Basic collecge mathematics 2e by miller

Santa Fe Focus Group Attendees Pauline Chow, Harrisburg Area Community College Alina Coronel, Miami-Dade College Anabel Darini, Suffolk County Community College Susan Dimick, Spokane Com

Miller O’Neill Hyde

SECONDEDITION

A Student Success Story

Middlesex County College (Edison, NJ) achieved an 80% pass rate in their developmental mathematics

course, an increase of 29% in student retention, using Miller/O’Neill/Hyde and ALEKS!

In fall semester 2007, 308 Middlesex students piloted ALEKS in their developmental course sequence

with pass rate of 80%, whereas the remaining sections, using only an online homework system,

resulted in a 51% pass rate

ALEKS has enabled our students to grow in both profi ciency and understanding of

the fundamental concepts needed to succeed in developmental math courses I am

delighted with how our students have embraced this learning tool

—Maria DeLucia, Chair, Middlesex County College

Success Rates at Middlesex County College

GET BETTER RESULTS with Miller/O’Neill/Hyde + ALEKS!

ALEKS (Assessment and LEarning in Knowledge Spaces) is an artifi cial intelligence-based system for mathematics learning, available online 24/7

individually and class-wide

Go to www.aleks.com/highered/math to learn more and register!

In spring semester 2008, Middlesex County

to achieve a pass rate of 79%

ISBN 978-0-07-340611-4 MHID 0-07-340611-2 Part of

ISBN 978-0-07-728113-7 MHID 0-07-728113-6

www.mhhe.com

BASIC COLLEGE MATHEMATICS, SECOND EDITION

Published by McGraw-Hill, a business unit of The McGraw-Hill Companies, Inc., 1221 Avenue of the Americas, New York, NY 10020 Copyright © 2009

by The McGraw-Hill Companies, Inc All rights reserved Previous edition © 2007 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 the prior written consent of The McGraw-Hill Companies, Inc., including, but not limited to, in any network or other electronic storage or transmission, or broadcast for distance learning.

Some ancillaries, including electronic and print components, may not be available to customers outside the United States.

This book is printed on acid-free paper

Editorial Director: Stewart K Mattson

Senior Sponsoring Editor: David Millage

Director of Development: Kristine Tibbetts

Senior Developmental Editor: Emilie J Berglund

Marketing Manager: Victoria Anderson

Lead Project Manager: Peggy J Selle

Senior Production Supervisor: Sherry L Kane

Lead Media Project Manager: Stacy A Patch

Designer: Laurie B Janssen

Cover Illustration: Imagineering Media Services, Inc.

Lead Photo Research Coordinator: Carrie K Burger

Supplement Producer: Mary Jane Lampe

Compositor: Aptara®, Inc.

Typeface: 10/12 Times Ten Roman

Printer: R R Donnelley Willard, OH

The credits section for this book begins on page C-1 and is considered an extension of the copyright page.

Library of Congress Cataloging-in-Publication Data

Miller, Julie, 1962–

Basic college mathematics / Julie Miller, Molly O’Neill, Nancy Hyde — 2nd ed.

p cm.

Includes index.

ISBN 978–0–07–340611–4 — ISBN 0–07–340611–2 (hard copy : alk paper)

ISBN 978–0–07–335806–2 — ISBN 0–07–335806–1 (inst ed : hard copy : alk paper) 1 Mathematics—Textbooks I O’Neill, Molly, 1953– II Hyde, Nancy III Title

QA37.3.M55 2009

510—dc22

2008019841

www.mhhe.com

The primary goal of our project was to create teaching and learning materials that would get better results.

At Daytona State College, our students were instrumental in helping us develop the clarity of writing, the step-by-step examples, and the pedagogical elements, such as Avoiding Mistakes, Concept Connections, and Problem Recognition Exercises, found in our textbooks When our text and course redesign were implemented at Daytona State College in 2006, our student success rates in developmental courses improved by 20% We think you will agree that these are the kinds of results we are all striving for in developmental mathematics courses

This project has been a true collaboration with our Board of Advisors and colleagues in developmental mathematics around the country We have been truly humbled by those

of you who adopted the first edition and the over 400 colleagues around the country who partnered with us providing valuable feedback and suggestions through reviews, symposia, focus groups, and being on our Board of Advisors You partnered with us to create materials that will help students get better results For that we are immeasurably grateful

As an author team, we have an ongoing commitment to provide the best possible materials for instructors and students With your continued help and suggestions we will continue the quest to help all of our students get better results

Sincerely,

millerj@DaytonaState.edu oneillm@DaytonaState.edu nhyde@montanasky.com

Julie, Molly, and Nancy:

“Dedicated to Getting Better Results”

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en-“My father is a medical researcher, and I got hooked on math and science when I was young and would visit his laboratory I can remember using graph paper to plot data points for his experiments and doing simple calculations He would then tell me what the peaks and features in the graph meant in the context of his experiment I think that applications and hands-on ex-perience made math come alive for me and I’d like to see math come alive for my students.”

