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Mathematically Seventh Edition

Robert Blitzer Miami Dade College

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Library of Congress Cataloging-in-Publication Data Names: Blitzer, Robert, author.

Title: Thinking mathematically / Robert F. Blitzer.

Description: Seventh edition. | Boston : Pearson, [2019]

Identifiers: LCCN 2017046337 | ISBN 9780134683713 (alk. paper) | ISBN 0134683714 (alk. paper)

Subjects: LCSH: Mathematics–Textbooks.

Classification: LCC QA39.3 .B59 2019 | DDC 510–dc23

LC record available at

ISBN-13: 978-0-13-468371-3

ISBN-10: 0-13-468371-4

Contents About the Author vi

Preface vii

Resources for Success ix

To the Student xi

Acknowledgments xii

Index of Applications xv

1 Problem Solving and Critical Thinking 1

1.1 Inductive and Deductive

Reasoning 2

1.2 Estimation, Graphs,

and Mathematical

Models 14

1.3 Problem Solving 30

Chapter Summary,

Review, and Test 43

Chapter 1 Test 46

2 Set Theory


2.1 Basic Set Concepts 50

2.2 Subsets 64

2.3 Venn Diagrams and

Set Operations 73

2.4 Set Operations and

Venn Diagrams with

Three Sets 87

2.5 Survey Problems 99

Chapter Summary,

Review, and Test 110

Chapter 2 Test 114

3 Logic 117

3.1 Statements, Negations,

and Quantified Statements 118

3.2 Compound Statements

and Connectives 126

3.3 Truth Tables for

Negation, Conjunction,

and Disjunction 139

3.4 Truth Tables for the Conditional

and the  Biconditional 154

3.5 Equivalent Statements and Variations

of Conditional Statements 166

3.6 Negations of Conditional Statements

and De Morgan’s Laws 176

3.7 Arguments and Truth Tables 184

3.8 Arguments and Euler Diagrams 199

Chapter Summary, Review, and Test 209

Chapter 3 Test 213

4 Number Representation

and Calculation 215

4.1 Our Hindu-Arabic

System and Early

Positional Systems 216

4.2 Number Bases in

Positional Systems 224

4.3 Computation in

Positional Systems 231

4.4 Looking Back at Early

Numeration Systems 240

Chapter Summary,

Review, and Test 247

Chapter 4 Test 250


iv Contents

5 Number Theory and

the Real Number



5.1 Number Theory:

Prime and Composite

Numbers 252

5.2 The Integers; Order

of Operations 262

5.3 The Rational

Numbers 276

5.4 The Irrational

Numbers 291

5.5 Real Numbers and

Their Properties;

