest form. Explain the probability. Count the number of sections labeled with an even number. Write this number in the numerator.

14

A bowl contains 24 marbles. Of these 24 marbles, 3 are white, 5 are pink, 4 are red, 6 are yellow, 2 are purple, and 4 are orange. If you reach into the bowl without looking and choose one marble, what is the probability of choosing a white or pink marble?

8

1

7

2

6

3 5

4

Count the total number of sections. Write this number in the denominator. 4 P(even number) = __ 8

The spinner has four even numbers. The spinner has 8 numbers.

4 . The ratio can be simplified. The ratio is __ 8 4_____ ÷ 4 __ =1 8÷4 2 1 means that 1 out of The probability of __ 2 every 2 spins should be an even number. Chapter 1 Study Guide

27

Chapter

Chapter Test

1

Use the diagram to write each ratio as a fraction in simplest form. 1

shaded squares to total number of squares

2

unshaded parts to shaded parts

Write the ratio of width to length for each rectangle as a fraction in simplest form. 3

4 6 cm 8 cm

11 ft

3 ft

Write each rate as a fraction. Find each unit rate. 5

60 miles in 2 hours

6

12 pounds in 3 weeks

7

21 out of 168 were not wearing team colors

8

7 of the 15 fish in the tank were goldfish

9

21 laps in 3 days

Copyright © by The McGraw-Hill Companies, Inc.

Write each ratio as a fraction in simplest form.

Which product has the lowest unit cost? Round to the nearest cent. 10

8-oz bag of chocolate chips for $1.99, a 12-oz bag of chocolate chips for $2.49, or a 16-oz bag of chocolate chips for $2.99

11

4 oranges for $1, 10 oranges for $2, or 24 oranges for $6

12

4 DVDs for $59.99, 6 DVDs for $74.99, or 10 DVDs for $99.99

28

Chapter 1 Test

GO ON

Use the spinner to find each probability. Write the probability as a fraction in simplest form. 13

P(odd)

14

P(number greater than 4)

15

P(1 or 7)

9 8 7

10 1

6 5

2 3 4

Solve. 16

READING Toby was reading a book for his social studies class. He read 144 of the book’s 200 pages. Write the pages Toby has read to the total number of pages as a ratio in simplest form.

17

TRAVEL Tom drove his truck 225 miles in 3 hours. What was his unit rate?

18

SPELLING Suppose the letters of the word mathematics are placed in a bag. A letter is pulled out without looking. What is the probability that the letter is an m?

Copyright © by The McGraw-Hill Companies, Inc.

Correct the mistakes. 19

At the Johnson Family Market, a sign in the window read: “8-oz bag of peanuts for $2.99. That’s less than $5 per pound!!” Is the sign correct?

20

When Joey and his sister Margaret were playing a game, Joey shuffled the entire deck of game cards and asked Margaret to draw one from the deck without looking. He told her that she had 2 chance of drawing a red card. (There were 4 red cards in the a ___ 25 deck of 52 game cards.)

Chapter 1 Test

29

Chapter

1

Standards Practice

Choose the best answer and fill in the corresponding circle on the sheet at right. 1

2

In Cassie’s bookshelf, there are 32 mysteries, 8 nonfiction titles, and 6 science fiction novels. What is the ratio of mysteries to nonfiction books? 16 , 16:3, or 16 to 3 A ___ 3 4 B __, 4:1, or 4 to 1 1 1 , 1:4, or 1 to 4 C __ 4 ___ D 16 , 16:7, or 16 to 7 7

5

6

Leon is driving to his grandmother’s house. He makes the 275-mile trip in 5 hours. What is Leon’s average speed? F 1,375 miles

H 55 miles/hour

G 5 hours

J 65 miles/hour 7

A store sells an 8-pack of water for $4. What is the cost of one bottle of water? A $0.50

C $2.00

B $1.00

D $4.00 8

4

30

Ms. Vierta and her 26 students are going to an art museum. Admission and lunch for everyone will cost $364.50. What is the price per person? F $13.50

H $14.50

G $14.02

J $15.00

Chapter 1 Standards Practice

A 48 pages

C 96 pages

B 64 pages

D 112 pages

Pedro is running in a 26.2-mile marathon. If he completes the marathon in 4 hours, what rate did he average? F 26.2 miles

H 5.15 miles/hour

G 4 hours

J 6.55 miles/hour

Julio is an avid biker. He rides about 140 miles every 4 days. At this rate, how many miles does he ride in 6 days? A 35 miles

C 210 miles

B 175 miles

D 840 miles

A bag contains 5 blue, 6 white, and 3 red marbles. A marble is drawn without looking. What is the probability of drawing a white marble? F 5 out of 14

H 6 out of 14

G 3 out of 14

J 6 out of 12 GO ON

Copyright © by The McGraw-Hill Companies, Inc.

3

Marta finished reading a novel in 8 days. The book was 384 pages. About how many pages did she read per day?

9

A number cube with six sides labeled 1 through 6 is rolled. What is the probability of landing on an even number? 3 or __ 1 1 C __ A __ 2 6 6 5 1 2 or __ D __ B __ 3 6 6

12

Orlando tosses a coin. What is the probability that the coin will land on heads? F 2 out of 2

H 2 out of 1

G 1 out of 2

J 1 out of 1

ANSWER SHEET Directions: Fill in the circle of each correct answer.

Copyright © by The McGraw-Hill Companies, Inc.

10

11

Write a ratio that compares the number of faces to the number of hearts.

F 1 to 1

H 4 to 1

G 1 to 4

J 1 to 5

Claire took a survey of the ways her classmates get to school. The results are shown in the table. Based on her survey, if a student from this class is asked at random, what is the probability that a student walks to school?