—Julie Miller

Molly O’Neill is also from Daytona State College, where she has taught for 21 years

in the Mathematics Department She has taught a variety of courses from

devel-opmental mathematics to calculus Before she came to ida, Molly taught as an adjunct instructor at the University of Michigan–Dearborn, Eastern Michigan University, Wayne State University, and Oakland Community College Molly earned

Flor-a bFlor-achelor of science in mFlor-athemFlor-atics Flor-and Flor-a mFlor-aster of Flor-arts Flor-and teaching from Western Michigan University in Kalamazoo, Michigan Besides this textbook, she has authored several course supplements for college algebra, trigonometry, and precalculus and has reviewed texts for developmental mathematics

“I differ from many of my colleagues in that math was not always easy for me But in seventh grade I had a teacher who taught me that if I follow the rules of mathematics, even I could solve math problems Once I understood this, I enjoyed math to the point of choosing it for my career I now have the greatest job because I get to

do math everyday and I have the opportunity to influence my students just as I was influenced Authoring these texts has given me another avenue to reach even more students.”

acan remember usin

oiMUatMsa

atsthe point of choosin

Depart-equations She received a bachelor of science degree in math education from Florida State University and master’s degree in math education from Florida Atlantic University She has con-ducted workshops and seminars for both students and teachers

on the use of technology in the classroom In addition to this textbook, she has authored a graphing calculator supplement for College Algebra

“I grew up in Brevard County, Florida, where my father worked

at Cape Canaveral I was always excited by mathematics and ics in relation to the space program As I studied higher levels of mathematics I became more intrigued by its abstract nature and infinite possibili-ties It is enjoyable and rewarding to convey this perspective to students while help-ing them to understand mathematics.”

phys-—Nancy Hyde

finfinite possibili-

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About the Cover

A mosaic is made up of pieces placed together to create a unified whole Similarly, a basic math course provides an array of materials that together create a solid mathematical foundation for the developmental mathematics student

The Miller/O’Neill/Hyde developmental mathematics series helps students to see the whole picture through the better pedagogy and supplemental materials In the second edition of their developmental mathematics series, Julie Miller, Molly O’Neill, and Nancy Hyde focused their efforts on guiding students successfully through core topics to build students’ mathematical proficiency and to get better results

“ We originally embarked on this textbook project because we were seeing a lack of student success in courses beyond our developmental sequence We wanted to build a better bridge between developmental algebra and higher level math courses Our goal has been to develop pedagogical features to help students achieve better results in mathematics.”

—Julie Miller, Molly O’Neill, Nancy Hyde

Get Better Results with Miller/O’Neill/Hyde

Better Clarity, Quality, and Accuracy

Julie Miller, Molly O’Neill, and Nancy Hyde know what students need to be successful in mathematics Better results come from clarity in their exposition, quality of step-by-step worked examples, and accuracy of their exercise sets, but it takes more than just great authors to build a textbook series

to help students achieve success in mathematics Our authors worked with a strong mathematical team of instructors from around the country to ensure clarity, quality, and accuracy

Better Exercise Sets!

A comprehensive set of exercises are available for every student level Julie Miller, Molly O’Neill, and Nancy Hyde worked with a national board of advisors from across the country to ensure the series will offer the appropriate depth

and breadth of exercises for your students New to this edition, Problem Recognition Exercises were created in

direct response to student need and resulted in improved student performance on tests

Our exercise sets help students progress from skill development to conceptual understanding

Student tested and instructor approved, the Miller/O’Neill/Hyde exercise sets will help your students get better results

▶ Problem Recognition Exercises

▶ Skill Practice Exercises

▶ Study Skills Exercises

▶ Mixed Exercises

▶ Expanding Your Skills Exercises

Better Step-By-Step Pedagogy!

The second edition provides enhanced step-by-step learning tools available to help students get better results

▶ Worked Examples provide an “easy-to-understand” approach, clearly guiding each student through a

step-by-step approach to master each practice exercise for better comprehension

▶ TIPS offer students extra cautious direction to help improve understanding through hints and further insight

▶ Avoiding Mistakes boxes alert students to common

errors and provide practical ways to avoid them

These learning aids will help students get better results by learning how to work through a problem using a clearly defined step-by-step methodology that has been class-tested and student approved

How Will Miller/O’Neill/Hyde Help Your Students

Get Better Results?

ries

“ The authors’ writing style is very straight forward and easy to follow The level of formality is just right for this level of math course.”

—Lynette King, Gadsden State College

“ Miller/O’Neill/Hyde has a very good pedagogy that is student-friendly It has

a plethora of problems and variety of them It allows success for all students.”

—Mark Marino, Erie Community College

“ I think that of all the textbooks that I have seen (or evaluated) they (MOH) have by far the most comprehensive sets of exercises

at every level (skill-based, study skills, etc.).”

—Juan Jimenez, Springfi eld Technical

Community College

viiGet Better Results

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Applying a Proportion to Environmental Science

A biologist wants to estimate the number of elk

in a wildlife preserve She sedates 25 elk and

clips a small radio transmitter onto the ear of

each animal The elk return to the wild, and

after 6 months, the biologist studies a sample of

120 elk in the preserve Of the 120 elk sampled,

4 have radio transmitters Approximately how

many elk are in the whole preserve?

4 To estimate the number of fish

in a lake, the park service catches 50 fish and tags them After several months the park service catches a sample

of 100 fish and finds that 6 are tagged Approximately how many fish are in the lake?

Solution:

Let n represent the number of elk in the whole preserve.

Sample Population

Equate the cross products.

Divide both sides by 4.

Divide There are approximately 750 elk in the wildlife preserve.

3000 ⫼ 4 ⫽ 750.

n ⫽ 750

41n

4 1

⫽30004

with radio transmitters

with radio transmitters

PROCEDURE Solving a Proportion

Step 1 Set the cross products equal to each other.

Step 2 Divide both sides of the equation by the number being multiplied

by the variable.

Step 3 Check the solution in the original proportion.

Step-by-Step Worked Examples

▶ Do you get the feeling that there is a disconnect between your students classwork and homework?

▶ Do your students have trouble finding worked examples that match the practice exercises?

▶ Wouldn’t you like your students to see examples in the textbook that match the ones you use in class?