Clock Addition 304

5.6 Exponents and Scientific

Notation 315

5.7 Arithmetic and Geometric

Sequences 326

Chapter Summary, Review, and Test 336

Chapter 5 Test 341

6 Algebra:

Equations and

Inequalities 343

6.1 Algebraic


and Formulas 344

6.2 Linear Equations

in One Variable and  Proportions 354

6.3 Applications of Linear Equations 369

6.4 Linear Inequalities in One Variable 380

6.5 Quadratic Equations 390

Chapter Summary, Review, and Test 405

Chapter 6 Test 409

7 Algebra: Graphs,


and Linear

Systems 411

7.1 Graphing and

Functions 412

7.2 Linear Functions

and Their Graphs 424

7.3 Systems of Linear

Equations in

Two Variables 438

7.4 Linear Inequalities

in Two Variables 453

7.5 Linear Programming 462

7.6 Modeling Data: Exponential,

Logarithmic, and Quadratic Functions 468

Chapter Summary, Review, and Test 484

Chapter 7 Test 490

8 Personal



8.1 Percent,

Sales Tax,

and Discounts 494

8.2 Income Tax 503

8.3 Simple

Interest 514

8.4 Compound Interest 519

8.5 Annuities, Methods of Saving,

and Investments 529

8.6 Cars 545

8.7 The Cost of Home Ownership 554

8.8 Credit Cards 563

Chapter Summary, Review, and Test 572

Chapter 8 Test 578

Contents v

9 Measurement


9.1 Measuring Length;

The Metric System 582

9.2 Measuring Area

and Volume 592

9.3 Measuring Weight

and Temperature 602

Chapter Summary, Review, and Test 611

Chapter 9 Test 614

10 Geometry


10.1 Points, Lines,

Planes, and Angles 616

10.2 Triangles 625

10.3 Polygons, Perimeter, and Tessellations 637

10.4 Area and Circumference 646

10.5 Volume and Surface Area 657

10.6 Right Triangle Trigonometry 666

10.7 Beyond Euclidean Geometry 676

Chapter Summary, Review, and Test 685

Chapter 10 Test 691


Counting Methods

and Probability



11.1 The Fundamental

Counting Principle 694

11.2 Permutations 700

11.3 Combinations 708

11.4 Fundamentals of Probability 715

11.5 Probability with the Fundamental

Counting Principle, Permutations,

and  Combinations 724

11.6 Events Involving Not and Or; Odds 731

11.7 Events Involving And; Conditional

Probability 744

11.8 Expected Value 756

Chapter Summary, Review, and Test 763

Chapter 11 Test 769

12 Statistics


12.1 Sampling, Frequency

Distributions, and

Graphs 772

12.2 Measures of

Central Tendency 786

12.3 Measures of Dispersion 800

12.4 The Normal Distribution 808

12.5 Problem Solving with the Normal

Distribution 822

12.6 Scatter Plots, Correlation, and

Regression Lines 827

Chapter Summary, Review, and Test 838

Chapter 12 Test 843

13 Voting and

Apportionment 845

13.1 Voting Methods 846

13.2 Flaws of Voting

Methods 858

13.3 Apportionment

Methods 869

13.4 Flaws of


Methods 883

Chapter Summary, Review, and Test 893

Chapter 13 Test 896

14 Graph Theory 897

14.1 Graphs, Paths,

and Circuits 898

14.2 Euler Paths and Euler Circuits 908

14.3 Hamilton Paths and Hamilton Circuits 920

14.4 Trees 930

Chapter Summary, Review, and Test 939

Chapter 14 Test 944

Answers to Selected Exercises AA1

Subject Index I1

Credits C1

About the Author

Bob Blitzer is a native of Manhattan and received a Bachelor of Arts degree with dual majors in mathematics and psychology (minor: English literature) from

the City College of New York. His unusual combination of academic interests

led him toward a Master of Arts in mathematics from the University of Miami

and a doctorate in behavioral sciences from Nova University. Bob’s love for

teaching mathematics was nourished for nearly 30 years at Miami Dade College,

where he received numerous teaching awards, including Innovator of the Year

from the League for Innovations

in the Community College and an

endowed chair based on excellence in

the classroom. In addition to Thinking Mathematically, Bob has written textbooks covering introductory

algebra, intermediate algebra, college

algebra, algebra and trigonometry,

precalculus, trigonometry, and liberal

arts mathematics for high school

students, all published by Pearson.

When not secluded in his Northern

California writer’s cabin, Bob can be

found hiking the beaches and trails

of Point Reyes National Seashore,

and tending to the chores required

by his beloved entourage of horses,

chickens, and irritable roosters.



Thinking Mathematically, Seventh Edition provides a general survey of mathematical topics that are useful

in our contemporary world. My primary purpose in

writing the book was to show students how mathematics

can be applied to their lives in interesting, enjoyable,

and meaningful ways. The book’s variety of topics and

flexibility of sequence make it appropriate for a one- or

two-term course in liberal arts mathematics, quantitative

reasoning, finite mathematics, as well as for courses

specifically designed to meet state-mandated requirements

in mathematics.

I wrote the book to help diverse students, with

different backgrounds and career plans, to succeed.