Bus

Number of Responses 10

Car

6

Walk

5

Bike

4

Form of Travel

2 , 0.4, or 40% A __ 5 1 , 0.2, or 20% B __ 5

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 Easier questions usually come before harder ones. For the more difficult questions, try to break the information down into smaller pieces. Make sure the answer is reasonable and matches the question asked.

6 , 0.24, or 24% C ___ 25 4 , 0.16, or 16% D ___ 25 Chapter 1 Standards Practice

31

Chapter

2

Percents, Fractions, and Decimals We use percents, fractions, and decimals to show values every day. For example, you earned a 97% on a test. 70 ____ of Earth’s surface is covered with water. 100 I need $0.75 to buy a snack.

Copyright © by The McGraw-Hill Companies, Inc.

32

Chapter 2 Percents, Fractions, and Decimals

Miles Miles Ertman/Masterfile Ertman/Masterfile

STEP

STEP

1 Quiz

2 Preview

Are you ready for Chapter 2? Take the Online Readiness Quiz at ca.mathtriumphs.com to find out. Get ready for Chapter 2. Review these skills and compare them with what you’ll learn in this chapter.

What You Know

What You Will Learn

You know how to understand grades.

Lesson 2-1

If you get 95 questions correct on a test with 100 one-point questions, then you get a 95%.

A percent is a ratio that compares a number to 100. 95 95% means 95 out of 100 or ____. 100

95/100 A

You know how to multiply whole numbers, fractions, and decimals.

Copyright © by The McGraw-Hill Companies, Inc.

The skateboard is on sale for half off. 1 of $60 to determine the You can find __ 2 discount amount. Use multiplication. $60 1 × $60 = ____ __ 2 2 = $30

Lesson 2-2 50 1 or 0.5. 50% is the same as ____ or __ 2 100 To find 50% of $60, multiply $60 by 0.5. $60 · 0.5 = $30 The discount is $30.

The discount is $30. You know how to find probability.

Lesson 2-3

Probability = number of favorable outcomes ____________________________ total number of outcomes

You can compare fractions, decimals, and probabilities. 3 1 is less than __ __ . 7 7 So, the probability of choosing a white sock is less than the probability of choosing a blue sock.

The probability of choosing a white 1. sock without looking is __ 7 The probability of choosing a blue 3 sock without looking is __. 7

33 (bkgd)Miles Ertman/Masterfile, (cl)Lon C. Diehl/PhotoEdit Inc.

Lesson

2-1 Introduction to Percents KEY Concept A ratio is a comparison of two numbers by division. A percent is a ratio that compares a number to 100. A percent is written using the symbol %, called the percent symbol. Percents can also be written as fractions or decimals. 45 squares out of 100 squares are shaded. read as forty-five percent read as forty-five hundredths

45

45% = 100 = 0.45

You can write some fractions as percents by finding an equivalent fraction with a denominator of 100. Any fraction can be converted to a decimal using division. Any decimal can be converted to a percent.

5NS1.2 Interpret percents as a part of a hundred; find decimal and percent equivalents for common fractions and explain why they represent the same value; compute a given percent of a whole number.

VOCABULARY percent a ratio that compares a number to 100; it uses the symbol % The word percent means hundredths or out of 100. equivalent fractions fractions that name the same number 3 6 Example: __ and __ 4 8 ratio a comparison of two numbers by division Example: The ratio of 2 to 3 can be stated as 2 out 2 of 3, 2 to 3, 2:3, or __. 3 (Lesson 1-1, p. 4)

You can use models to show percents, fractions, and decimals.

Identify the percent that is modeled.

1. The model has 100 squares.

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Example 1

YOUR TURN! Identify the percent that is modeled.

1. The model has

squares.

2. There are 15 squares shaded.

2. There are

3. The ratio as a fraction of shaded squares 15 to total squares is ____. 100 4. The fraction is equivalent to 15%.

3. The ratio as a fraction of shaded squares to total squares is .

34

Chapter 2 Percents, Fractions, and Decimals

squares shaded.

4. The fraction is equivalent to

.

Example 2

YOUR TURN!

By the time a child reaches the age of 16 years, about 39% of his or her life has been spent sleeping. Write this percent as a fraction and as a decimal.

Janice works as an office manager. Her company recently moved to a new building with 133% more square feet of space. Write this percent as a fraction and as a decimal.

1. To write 39% as a fraction, drop the percent sign and write a fraction with 39 in the numerator and 100 in the denominator.

1. Write 133% as a fraction. 2. Write 133% as a decimal.

39 _ 100 2. To write the fraction as a decimal, divide 39 by 100. 0.39 Remember, to divide by 100, move the decimal point of the dividend two places to the left.

Copyright © by The McGraw-Hill Companies, Inc.

Who is Correct? 1 Write __ as a percent. 5

Evelina

25_ 1 __ = ___0 = 25% 10 5

Rodney

Anne

20_ 1 __ = ___0 = 200% 10 5

20_ 1 __ = ___0 = 20% 10 5

Circle correct answer(s). Cross out incorrect answer(s).

Guided Practice Identify each percent that is modeled. 1

2

GO ON Lesson 2-1 Introduction to Percents

35

Step by Step Practice 3

Write the ratio of red circles to total circles as a fraction in simplest form. Then write the ratio as a decimal and as a percent. Step 1 Write the ratio of red circles to total circles as a fraction in simplest form. red circles ÷2 ___________ = ______ = _________ = ______ total circles

÷2

Step 2 Write an equivalent fraction with a denominator of 100. × 20 ________ ______ = __________ = 100

× 20

Step 3 Write this fraction as a percent and as a decimal. ________ =

100

________ =

.

100

4

a classroom of 12 girls and 13 boys number of girls to total number of students fraction: ______ in simplest form is ______

× percent: ______________ = ________ = × decimal:

36

Chapter 2 Percents, Fractions, and Decimals

%

Copyright © by The McGraw-Hill Companies, Inc.

