Fundamentals of Mathematics Textbook PDF

FUNDAMENTALS OF MATHEMATICS Denny Burzynski & Wade Ellis, Jr. College of Southern Nevada Default Text This text is disseminated via the Open Education Resource (OER) LibreTexts Project (https://LibreTexts.org) and like the hundreds of other texts available within this powerful platform, it is fre...

FUNDAMENTALS OF MATHEMATICS Denny Burzynski & Wade Ellis, Jr. College of Southern Nevada Default Text This text is disseminated via the Open Education Resource (OER) LibreTexts Project (https://LibreTexts.org) and like the hundreds of other texts available within this powerful platform, it is freely available for reading, printing and "consuming." Most, but not all, pages in the library have licenses that may allow individuals to make changes, save, and print this book. Carefully consult the applicable license(s) before pursuing such effects. Instructors can adopt existing LibreTexts texts or Remix them to quickly build course-specific resources to meet the needs of their students. Unlike traditional textbooks, LibreTexts’ web based origins allow powerful integration of advanced features and new technologies to support learning. 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This text was compiled on 06/08/2024 About the Authors Denny Burzynski is a mathematics professor at College of Southern Nevada located in Las Vegas, Nevada. Wade Ellis, Jr., has been a mathematics instructor at West Valley Community College in Saratoga, California for 20 years. Wade is currently Second Vice President of the Mathematical Association of America. He is a past president of the California Mathematics Council, Community College and has served as a member of the Mathematical Sciences Education Board. He is the coauthor of numerous books on the use of computers in teaching and learning mathematics. Among his many honors are the AMATYC Mathematics Excellence Award, the Outstanding Civilian Service Medal of the United States Army, the Hayward Award for Excellence in Education from the California Academic Senate, and the Distinguished Service Award from the California Mathematics Council, Community College. 1 https://math.libretexts.org/@go/page/52129 TABLE OF CONTENTS About the Authors Acknowledgements Licensing Preface 1: Addition and Subtraction of Whole Numbers 1.1: Whole Numbers 1.2: Reading and Writing Whole Numbers 1.3: Rounding Whole Numbers 1.4: Addition of Whole Numbers 1.5: Subtraction of Whole Numbers 1.6: Properties of Addition 1.7: Summary of Key Concepts 1.8: Exercise Supplement 1.9: Proficiency Exam 2: Multiplication and Division of Whole Numbers 2.1: Multiplication of Whole Numbers 2.2: Concepts of Division of Whole Numbers 2.3: Division of Whole Numbers 2.4: Some Interesting Facts about Division 2.5: Properties of Multiplication 2.6: Summary of Key Concepts 2.7: Exercise Supplement 2.8: Proficiency Exam 3: Exponents, Roots, and Factorization of Whole Numbers 3.1: Exponents and Roots 3.2: Grouping Symbols and the Order of Operations 3.3: Prime Factorization of Natural Numbers 3.4: The Greatest Common Factor 3.5: The Least Common Multiple 3.6: Summary of Key Concepts 3.7: Exercise Supplement 3.8: Proficiency Exam 4: Introduction to Fractions and Multiplication and Division of Fractions 4.1: Factions of Whole Numbers 4.2: Proper Fractions, Improper Fractions, and Mixed Numbers 4.3: Equivalent Fractions, Reducing Fractions to Lowest Terms, and Raising Fractions to Higher Terms 4.4: Multiplication of Fractions 4.5: Division of Fractions 4.6: Applications Involving Fractions 1 https://math.libretexts.org/@go/page/53164  4.7: Summary of Key Concepts 4.8: Exercise Supplement 4.9: Proficiency Exam 5: Addition and Subtraction of Fractions, Comparing Fractions, and Complex Fractions 5.1: Addition and Subtraction of Fractions with Like Denominators 5.2: Addition and Subtraction of Fractions with Unlike Denominators 5.3: Addition and Subtraction of Mixed Numbers 5.4: Comparing Fractions 5.5: Complex Fractions 5.6: Combinations of Operations with Fractions 5.7: Summary of Key Concepts 5.8: Exercise Supplement 5.9: Proficiency Exam 6: Decimals 6.1: Reading and Writing Decimals 6.2: Converting a Decimal to a Fraction 6.3: Rounding Decimals 6.4: Addition and Subtraction of Decimals 6.5: Multiplication of Decimals 6.6: Division of Decimals 6.7: Nonterminating Divisions 6.8: Converting a Fraction to a Decimal 6.9: Combinations of Operations with Decimals and Fractions 6.10: Summary of Key Concepts 6.11: Exercise Supplement 6.12: Proficiency Exam 7: Ratios and Rates 7.1: Ratios and Rates 7.2: Proportions 7.3: Applications of Proportions 7.4: Percent 7.5: Fractions of One Percent 7.6: Applications of Percents 7.7: Summary of Key Concepts 7.8: Exercise Supplement 7.9: Proficiency Exam 8: Techniques of Estimation 8.1: Estimation by Rounding 8.2: Estimation by Clustering 8.3: Mental Arithmetic—Using the Distributive Property 8.4: Estimation by Rounding Fractions 8.5: Summary of Key Concepts 8.6: Exercise Supplement 8.7: Proficiency Exam 2 https://math.libretexts.org/@go/page/53164 9: Measurement and Geometry 9.1: Measurement and the United States System 9.2: The Metric System of Measurement 9.3: Simplification of Denominate Numbers 9.4: Perimeter and Circumference of Geometric Figures 9.5: Area and Volume of Geometric Figures and Objects 9.6: Summary of Key Concepts 9.7: Exercise Supplement 9.8: Proficiency Exam 10: Signed Numbers 10.1: Variables, Constants, and Real Numbers 10.2: Signed Numbers 10.3: Absolute Value 10.4: Addition of Signed Numbers 10.5: Subtraction of Signed Numbers 10.6: Multiplication and Division of Signed Numbers 10.7: Summary of Key Concepts 10.8: Exercise Supplement 10.9: Proficiency Exam 11: Algebraic Expressions and Equations 11.1: Algebraic Expressions 11.2: Combining Like Terms Using Addition and Subtraction 11.3: Solving Equations of the Form x + a = b and x - a = b 11.4: Solving Equations of the Form ax = b and x/a = b 11.5: Applications I- Translating Words to Mathematical Symbols 11.6: Applications II- Solving Problems 11.7: Summary of Key Concepts 11.8: Exercise Supplement 11.9: Proficiency Exam Index Glossary Detailed Licensing 3 https://math.libretexts.org/@go/page/53164 Acknowledgements Many extraordinarily talented people are responsible for helping to create this text. We wish to acknowledge the efforts and skill of the following mathematicians. Their contributions have been invaluable. Barbara Conway, Berkshire Community College Bill Hajdukiewicz, Miami-Dade Community College Virginia Hamilton, Shawnee State University David Hares, El Centro College Norman Lee, Ball State University Ginger Y. Manchester, Hinds Junior College John R. Martin, Tarrant County Junior College Shelba Mormon, Northlake College Lou Ann Pate, Pima Community College Gus Pekara, Oklahoma City Community College David Price, Tarrant County Junior College David Schultz, Virginia Western Community College Sue S. Watkins, Lorain County Community College Elizabeth M. Wayt, Tennessee State University Prentice E. Whitlock, Jersey City State College Thomas E. Williamson, Montclair State College Special thanks to the following individuals for their careful accuracy reviews of manuscript, galleys, and page proofs: Steve Blasberg, West Valley College; Wade Ellis, Sr., University of Michigan; John R. Martin, Tarrant County Junior College; and Jane Ellis. We would also like to thank Amy Miller and Guy Sanders, Branham High School. Our sincere thanks to Debbie Wiedemann for her encouragement, suggestions concerning psychobiological examples, proofreading much of the manuscript, and typing many of the section exercises; Sandi Wiedemann for collating the annotated reviews, counting the examples and exercises, and untiring use of "white-out"; and Jane Ellis for solving and typing all of the exercise solutions. We thank the following people for their excellent work on the various ancillary items that accompany Fundamentals of Mathematics: Steve Blasberg, West Valley College; Wade Ellis, Sr., University of Michigan; and Jane Ellis ( Instructor's Manual); John R. Martin, Tarrant County Junior College (Student Solutions Manual and Study Guide); Virginia Hamilton, Shawnee State University (Computerized Test Bank); Patricia Morgan, San Diego State University (Prepared Tests); and George W. Bergeman, Northern Virginia Community College (Maxis Interactive Software). We also thank the talented people at Saunders College Publishing whose efforts made this text run smoothly and less painfully than we had imagined. Our particular thanks to Bob Stern, Mathematics Editor, Ellen Newman, Developmental Editor, and Janet Nuciforo, Project Editor. Their guidance, suggestions, open minds to our suggestions and concerns, and encouragement have been extraordinarily helpful. Although there were times we thought we might be permanently damaged from rereading and rewriting, their efforts have improved this text immensely. It is a pleasure to work with such high-quality professionals. Denny Burzynski Wade Ellis, Jr. San Jose, California December 1988 I would like to thank Doug Campbell, Ed Lodi, and Guy Sanders for listening to my frustrations and encouraging me on. Thanks also go to my cousin, David Raffety, who long ago in Sequoia National Forest told me what a differential equation is. Particular thanks go to each of my colleagues at West Valley College. Our everyday conversations regarding mathematics instruction have been of the utmost importance to the development of this text and to my teaching career. D.B. 1 https://math.libretexts.org/@go/page/52131 Licensing A detailed breakdown of this resource's licensing can be found in Back Matter/Detailed Licensing. 