Thinking Mathematically, Seventh Edition, has four major goals:

1. To help students acquire knowledge of fundamental mathematics.

2. To show students how mathematics can solve authentic problems that apply to their lives.

3. To enable students to understand and reason with quantitative issues and mathematical ideas they are

likely to encounter in college, career, and life.

4. To enable students to develop problem-solving skills, while fostering critical thinking, within an interesting


One major obstacle in the way of achieving these

goals is the fact that very few students actually read their

textbook. This has been a regular source of frustration

for me and my colleagues in the classroom. Anecdotal

evidence gathered over years highlights two basic reasons

why students do not take advantage of their textbook:

“I’ll never use this information.”

“I can’t follow the explanations.”

I’ve written every page of the Seventh Edition with the

intent of eliminating these two objections. The ideas and

tools I’ve used to do so are described for the student in

“A Brief Guide to Getting the Most from This Book,”

which appears inside the front cover.

What’s New in the Seventh Edition?

• New and Updated Applications and Real-World Data. I’m on a constant search for real-world data that can

be used to illustrate unique mathematical applications.

I researched hundreds of books, magazines,

newspapers, almanacs, and online sites to prepare the

Seventh Edition. This edition contains 110 worked-out

examples and exercises based on new data sets and

104 examples and exercises based on updated data.

New applications include student-loan debt (Exercise

Set 1.2), movie rental options (Exercise  Set  1.3),

impediments to academic performance (Section  2.1),

measuring racial prejudice, by age (Exercise Set 2.1),

generational support for legalized adult marijuana

use (Exercise Set 2.3), different cultural values among

nations (Exercise Set 2.5), episodes from the television

series The Twilight Zone (Section 3.6) and the film Midnight Express (Exercise Set 3.7), excuses by college students for not meeting assignment deadlines

(Exercise Set 5.3), fraction of jobs requiring various

levels of education by 2020 (Exercise Set 5.3), average

earnings by college major (Exercise Set 6.5), the pay gap

(Exercise Set 7.2), inmates in federal prisons for drug

offenses and all other crimes (Exercise Set 7.3), time

breakdown for an average 90-minute NFL  broadcast

(Section 11.6), Scrabble tiles (Exercise Set 11.5), and

are inventors born or made? (Section 12.2).

• New Blitzer Bonuses. The Seventh Edition contains a variety of new but optional enrichment essays. There

are more new Blitzer Bonuses in this edition than in any

previous revision of Thinking Mathematically. These include “Surprising Friends with Induction” (Section 1.1),

“Predicting Your Own Life Expectancy” (Section 1.2),

“Is College Worthwhile?” (Section 1.2), “Yogi-isms”

(Section  3.4), “Quantum Computers” (Section 4.3),

“Slope and Applauding Together” (Section  7.2),

“A Brief History of U.S. Income Tax” (Section 8.2)

“Three Decades of Mortgages” (Section  8.7), “Up to

Our Ears in Debt” (Section 8.8), “The Best Financial

Advice for College Graduates” (Section 8.8), “Three

Weird Units of Measure” (Section 9.1), “Screen Math”

(Section 10.2), “Senate Voting Power” (Section 13.3),

“Hamilton Mania” (Section 13.3), “Dirty Presidential

Elections” (Section  13.3), “Campaign Posters as Art”

(Section  13.4), and “The 2016 Presidential Election”

(Section 13.4).

• New Graphing Calculator Screens. All screens have been updated using the TI-84 Plus C.

• Updated Tax Tables. Section 8.2 (Income Tax) contains the most current federal marginal tax tables

and FICA tax rates available for the Seventh Edition.

• New MyLabTM Math. In addition to the new functionalities within an updated MyLab Math,

the new items specific to Thinking Mathematically, Seventh Edition MyLab Math include

~ All new objective-level videos with assessment

~ Interactive concept videos with assessment

~ Animations with assessment

~ StatCrunch integration.


viii Preface

What Familiar Features Have Been Retained in the

Seventh Edition?