Miller/O’Neill/Hyde’s worked examples offer a clear, concise methodology that replicates the

mathematical processes used in the authors’ classroom lectures!

Formula for Student Success

Applying a Proportion to Environ

Example 4

“In the year we’ve used this text I’ve noticed

that students seem to be able to learn the

material without diffi culties I attribute a lot of

that to the fact the text contains examples that

are worked out clearly and able to follow.”

—Rod Oberdick, Durham Tech Comm Coll

—Susan Haley, Florence-Darlington Technical College

e s se“ Miller/O’Neill/Hyde presents each concept in ve.

clear language Multiple examples covering various forms of problems are included and explained step by step.”

—Susan Harrison, University of

Wisconsin-Eau Claire

viii

Better Learning Tools

Simplify to lowest terms.

Simplify to lowest terms.

⫽23

⫽1015

⫽63

4 2

Avoiding Mistakes

Notice that when adding fractions,

we do not add the denominators.

We add only the numerators.

Write the sum over the common denominator.

TIP: To use the prefix line effectively, you must know the order of the metric prefixes Sometimes a mnemonic (memory device) can help Consider the following sentence The first letter of each word represents one of the metric prefixes.

kids have doughnuts until dad calls mom.

kilo- hecto- deka- unit deci- centi- m

illi-represents the main unit of measurement (meter, liter, or gram)

Tips Boxes

Teaching Tips are usually only revealed in the classroom Not anymore Tip boxes offer students helpful hints and extra direction to help improve understanding and further insight

Avoiding Mistakes Boxes

Avoiding Mistakes boxes are integrated throughout the textbook to alert students to common errors and how to avoid them

⫽23

⫽1015

1

“ Loving these—students make so many mental mistakes we are not always mindful of, so these were very intentionally placed and benefi cial

—Ena Salter, Manatee Community College

Concept Connections

7 From Figure 7-2, determine

how many cups are in 1 gal.

8 From Figure 7-2, determine

how many pints are in 1 gal.

Figure 7-2

1 cup (c) 1 pint (pt) 1 quart (qt) 1 gallon (gal)

8 fl oz ⫽

Concept Connection Boxes

Concept Connections help students understand the conceptual meaning of the problems they are solving—a vital skill in mathematics

Get Better Results

ix

8 fl

the problems they ar

“This feature is one of my favorite

parts in the textbook It is useful

when trying to get students to think

critically about types of problems.”

—Sue Duff, Guilford Technical

Community College

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2000 冄5,400,000

6400 ⫼ 0.001

20 ⫻ 0.05 54⫻ 9.2 496.8 ⫼ 9.2 0.5192.52

2 192.52 0.8 ⫻ 74.23 74.23 ⫻ 0.8 98.0034 ⫹ 632.46 632.46 ⫹ 98.0034 5078.3 ⫺ 0.001 5078.3 ⫼ 0.001 5078.3 ⫻ 0.001 5078.3 ⫹ 0.001 5078.3⫼ 1000 5078.3⫺ 1000 5078.3⫻ 1000 5078.3 ⫹ 1000 For Exercises 1–20, perform the indicated operations.

490 冄98,000,000

280 ⫼ 0.07

8 ⫻ 0.125 5.6 ⫻ 80

448 ⫼ 5.6 0.25121.62

4 121.62 1.6 ⫻ 32.9 32.9 ⫻ 1.6 2.391 ⫹ 4.8 4.8 ⫹ 2.391 123.04 ⫺ 0.01 123.04 ⫼ 0.01 123.04 ⫻ 0.01 123.04 ⫹ 0.01 123.04 ⫼ 100 123.04⫺ 100 123.04 ⫻ 100 123.04 ⫹ 100

Problem Recognition Exercises Operations on Decimals

Problem Recognition Exercises!

Problem Recognition Exercises present a collection of problems that look similar to a student

upon first glance, but are actually quite different in the manner of their individual solutions

Students sharpen critical thinking skills and better develop their “solution recall” to help them

distinguish the method needed to solve an

exercise—an essential skill in mathematics

Problem Recognition Exercises, tested in

a developmental mathematics classroom,

were created in direct response to student

need to improve performance in testing

where different problem types are mixed

x

New to this Edition

▶ Do your students have trouble with problem solving?

▶ Do you want to help students overcome math anxiety?

▶ Do you want to help your students improve performance on math assessments?

4 121.62 1.6 ⫻ 32.9 32.9 ⫻ 1.6 2.391 ⫹ 4.8 4.8 ⫹ 2.391

123 04 ⫺ 0 01

“ This is excellent Great drill that could

be used in a variety of ways, i.e., group, calculator, individual, speed, extra credit—endless options!”

—Betty Vix Weinberger, Delgado

Community College

“ The MOH chapter does an excellent job giving practice with these special types of problems I found this approach interesting and enlightening.”

—Valerie Melvin, Cape Fear Community College

s 1–20, perform the indicated operations.

cognition Exercises e

em m m R R Re e ec c

ons on Decimals

“ This is a GREAT idea This “pattern recognition” is something that I go through in my classroom, and really helps students to fl esh out the idea and look at specifi c differences and similarities in problems.”

—Matthew Robinson, Tallahassee Community College

Get Better Results

New and Improved Applications!

Class-Tested and Student Approved!

New and improved applications have been developed

by an advisory team The Miller/O’Neill/Hyde Board of Advisor Team partnered with

our authors to bring you the

best applications from every

region of the country! These applications include real data and topics which are more relevant and interesting to today’s student

NEW Group Activities!

Each chapter concludes with a Group Activity selected by objective to promote classroom discussion and collaboration—helping students not only to solve problems but to explain their solutions for better mathematical mastery Group Activities are great for instructors

and adjuncts—bringing a more interactive approach

to teaching mathematics!