1 https://math.libretexts.org/@go/page/115535 Preface To the next generation of explorers: Kristi, BreAnne, Lindsey, Randi, Piper, Meghan, Wyatt, Lara, Mason, and Sheanna. Fundamentals of Mathematics is a work text that covers the traditional topics studied in a modern prealgebra course, as well as the topics of estimation, elementary analytic geometry, and introductory algebra. It is intended for students who 1. have had a previous course in prealgebra, 2. wish to meet the prerequisite of a higher level course such as elementary algebra, and 3. need to review fundamental mathematical concepts and techniques. This text will help the student develop the insight and intuition necessary to master arithmetic techniques and manipulative skills. It was written with the following main objectives: 1. to provide the student with an understandable and usable source of information, 2. to provide the student with the maximum opportunity to see that arithmetic concepts and techniques are logically based, 3. to instill in the student the understanding and intuitive skills necessary to know how and when to use particular arithmetic concepts in subsequent material, courses, and nonclassroom situations, and 4. to give the student the ability to correctly interpret arithmetically obtained results. We have tried to meet these objectives by presenting material dynamically, much the way an instructor might present the material visually in a classroom. (See the development of the concept of addition and subtraction of fractions in [link], for example.) Intuition and understanding are some of the keys to creative thinking; we believe that the material presented in this text will help the student realize that mathematics is a creative subject. This text can be used in standard lecture or self-paced classes. To help meet our objectives and to make the study of prealgebra a pleasant and rewarding experience, Fundamentals of Mathematics is organized as follows. Pedagogical Features The work text format gives the student space to practice mathematical skills with ready reference to sample problems. The chapters are divided into sections, and each section is a complete treatment of a particular topic, which includes the following features: Sample Sets Practice Sets Section Exercises Exercises for Review Answers to Practice Sets The chapters begin with Objectives and end with a Summary of Key Concepts, an Exercise Supplement, and a Proficiency Exam. Objectives Each chapter begins with a set of objectives identifying the material to be covered. Each section begins with an overview that repeats the objectives for that particular section. Sections are divided into subsections that correspond to the section objectives, which makes for easier reading. Sample Sets Fundamentals of Mathematics contains examples that are set off in boxes for easy reference. The examples are referred to as Sample Sets for two reasons: 1. They serve as a representation to be imitated, which we believe will foster understanding of mathematical concepts and provide experience with mathematical techniques. 2. Sample Sets also serve as a preliminary representation of problem-solving techniques that may be used to solve more general and more complicated problems. The examples have been carefully chosen to illustrate and develop concepts and techniques in the most instructive, easily remembered way. Concepts and techniques preceding the examples are introduced at a level below that normally used in similar texts and are thoroughly explained, assuming little previous knowledge. 1 https://math.libretexts.org/@go/page/52130 Practice Sets A parallel Practice Set follows each Sample Set, which reinforces the concepts just learned. There is adequate space for the student to work each problem directly on the page. Answers to Practice Sets The Answers to Practice Sets are given at the end of each section and can be easily located by referring to the page number, which appears after the last Practice Set in each section. Section Exercises The exercises at the end of each section are graded in terms of difficulty, although they are not grouped into categories. There is an ample number of problems, and after working through the exercises, the student will be capable of solving a variety of challenging problems. The problems are paired so that the odd-numbered problems are equivalent in kind and difficulty to the even-numbered problems. Answers to the odd-numbered problems are provided at the back of the book. Exercises for Review This section consists of five problems that form a cumulative review of the material covered in the preceding sections of the text and is not limited to material in that chapter. The exercises are keyed by section for easy reference. Since these exercises are intended for review only, no work space is provided. Summary of Key Concepts A summary of the important ideas and formulas used throughout the chapter is included at the end of each chapter. More than just a list of terms, the summary is a valuable tool that reinforces concepts in preparation for the Proficiency Exam at the end of the chapter, as well as future exams. The summary keys each item to the section of the text where it is discussed. Exercise Supplement In addition to numerous section exercises, each chapter includes approximately 100 supplemental problems, which are referenced by section. Answers to the odd-numbered problems are included in the back of the book. Proficiency Exam Each chapter ends with a Proficiency Exam that can serve as a chapter review or evaluation. The Proficiency Exam is keyed to sections, which enables the student to refer back to the text for assistance. Answers to all the problems are included in the Answer Section at the end of the book. Content The writing style used in Fundamentals of Mathematics is informal and friendly, offering a straightforward approach to prealgebra mathematics. We have made a deliberate effort not to write another text that minimizes the use of words because we believe that students can best study arithmetic concepts and understand arithmetic techniques by using words and symbols rather than symbols alone. It has been our experience that students at the prealgebra level are not nearly experienced enough with mathematics to understand symbolic explanations alone; they need literal explanations to guide them through the symbols. We have taken great care to present concepts and techniques so they are understandable and easily remembered. After concepts have been developed, students are warned about common pitfalls. We have tried to make the text an information source accessible to prealgebra students. Addition and Subtraction of Whole Numbers This chapter includes the study of whole numbers, including a discussion of the Hindu-Arabic numeration and the base ten number systems. Rounding whole numbers is also presented, as are the commutative and associative properties of addition. Multiplication and Division of Whole Numbers The operations of multiplication and division of whole numbers are explained in this chapter. Multiplication is described as repeated addition. Viewing multiplication in this way may provide students with a visualization of the meaning of algebraic terms such as 8x when they start learning algebra. The chapter also includes the commutative and associative properties of multiplication. 