• Chapter-Opening and Section-Opening Scenarios. Every chapter and every section open with a scenario

presenting a unique application of mathematics in

students’ lives outside the classroom. These scenarios

are revisited in the course of the chapter or section

in an example, discussion, or exercise. The often

humorous tone of these openers is intended to help

fearful and reluctant students overcome their negative

perceptions about math. A feature called “Here’s

Where You’ll Find These Applications” is included

with each chapter opener.

• Section Objectives (What Am I Supposed to Learn?). Learning objectives are clearly stated at the beginning

of each section. These objectives help students

recognize and focus on the section’s most important

ideas. The objectives are restated in the margin at their

point of use.

• Detailed Worked-Out Examples. Each example is titled, making the purpose of the example clear.

Examples are clearly written and provide students with

detailed step-by-step solutions. No steps are omitted

and each step is thoroughly explained to the right of

the mathematics.

• Explanatory Voice Balloons. Voice balloons are used in a variety of ways to demystify mathematics.

They translate mathematical language into everyday

English, help clarify problem-solving procedures,

present alternative ways of understanding concepts,

and connect problem solving to concepts students

have already learned.

• Check Point Examples. Each example is followed by a similar matched problem, called a Check Point,

offering students the opportunity to test for conceptual

understanding by working a similar exercise. The

answers to the Check Points are provided in the answer

section in the back of the book. Worked-out video

solutions for many Check Points are in the MyLab

Math course.

• Great Question! This feature presents study tips in the context of students’ questions. Answers to the

questions offer suggestions for problem solving, point

out common errors to avoid, and provide informal

hints and suggestions. As a secondary benefit, this

feature should help students not to feel anxious or

threatened when asking questions in class.

• Brief Reviews. The book’s Brief Review boxes summarize mathematical skills that students should

have learned previously, but which many students

still need to review. This feature appears whenever a

particular skill is first needed and eliminates the need

to reteach that skill.

• Concept and Vocabulary Checks. The Seventh Edition contains 653 short-answer exercises, mainly fill-in-

the blank and true/false items, that assess students’

understanding of the definitions and concepts

presented in each section. The Concept and Vocabulary

Checks appear as separate features preceding the

Exercise Sets. These are assignable in the MyLab Math


• Extensive and Varied Exercise Sets. An abundant collection of exercises is included in an Exercise Set at

the end of each section. Exercises are organized within

seven category types: Practice Exercises, Practice

Plus Exercises, Application Exercises, Explaining the

Concepts, Critical Thinking Exercises, Technology

Exercises, and Group Exercises.

• Practice Plus Problems. This category of exercises contains practice problems that often require students

to combine several skills or concepts, providing

instructors the option of creating assignments that

take Practice Exercises to a more challenging level.

• Chapter Summaries. Each chapter contains a review chart that summarizes the definitions and concepts in

every section of the chapter. Examples that illustrate

these key concepts are also referenced in the chart.

• End-of-Chapter Materials. A comprehensive collection of review exercises for each of the chapter’s sections

follows the Summary. This is followed by a Chapter

Test that enables students to test their understanding of

the material covered in the chapter. Worked-out video

solutions are available for every Chapter Test Prep

problem in the MyLab Math course or on YouTube.

• Learning Guide. This study aid is organized by objective  and provides support for note-taking,

practice,  and video review. The Learning Guide

is available as PDFs in MyLab Math. It can also

be packaged with the textbook and MyLab Math

access code.

I hope that my love for learning, as well as my respect

for the diversity of students I have taught and learned from

over the years, is apparent throughout this new edition.

By connecting mathematics to the whole spectrum of

learning, it is my intent to show students that their world is

profoundly mathematical, and indeed, p is in the sky.

Robert Blitzer

Resources for Success MyLab

TM Math Online Course for

Thinking Mathematically, Seventh Edition

by Robert Blitzer (access code required) MyLab Math is available to accompany Pearson’s market leading text offerings. To give

students a consistent tone, voice, and teaching method each text’s flavor and approach

are tightly integrated throughout the accompanying MyLab Math course, making

learning the material as seamless as possible.