All required materials, activity time, and suggested group sizes are provided

in the directions of the activity Activities include:

Investigating Probability, Tracking Stocks, Using Card Games with Fractions, and more!

Group Activity

Investigating Probability

Materials: Paper bags containing 10 white poker chips, 6 red poker chips, and 4 blue poker chips.

Estimated time: 15 minutes Group Size: 3

1 Each group will receive a bag of poker chips, with 10 white, 6 red, and 4 blue chips.

2 a Write the ratio of red chips in the bag to the total number of chips in the bag. This value represents the probability of randomly selecting a red chip from the bag.

b Write this fraction in decimal form.

c Write the decimal from step (b) as a percent.

A probability value indicates the likeliness of an event to occur For example, to interpret this probability, one might say that there is a 30% chance of selecting a red chip at random from the bag.

p

87 The drug cyanocobalamin is prescribed by one

doctor in the amount of 1000 mcg How many milligrams is this?

89 A nurse must administer 45 mg of a drug The drug

is available in a liquid form with a concentration

of 15 mg per milliliter of the solution How many milliliters of the solution should the nurse give?

Expanding Your Skills

91 A normal value of hemoglobin in the blood for an adult male is 18 gm/dL

(that is, 18 grams per deciliter) How much hemoglobin would be expected

in 20 mL of a males’s blood?

92 A normal value of hemoglobin in the blood for an adult female is 15 gm/dL

(that is, 15 gm per deciliter) How much hemoglobin would be expected in

40 mL of a female’s blood?

88 An injection of naloxone is given in the

amount of 800 mcg How many milligrams

is this?

90 A patient must receive 500 mg of medication

in a solution that has a strength of 250 mg per 5 milliliter of solution How many milliliters of solution should be given?

Objective 4: Medical Applications

ibed by one H

88 An injection of naloxone is given in th

“ My students would fi nd the application problems very relevant to their world, particularly the problems in the U S

Customary Units sections.”

—Pat Rome, Delgado Community College

y v viiitttty t

“ What I liked the most was how the applications required students to fi gure out the appropriate operation to use—I also appreciate the variety of applications from those dealing with simplifying fractions to the area of rectangles and composite fi gures.”

—Vernon Bridges, Durham Technical Community College

xi

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What makes this new edition special?

Better Development!

Question: How do you build a better developmental mathematics textbook series?

Answer: Employ a developmental mathematics instructor from the classroom to become a McGraw-Hill editor!

Emilie Berglund joined the developmental mathematics team at McGraw-Hill

bringing her extensive classroom experience to the Miller/O’Neill/Hyde textbook

series A former developmental mathematics instructor at Utah Valley State College,

Ms Berglund has won numerous teaching awards and has served as the

beginning algebra course coordinator for the department Ms Berglund’s experience

teaching developmental mathematics students from the Miller/O’Neill/Hyde textbook

translates into more well-developed pedagogy throughout the textbook series and

can be seen in everything from the updated Worked Examples to the Exercise Sets

Listening to You

This textbook has been reviewed by over 300 teachers across the country Our textbook is a commitment to your students, providing a clear explanation, concise writing style, step-by-step learning tools, and the best exercises and

applications in developmental mathematics How do we know? You told us so!

Teachers Just Like You are saying great things about the Miller/O’Neill/Hyde

devel-opmental mathematics series:

“ This text provides a comprehensive presentation

of topics to students Their use of well explained examples, concept connections, and variety

of exercise material is ideally adapted to the developmental student.”

—Susan D Caire, Delgado Community College

“ The authors’ writing style is very straight forward and easy to follow The level of formality is just right for this level of math course.”

—Lynette King, Gadsden State College

““TThe authors’writing style is very

“ It reads as a teacher who tries explaining using everyday language and everyday examples.”

—Pat Rome, Delgado Community College

“ When adopting a new text a year ago, we evaluated

the MOH text at the top and adopted it as our text for

our Basic and Review of Math courses I truly believed

that the book was the best for our purposes Upon

reviewing the latest iteration of the book, my opinion

is still the same It is the best book out there on the

market in my opinion.”

—Rod Oberdick, Durham Technical Community College

“ The topics are clear and understandable It is probably the most complete textbook I have ever reviewed in terms of clarity and understandability Nothing needs to change.”

—Sonny Kirby, Gadsden State College

“ I really like the “avoiding mistakes” and “tips”

areas I refer to these in class all the time.”

—Joe Howe, Saint Charles Community College

Get Better Results

McGraw-Hill’s 360° Development Process is an ongoing, never-ending, market-oriented approach to

building accurate and innovative print and digital products It is dedicated to continual large-scale and incremental improvement driven by multiple customer feedback loops and checkpoints The process is initiated during the early planning stages of our new products, and is intensified during development and production

Then the process begins again upon publication in anticipation of the next edition

A key principle in the development of any mathematics text is its ability to adapt to teaching specifications in a universal way The only way to do so is by contacting those universal voices—and learning from their suggestions

We are confident that our book has the most current content the industry has to offer, thus pushing our desire for accuracy to the highest standard possible In order to accomplish this, we have moved through an arduous road to production Extensive and open-minded advice is critical in the production of a superior text

Here is a brief overview of the initiatives included in the Basic College Mathematics, Second Edition, 360°

Development Process:

Board of Advisors

A hand-picked group of trusted teachers active in the basic math course served as chief advisors and consultants to the authors and editorial team with regards to manuscript development The Board of Advisors reviewed parts of the manuscript;

served as a sounding board for pedagogical, media, and design concerns; consulted on organizational changes; and attended a focus group to confirm the manuscript’s readiness for publication

xiii

G

360° Development Process

Basic College Mathematics

Vernon Bridges, Durham Technical

Rod Oberdick, Delaware Technical

and Community College

Matthew Robinson, Tallahassee

Would you like to inquire about becoming a BOA member?