2 https://math.libretexts.org/@go/page/52130 Exponents, Roots, and Factorizations of Whole Numbers The concept and meaning of the word root is introduced in this chapter. A method of reading root notation and a method of determining some common roots, both mentally and by calculator, is then presented. We also present grouping symbols and the order of operations, prime factorization of whole numbers, and the greatest common factor and least common multiple of a collection of whole numbers. Introduction to Fractions and Multiplication and Division of Fractions We recognize that fractions constitute one of the foundations of problem solving. We have, therefore, given a detailed treatment of the operations of multiplication and division of fractions and the logic behind these operations. We believe that the logical treatment and many practice exercises will help students retain the information presented in this chapter and enable them to use it as a foundation for the study of rational expressions in an algebra course. Addition and Subtraction of Fractions, Comparing Fractions, and Complex Fractions A detailed treatment of the operations of addition and subtraction of fractions and the logic behind these operations is given in this chapter. Again, we believe that the logical treatment and many practice exercises will help students retain the information, thus enabling them to use it in the study of rational expressions in an algebra course. We have tried to make explanations dynamic. A method for comparing fractions is introduced, which gives the student another way of understanding the relationship between the words denominator and denomination. This method serves to show the student that it is sometimes possible to compare two different types of quantities. We also study a method of simplifying complex fractions and of combining operations with fractions. Decimals The student is introduced to decimals in terms of the base ten number system, fractions, and digits occurring to the right of the units position. A method of converting a fraction to a decimal is discussed. The logic behind the standard methods of operating on decimals is presented and many examples of how to apply the methods are given. The word of as related to the operation of multiplication is discussed. Nonterminating divisions are examined, as are combinations of operations with decimals and fractions. Ratios and Rates We begin by defining and distinguishing the terms ratio and rate. The meaning of proportion and some applications of proportion problems are described. Proportion problems are solved using the "Five-Step Method." We hope that by using this method the student will discover the value of introducing a variable as a first step in problem solving and the power of organization. The chapter concludes with discussions of percent, fractions of one percent, and some applications of percent. Techniques of Estimation One of the most powerful problem-solving tools is a knowledge of estimation techniques. We feel that estimation is so important that we devote an entire chapter to its study. We examine three estimation techniques: estimation by rounding, estimation by clustering, and estimation by rounding fractions. We also include a section on the distributive property, an important algebraic property. Measurement and Geometry This chapter presents some of the techniques of measurement in both the United States system and the metric system. Conversion from one unit to another (in a system) is examined in terms of unit fractions. A discussion of the simplification of denominate numbers is also included. This discussion helps the student understand more clearly the association between pure numbers and dimensions. The chapter concludes with a study of perimeter and circumference of geometric figures and area and volume of geometric figures and objects. Signed Numbers A look at algebraic concepts and techniques is begun in this chapter. Basic to the study of algebra is a working knowledge of signed numbers. Definitions of variables, constants, and real numbers are introduced. We then distinguish between positive and negative numbers, learn how to read signed numbers, and examine the origin and use of the double-negative property of real numbers. The concept of absolute value is presented both geometrically (using the number line) and algebraically. The algebraic definition is followed by an interpretation of its meaning and several detailed examples of its use. Addition, subtraction, multiplication, and division of signed numbers are presented first using the number line, then with absolute value. 3 https://math.libretexts.org/@go/page/52130 Algebraic Expressions and Equations The student is introduced to some elementary algebraic concepts and techniques in this final chapter. Algebraic expressions and the process of combining like terms are discussed in [link] and [link]. The method of combining like terms in an algebraic expression is explained by using the interpretation of multiplication as a description of repeated addition (as in [link]). 4 https://math.libretexts.org/@go/page/52130  CHAPTER OVERVIEW 1: Addition and Subtraction of Whole Numbers 1.1: Whole Numbers 1.2: Reading and Writing Whole Numbers 1.3: Rounding Whole Numbers 1.4: Addition of Whole Numbers 1.5: Subtraction of Whole Numbers 1.6: Properties of Addition 1.7: Summary of Key Concepts 1.8: Exercise Supplement 1.9: Proficiency Exam This page titled 1: Addition and Subtraction of Whole Numbers is shared under a CC BY license and was authored, remixed, and/or curated by Denny Burzynski & Wade Ellis, Jr. (OpenStax CNX) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request. 1 1.1: Whole Numbers  Learning Objectives know the difference between numbers and numerals know why our number system is called the Hindu-Arabic numeration system understand the base ten positional number system be able to identify and graph whole number Numbers and Numerals We begin our study of introductory mathematics by examining its most basic building block, the number.  Definition: Number A number is a concept. It exists only in the mind. The earliest concept of a number was a thought that allowed people to mentally picture the size of some collection of objects. To write down the number being conceptualized, a numeral is used.  Definition: Numeral A numeral is a symbol that represents a number. In common usage today we do not distinguish between a number and a numeral. In our study of introductory mathematics, we will follow this common usage.  Sample Set A The following are numerals. In each case, the first represents the number four, the second represents the number one hundred twenty-three, and the third, the number one thousand five. These numbers are represented in different ways. Hindu-Arabic numerals 4, 123, 1005 Roman numerals IV, CXXIII, MV Egyptian numerals: Practice Set A Do the phrases "four," "one hundred twenty-three," and "one thousand five" qualify as numerals? Yes or no? Answer Yes. Letters are symbols. Taken as a collection (a written word), they represent a number. The Hindu-Arabic Numeration System 1.1.1 https://math.libretexts.org/@go/page/48774   Definition: Hindu-Arabic Numeration System Our society uses the Hindu-Arabic numeration system. This system of numeration began shortly before the third century when the Hindus invented the numerals 0123456789  Definition: Lenoardo Fibonacci About a thousand years later, in the thirteenth century, a mathematician named Leonardo Fibonacci of Pisa introduced the system into Europe. It was then popularized by the Arabs. Thus, the name, Hindu-Arabic numeration system. The Base Ten Positional Number System  Definition: Digits The Hindu-Arabic numerals 0 1 2 3 4 5 6 7 8 9 are called digits. We can form any number in the number system by selecting one or more digits and placing them in certain positions. Each position has a particular value. The Hindu mathematician who devised the system about A.D. 500 stated that "from place to place each is ten times the preceding."  Definition: Base Ten Positional Systems It is for this reason that our number system is called a positional number system with base ten.  Definition: Commas When numbers are composed of more than three digits, commas are sometimes used to separate the digits into groups of three.  Definition: Periods These groups of three are called periods and they greatly simplify reading numbers. In the Hindu-Arabic numeration system, a period has a value assigned to each or its three positions, and the values are the same for each period. The position values are Thus, each period contains a position for the values of one, ten, and hundred. Notice that, in looking from right to left, the value of each position is ten times the preceding. Each period has a particular name. As we continue from right to left, there are more periods. The five periods listed above are the most common, and in our study of introductory mathematics, they are sufficient. The following diagram illustrates our positional number system to trillions. (There are, to be sure, other periods.) 1.1.2 https://math.libretexts.org/@go/page/48774 In our positional number system, the value of a digit is determined by its position in the number.  Sample Set B Find the value of 6 in the number 7,261. Solution Since 6 is in the tens position of the units period, its value is 6 tens. 6 tens = 60  Sample Set B Find the value of 9 in the number 86,932,106,005. Solution Since 9 is in the hundreds position of the millions period, its value is 9 hundred millions. 