Write each ratio as a fraction in simplest form. Then write the ratio as a percent and as a decimal.

5

a classroom of 12 girls and 13 boys number of boys to total number of students fraction: ______ in simplest form is ______

× percent: ______________ = ________ =

%

× decimal:

6

the letters in the word reindeer number of r’s to number of letters fraction: ______ in simplest form is ______

× percent: ______________ = ________ =

%

×

Copyright © by The McGraw-Hill Companies, Inc.

decimal:

7

the letters in the word reindeer number of e’s to number of letters fraction:

in simplest form is

percent: decimal: Write each percent as a fraction with 100 in the denominator and as a decimal. 8

14%

9

8%

10

65%

11

98%

12

72%

13

43% GO ON

Lesson 2-1 Introduction to Percents

37

Step by Step Problem-Solving Practice Solve. 14

MUSIC In Kareem’s class, 7 out of 25 students like country music. Did more or less than 50% of the students like country music? Understand

Read the problem. Write what you know. In Kareem’s class, out of like country music.

Plan

Pick a strategy. One strategy is to use a table.

Solve

Complete the table to name equivalent fractions 7. for ___ 25 Kareem’s class

7

14

25

50

75

Problem-Solving Strategies ✓ Use a table. Look for a pattern. Guess and check. Act it out. Write an equation.

100

Seven out of 25 students is equivalent to out of 100 students. Write this ratio as a fraction with a denominator of 100. Write the fraction as a percent.

So than 50% of the students like country music. You can use a 100 square to model this percent. Is over half of the square shaded?

Check

15

DOGS In 2005, 3 out of every 20 dogs registered with the American Kennel Club were Labrador retrievers. About 4 out of every 80 were German shepherds. Write these ratios as percents. Check off each step. Understand Plan Solve Check

38

Chapter 2 Percents, Fractions, and Decimals

Copyright © by The McGraw-Hill Companies, Inc.

Did more than 50% of Kareem’s class like country music?

16

BASKETBALL Genevieve made 8 out of 20 free throws in basketball practice. In the same practice, Tess made 20 out of 50 free throws. Write the ratios as percents.

Name the decimal, percent, and fraction shown in the model. Explain your answer.

17

Skills, Concepts, and Problem Solving Identify each percent that is modeled. 18

Copyright © by The McGraw-Hill Companies, Inc.

19

Write each percent as a fraction with 100 in the denominator and as a decimal. 20

16%

21

75%

22

80%

23

50%

24

150%

25

112%

GO ON Lesson 2-1 Introduction to Percents

39

Solve. 26

EDUCATION In a fifth-grade classroom of 25 students, 11 students have A’s in math. Write the ratio of A students to total students as a decimal and a percent.

27

FOOD The cafeteria kept track of the food selections of 400 students. The results are shown in the table at the right. Write each percent as a decimal.

Vocabulary Check each sentence.

chicken

Percent of Students 14%

taco

26%

pasta

20%

pizza

40%

Food

Write the vocabulary word that completes

28

A(n)

is a ratio that compares a quantity to 100.

29

Fractions that represent the same number are

30

Writing in Math To write a percent as a decimal, you can write the percent as a fraction with a denominator of 100. For example, 35 35% = ____ = 0.35. So, 35 divided by 100 is 0.35. 100

.

How can you write a decimal as a percent using multiplication?

Find each probability. Write the probability as a fraction in simplest form. (Lesson 1-3, p. 19) 31

choosing a green marble from a bag of 6 red, 5 green, and 4 blue marbles

32

choosing a medium or large T-shirt from a box containing 8 small, 5 medium, and 9 large T-shirts

33

FITNESS Jack and Ti played 12 rounds of golf this season. Ti won 7 rounds. Write a ratio of Ti’s wins to the total number of rounds in simplest form. (Lesson 1-1, p. 4)

40

Chapter 2 Percents, Fractions, and Decimals

Copyright © by The McGraw-Hill Companies, Inc.

Spiral Review

Lesson

2-2

Percents, Fractions, and Decimals

KEY Concept A fraction in simplest form has a numerator and denominator that do not have a common factor. Recall that to write fractions in simplest form, you have to divide the numerator and denominator by their greatest common factor (GCF). Half of all percents, when written as a fraction with 100 in the denominator, can be reduced.

An even number and 100 have a common factor of at least 2.

5NS1.2 Interpret percents as a part of a hundred; find decimal and percent equivalents for common fractions and explain why they represent the same value; compute a given percent of a whole number.

VOCABULARY percent a ratio that compares a number to 100; it uses the symbol % The word percent means hundredths or out of 100. (Lesson 2-1, p. 34)

34 ÷ __ 2 3 34 = ________ 17 34% = ____ = ___ 50 100 100 ÷ 42

50 ÷ 50 3 50 = __________ 1 __ 50% = ____ = __ 4 2 100 100 ÷ 50

When you convert a fraction to a decimal and a percent, divide the numerator by the denominator. 0.8 4 __ 0.8 = 80% 5 5 4.0 -40 ____ 0

Copyright © by The McGraw-Hill Companies, Inc.

For percents that are commonly used, you should become familiar with the equivalent decimals and fractions in simplest form.

Example 1

YOUR TURN!

Write 8% as a decimal and as a fraction in simplest form. 1. The % sign means out of 100. Write 8% as a fraction using this definition. 8% = 8 100 8 2. ____ is read as 8 hundredths. 100 Write this as a decimal.

_

0.08 3. Simplify the fraction, if possible. 8 ÷ __ 4 3 2 _ =_ 100 ÷ 44

25

Write 115% as a decimal and as a fraction in simplest form. 1. Write 115% as a fraction. 115% = ____ or 1____ 100 100 15 2. 1____ is read as 1 and 15 hundredths. 100 Write this as a decimal. 3. Simplify the fraction, if possible. 15 ÷ 15 1____ = 1 ____________ = 1___ 100 100 ÷ GO ON Lesson 2-2 Percents, Fractions, and Decimals

41

Example 2 Write

YOUR TURN!