If so, email the editor, David Millage at david_millage@mcgraw-hill.com.

miL58061_fm_i-xxviii.indd Page xiii 9/4/08 3:20:47 PM user

The development of this textbook series would never have been possible without the creative ideas and

feedback offered by many reviewers We are especially thankful to the following instructors for their careful review of the manuscript.

not otherwise met

The development of this textbook series would never have been possible without the creative ideas and

feedback offered by many reviewers We are especially thankful to the following instructors for their careful

Acknowledgments and Reviewers

Focus Groups

In addition to the symposia, we held two specific focus groups for this book—on the overall project and on the art These selected mathematics professors provided ideas on improvements and suggestions for fine tuning the content, pedagogy, and problems

Tina Levy, Diablo Valley College Barbara Lott, Seminole Community College Diane McHugh, Longview Community College Valarie Melvin, Cape Fear Community College Janis Orinson, Central Piedmont Community College Mari Peddycoart, Lone Star College–Kingwood Trudy Streiliein, Northern Virginia Community College Jane Wyatt, Longview Community College

Advisors Symposium

Class Tests

Six student class tests provided the editorial team with an understanding of how content and the design of a textbook impacts a student’s homework and study habits in the general mathematics course area

Santa Fe Focus Group Attendees

Pauline Chow, Harrisburg Area Community College

Alina Coronel, Miami-Dade College

Anabel Darini, Suffolk County Community College

Susan Dimick, Spokane Community College

Barbara Elzey-Miller, Bluegrass Community & Technical

College

Lori Grady, University of Wisconsin-Whitewater

Lynette King, Gadsden State Community College Mike Kirby, Tidewater Community College Charlotte Newsom, Tidewater Community College Matthew Robinson, Tallahassee Community College Patricia Rome, Delgado Community College Suzanne Rosenberger, Harrisburg Area College Patricia Roux, Delgado Community College

Rajeed Carriman, Miami-Dade College

Nancy Chell, Anne Arundel Community College

Alina Coronel, Miami-Dade College–Kendall

Sarah Fallis, Tarrant County College

Nancy Graham, Rose State College

Jane Hammontree, Tulsa Community College

Greta Harris-Hardland, Tarrant County College

Kristie Johnson, Tarrant County College

Chicago Digital Focus Group Attendees

Antonio Alfonso, Miami-Dade College

Eric Bennett, Lansing Community College

David DelRossi, Tallahassee Community College

Maria DeLucia, Middlesex County College

Patricia D’Emidio, Montclair State University

Brandie Faulkner, Tallahassee Community College

Mary Lou Hammond, Spokane Community College

Nicole Lloyd, Lansing Community College

Bill Morrow, Delaware Technical College Mari Peddycoart, Lone Star College–Kingwood Adelaida Quesada, Miami-Dade College–Kendall Patricia Roux, Delgado Community College Sharon Sledge, San Jacinto College Kathryn Wetzel, Amarillo College Bridget Young, Suffolk County Community College Beverly Vredevelty, Spokane Falls Community College

Get Better Results

xv

Reviewers of the Miller/O’Neill/Hyde Developmental Mathematics Series

Natalie Creed, Gaston College Anabel Darini, Suffolk County Community College–

Brentwood

Antonio David, Del Mar College Ron Davis, Kennedy-King College–Chicago Laurie Delitsky, Nassau Community College Patti D’Emidio, Montclair State University Bob Denton, Orange Coast College Robert Diaz, Fullerton College Robert Doran, Eileen, Palm Beach Community College Deborah Doucette, Erie Community College–

North Campus—Williamsville

Thomas Drucker, University of Wisconsin–Whitewater Michael Dubrowsky, Wayne Community College Barbara Duncan, Hillsborough Community College–Dale

Mabry

Jeffrey Dyess, Bishop State Community College Elizabeth Eagle, University of North Carolina–Charlotte Sabine Eggleston, Edison College–Fort Myers

Lynn Eisenberg, Rowan-Cabarrus Community College Barb Elzey, Bluegrass Community and Technical College Nerissa Felder, Polk Community College

Mark Ferguson, Chemeketa Community College Diane Fisher, Lousiana State University–Eunice David French, Tidewater Community College–Chesapeake Dot French, Community College of Philadelphia

Deborah Fries, Wor-Wic Community College Robert Frye, Polk Community College Jesse M Fuson, Mountain State University Patricia Gary, North Virginia Community College–

Manassas

Calvin Gatson, Alabama State Univetsity Donna Gerken, Miami Dade College–Kendall Mehrnaz Ghaffarian, Tarrant County College South Mark Glucksman, El Camino College

Judy Godwin, Collin County Community College William Graesser, Ivy Tech Community College Victoria Gray, Scott Community College

Manuscript Review Panels

Over 400 teachers and academics from across the country reviewed the various drafts of the manuscript to give feedback on content, design, pedagogy, and organization This feedback was summarized by the book team and used to guide the direction of the text

Special “thank you” to our Manuscript Class-Testers

Vernon Bridges, Durham Technical Community College Susan Dimick, Spokane Community College

Lori Grady, University of Wisconsin–Whitewater

Rod Oberdick, Delaware Technical and Community College Matthew Robinson, Tallahassee Community College Pat Rome, Delgado Community College–City Park

Darla Aguilar, Pima Community College–Desert Vista Ebrahim Ahmadizadeh, Northampton Community College Sara Alford, North Central Texas College