9 hundred millions = 9 hundred million  Sample Set B Find the value of 2 in the number 102,001. Solution Since 2 is in the ones position of the thousands period, its value is 2 one thousands. 2 one thousands = 2 thousand Practice Set B Find the value of 5 in the number 65,000. Answer five thousand Practice Set B Find the value of 4 in the number 439,997,007,010. Answer four hundred billion Practice Set B Find the value of 0 in the number 108. Answer zero tens, or zero 1.1.3 https://math.libretexts.org/@go/page/48774 Whole Numbers  Definition: Whole Numbers Numbers that are formed using only the digits 0123456789 are called whole numbers. They are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, … The three dots at the end mean "and so on in this same pattern." Graphing Whole Numbers  Definition: Number Line Whole numbers may be visualized by constructing a number line. To construct a number line, we simply draw a straight line and choose any point on the line and label it 0.  Definition: Origin This point is called the origin. We then choose some convenient length, and moving to the right, mark off consecutive intervals (parts) along the line starting at 0. We label each new interval endpoint with the next whole number.  Definition: Graphing We can visually display a whole number by drawing a closed circle at the point labeled with that whole number. Another phrase for visually displaying a whole number is graphing the whole number. The word graph means to "visually display."  Sample Set C Graph the following whole numbers: 3, 5, 9.  Sample Set C Specify the whole numbers that are graphed on the following number line. The break in the number line indicates that we are aware of the whole numbers between 0 and 106, and 107 and 872, but we are not listing them due to space limitations. Solution The numbers that have been graphed are 0, 106, 873, 874 1.1.4 https://math.libretexts.org/@go/page/48774  Practice Set C Graph the following whole numbers: 46, 47, 48, 325, 327. Answer Practice Set C Specify the whole numbers that are graphed on the following number line. Answer 4, 5, 6, 113, 978 A line is composed of an endless number of points. Notice that we have labeled only some of them. As we proceed, we will discover new types of numbers and determine their location on the number line. Exercises Exercise 1.1.1 What is a number? Answer concept Exercise 1.1.2 What is a numeral? Does the word "eleven" qualify as a numeral? Answer Yes, since it is a symbol that represents a number. Exercise 1.1.3 How many different digits are there? Our number system, the Hindu-Arabic number system, is a number system with base ? Answer positional; 10 1.1.5 https://math.libretexts.org/@go/page/48774  Exercise 1.1.4 Numbers composed of more than three digits are sometimes separated into groups of three by commas. These groups of three are called. In our number system, each period has three values assigned to it. These values are the same for each period. From right to left, what are they? Answer ones, tens, hundreds Exercise 1.1.5 Each period has its own particular name. From right to left, what are the names of the first four? In the number 841, how many tens are there? Answer 4 Exercise 1.1.6 In the number 3,392, how many ones are there? In the number 10,046, how many thousands are there? Answer 0 Exercise 1.1.7 In the number 779,844,205, how many ten millions are there? In the number 65,021, how many hundred thousands are there? Answer 0 For following problems, give the value of the indicated digit in the given number. Exercise 1.1.8 5 in 599 1 in 310,406 Answer ten thousand Exercise 1.1.9 9 in 29,827 6 in 52,561,001,100 Answer 1.1.6 https://math.libretexts.org/@go/page/48774  6 ten millions = 60 million Exercise 1.1.10 Write a two-digit number that has an eight in the tens position. Write a four-digit number that has a one in the thousands position and a zero in the ones position. Answer 1,340 (answers may vary) Exercise 1.1.11 How many two-digit whole numbers are there? How many three-digit whole numbers are there? Answer 900 Exercise 1.1.12 How many four-digit whole numbers are there? Is there a smallest whole number? If so, what is it? Answer yes; zero Exercise 1.1.13 Is there a largest whole number? If so, what is it? Another term for "visually displaying" is ? Answer graphing Exercise 1.1.14 The whole numbers can be visually displayed on a. Graph (visually display) the following whole numbers on the number line below: 0, 1, 31, 34. Answer 1.1.7 https://math.libretexts.org/@go/page/48774  Exercise 1.1.15 Construct a number line in the space provided below and graph (visually display) the following whole numbers: 84, 85, 901, 1006, 1007. Specify, if any, the whole numbers that are graphed on the following number line. Answer 61, 99, 100, 102 Exercise 1.1.16 Specify, if any, the whole numbers that are graphed on the following number line. This page titled 1.1: Whole Numbers is shared under a CC BY license and was authored, remixed, and/or curated by Denny Burzynski & Wade Ellis, Jr. (OpenStax CNX) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request. 1.1.8 https://math.libretexts.org/@go/page/48774 1.2: Reading and Writing Whole Numbers  Learning Objectives be able to read and write a whole number Because our number system is a positional number system, reading and writing whole numbers is quite simple. Reading Whole Numbers To convert a number that is formed by digits into a verbal phrase, use the following method: 1. Beginning at the right and working right to left, separate the number into distinct periods by inserting commas every three digits. 2. Beginning at the left, read each period individually, saying the period name.  Sample Set A Write the following numbers as words. Read 42958. Solution Step 1: Beginning at the right, we can separate this number into distinct periods by inserting a comma between the 2 and 9. 42, 958 Step 2: Beginning at the left, we read each period individually: Forty-two thousand, nine hundred fifty-eight.  Sample Set A Read 307991343. Solution Step 1: Beginning at the right, we can separate this number into distinct periods by placing commas between the 1 and 3 and the 7 and 9. 307, 991, 343 Step 2: Beginning at the left, we read each period individually. Three hundred seven million, nine hundred ninety-one thousand, three hundred forty-three. 1.2.1 https://math.libretexts.org/@go/page/48775   Sample Set A Read 36000000000001. Solution Step 1: Beginning at the right, we can separate this number into distinct periods by placing commas. 36, 000, 000, 000, 001 Step 2: Beginning at the left, we read each period individually. Thirty-six trillion, one. Practice Set A Write each number in words. 12,542 Answer Twelve thousand, five hundred forty-two Practice Set A 101,074,003 Answer One hundred one million, seventy-four thousand, three Practice Set A 1,000,008 Answer One million, eight Writing Whole Numbers To express a number in digits that is expressed in words, use the following method: 1. Notice first that a number expressed as a verbal phrase will have its periods set off by commas. 2. Starting at the beginning of the phrase, write each period of numbers individually. 1.2.2 https://math.libretexts.org/@go/page/48775 3. Using commas to separate periods, combine the periods to form one number.  Sample Set B Write each number using digits. Seven thousand, ninety-two. Solution Using the comma as a period separator, we have 7,092  Sample Set B Fifty billion, one million, two hundred thousand, fourteen. Solution Using the commas as period separators, we have 50,001,200,014  Sample Set B Ten million, five hundred twelve. Solution The comma sets off the periods. We notice that there is no thousands period. We'll have to insert this ourselves. 