_3 as a percent and as a decimal. 5

Write

_3 as a percent and as a decimal. 4

1. Identify the denominator. 5

1. Identify the denominator.

2. What number multiplied by 5 is 100? 20

2. What number multiplied by 100?

3. Write a fraction with a denominator of 20 that is equivalent to 1. 20 20 20 3 4. Multiply __ by ___ to obtain a fraction with 5 20 a denominator of 100.

_

20 3 3 × ___ × 20 = ____ 60 __ = _______ 20

5

5 × 20

100

5. Write the fraction as a percent and as a decimal. 60 = 60% _3 = _ 5

60 = 0.60 _3 = _ 5

100

100

To write 0.60 as a percent, you can move the decimal point two places to the right and add a % symbol. 0.60 = 60%

Example 3 8

1. Divide to write the fraction as a decimal. 1 __ 8

0.125   ➡ 8 1.000 ____8 20 - 16 _____ 40 40 _____ 0

2. Write the decimal as a percent by moving the decimal point two places to the right and adding a % sign. 1 = 0.125 = 12.5% 8

_

42

that is equivalent to 1. 3 4. Multiply __ by to obtain a 4 fraction with a denominator of 100. 3 ______ _______ __ × = ____ =3× 4

100

4×

5. Write the fraction as a percent and as a decimal. 3 ____ 3 __ = = % __ = ____ = 4 100 4 100

YOUR TURN!

_1 as a decimal and as a percent.

1 Read __ as 8 1 divided by 8.

3. Write a fraction with a denominator of

Chapter 2 Percents, Fractions, and Decimals

Write

_5 as a decimal and as a percent. 8

1. Divide to write the fraction as a decimal. 5 Read __ as 8 5 divided by 8.

5 __ 8

➡

0.   8 5.000

2. Write the decimal as a percent by moving the decimal point two places to the right and adding a % sign. 5 __ = . % = 0. 8

Copyright © by The McGraw-Hill Companies, Inc.

Write

is

Who is Correct? Write

1 _ as a decimal. 16

Anissa 0.06225 00 1.0 16  -64 360 -320 400 -320 80 -80 0

Delmar

Ken

6.25 = 0.0625 1 × 6.25 = _ _1 = _ 100 5 × 6.2

0.0625  00 1.0 16  -96 40 -32 80 -80 0

16

16

Circle correct answer(s). Cross out incorrect answer(s).

Guided Practice

Copyright © by The McGraw-Hill Companies, Inc.

Write each percent as a fraction or mixed number in simplest form and as a decimal. 1

4%

2

16%

3

126%

4

145%

Step by Step Practice 5

5 Write ___ as a percent and as a decimal. 16 Step 1 Divide 5 by 16.

16  5.000

Step 2 To write as a percent, move the decimal point two places to the right and add the percent sign.

GO ON Lesson 2-2 Percents, Fractions, and Decimals

43

Write each fraction as a decimal and as a percent. 8 6 ___ 7 25 6 8 ___ 9 32

2 ___ 25 7 ___ 16

Step by Step Problem-Solving Practice

Problem-Solving Strategies Use a table.

Solve. 10

✓ Look for a pattern.

CHEMISTRY During a chemistry experiment, Taina filled four beakers with liquids in different amounts. She must mark each beaker with the percent of liquid with which it was filled. How should she mark the beakers? Understand

Guess and check. ✓ Use logical reasoning. Work backward.

Read the problem. Write what you know. The beakers must be marked with a

Plan

Pick a strategy. Two strategies are to look for a pattern and use logical reasoning.

Solve

Look at the first beaker. How full is it?

.

How does the second beaker compare to the full one? How does the third beaker compare with the second one? How does the fourth beaker compare with the other three beakers?

1

2

3

4

Percent Fraction Decimal Check

44

Is the greatest percent under the beaker with the largest amount of liquid? Is the least percent under the beaker with the least amount of liquid?

Chapter 2 Percents, Fractions, and Decimals

Copyright © by The McGraw-Hill Companies, Inc.

Label each beaker with the correct percent, fraction, and decimal.

11

MOVIES The graph shows the percents of students surveyed who enjoy science fiction, action, foreign, and animated films. In a circle graph, the percents add up to 100%. Label the four categories with their approximate percents.

Favorite Types of Movies animated action foreign sci-fi

Check off each step. Understand Plan Solve Check

Copyright © by The McGraw-Hill Companies, Inc.

12

When you change a percent to a decimal, you move the decimal point two place values to the left. Why is it always two places?

Skills, Concepts, and Problem Solving Write each percent as a fraction or mixed number in simplest form and as a decimal. 13

12%

14

6%

15

115%

16

130%

17

40%

18

75%

19

220%

20

3%

21

33%

22

125% GO ON Lesson 2-2 Percents, Fractions, and Decimals

45

Write each fraction as a percent and as a decimal. 1 2 23 __ 24 __ 3 3 25

3 __

27

3 __

29

23 ___

31

3 __

2 5 10 4

26

9 __

28

5 __

30

5 __

32

8 __

4 8 6 5

Write each percent as a fraction and decimal to complete this chart of common percents. Meaning

33

10%

10 out of 100

34

20%

20 out of 100

35

25%

25 out of 100

36

50%

50 out of 100

37

75%

75 out of 100

Fraction

Decimal

Solve. 38

2 pineapple juice. What FOOD Cheryl is making punch that is __ 5 percent of the punch is pineapple juice?

39

FOOD James is making trail mix. The recipe says 2 parts crunchy cereal, 1 part peanuts, 1 part raisins, and 1 part pretzels. What fraction of the mix is peanuts? What percent of the mix is raisins?