Theresa Allen, University of Idaho Sheila Anderson, Housatonic Community College Lane Andrew, Arapahoe Community College Jan Archibald, Ventura College

Yvonne Aucoin, Tidewater Community College–Norfolk Eric Aurand, Mohave Community College

Sohrab Bakhtyari, St Petersburg College Anna Bakman, Los Angeles Trade Technical Andrew Ball, Durham Technical Community College Russell Banks, Guilford Technical Community College Suzanne Battista, St Petersburg College

Kevin Baughn, Kirtland Community College Sarah Baxter, Gloucester County College Lynn Beckett-Lemus, El Camino College Edward Bender, Century College Emilie Berglund, Utah Valley State College Rebecca Berthiaume, Edison College–Fort Myers John Beyers, Miami Dade College–Hialeah Leila Bicksler, Delgado Community College–West Bank Norma Bisulca, University of Maine–Augusta

Kaye Black, Bluegrass Community and Technical College Deronn Bowen, Broward College–Central

Timmy Bremer, Broome Community College Donald Bridgewater, Broward College Peggy Brock, TVI Community College Kelly Brooks, Pierce College

Susan D Caire, Delgado Community College–West Bank Peter Carlson, Delta College

Judy Carter, North Shore Community College Veena Chadha, University of Wisconsin–Eau Claire Zhixiong Chen, New Jersey City University Tyrone Clinton, Saint Petersburg College–Gibbs Eugenia Cox, Palm Beach Community College Julane Crabtree, Johnson Community College Mark Crawford, Waubonsee Community College

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to Cindy Reed for her work in the video series, and to Ethel Wheland for advising us on the Instructor Notes.

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xxi

Chapter 1 Whole Numbers 1

Additional Topics Appendix

A.1 Energy and Power A–1

A.2 Scientific Notation A–6

A.3 Rectangular Coordinate System A–10

xxii

Brief Contents

Chapter 1 Whole Numbers 1

1.2 Addition of Whole Numbers and Perimeter 9

1.5 Multiplication of Whole Numbers and Area 38

1.7 Exponents, Square Roots, and the Order of Operations 62

2.1 Introduction to Fractions and Mixed Numbers 96

2.2 Prime Numbers and Factorization 106

2.3 Simplifying Fractions to Lowest Terms 114

2.4 Multiplication of Fractions and Applications 123

2.5 Division of Fractions and Applications 133

of Fractions 143

2.6 Multiplication and Division of Mixed Numbers 144

Group Activity: Cooking for Company 151

Chapter 3 Fractions and Mixed Numbers: Addition

and Subtraction 163

3.1 Addition and Subtraction of Like Fractions 164

3.2 Least Common Multiple 171

3.3 Addition and Subtraction of Unlike Fractions 179

3.4 Addition and Subtraction of Mixed Numbers 188

and Mixed Numbers 198

3.5 Order of Operations and Applications of Fractions and Mixed Numbers 199

Group Activity: Card Games with Fractions 208

4.1 Decimal Notation and Rounding 218

4.2 Addition and Subtraction of Decimals 228

4.3 Multiplication of Decimals 238

4.4 Division of Decimals 246

4.5 Fractions as Decimals 258

4.6 Order of Operations and Applications of Decimals 266

Group Activity: Purchasing from a Catalog 277

5.1 Ratios 292

5.2 Rates 299

5.3 Proportions 306

5.4 Applications of Proportions and Similar Figures 313

Group Activity: Investigating Probability 322

6.1 Percents and Their Fraction and Decimal Forms 334

6.2 Fractions and Decimals and Their Percent Forms 342

6.3 Percent Proportions and Applications 349

6.4 Percent Equations and Applications 360

6.5 Applications Involving Sales Tax, Commission, Discount, and Markup 369

6.6 Percent Increase and Decrease 379

6.7 Simple and Compound Interest 384

Group Activity: Tracking Stocks 393

7.1 Converting U.S Customary Units of Length 410

7.2 Converting U.S Customary Units of Time, Weight, and Capacity 419

7.3 Metric Units of Length 425

7.4 Metric Units of Mass and Capacity and Medical Applications 433

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8.1 Lines and Angles 464

8.2 Triangles and Pythagorean Theorem 473

8.3 Quadrilaterals, Perimeter, and Area 483

8.4 Circles, Circumference, and Area 494

9.1 Tables, Bar Graphs, Pictographs, and Line Graphs 530

9.2 Frequency Distributions and Histograms 541

10.1 Real Numbers and the Real Number Line 586

10.2 Addition of Real Numbers 593

10.3 Subtraction of Real Numbers 602

of Real Numbers 609

10.4 Multiplication and Division of Real Numbers 610

11.1 Properties of Real Numbers 634

11.2 Simplifying Expressions 641

11.3 Addition and Subtraction Properties of Equality 646

11.4 Multiplication and Division Properties of Equality 652

11.5 Solving Equations with Multiple Steps 658

11.6 Applications and Problem Solving 666

Group Activity: Constructing Linear Equations 674

Additional Topics Appendix A-1

A.1 Energy and Power A–1

A.2 Scientific Notation A–6

A.3 Rectangular Coordinate System A–10

Student Answer Appendix SA–1Credits C–1

Index I–1

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CHAPTER OUTLINE

1.1 Introduction to Whole Numbers 2

1.2 Addition of Whole Numbers and Perimeter 9

1.3 Subtraction of Whole Numbers 21

1.4 Rounding and Estimating 31

1.5 Multiplication of Whole Numbers and Area 38

1.6 Division of Whole Numbers 50

Chapter 1 begins with adding, subtracting, multiplying, and dividing whole numbers