10,000,512 Practice Set B Express each number using digits. One hundred three thousand, twenty-five. Answer 103,025 1.2.3 https://math.libretexts.org/@go/page/48775  Practice Set B Six million, forty thousand, seven. Answer 6,040,007 Practice Set B Twenty trillion, three billion, eighty million, one hundred nine thousand, four hundred two. Answer 20,003,080,109,402 Practice Set B Eighty billion, thirty-five. Answer 80,000,000,035 Exercises For the following problems, write all numbers in words. Exercise 1.2.1 912 Answer nine hundred twelve Exercise 1.2.2 84 Exercise 1.2.3 1491 Answer one thousand, four hundred ninety-one Exercise 1.2.4 8601 Exercise 1.2.5 35,223 Answer thirty-five thousand, two hundred twenty-three 1.2.4 https://math.libretexts.org/@go/page/48775 Exercise 1.2.6 71,006 Exercise 1.2.7 437,105 Answer four hundred thirty-seven thousand, one hundred five Exercise 1.2.8 201,040 Exercise 1.2.9 8,001,001 Answer eight million, one thousand, one Exercise 1.2.10 16,000,053 Exercise 1.2.11 770,311,101 Answer seven hundred seventy million, three hundred eleven thousand, one hundred one Exercise 1.2.12 83,000,000,007 Exercise 1.2.13 106,100,001,010 Answer one hundred six billion, one hundred million, one thousand ten Exercise 1.2.14 3,333,444,777 Exercise 1.2.15 800,000,800,000 Answer 1.2.5 https://math.libretexts.org/@go/page/48775  eight hundred billion, eight hundred thousand Exercise 1.2.16 A particular community college has 12,471 students enrolled. Exercise 1.2.17 A person who watches 4 hours of television a day spends 1460 hours a year watching T.V. Answer four; one thousand, four hundred sixty Exercise 1.2.18 Astronomers believe that the age of the earth is about 4,500,000,000 years. Exercise 1.2.19 Astronomers believe that the age of the universe is about 20,000,000,000 years. Answer twenty billion Exercise 1.2.20 There are 9690 ways to choose four objects from a collection of 20. Exercise 1.2.21 If a 412 page book has about 52 sentences per page, it will contain about 21,424 sentences. Answer four hundred twelve; fifty-two; twenty-one thousand, four hundred twenty-four Exercise 1.2.22 In 1980, in the United States, there was $1,761,000,000,000 invested in life insurance. Exercise 1.2.23 In 1979, there were 85,000 telephones in Alaska and 2,905,000 telephones in Indiana. Answer one thousand, nine hundred seventy-nine; eighty-five thousand; two million, nine hundred five thousand Exercise 1.2.24 In 1975, in the United States, it is estimated that 52,294,000 people drove to work alone. 1.2.6 https://math.libretexts.org/@go/page/48775  Exercise 1.2.25 In 1980, there were 217 prisoners under death sentence that were divorced. Answer one thousand, nine hundred eighty; two hundred seventeen Exercise 1.2.26 In 1979, the amount of money spent in the United States for regular-session college education was $50,721,000,000,000. Exercise 1.2.27 In 1981, there were 1,956,000 students majoring in business in U.S. colleges. Answer one thousand, nine hundred eighty one; one million, nine hundred fifty-six thousand Exercise 1.2.28 In 1980, the average fee for initial and follow up visits to a medical doctors office was about $34. Exercise 1.2.29 In 1980, there were approximately 13,100 smugglers of aliens apprehended by the Immigration border patrol. Answer one thousand, nine hundred eighty; thirteen thousand, one hundred Exercise 1.2.30 In 1980, the state of West Virginia pumped 2,000,000 barrels of crude oil, whereas Texas pumped 975,000,000 barrels. Exercise 1.2.31 The 1981 population of Uganda was 12,630,000 people. Answer twelve million, six hundred thirty thousand Exercise 1.2.32 In 1981, the average monthly salary offered to a person with a Master's degree in mathematics was $1,685. For the following problems, write each number using digits. Exercise 1.2.33 Six hundred eighty-one Answer 681 1.2.7 https://math.libretexts.org/@go/page/48775 Exercise 1.2.34 Four hundred ninety Exercise 1.2.35 Seven thousand, two hundred one Answer 7,201 Exercise 1.2.36 Nineteen thousand, sixty-five Exercise 1.2.37 Five hundred twelve thousand, three Answer 512,003 Exercise 1.2.38 Two million, one hundred thirty-three thousand, eight hundred fifty-nine Exercise 1.2.39 Thirty-five million, seven thousand, one hundred one Answer 35,007,101 Exercise 1.2.40 One hundred million, one thousand Exercise 1.2.41 Sixteen billion, fifty-nine thousand, four Answer 16,000,059,004 Exercise 1.2.42 Nine hundred twenty billion, four hundred seventeen million, twenty-one thousand Exercise 1.2.43 Twenty-three billion Answer 1.2.8 https://math.libretexts.org/@go/page/48775  23,000,000,000 Exercise 1.2.44 Fifteen trillion, four billion, nineteen thousand, three hundred five Exercise 1.2.45 One hundred trillion, one Answer 100,000,000,000,001 Exercises for Review Exercise 1.2.46 ([link]) How many digits are there? Exercise 1.2.47 ([link]) In the number 6,641, how many tens are there? Answer 4 Exercise 1.2.48 ([link]) What is the value of 7 in 44,763? Exercise 1.2.49 ([link]) Is there a smallest whole number? If so, what is it? Answer yes, zero Exercise 1.2.50 ([link]) Write a four-digit number with a 9 in the tens position. This page titled 1.2: Reading and Writing Whole Numbers is shared under a CC BY license and was authored, remixed, and/or curated by Denny Burzynski & Wade Ellis, Jr. (OpenStax CNX) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request. 1.2.9 https://math.libretexts.org/@go/page/48775 1.3: Rounding Whole Numbers  Learning Objectives understand that rounding is a method of approximation be able to round a whole number to a specified position Rounding as an Approximation A primary use of whole numbers is to keep count of how many objects there are in a collection. Sometimes we're only interested in the approximate number of objects in the collection rather than the precise number. For example, there are approximately 20 symbols in the collection below. The precise number of symbols in the above collection is 18.  Definition: Rounding We often approximate the number of objects in a collection by mentally seeing the collection as occurring in groups of tens, hundreds, thousands, etc. This process of approximation is called rounding. Rounding is very useful in estimation. We will study estimation in Chapter 8. When we think of a collection as occurring in groups of tens, we say we're rounding to the nearest ten. When we think of a collection as occurring in groups of hundreds, we say we're rounding to the nearest hundred. This idea of rounding continues through thousands, ten thousands, hundred thousands, millions, etc. The process of rounding whole numbers is illustrated in the following examples.  Example 1.3.1 Round 67 to the nearest ten. Solution On the number line, 67 is more than halfway from 60 to 70. The digit immediately to the right of the tens digit, the round-off digit, is the indicator for this. Thus, 67, rounded to the nearest ten, is 70.  Example 1.3.2 Round 4,329 to the nearest hundred. Solution On the number line, 4,329 is less than halfway from 4,300 to 4,400. The digit to the immediate right of the hundreds digit, the round-off digit, is the indicator. 1.3.1 https://math.libretexts.org/@go/page/48776  Thus, 4,329, rounded to the nearest hundred is 4,300.  Example 1.3.3 Round 16,500 to the nearest thousand. Solution On the number line, 16,500 is exactly halfway from 16,000 to 17,000. By convention, when the number to be rounded is exactly halfway between two numbers, it is rounded to the higher number. Thus, 16,500, rounded to the nearest thousand, is 17,000.  Example 1.3.4 A person whose salary is $41,450 per year might tell a friend that she makes $41,000 per year. She has rounded 41,450 to the nearest thousand. The number 41,450 is closer to 41,000 than it is to 42,000. The Method of Rounding Whole Numbers From the observations made in the preceding examples, we can use the following method to round a whole number to a particular position. 1. Mark the position of the round-off digit. 2. Note the digit to the immediate right of the round-off digit. a. If it is less than 5, replace it and all the digits to its right with zeros. Leave the round-off digit unchanged. b. If it is 5 or larger, replace it and all the digits to its right with zeros. Increase the round-off digit by 1.  Sample Set A Use the method of rounding whole numbers to solve the following problems. Round 3,426 to the nearest ten. Solution We are rounding to the tens position. Mark the digit in the tens position Observe the digit immediately to the right of the tens position. It is 6. Since 6 is greater than 5, we round up by replacing 6 with 0 and adding 1 to the digit in the tens position (the round-off position): 2+1=32+1=3. 3,430 Thus, 3,426 rounded to the nearest ten is 3,430. 1.3.2 https://math.libretexts.org/@go/page/48776  Sample Set A Round 9,614,018,007 to the nearest ten million. Solution 1. We are rounding to the nearest ten million. 2. Observe the digit immediately to the right of the ten millions position. It is 4. Since 4 is less than 5, we round down by replacing 4 and all the digits to its right with zeros. 9,610,000,000 Thus, 9,614,018,007 rounded to the nearest ten million is 9,610,000,000.  Sample Set A Round 148,422 to the nearest million. Solution 1. Since we are rounding to the nearest million, we'll have to imagine a digit in the millions position. We'll write 148,422 as 0,148,422. 2. The digit immediately to the right is 1. Since 1 is less than 5, we'll round down by replacing it and all the digits to its right with zeros. 0,000,000 This number is 0. Thus, 148,422 rounded to the nearest million is 0.  Sample Set A Round 397,000 to the nearest ten thousand. Solution We are rounding to the nearest ten thousand. The digit immediately to the right of the ten thousand position is 7. Since 7 is greater than 5, we round up by replacing 7 and all the digits to its right with zeros and adding 1 to the digit in the ten thousands position. But 9+1=109+1=10 and we must carry the 1 to the next (the hundred thousands) position. 