46

Chapter 2 Percents, Fractions, and Decimals

Copyright © by The McGraw-Hill Companies, Inc.

Percent

Vocabulary Check sentence.

Write the vocabulary word that completes each means “out of one hundred.”

40 41

A percent is a(n)

42

Writing in Math

that compares a number to 100. Explain how to change a fraction to a percent.

Spiral Review Identify the percent that is modeled. 43

44

Solve. Copyright © by The McGraw-Hill Companies, Inc.

(Lesson 2-1, p. 34)

45

(Lesson 1-3, p. 19)

46

ARCHEOLOGY Read the photo caption at the right. Write the ratio that represents Anica and JaLisa’s findings. What is the probability they will not find a trilobite next?

47

NATURE The after-school study group collected 75 leaves one afternoon and placed them in a bag. If 30 of those leaves are maple leaves, what is the probability that a leaf When Anica and JaLisa dug for fossils, they selected from the bag without looking found 3 trilobites for every 4 fossils. will be a maple leaf?

Write the ratio as a fraction in simplest form.

(Lesson 1-1, p. 4)

48

A bag of 12 marbles has 5 marbles that are not white. Write the ratio in simplest form of white marbles to nonwhite marbles.

49

In the classroom, there are 36 students to 4 computers. Write the ratio of students to computers in simplest form. Lesson 2-2 Percents, Fractions, and Decimals

(l)Getty Images, (r)Mark A. Schneider/Photo Researchers

47

Chapter

2

Progress Check 1

(Lessons 2-1 and 2-2)

Identify the percent that is modeled. 1

2

Write each percent as a decimal and as a fraction in simplest form. 3

25%

4

66%

5

150%

6

101%

A bag of nuts has 8 peanuts, 4 cashews, and 8 walnuts. Write each ratio as a fraction in simplest form, a percent, and a decimal. 7

number of peanuts to total number of nuts

8

number of cashews to total number of nuts

11

11 ___

2 7 14 __ 8

Solve. 15

FOOD William is making fruit baskets. Each basket contains 3 apples, 2 oranges, 1 banana, 1 grapefruit, and 1 pear. What percent of the fruit is oranges?

16

EXERCISE Janelle works out for 40 minutes each morning. She does stretching exercises for 5 minutes, weight training for 15 minutes, and a cardio workout for 20 minutes. What percent of her workout is cardio?

17

ENTERTAINMENT In one town, 66% of the residents read the newspaper. Before an election, 87% of the same residents read the paper. Write each percent as a decimal and a fraction in simplest form.

48

Chapter 2 Percents, Fractions, and Decimals

Copyright © by The McGraw-Hill Companies, Inc.

Write each fraction as a percent and as a decimal. 3 1 9 __ 10 __ 4 4 6 3 12 __ 13 __ 8 4

Lesson

2-3 Compare Data Sets of

_

Different Sizes KEY Concept Percents can be used to compare ratios from sets of data that have different sizes. When you compare ratios written as percents, you are comparing numerators of fractions that have the same denominator of 100. Consider the ratios of 2 out of 5 and 1 out of 2. 3 × 20 40 2 = 2_______ __ __ = ____ = 40% 5

4 5 × 20

100

3 × 50 50 1 = 1_______ __ __ = ____ = 50% 2

4 2 × 50

6NS1.2 Interpret and use ratios in different contexts to show the relative sizes of two quantities using appropriate a notations ( , a to b, a:b). b 5SDAP1.3 Use fractions and percentages to compare data sets of different sizes.

VOCABULARY compare to look closely at numbers or objects to find similarities and differences Example: 5 is less than 7 (difference). 5 and 7 are numbers (similarity)

100

Since 40% < 50%, 2 < __ 1 __ 5 2

Copyright © by The McGraw-Hill Companies, Inc.

When you compare numbers, write them in the same form.

Example 1

YOUR TURN!

Express the circled data as a fraction and a percent of the entire data set.

Express the circled data as a fraction and a percent of the entire data set.

2, 4, 7, 8, 9 1. How many numbers are in the data set? 5 2. How many numbers are circled? 3 3 3. Write the ratio as a fraction. __ 5 4. Write the ratio as a percent. 60%

1, 3, 5, 6, 10, 11, 12, 15 1. How many numbers are in the data set? 2. How many numbers are circled? 3. Write the ratio as a fraction. 4. Write the ratio as a percent.

GO ON Lesson 2-3 Compare Data Sets of Different Sizes

49

Example 2

YOUR TURN!

Use fractions to compare the ratios of the number of red triangles to the number of blue triangles for each set of figures. Set A

Set B

1. Write the ratio of red triangles to blue triangles for Set A as a fraction in simplest 2 form. __ 3 2. Write the ratio of red triangles to blue triangles for Set B as a fraction in simplest 3 form. __ 5 3 2 > __ 3. Compare the fractions. __ 3 5 So, Set A has the greater ratio of red triangles to blue triangles.

Use fractions to compare the number of green circles to the number of red circles for each set of figures. Set A

Set B

1. Write the ratio of the number of green circles to the number ______ of red circles for Set A as a fraction in simplest form. 2. Write the ratio of the number of green circles to the number ______ of red circles for Set B as a fraction in simplest form. 3. Compare the fractions. ______ > ______ has a greater ratio of the So, number of green circles to the number of red circles.

One bag contains 50 marbles. Thirty of them are red. Another bag contains 35 marbles and 28 of them are red. Use percents to compare the ratios of the red marbles to the entire bag of marbles to determine which bag has the greater percentage of red marbles. 1. Write the ratio of the red marbles to the entire bag of marbles as a 30 percent for the first bag. ___ = 60% 50 2. Write the ratio of the red marbles to the entire bag of marbles as a 28 percent for the second bag. ___ = 80% 35 3. Compare the percents. 80% > 60% 4. The second bag has a greater percentage of red marbles. 50

Chapter 2 Percents, Fractions, and Decimals

Copyright © by The McGraw-Hill Companies, Inc.