NEW! Video Program

All new objective-level videos provide

a new level of coverage throughout the

text. Videos at the objective level allow

students to get support just where they

need it. Instructors can assign these as

media assignments or use the provided

assessment questions for each video.

NEW! Interactive

Concept Videos

New Interactive Concept Videos are also

available in MyLab Math. After a brief

explanation, the video pauses to ask

students to try a problem on their own.

Incorrect answers are followed by further

explanation, taking into consideration what

may have led to the student selecting

that particular wrong answer. Incorrect

answer ‘A’ goes down one path while

incorrect answer ‘B’ provides a different

explanation based on why the student may

have selected that option.

NEW! Animations

New animations let students interact with

the math in a visual, tangible way. These

animations allow students to explore and

manipulate the mathematical concepts,

leading to more durable understanding.

Corresponding exercises in MyLab Math

make these truly assignable.

StatCrunch Newly integrated StatCrunch allows

students to harness technology to

perform complex analyses on data.

Resources for Success ix

Instructor Resources

Annotated Instructor’s Edition (AIE) ISBN-10: 0-13-468454-0

ISBN-13: 978-0-13-468454-3

The AIE includes answers to all exercises presented in

the book, most on the page with the exercise and the

remainder in the back of the book.

The following resources can be downloaded from

MyLab Math or the Instructor’s Resource Center on

MyLab Math with Integrated Review Provides a full suite of supporting resources for the

collegiate course content plus additional assignments

and study aids for students who will benefit from

remediation. Assignments for the integrated review

content are preassigned in MyLab™ Math, making it

easier than ever to create your course.

Instructor’s Solutions Manual This manual contains detailed, worked-out solutions to

all the exercises in the text.

PowerPoint Lecture Presentation These editable slides present key concepts and

definitions from the text. Instructors can add art from

the text located in the Image Resource Library in MyLab

Math or slides that they create on their own. PointPoint

slides are fully accessible.

Image Resource Library This resource in MyLab Math contains all art from the

text, for instructors to use in their own presentations

and handouts.

Instructor’s Testing Manual The Testing Manual includes two alternative tests per

chapter. These items may be used as actual tests or as

references for creating actual tests.

TestGen TestGen® ( enables

instructors to build, edit, print, and administer tests

using a computerized bank of questions developed

to cover all the objectives of the text. TestGen is

algorithmically based, allowing instructors to create

multiple but equivalent versions of the same question

or test with the click of a button. Instructors can also

modify test bank questions or add new questions. The

software are available for download from Pearson’s

Instructor Resource Center.

Student Resources Learning Guide with Integrated Review Worksheets

ISBN 10: 0-13-470508-4

ISBN 13: 978-0-13470508-8

Bonnie Rosenblatt, Reading Area Community College

This workbook is organized by objective and provides

support for note-taking, practice, and video review and

includes the Integrated Review worksheets from the

Integrated Review version of the MyLab Math course.

The Learning Guide is also available as PDFs in MyLab

Math. It can also be packaged with the textbook and

MyLab Math access code.

Student’s Solutions Manual ISBN 10: 0-13-468650-0

ISBN 13: 978-0-13-468650-9

Daniel Miller, Niagara County Community College

This manual provides detailed, worked-out solutions

to odd-numbered exercises, as well as solutions to all

Check Points, Concept and Vocabulary Checks, Chapter

Reviews, and Chapter Tests.

Resources for Success

x Resources for Success

To the Student The bar graph shows some of the qualities that students say make a great teacher.

It was my goal to incorporate each of these qualities throughout the pages of this

book to help you gain control over the part of your life that involves numbers and

mathematical ideas.