400,000 Thus, 397,000 rounded to the nearest ten thousand is 400,000. Practice Set A Use the method of rounding whole numbers to solve each problem. Round 3387 to the nearest hundred. Answer 3400 1.3.3 https://math.libretexts.org/@go/page/48776  Practice Set A Round 26,515 to the nearest thousand. Answer 27,000 Practice Set A Round 30,852,900 to the nearest million. Answer 31,000,000 Practice Set A Round 39 to the nearest hundred. Answer 0 Practice Set A Round 59,600 to the nearest thousand. Answer 60,000 Exercises For the following problems, complete the table by rounding each number to the indicated positions. Exercise 1.3.1 1,642 hundred thousand ten thousand million Answer hundred thousand ten thousand million 1,600 2000 0 0 Exercise 1.3.2 5,221 hundred thousand ten thousand million 1.3.4 https://math.libretexts.org/@go/page/48776 Exercise 1.3.3 91,803 hundred thousand ten thousand million Answer hundred thousand ten thousand million 91,800 92,000 90,000 0 Exercise 1.3.4 106,007 hundred thousand ten thousand million Exercise 1.3.5 208 hundred thousand ten thousand million Answer hundred thousand ten thousand million 200 0 0 0 Exercise 1.3.6 199 hundred thousand ten thousand million Exercise 1.3.7 863 hundred thousand ten thousand million Answer hundred thousand ten thousand million 900 1,000 0 0 1.3.5 https://math.libretexts.org/@go/page/48776 Exercise 1.3.8 794 hundred thousand ten thousand million Exercise 1.3.9 925 hundred thousand ten thousand million Answer hundred thousand ten thousand million 900 1,000 0 0 Exercise 1.3.10 909 hundred thousand ten thousand million Exercise 1.3.11 981 hundred thousand ten thousand million Answer hundred thousand ten thousand million 1,000 1,000 0 0 Exercise 1.3.12 965 hundred thousand ten thousand million Exercise 1.3.13 551,061,285 hundred thousand ten thousand million 1.3.6 https://math.libretexts.org/@go/page/48776  Answer hundred thousand ten thousand million 551,061,300 551,061,000 551,060,000 551,000,000 Exercise 1.3.14 23,047,991,521 hundred thousand ten thousand million Exercise 1.3.15 106,999,413,206 hundred thousand ten thousand million Answer hundred thousand ten thousand million 106,999,413,200 106,999,413,000 106,999,410,000 106,999,000,000 Exercise 1.3.16 5,000,000 hundred thousand ten thousand million Exercise 1.3.17 8,006,001 hundred thousand ten thousand million Answer hundred thousand ten thousand million 8,006,000 8,006,000 8,010,000 8,000,000 Exercise 1.3.18 94,312 hundred thousand ten thousand million 1.3.7 https://math.libretexts.org/@go/page/48776 Exercise 1.3.19 33,486 hundred thousand ten thousand million Answer hundred thousand ten thousand million 33,500 33,000 30,000 0 Exercise 1.3.20 560,669 hundred thousand ten thousand million Exercise 1.3.21 388,551 hundred thousand ten thousand million Answer hundred thousand ten thousand million 388,600 389,000 390,000 0 Exercise 1.3.22 4,752 hundred thousand ten thousand million Exercise 1.3.23 8,209 hundred thousand ten thousand million Answer hundred thousand ten thousand million 8,200 8,000 10,000 0 1.3.8 https://math.libretexts.org/@go/page/48776  Exercise 1.3.24 In 1950, there were 5,796 cases of diphtheria reported in the United States. Round to the nearest hundred. Exercise 1.3.25 In 1979, 19,309,000 people in the United States received federal food stamps. Round to the nearest ten thousand. Answer 19,310,000 Exercise 1.3.26 In 1980, there were 1,105,000 people between 30 and 34 years old enrolled in school. Round to the nearest million. Exercise 1.3.27 In 1980, there were 29,100,000 reports of aggravated assaults in the United States. Round to the nearest million. Answer 29,000,000 For the following problems, round the numbers to the position you think is most reasonable for the situation. Exercise 1.3.28 In 1980, for a city of one million or more, the average annual salary of police and firefighters was $16,096. Exercise 1.3.29 The average percentage of possible sunshine in San Francisco, California, in June is 73%. Answer 70% or 75% Exercise 1.3.30 In 1980, in the state of Connecticut, $3,777,000,000 in defense contract payroll was awarded. Exercise 1.3.31 In 1980, the federal government paid $5,463,000,000 to Viet Nam veterans and dependants. Answer $5,500,000,000 Exercise 1.3.32 In 1980, there were 3,377,000 salespeople employed in the United States. 1.3.9 https://math.libretexts.org/@go/page/48776  Exercise 1.3.33 In 1948, in New Hampshire, 231,000 popular votes were cast for the president. Answer 230,000 Exercise 1.3.34 In 1970, the world production of cigarettes was 2,688,000,000,000. Exercise 1.3.35 In 1979, the total number of motor vehicle registrations in Florida was 5,395,000. Answer 5,400,000 Exercise 1.3.36 In 1980, there were 1,302,000 registered nurses the United States. Exercises for Review Exercise 1.3.37 There is a term that describes the visual displaying of a number. What is the term? Answer graphing Exercise 1.3.38 What is the value of 5 in 26,518,206? Exercise 1.3.39 Write 42,109 as you would read it. Answer Forty-two thousand, one hundred nine Exercise 1.3.40 Write "six hundred twelve" using digits. Exercise 1.3.41 Write "four billion eight" using digits. Answer 4,000,000,008 1.3.10 https://math.libretexts.org/@go/page/48776 This page titled 1.3: Rounding Whole Numbers is shared under a CC BY license and was authored, remixed, and/or curated by Denny Burzynski & Wade Ellis, Jr. (OpenStax CNX) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request. 1.3.11 https://math.libretexts.org/@go/page/48776 1.4: Addition of Whole Numbers  Learning Objectives understand the addition process be able to add whole numbers be able to use the calculator to add one whole number to another Addition Suppose we have two collections of objects that we combine together to form a third collection. For example, We are combining a collection of four objects with a collection of three objects to obtain a collection of seven objects.  Definition: Addition The process of combining two or more objects (real or intuitive) to form a third, the total, is called addition. In addition, the numbers being added are called addends or terms, and the total is called the sum. The plus symbol (+) is used to indicate addition, and the equal symbol (=) is used to represent the word "equal." For example, 4 + 3 = 7 means "four added to three equals seven." Addition Visualized on the Number Line Addition is easily visualized on the number line. Let's visualize the addition of 4 and 3 using the number line. To find 4 + 3 1. Start at 0. 2. Move to the right 4 units. We are now located at 4. 3. From 4, move to the right 3 units. We are now located at 7. Thus, 4 + 3 = 7 The Addition Process We'll study the process of addition by considering the sum of 25 and 43. We write this as 68. We can suggest the following procedure for adding whole numbers using this example.  The Process of Adding Whole Numbers To add whole numbers, The process: 1.4.1 https://math.libretexts.org/@go/page/48777  1. Write the numbers vertically, placing corresponding positions in the same column. 25 +43 –––– 2. Add the digits in each column. Start at the right (in the ones position) and move to the left, placing the sum at the bottom. 25 +43 –––– 68  Caution Confusion and incorrect sums can occur when the numbers are not aligned in columns properly. Avoid writing such additions as 25 +43 –––– 25 +43 –––––  Sample Set A Add 276 and 103. Solution 276 6 + 3 = 9. +103 7 + 0 = 7. ––––– 379 2 + 1 = 3.  Sample Set A Add 1459 and 130 Solution 9 + 0 = 9. 1459 5 + 3 = 8. +130 ––––– 4 + 1 = 5. 1589 1 + 0 = 1. In each of these examples, each individual sum does not exceed 9. We will examine individual sums that exceed 9 in the next section. Practice Set A Perform each addition. Show the expanded form in problems 1 and 2. Add 63 and 25. Answer 88 1.4.2 https://math.libretexts.org/@go/page/48777  Practice Set A Add 4,026 and 1,501. Answer 5,527 Practice Set A Add 231,045 and 36,121. Answer 267,166 Addition Involving Carrying It often happens in addition that the sum of the digits in a column will exceed 9. This happens when we add 18 and 34. We show this in expanded form as follows. Notice that when we add the 8 ones to the 4 ones we get 12 ones. We then convert the 12 ones to 1 ten and 2 ones. In vertical addition, we show this conversion by carrying the ten to the tens column. We write a 1 at the top of the tens column to indicate the carry. This same example is shown in a shorter form as follows: 8 + 4 = 12 Write 2, carry 1 ten to the top of the next column to the left.  Sample Set B Perform the following additions. Use the process of carrying when needed. Add 1875 and 358. Solution 5 + 8 = 13 Write 3, carry 1 ten. 1 + 7 + 5 = 13 Write 3, carry 1 hundred. 1 + 8 + 3 = 12 Write 2, carry 1 thousand. 1 +1 = 2 The sum is 2233. 