Example 3

YOUR TURN! One package of 8 socks contains 3 blue socks. A second package of 15 socks contains 5 blue socks. Use percents to compare the ratios of the number of blue socks to the entire package to determine which package has the greater percentage of blue socks. 1. Write the ratio of the number of blue socks to the entire package as a percent for the first package.

______ =

%

2. Write the ratio of the number of blue socks to the entire package as a percent for the second package.

______ =

%

>

3. Compare the percents.

4. The package has a greater percentage of blue socks to the entire package.

Who is Correct? Compare

28 _ and 55%.

Sara 14 28 = ___ ___

50

25 0.56  .00 14 25  25 -1 150 -150 28 > 55% ___ 50

50

Copyright © by The McGraw-Hill Companies, Inc.

Paquito

A.J.

56_ × 2_ = ___ 28 = 28 ______ ___ 0 10 2 × 50 50

56_ × 2_ = ___ 28 = 28 ______ ___ 0 10 2 × 50 50

28 > 55% ___

28 < 55% ___

50

50

Circle correct answer(s). Cross out incorrect answer(s).

Guided Practice Express the shaded areas of the figures as a fraction and as a percent of the area of the entire figure. 1

0.6

2

GO ON Lesson 2-3 Compare Data Sets of Different Sizes

51

Step by Step Practice 3

Your sister gave you two bags of pens. One bag contained 4 blue pens and 6 red pens. The other bag contained 9 red pens and 3 green pens. Which bag had the greater percentage of red pens? red pens and a total

Step 1 In the first bag, there were of pens.

red pens and a

In the second bag, there were total of pens. Step 2 Write each ratio as a percent. ______ =

%

______ =

<

Step 3 Compare the percents. Step 4 The of red pens.

4

%

had the greater percentage

In Amira’s box of 15 party hats, 9 of them are silver. In Lupe’s box of 30 hats, 21 of them are silver. Which girl has the greater ratio of the number of silver party hats to the total number of hats?

What is the ratio in simplest form of the number of silver hats to the total number of hats in Lupe’s box?

has the greater ratio of the number of silver hats to the total number of hats in her box. 5

52

Jaime has two binders that are filled with CDs. One can hold 50 CDs, and the other can hold 30 CDs. The first one contains 26 hip-hop CDs, and the second one contains 16 hip-hop CDS. Which binder has the greater ratio of the number of hip-hop CDs to the total number of CDs?

Chapter 2 Percents, Fractions, and Decimals

Copyright © by The McGraw-Hill Companies, Inc.

What is the ratio in simplest form of the number of silver hats to the total number of hats in Amira’s box?

Step by Step Problem-Solving Practice

Problem-Solving Strategies

Solve. 6

SPORTS Bianca and Chris played darts. Bianca hit the bull’s-eye 5 out of 18 times. Chris missed the bull’s-eye 5 out of 15 times. Who hit the bull’s-eye a greater percent of the time?

Use a table. Look for a pattern. Write an equation. Guess and check. ✓ Use logical reasoning.

Understand

Read the problem. Write what you know. Bianca hit the bull’s-eye out of times. Chris did not hit the bull’s-eye out of times.

Plan

Pick a strategy. One strategy is to use logical reasoning.

Solve

Write the fraction of Bianca’s hits to attempts. Write the fraction of Chris’s misses to attempts. Write each fraction as a decimal rounded to the nearest hundredth, then as a percent. .28

5 ___ = 18  5.00 = 28% 18

.33

5 ___ = 15  5.00 = 33% 15

Copyright © by The McGraw-Hill Companies, Inc.

Chris missed the bull’s-eye 33% of the time, so he hit the bull’s-eye 100% - 33% = % of the time. Compare the percents. hit the bull’s-eye a greater percent of the time. Does the answer make sense? Look over your solution. Did you answer the question?

Check

7

WEATHER The table shows how much rain fell in Albuquerque and Denver for the given number of days. Which city had the greater fraction of inches of rain per day? Check off each step.

City

Rain (in.)

Days (#)

Albuquerque, NM

9

60

Denver, CO

15

90

Understand Plan Solve Check

GO ON Lesson 2-3 Compare Data Sets of Different Sizes

53

8

Nestor is using two different recipes for a cake. The first recipe calls for 1 cup of sugar and 3 eggs. The other recipe calls for 1 cups of sugar and 4 eggs. Name the ratios of eggs to sugar for 1__ 2 each recipe. Which recipe has the smaller ratio of eggs to sugar?

9

Mrs. Rodriguez just returned from the grocery store. Her first bag of groceries contains 12 items, and 6 of them are frozen. Her second bag of groceries contains 18 items, and 10 of them are frozen. Which bag has the greater percentage of frozen items in it?

Do you prefer to compare sets of data of different sizes by using ratios written as fractions or as percents? Explain.

10

Skills, Concepts, and Problem Solving Compare. Circle the greater value. 11

1 and 40% ___

5 12 __

8 13 ___

and 62%

10

and 81%

Express the shaded area of the figures as a fraction and a percent of the area of the entire figure. 14

15

Express the circled data as a fraction and as a percent of the entire data set. 16

1, 3, 5, 6, 10, 11

18

54

17

19

Chapter 2 Percents, Fractions, and Decimals

10, 20, 30, 40, 50, 60, 70, 80

Copyright © by The McGraw-Hill Companies, Inc.

25

8

Solve. 20

For the fall school concert, the music director purchased 60 bottles of juice for refreshments. Of those, 25 bottles were apple juice. For the spring concert, she purchased 75 bottles of juice, and 35 bottles were apple juice. Which concert had the greater ratio of the number of bottles of apple juice to the total number of bottles of juice?