Explains Things Clearly

I understand that your primary purpose in reading Thinking Mathematically is to acquire a solid understanding of the required topics in your

liberal arts math course. In order to achieve this goal, I’ve carefully

explained each topic. Important definitions and procedures are

set off in boxes, and worked-out examples that present solutions

in a step-by-step manner appear in every section. Each example is

followed by a similar matched problem, called a Check Point, for you

to try so that you can actively participate in the learning process as

you read the book. (Answers to all Check Points appear in the back

of the book and video solutions are in MyLab Math.)

Funny & Entertaining

Who says that a math textbook can’t be entertaining? From our

engaging cover to the photos in the chapter and section openers, prepare

to expect the unexpected. I hope some of the book’s enrichment essays,

called Blitzer Bonuses, will put a smile on your face from time to time.


I designed the book’s features to help you acquire knowledge of

fundamental mathematics, as well as to show you how math can solve authentic

problems that apply to your life. These helpful features include

• Explanatory Voice Balloons: Voice balloons are used in a variety of ways to make math less intimidating. They translate mathematical language into everyday English,

help clarify problem-solving procedures, present alternative ways of understanding

concepts, and connect new concepts to concepts you have already learned.

• Great Question!: The book’s Great Question! boxes are based on questions students ask in class. The answers to these questions give suggestions for problem solving,

point out common errors to avoid, and provide informal hints and suggestions.

• Chapter Summaries: Each chapter contains a review chart that summarizes the definitions and concepts in every section of the chapter. Examples from the

chapter that illustrate these key concepts are also referenced in the chart. Review

these summaries and you’ll know the most important material in the chapter!

Passionate about the Subject

I passionately believe that no other discipline comes close to math in offering a more

extensive set of tools for application and development of your mind. I wrote the book

in Point Reyes National Seashore, 40 miles north of San Francisco. The park consists

of 75,000 acres with miles of pristine surf-washed beaches, forested ridges, and bays

bordered by white cliffs. It was my hope to convey the beauty and excitement of

mathematics using nature’s unspoiled beauty as a source of inspiration and creativity.

Enjoy the pages that follow as you empower yourself with the mathematics needed

to succeed in college, your career, and in your life.


Bob Robert Blitzer


An enormous benefit of authoring a successful textbook

is the broad-based feedback I receive from students,

dedicated users, and reviewers. Every change to this

edition is the result of their thoughtful comments and

suggestions. I would like to express my appreciation to all

the reviewers, whose collective insights form the backbone

of this revision. In particular, I would like to thank the

following people for reviewing Thinking Mathematically for this Seventh Edition.

Deana Alexander, Indiana University—Purdue University Nina Bohrod, Anoka-Ramsey Community College Kim Caldwell, Volunteer State Community College Kevin Charlwood, Washburn University Elizabeth T. Dameron, Tallahassee Community College Darlene O. Diaz, Santiago Canyon College Cornell Grant, Georgia Piedmont Technical College Theresa Jones, Texas State University Elizabeth Kiedaisch, College of DuPage Lauren Kieschnick, Mineral Area College Alina Klein, University of Dubuque Susan Knights, College of Western Idaho Isabelle Kumar, Miami Dade College Dennine LaRue, Farmont State University David Miller, William Paterson University Carla A. Monticelli, Camden County College Tonny Sangutei, North Carolina Central University Cindy Vanderlaan, Indiana Purdue University —Fort Wayne Alexandra Verkhovtseva, Anoka-Ramsey Community College

Each reviewer from every edition has contributed to

the success of this book and I would like to also continue

to offer my thanks to them.

David Allen, Iona College; Carl P. Anthony, Holy Family University; Laurel Berry, Bryant and Stratton College; Kris Bowers, Florida State University; Gerard Buskes, University of Mississippi; Fred Butler, West Virginia University; Jimmy Chang, St. Petersburg College; Jerry Chen, Suffolk County Community College; Ivette Chuca, El Paso Community College; David Cochener, Austin Peay State University; Stephanie Costa, Rhode Island College; Tristen Denley, University of Mississippi; Suzanne Feldberg, Nassau Community College; Margaret Finster, Erie Community College; …