1.4.3 https://math.libretexts.org/@go/page/48777  Sample Set B Add 89,208 and 4,946. Solution 8 + 6 = 14 Write 4, carry 1 ten. 1 +0 +4 = 5 Write the 5 (nothing to carry). 2 + 9 = 11 Write 1, carry one thousand. 1 + 9 + 4 = 14 Write 4, carry one ten thousand. 1 +8 = 9 The sum is 94,154.  Sample Set B Add 38 and 95. Solution 8 + 5 = 13 Write 3, carry 1 ten. 1 + 3 + 9 = 13 Write 3, carry 1 hundred. 1 +0 = 1 As you proceed with the addition, it is a good idea to keep in mind what is actually happening. The sum is 133.  Sample Set B Find the sum 2648, 1359, and 861. Solution 1.4.4 https://math.libretexts.org/@go/page/48777  8 + 9 + 1 = 18 Write 8, carry 1 ten. 1 + 4 + 5 + 6 = 16 Write 6, carry 1 hundred. 1 + 6 + 3 + 8 = 18 Write 8, carry 1 thousand. 1 +2 +1 = 4 The sum is 4,868. Numbers other than 1 can be carried as illustrated in next example.  Sample Set B Find the sum of the following numbers. Solution 6 + 5 + 1 + 7 = 19 Write 9, carry the 1. 1 +1 +0 +5 +1 = 8 Write 8. 0 + 9 + 9 + 8 = 26 Write 6, carry the 2. 2 + 8 + 9 + 8 + 6 = 33 Write 3, carry the 3. 3 + 7 + 3 + 5 = 18 Write 8, carry the 1. 1 +8 = 9 Write 9. The sum is 983,689.  Sample Set B The number of students enrolled at Riemann College in the years 1984, 1985, 1986, and 1987 was 10,406, 9,289, 10,108, and 11,412, respectively. What was the total number of students enrolled at Riemann College in the years 1985, 1986, and 1987? Solution We can determine the total number of students enrolled by adding 9,289, 10,108, and 11,412, the number of students enrolled in the years 1985, 1986, and 1987. The total number of students enrolled at Riemann College in the years 1985, 1986, and 1987 was 30,809. Practice Set B Perform each addition. For the next three problems, show the expanded form. Add 58 and 29. Answer 87 1.4.5 https://math.libretexts.org/@go/page/48777  = 7 tens + 1 ten + 7 ones = 8 tens + 7 ones = 87 Practice Set B Add 476 and 85. Answer 561 = 4 hundreds + 15 tens + 1 ten + 1 one = 4 hundreds + 16 tens + 1 one = 4 hundreds + 1 hundred + 6 tens + 1 one = 5 hundreds + 6 tens + 1 one = 561 Practice Set B Add 27 and 88. Answer 115 = 10 tens + 1 ten + 5 ones = 11 tens + 5 ones = 11 hundred + 1 ten + 5 ones = 115 Practice Set B Add 67,898 and 85,627. Answer 153,525 For the next three problems, find the sums. Practice Set B 57 26 84 –––– Answer 1.4.6 https://math.libretexts.org/@go/page/48777  167 Practice Set B 847 825 796 ––––– Answer 2,468 Practice Set B 16, 945 8, 472 387, 721 21, 059 629 –––––––– Answer 434,826 Calculators Calculators provide a very simple and quick way to find sums of whole numbers. For the two problems in Sample Set C, assume the use of a calculator that does not require the use of an ENTER key (such as many Hewlett-Packard calculators).  Sample Set C Use a calculator to find each sum. 34 + 21 Display Reads Type 34 34 Press + 34 Type 21 21 Press = 55 Solution The sum is 55. Sample Set C 106 + 85 + 322 + 406 Display Reads The calculator keeps a running Type 106 106 subtotal Press + 106 Type 85 85 Press = 191 ← 106 + 85 Type 322 322 Press + 513 ← 191 + 322 1.4.7 https://math.libretexts.org/@go/page/48777  Type 406 406 Press = 919 ← 513 + 406 Answer The sum is 919. Practice Set C Use a calculator to find the following sums. 62 + 81 + 12 Answer 155 Practice Set C 9,261 + 8,543 + 884 + 1,062 Answer 19,750 Practice Set C 10,221 + 9,016 + 11,445 Answer 30,682 Exercises For the following problems, perform the additions. If you can, check each sum with a calculator. Exercise 1.4.1 14 + 5 Answer 19 Exercise 1.4.2 12 + 7 Exercise 1.4.3 46 + 2 Answer 48 1.4.8 https://math.libretexts.org/@go/page/48777 Exercise 1.4.4 83 + 16 Exercise 1.4.5 77 + 21 Answer 98 Exercise 1.4.6 321 + 84 ––––– Exercise 1.4.7 916 + 62 ––––– Answer 978 Exercise 1.4.8 104 +561 ––––– Exercise 1.4.9 265 +103 ––––– Answer 368 Exercise 1.4.10 552 + 237 Exercise 1.4.11 8,521 + 4,256 Answer 12,777 1.4.9 https://math.libretexts.org/@go/page/48777 Exercise 1.4.12 16, 408 + 3, 101 ––––––––– Exercise 1.4.13 16, 515 +42, 223 ––––––––– Answer 58,738 Exercise 1.4.14 616,702 + 101,161 Exercise 1.4.15 43,156,219 + 2,013,520 Answer 45,169,739 Exercise 1.4.16 17 + 6 Exercise 1.4.17 25 + 8 Answer 33 Exercise 1.4.18 84 + 7 –––– Exercise 1.4.19 75 + 6 –––– Answer 81 1.4.10 https://math.libretexts.org/@go/page/48777 Exercise 1.4.20 36 + 48 Exercise 1.4.21 74 + 17 Answer 91 Exercise 1.4.22 486 + 58 Exercise 1.4.23 743 + 66 Answer 809 Exercise 1.4.24 381 + 88 Exercise 1.4.25 687 +175 ––––– Answer 862 Exercise 1.4.26 931 +853 ––––– Exercise 1.4.27 1,428 + 893 Answer 2,321 Exercise 1.4.28 12,898 + 11,925 1.4.11 https://math.libretexts.org/@go/page/48777 Exercise 1.4.29 631, 464 +509, 740 –––––––––– Answer 1,141,204 Exercise 1.4.30 805, 996 + 98, 516 –––––––––– Exercise 1.4.31 38, 428, 106 +522, 936, 005 –––––––––––––– Answer 561,364,111 Exercise 1.4.32 5,288,423,100 + 16,934,785,995 Exercise 1.4.33 98,876,678,521,402 + 843,425,685,685,658 Answer 942,302,364,207,060 Exercise 1.4.34 41 + 61 + 85 + 62 Exercise 1.4.35 21 + 85 + 104 + 9 + 15 Answer 234 Exercise 1.4.36 116 27 110 110 + 8 ––––– 1.4.12 https://math.libretexts.org/@go/page/48777  Exercise 1.4.37 75, 206 4, 152 +16, 007 ––––––––– Answer 95,365 Exercise 1.4.38 8, 226 143 92, 015 8 487, 553 + 5, 218 –––––––––– Exercise 1.4.39 50, 006 1, 005 100, 300 20, 008 1, 000, 009 + 800, 800 ––––––––––– Answer 1,972,128 Exercise 1.4.40 616 42, 018 1, 687 225 8, 623, 418 12, 506, 508 19 2, 121 195, 643 ––––––––––– For the following problems, perform the additions and round to the nearest hundred. Exercise 1.4.41 1, 468 2, 183 –––––– Answer 3,700 1.4.13 https://math.libretexts.org/@go/page/48777 Exercise 1.4.42 928, 725 15, 685 ––––––––– Exercise 1.4.43 82, 006 3, 019, 528 ––––––––––– Answer 3,101,500 Exercise 1.4.44 18, 621 5, 059 –––––––– Exercise 1.4.45 92 48 –––– Answer 100 Exercise 1.4.46 16 37 –––– Exercise 1.4.47 21 16 –––– Answer 0 Exercise 1.4.48 11, 171 22, 749 12, 248 –––––––– Exercise 1.4.49 240 280 210 310 ––––– 1.4.14 https://math.libretexts.org/@go/page/48777  Answer 1000 Exercise 1.4.50 9, 573 101, 279 122, 581 ––––––––– For the next five problems, replace the letter mm with the whole number that will make the addition true. Exercise 1.4.51 62 + m –––––– 67 Answer 5 Exercise 1.4.52 106 + m –––––– 113 Exercise 1.4.53 432 + m –––––– 451 Answer 19 Exercise 1.4.54 803 + m –––––– 830 Exercise 1.4.55 1, 893 + m ––––––– 1, 981 Answer 88 1.4.15 https://math.libretexts.org/@go/page/48777 Exercise 1.4.56 The number of nursing and related care facilities in the United States in 1971 was 22,004. In 1978, the number was 18,722. What was the total number of facilities for both 1971 and 1978? Exercise 1.4.57 The number of persons on food stamps in 1975, 1979, and 1980 was 19,179,000, 19,309,000, and 22,023,000, respectively. What was the total number of people on food stamps for the years 1975, 1979, and 1980? Answer 60,511,000 Exercise 1.4.58 The enrollment in public and nonpublic schools in the years 1965, 1970, 1975, and 1984 was 54,394,000, 59,899,000, 61,063,000, and 55,122,000, respectively. What was the total enrollment for those years? Exercise 1.4.59 The area of New England is 3,618,770 square miles. The area of the Mountain states is 863,563 square miles. The area of the South Atlantic is 278,926 square miles. The area of the Pacific states is 921,392 square miles. What is the total area of these regions? Answer 5,682,651 square miles Exercise 1.4.60 In 1960, the IRS received 1,188,000 corporate income tax returns. In 1965, 1,490,000 returns were received. In 1970, 1,747,000 returns were received. In 1972 —1977, 1,890,000; 1,981,000; 2,043,000; 2,100,000; 2,159,000; and 2,329,000 re‐ turns were received, respectively. What was the total number of corporate tax returns received by the IRS during the years 1960, 1965, 1970, 1972 —1977? Exercise 1.4.61 Find the total number of scientists employed in 1974. 1.4.16 https://math.libretexts.org/@go/page/48777  Answer 1,190,000 Exercise 1.4.62 Find the total number of sales for space vehicle systems for the years 1965-1980. Exercise 1.4.63 Find the total baseball attendance for the years 1960-1980. Answer 271,564,000 Exercise 1.4.64 Find the number of prosecutions of federal officials for 1970-1980. 1.4.17 https://math.libretexts.org/@go/page/48777 For the following problems, try to add the numbers mentally. Exercise 1.4.65 5 5 3 7 –– – Answer 20 Exercise 1.4.66 8 2 6 4 –– – Exercise 1.4.67 9 1 8 5 2 –– – Answer 25 Exercise 1.4.68 5 2 5 8 3 7 –– – 1.4.18 https://math.libretexts.org/@go/page/48777 Exercise 1.4.69 6 4 3 1 6 7 9 4 –– – Answer 40 Exercise 1.4.70 20 30 –––– Exercise 1.4.71 15 35 –––– Answer 50 Exercise 1.4.72 16 14 –––– Exercise 1.4.73 23 27 –––– Answer 50 Exercise 1.4.74 82 18 –––– Exercise 1.4.75 36 14 –––– Answer 1.4.19 https://math.libretexts.org/@go/page/48777  50 Exercises for Review (link) Exercise 1.4.76 Each period of numbers has its own name. From right to left, what is the name of the fourth period? Exercise 1.4.77 In the number 610,467, how many thousands are there? Answer 0 Exercise 1.4.78 Write 8,840 as you would read it. Exercise 1.4.79 Round 6,842 to the nearest hundred. Answer 6,800 Exercise 1.4.80 Round 431,046 to the nearest million. This page titled 1.4: Addition of Whole Numbers is shared under a CC BY license and was authored, remixed, and/or curated by Denny Burzynski & Wade Ellis, Jr. (OpenStax CNX) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request. 