21

One spinner is divided into 6 equal parts. Two of the parts are green, 3 of the parts are red, and 1 part is yellow. Another spinner is divided into 5 equal parts. Two of the parts are green, 2 of the parts are red and 1 part is yellow. Which spinner has the greater percentage of red parts?

Copyright © by The McGraw-Hill Companies, Inc.

Vocabulary Check sentence.

Write the vocabulary word that completes each

22

When you , you look closely at numbers or objects to find similarities and differences.

23

If numbers ascend from least to greatest, or descend from greatest to least, they are said to be in .

24

Writing in Math Which of these ratios does not equal the value of the others? How do you know? 3 __ 9 out of 15 0.06 60% 5

Spiral Review Solve.

(Lesson 2-1, p. 34)

25

FITNESS Miriam ran 5 miles in 35 minutes. Terry ran 6 miles in 36 minutes. Who ran at a slower rate?

26

NATURE During a 30-minute walk, Alma identified 6 bird species. During an hour walk, Kota identified 10 bird species. Who identified birds at a faster rate?

Find each unit rate.

(Lesson 1-2, p. 11)

27

15 inches every 30 seconds

28

150 miles every 2 hours Lesson 2-3 Compare Data Sets of Different Sizes

55

Chapter

2

Study Guide

Vocabulary and Concept Check compare, p. 49

Write the vocabulary word that completes each sentence.

equivalent fractions, p. 34

1

order, p. 49 percent, p. 34

To is to look closely at numbers or objects to find similarities and differences. means hundredths or out of 100.

2

ratio, p. 34 3

When numbers ascend from least to greatest, or descend from greatest to least, they are in .

4

A(n) numbers by division.

is a comparison of two

Write the correct vocabulary term in each blank. 5

6

6 2 and __ __ 3

1 < 0.5 20% < __ 4

9

Lesson Review Introduction to Percents

Identify each percent that is modeled.

(pp. 34–40)

Example 1 Identify the percent that is modeled.

7

The model has 100 squares. 20 squares are shaded.

8

The ratio as a fraction of shaded squares to 20 total squares is ____. 100 The fraction written as a percent is 20%.

56

Chapter 2 Study Guide

Copyright © by The McGraw-Hill Companies, Inc.

2-1

Write each percent as a fraction with a denominator of 100 and as a decimal. 9

44%

Write 84% as a fraction with a denominator of 100 and as a decimal.

10

13%

11

26%

Write 84% as a fraction. 84 100 Write 84% as a decimal. 0.84

2-2

Percents, Fractions, and Decimals (pp. 41–47)

Write each fraction as a decimal and as a percent. 1 12 __ 2 13

9 __

14

1 __

_

Example 3 Write

_5 as a decimal and as a percent. 8

0.625 8  5.000 -48 20 -16 40 -40 0

1. Divide to convert to a decimal. 2. Move the decimal point two places to the right and add the percent symbol. 62.5%

4 5

2-3

Copyright © by The McGraw-Hill Companies, Inc.

Example 2

Compare Data Sets of Different Sizes (pp. 49–55)

Compare the ratios described as a fraction in simplest form and as a percent. Use the following data sets. Set A 2, 3, 5, 7, 11, 13

Example 4 Compare the ratio of the number of blue squares to the total number of squares as a fraction in simplest form and as a percent. Set A

Set B

Set B 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 15

16

17

The number of even numbers to the number of odd numbers for each data set.

The number of blue numbers to the total numbers for each data set.

Write as a fraction and as a percent the ratio of the number of blue squares to the total number of squares for each set. Set A 2 = 66__ 2% __ 3 3

Set B 4 = 80% __ 5

2 %, so Set B has a greater Compare. 80% > 66__ 3 ratio of the number of blue squares to the total number of squares.

The number of odd numbers to the total numbers for each data set.

Chapter 2 Study Guide

57

Chapter

2

Chapter Test

Identify the percent that is modeled. 1

2

Write each percent as a decimal and as a fraction in simplest form. 3

55%

4

28%

5

125%

6

107%

A small bag of snack mix has 8 pretzels, 12 peanuts, 10 cheese crackers, and 2 raisins. Write the ratios of the pretzels to the total number of pieces and peanuts to the total number of pieces as fractions in simplest form, as percents, and as decimals. 7

number of pretzels to total number of pieces

8

number of peanuts to total number of pieces

5 __ 4 12 ___ 5 6 ___ 15

Copyright © by The McGraw-Hill Companies, Inc.

Write each fraction as a decimal and as a percent. 1 9 __ 10 2 9 11 __ 12 8 3 13 __ 14 5

Use the spinner to write each ratio as a fraction in simplest form. Then write the ratio as a percent and as a decimal. 15

number of orange sections to total number of sections fraction

16

percent

decimal

number of blue sections to number of nonblue sections fraction

58

decimal

number of yellow sections to the number of orange sections fraction

17

percent

Chapter 2 Test

percent

decimal

GO ON

Express the circled data as a fraction in simplest form and as a percent of the entire data set. 18

a, b, c, d, e, f, g, h, i, j

19 20

LeBron made 49 free throws out of 70 attempts. Donyell made 39 free throws out of 59 attempts. Which player made the greater percentage of free throws?

Solve. 21

GAMES At one video-game store, 29% of the sales of the newrelease video games were sold by presale orders. At a larger store in town, 54% of the new-release sales were sold to customers who were waiting in line outside the store for several hours. Write each percent as a decimal and as a fraction in simplest form.

22

BASEBALL Matthew got 24 hits out of his last 75 at bats. Write his batting average as a decimal and as a fraction in simplest form.

Copyright © by The McGraw-Hill Companies, Inc.