We also include rounding, estimating, and applying the order of operations As youwork through the chapter, you can check your skills by filling in this puzzle

Figure 1-1

1 Place Value

Numbers provide the foundation that is used in mathematics We begin this ter by discussing how numbers are represented and named All numbers in our

chap-numbering system are composed from the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 In

mathematics, the numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, are called the whole numbers (The three dots are called ellipses and indicate that the list goes on

indefinitely.)For large numbers, commas are used to separate digits into groups of three

called periods For example, the number of live births in the United States in a

recent year was 4,058,614 (Source: The World Almanac) Numbers written in this

way are said to be in standard form The position of each digit within a number

determines the place value of the digit To interpret the number of births in theUnited States, refer to the place value chart (Figure 1-1)

Answers

1 First 3 (on the left) represents

3 hundreds, while the second 3

(on the right) represents 3 ones.

2 Ten-millions

3 Thousands

4 Hundreds

Billions Period

Millions Period

Thousands Period

Ones Period

Hundred-billionsTen-billions Billions Hundred-millionsTen-millions Millions Hundred-thousandsTen-thousands Thousands Hundreds Tens Ones

4, 0 5 8, 6 1 4

The digit 5 in the number represents 5 ten-thousands because it is in theten-thousands place The digit 4 on the left represents 4 millions, whereas the digit 4

on the right represents 4 ones

Determining Place ValueDetermine the place value of the digit 2 in each number

417,216,900

502,000,700724

3 Writing Numbers in Words

4 The Number Line and Order

Concept Connections

1 Explain the difference

between the two 3’s in

the number 303.

Skill Practice

Determine the place value of the

digit 4 in each number.

Section 1.1 Introduction to Whole Numbers 3

Determining Place ValueMount Everest, the highest mountain on earth, is29,035 feet (ft) tall Give the place value for eachdigit in this number

Solution:

onestenshundredsthousandsten-thousands

2 Standard Notation and Expanded Notation

A number can also be written in an expanded form by writing each digit with itsplace value unit For example, the number 287 can be written as

This is called expanded form.

Converting Standard Form to Expanded FormConvert to expanded form

1 thousand ⫹ 0 hundreds ⫹ 2 tens ⫹ 5 ones ⫽ 1,025

2 hundreds ⫹ 5 tens ⫹ 9 ones ⫽ 259

1 thousand ⫹ 2 tens ⫹ 5 ones

2 hundreds ⫹ 5 tens ⫹ 9 ones

5 Alaska is the largest state

geographically Its land area is 571,962 square miles Give the place value for each digit.

3 Writing Numbers in Words

The word names of some two-digit numbers appear with a hyphen, while others

do not For example:

Answers

10 One billion, four hundred fifty million,

three hundred twenty-seven

thousand, two hundred fourteen

11 14,609

To write a three-digit or larger number, begin at the leftmost group of digits Thenumber named in that group is followed by the period name, followed by a comma.Then the next period is named, and so on

Writing a Number in WordsWrite the number 621,417,325 in words

Solution:

621,417,325

six hundred twenty-one million,

four hundred seventeen thousand,

three hundred twenty-five

Notice from Example 5 that when naming numbers, the name of the ones period

is not attached to the last group of digits Also note that for whole numbers, the

word and should not appear in word names For example, the number 405 should

be written as four hundred five

Writing a Number in Standard FormWrite the number in standard form

Six million, forty-six thousand, nine hundred three

11 Write the number in standard

form: fourteen thousand, six

hundred nine.

The whole numbers begin at 0 and are ordered from left to right by increasingvalue.

A number is graphed on a number line by placing a dot at the ing point For any two numbers graphed on a number line, the number to theleft is less than the number to the right Similarly, a number to the right isgreater than the number to the left In mathematics, the symbol is used todenote “is less than,” and the symbol means “is greater than.” Therefore,

correspond-Determining Order Between Two NumbersFill in the blank with the symbol or

67

7.6

Example 7

5 7 3 means 5 is greater than 3

3 6 5 means 3 is less than 5

7

6

Section 1.1 Introduction to Whole Numbers 5

We have seen several examples of writing a number in standard form, in expandedform, and in words Standard form is the most concise representation Also notethat when we write a four-digit number in standard form, the comma is often omit-ted For example, the number 4,389 is often written as 4389

4 The Number Line and Order

Whole numbers can be visualized as equally spaced points on a line called a ber line (Figure 1-2)

num-Figure 1-2

12 11 10 9 8 7 6 5 4 3 2 1 0

8 7 6 5 4 3 2 1 0

Answers

BASIC—

8 9 10 7

6 5 4 3 2 1 0

80 90 100 70

60 50 40 30 20 10 0

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Study Skills Exercises

In this text, we provide skills for you to enhance your learning experience Each set of practice exercises beginswith an activity that focuses on one of eight areas: learning about your course, using your text, taking notes,doing homework, taking an exam (test and math anxiety), managing your time, recognizing your learning style,and studying for the final exam

Each activity requires only a few minutes and will help you to pass this class and become a better mathstudent Many of these skills can be carried over to other disciplines and help you become a model collegestudent

1 To begin, write down the following information.

a Instructor’s name b Instructor’s office number

c Instructor’s telephone number d Instructor’s email address

e Instructor’s office hours f Days of the week that the class meets

g The room number in which the h Is there a lab requirement for this course?

class meets If so, where is the lab located and how often must you go?

2 Define the key terms.

a Digit b Standard form c Periods d Expanded form

Objective 1: Place Value

3 Name the place values for each of the digits in the number 8,213,457.

4 Name the place values for each of the digits in the number 103,596.

For Exercises 5–24, determine the place value for each underlined digit.(See Example 1.)