Correct the mistakes. 23

Thomas wanted to purchase an MP3 player. The regular price was $59.99. It was on sale for 10% off. When Thomas’s mother asked him about the sale price, he told her that the discount perentage was equal to 0.12. What did Thomas do wrong?

24

Jamal plays soccer in a league with 100 other students, 53 of which are girls. Ito plays in a different league, and 27 out of 80 students are girls. Heather said the percentage of girls in each league was equal since 100 - 53 = 47 and 80 - 27 = 47. What did Heather do wrong?

Chapter 2 Test

59

Chapter

2

Standards Practice

Choose the best answer and fill in the corresponding circle on the sheet at right. 1

Which model represents 74%?

3

Which statement is correct? 2 = 0.24 = 24% 1 = 0.20 = 20% C __ A __ 5 4 6 = 0.75 = 7.5% 1 = 1.0 = 10% D __ B ___ 8 10

4

Dominic watches a quiz show on TV every night. He guesses correctly on 70% of the questions. What fraction of the questions does Dominic answer correctly? 7 7 H ___ F __ 5 10 3 2 G __ J __ 4 4

A

B

C

D 5

A 70%

C 80%

B 75%

D 90%

Which percent represents the shaded portion of the model? 6

Damian read 320 pages of his novel this week. Eli read 80% of Damian’s total pages. How many pages did Eli read this week?

F 50%

H 112%

F 26 pages

H 256 pages

G 100%

J 150%

G 240 pages

J 260 pages

GO ON 60

Chapter 2 Standards Practice

Copyright © by The McGraw-Hill Companies, Inc.

2

_

Ayden scored 16 on his math quiz. 20 What is his percent?

7

A meal totals $22.60, including tax. After a 20% tip is added, what is the total cost of the meal?

10

A $4.52

Tisha flew 2,496 miles across the country. The flight lasted 6 hours. At what rate was the plane traveling? F 6 hours

B $18.08

G 416 miles/hour

C $22.60

H 419 miles/hour

D $27.12

J 2,496 miles

8

This sweater is priced at $38. What is the price after the discount?

Copyright © by The McGraw-Hill Companies, Inc.

Sweaters 25% off

9

ANSWER SHEET Directions: Fill in the circle of each correct answer. 1

A

B

C

D

2

F

G

H

J

3

A

B

C

D

F $9.50

4

F

G

H

J

G $25

5

A

B

C

D

H $28.50

6

F

G

H

J

J $950

7

A

B

C

D

8

F

G

H

J

9

A

B

C

D

10

F

G

H

J

What is the ratio of striped marbles to solid-colored marbles?

Success Strategy Read the entire question before looking at the answer choices. Make sure you know what the question is asking.

5 , 5:3, or 5 to 3 A __ 3 4 , 4:5, or 4 to 5 B __ 5

4 , 4:3, or 4 to 3 C __ 3 3 , 3:5, or 3 to 5 D __ 5 Chapter 2 Standards Practice

61

Index A Algebra and Functions, 11 Answer sheet, 31, 61 Assessment, 28–29, 58–59

C California Mathematics Content Standards, 4, 11, 19, 34, 41, 49 Chapter Preview, 3, 33 Chapter Test, 28–29, 58–59

M Manipulatives base-ten blocks, 34, 35, 39, 47, 48, 54, 56, 58, 60 fraction circles and strips, 51, 54 Mathematical Reasoning. See Step-by-Step Problem Solving

N Number Sense, 4, 11, 34, 41, 49

compare, 49–55 Correct the Mistakes, 29, 59

D decimals, 41–47

E

O outcomes, 19–25

P probability, 19–25

Standards Practice, 30–31, 60–61

event, 19–25

Progress Check, 18, 48

K Key Concept, 4, 19, 34, 41, 49

62

Index

R rate, 4–10, 11–17 ratio, 4–10, 19–25, 34–40 Real-World Applications ages, 8 archeology, 47 baseball, 59 basketball, 39 business, 15 chemistry, 44 cooking, 54 dogs, 38 education, 40 entertainment, 48 exercise, 48 fitness, 10, 40, 55 food, 23, 40, 46, 48

Statistics, Data Analysis, and Probability, 19, 34, 49 Step-by-Step Practice, 7, 14, 21, 36, 43, 52 Step-by-Step Problem Solving Practice, 8–9, 15, 23, 38, 44, 53 Look for a pattern, 44 Solve a simpler problem, 8, 15 Use a table, 38 Use logical reasoning, 23, 44, 53 Study Guide, 26–27, 56–57 Success Strategy, 31, 61

U unit cost, 11–17 unit rate, 11–17

Copyright © by The McGraw-Hill Companies, Inc.

fraction comparing, 49–55 decimals to, 41–47 equivalent, 4–10, 11–17, 34–40 percents to, 41–47 probability, 19–25 ratios as, 4–10, 11–17 to decimals, 41–47 to percents, 41–47

S Spiral Review, 17, 25, 40, 47, 55

equivalent ratios, 4–10, 11–17

F

Reflect, 9, 16, 24, 39, 45, 54

percent, 34–40, 41–47, 49–55 Problem-Solving. See Step-byStep Problem Solving

equivalent fractions, 34–40

football, 9 fund-raiser, 16 games, 24 genetics, 23 health, 25 insects, 25 language, 25 life science, 17 movies, 45 music, 38 nature, 15, 47, 55 population, 16, 17 reading, 29 sewing, 25 spelling, 18, 29 sports, 10, 18, 53 tennis, 9 travel, 29 weather, 53

V Vocabulary, 4, 19, 34, 41, 49 Vocabulary and Concept Check, 26, 56 Vocabulary Check, 10, 17, 25, 40, 47, 55

W Who is Correct?, 6, 13, 21, 35, 43, 51

Copyright © by The McGraw-Hill Companies, Inc.

Writing in Math, 10, 17, 25, 40, 47, 55

Index

63