21 The number of U.S travelers abroad in a recent year was (See Example 2.)

22 The area of Lake Superior is 31,820mi2

10,677,881

93,971,22451,033,201

58,10622,422

3,111,901,2111,023,676,207

655,878452,723

3,1011,430

2,2938,710

738214

689321

• Practice Problems • e-Professors

• Self-Tests • Videos

• NetTutor

ALEKS.com!

Section 1.1 Introduction to Whole Numbers 7

23. For a recent year, the total number of U.S $1 bills in circulation was

24. For a certain flight, the cruising altitude of a commercial jet is ft

Objective 2: Standard Notation and Expanded Notation

For Exercises 25–32, convert the numbers to expanded form.(See Example 3.)

41. Name the first four periods of a number 42. Name the first four place values of a number

Objective 3: Writing Numbers in Words

For Exercises 43–50, write the number in words.(See Example 5.)

51. The Shuowen jiezi dictionary, an ancient 52. Researchers calculate that about 590,712 stoneChinese dictionary that dates back to the blocks were used to construct the Great Pyramid.year 100, contained 9,535 characters Write the number 590,712 in words

Write the number 9,535 in words

53. Mt McKinley in Alaska is 20,320 ft high 54. There are 1,800 seats in the Regal ChamplainWrite the number 20,320 in words Theater in Plattsburgh, New York Write the

number 1,800 in words

55. Interstate I-75 is 1,377 miles (mi) long Write the number 1,377 in words

56. In the United States, there are approximately 60,000,000 cats living in households Write the number60,000,000 in words

2 ten-thousands ⫹ 6 thousands ⫹ 2 ones

8 ten-thousands ⫹ 5 thousands ⫹ 7 ones

4 thousands ⫹ 2 hundreds ⫹ 1 one

1 thousand ⫹ 9 hundreds ⫹ 6 ones

6 hundreds ⫹ 2 tens

1 hundred ⫹ 5 tens

3 hundreds ⫹ 1 ten ⫹ 8 ones

5 hundreds ⫹ 2 tens ⫹ 4 ones

31,0007,653,468,440

BASIC—

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For Exercises 57–62, convert the number to standard form.(See Example 6.)

59. Six hundred seventy-two thousand 60. Two hundred forty-eight thousand

61. One million, four hundred eighty-four 62. Two million, six hundred forty-seven thousand,

Objective 4: The Number Line and Order

For Exercises 63–64, graph the numbers on the number line

65. On a number line, what number is 4 units to 66. On a number line, what number is 8 units to

67. On a number line, what number is 3 units to 68. On a number line, what number is 5 units to the

For Exercises 69–72, translate the inequality to words

Expanding Your Skills

85. Answer true or false The number 12 is a digit 86. Answer true or false The number 26 is a digit

87. What is the greatest two-digit number? 88. What is the greatest three-digit number?

89. What is the greatest whole number? 90. What is the least whole number?

91. How many zeros are there in the number 92. How many zeros are there in the number

93. What is the greatest three-digit number that 94. What is the greatest three-digit number thatcan be formed from the digits 6, 9, and 4? can be formed from the digits 0, 4, and 8?

7.6

6 5 4 3 2 1 0

8 9 10 11 12 13 7

6 5 4 3 2 1 0

Section 1.2 Addition of Whole Numbers and Perimeter 9

The result of an addition problem is called the sum, and the numbers being added are called addends Thus,

The number line is a useful tool to visualize the operation of addition To add

5 and 3 on a number line, begin at 0 and move 5 units to the right Then move anadditional 3 units to the right The final location indicates the sum

6 5 4 3 2 1 0

The sum

is 8.

You can use a number line to find the sum of any pair of digits The sums for allpossible pairs of one-digit numbers should be memorized (see Exercise 9) Memo-rizing these basic addition facts will make it easier for you to add larger numbers

2 Addition of Whole Numbers

To add whole numbers, line up the numbers vertically by place value Then addthe digits in the corresponding place positions

5 Perimeter

Adding Whole NumbersAdd

Add digits in ones column.

Add digits in tens column.

Add digits in hundreds column.

The sum of the digits in the ones placeexceeds 9 But 13 ones is the same as

1 ten and 3 ones We can carry 1 ten to

the tens column while leaving the 3 ones

in the ones column Notice that we placedthe carried digit above the tens column

Add the digits in the tens column (including the carry):

Write the 2 in the tens column, and carry the

1 to the hundreds column

1⫹ 5 ⫹ 6 ⫽ 12

Add the digits in the hundreds column

Adding Whole NumbersAdd

Solution:

Sometimes when adding numbers, the sum of the digits in a given place position

is greater than 9 If this occurs, we must do what is called carrying or regrouping.

Example 3 illustrates this process

Adding Whole Numbers with CarryingAdd

45

1

8

⫹ 675

261⫹ 28

Example 2

Section 1.2 Addition of Whole Numbers and Perimeter 11

Addition of numbers may include more than two addends

Adding Whole NumbersAdd

Solution:

3 Properties of Addition

We present three properties of addition that you may have already discovered

PROPERTY Addition Property of 0

The sum of any number and 0 is that number

Examples:

PROPERTY Commutative Property of Addition

Changing the order of two addends does not affect the sum

PROPERTY Associative Property of Addition

The manner in which addends are grouped does not affect the sum

⫹ 2,419107,653

21,076⫹ 84,158 ⫹ 2419

Example 5

Answer

6 71,147

In this example, the sum of the digits in the ones column

is 23 Therefore, we write the 3 and carry the 2

Skill Practice

6 Add.

57,296 4,089