1.4.20 https://math.libretexts.org/@go/page/48777 1.5: Subtraction of Whole Numbers  Learning Objectives understand the subtraction process be able to subtract whole numbers be able to use a calculator to subtract one whole number from another whole number Subtraction  Definition: Subtraction Subtraction is the process of determining the remainder when part of the total is removed. Suppose the sum of two whole numbers is 11, and from 11 we remove 4. Using the number line to help our visualization, we see that if we are located at 11 and move 4 units to the left, and thus remove 4 units, we will be located at 7. Thus, 7 units remain when we remove 4 units from 11 units.  Definition: The Minus Symbol The minus symbol (-) is used to indicate subtraction. For example, 11 − 4 indicates that 4 is to be subtracted from 11.  Definition: Minuend The number immediately in front of or the minus symbol is called the minuend, and it represents the original number of units.  Definition: Subtrahend The number immediately following or below the minus symbol is called the subtrahend, and it represents the number of units to be removed.  Definition: Difference The result of the subtraction is called the difference of the two numbers. For example, in 11 − 4 = 7 Subtraction as the Opposite of Addition Subtraction can be thought of as the opposite of addition. We show this in the problems in Sample Set A.  Sample Set A 8 - 5 = 3 since 3 + 5 = 8.  Sample Set A 9 - 3 = 6 since 6 + 3 = 9. 1.5.1 https://math.libretexts.org/@go/page/48778  Practice Set A Complete the following statements. 7 - 5 = since + 5 = 7 Answer 7 - 5 = 2 since 2 + 5 = 7 Practice Set A 9 - 1 = since + 1 = 9 Answer 9 - 1 = 8 since 8 + 1 = 9 Practice Set A 17 - 8 = since + 8 = 17 Answer 17 - 8 = 9 since 9 + 8 = 17 The Subtraction Process We'll study the process of the subtraction of two whole numbers by considering the difference between 48 and 35. which we write as 13.  The Process of Subtracting Whole Numbers To subtract two whole numbers, The process 1. Write the numbers vertically, placing corresponding positions in the same column. 48 −35 –––– 2. Subtract the digits in each column. Start at the right, in the ones position, and move to the left, placing the difference at the bottom. 48 −35 –––– 13  Sample Set B Perform the following subtractions. 275 −142 ––––– 133 1.5.2 https://math.libretexts.org/@go/page/48778  5 − 2 = 3. 7 − 4 = 3. 2 − 1 = 1.  Sample Set B 46, 042 − 1, 031 ––––––––– 45, 011 2 − 1 = 1. 4 − 3 = 1. 0 − 0 = 0. 6 − 1 = 5. 4 − 0 = 4.  Sample Set B Find the difference between 977 and 235. Write the numbers vertically, placing the larger number on top. Line up the columns properly. 977 −235 ––––– 742 The difference between 977 and 235 is 742.  Sample Set B In Keys County in 1987, there were 809 cable television installations. In Flags County in 1987, there were 1,159 cable television installations. How many more cable television installations were there in Flags County than in Keys County in 1987? We need to determine the difference between 1,159 and 809. There were 350 more cable television installations in Flags County than in Keys County in 1987. Practice Set B Perform the following subtractions. 534 −203 ––––– Answer 331 1.5.3 https://math.libretexts.org/@go/page/48778  Practice Set B 857 − 43 ––––– Answer 814 Practice Set B 95, 628 −34, 510 ––––––––– Answer 61,118 Practice Set B 11, 005 − 1, 005 ––––––––– Answer 10,000 Practice Set B Find the difference between 88,526 and 26,412. Answer 62,114 In each of these problems, each bottom digit is less than the corresponding top digit. This may not always be the case. We will examine the case where the bottom digit is greater than the corresponding top digit in the next section. Subtraction Involving Borrowing  Definition: Minuend and Subtrahend It often happens in the subtraction of two whole numbers that a digit in the minuend (top number) will be less than the digit in the same position in the subtrahend (bottom number). This happens when we subtract 27 from 84. 84 −27 –––– We do not have a name for 4 − 7. We need to rename 84 in order to continue. We'll do so as follows: Our new name for 84 is 7 tens + 14 ones. = 57 1.5.4 https://math.libretexts.org/@go/page/48778  Notice that we converted 8 tens to 7 tens + 1 ten, and then we converted the 1 ten to 10 ones. We then had 14 ones and were able to perform the subtraction.  Definition: Borrowing The process of borrowing (converting) is illustrated in the problems of Sample Set C.  Sample Set C 1. Borrow 1 ten from the 8 tens. This leaves 7 tens. 2. Convert the 1 ten to 10 ones. 3. Add 10 ones to 4 ones to get 14 ones.  Sample Set C 1. Borrow 1 hundred from the 6 hundreds. This leaves 5 hundreds. 2. Convert the 1 hundred to 10 tens. 3. Add 10 tens to 7 tens to get 17 tens. Practice Set C Perform the following subtractions. Show the expanded form for the first three problems. 53 −35 –––– Answer Practice Set C 76 −28 –––– Answer 1.5.5 https://math.libretexts.org/@go/page/48778  Practice Set C 872 −565 ––––– Answer Practice Set C 441 −356 ––––– Answer 85 Practice Set C 775 − 66 ––––– Answer 709 Practice Set C 5, 663 −2, 559 –––––––– Answer 3,104 Borrowing More Than Once Sometimes it is necessary to borrow more than once. This is shown in the problems in Sample Set D. 1.5.6 https://math.libretexts.org/@go/page/48778  Sample Set D Perform the Subtractions. Borrowing more than once if necessary 1. Borrow 1 ten from the 4 tens. This leaves 3 tens. 2. Convert the 1 ten to 10 ones. 3. Add 10 ones to 1 one to get 11 ones. We can now perform 11 − 8. 4. Borrow 1 hundred from the 6 hundreds. This leaves 5 hundreds. 5. Convert the 1 hundred to 10 tens. 6. Add 10 tens to 3 tens to get 13 tens. 7. Now 13 − 5 = 8. 8. 5 − 3 = 2.  Sample Set D 1. Borrow 1 ten from the 3 tens. This leaves 2 tens. 2. Convert the 1 ten to 10 ones. 3. Add 10 ones to 4 ones to get 14 ones. We can now perform 14 − 5. 4. Borrow 1 hundred from the 5 hundreds. This leaves 4 hundreds. 5. Convert the 1 hundred to 10 tens. 6. Add 10 tens to 2 tens to get 12 tens. We can now perform 12 − 8 = 4. 7. Finally, 4 − 0 = 4.  Sample Set D 71529 − 6952 –––––––– After borrowing, we have Practice Set D Perform the following subtractions. 526 −358 ––––– Answer 1.5.7 https://math.libretexts.org/@go/page/48778  168 Practice Set D 63, 419 − 7, 779 ––––––––– Answer 55,640 Practice Set D 4, 312 −3, 123 –––––––– Answer 1,189 Borrowing from Zero It often happens in a subtraction problem that we have to borrow from one or more zeros. This occurs in problems such as 503 − 37 ––––– and 5000 − 37 ––––––– We'll examine each case.  Borrowing from a single zero. 503 Consider the problem − 37 ––––– Since we do not have a name for 3 − 7, we must borrow from 0. Since there are no tens to borrow, we must borrow 1 hundred. One hundred = 10 tens. We can now borrow 1 ten from 10 tens (leaving 9 tens). One ten = 10 ones and 10 ones + 3 ones = 13 ones. Now we can suggest the following method for borrowing from a single zero. Borrowing from a Single Zero To borrow from a single zero, 1. Decrease the digit to the immediate left of zero by one. 2. Draw a line through the zero and make it a 10. 3. Proceed to subtract as usual. 1.5.8 https://math.libretexts.org/@go/page/48778  Sample Set E Perform this subtraction. 503 − 37 ––––– The number 503 contains a single zero 1. The number to the immediate left of 0 is 5. Decrease 5 by 1. 5-1=4 2. Draw a line through the zero and make it a 10. 3. Borrow from the 10 and proceed. 1 ten + 10 ones 10 ones + 3 ones = 13 ones Practice Set E Perform each subtraction. 906 − 18 ––––– Answer 888 Practice Set E 5102 − 559 ––––––– Answer 4,543 Practice Set E 9055 − 386 ––––––– Answer 8,669  Borrowing from a group of zeros 5000 Consider the problem − 37 ––––––– In this case, we have a group of zeros. 1.5.9 https://math.libretexts.org/@go/page/48778  Since we cannot borrow any tens or hundreds, we must borrow 1 thousand. One thousand = 10 hundreds. We can now borrow 1 hundred from 10 hundreds. One hundred = 10 tens. We can now borrow 1 ten from 10 tens. One ten = 10 ones. From observations made in this procedure we can suggest the following method for borrowing from a group of zeros. Borrowing from a Group of zeros To borrow from a group of zeros, 1. Decrease the digit to the immediate left of the group of zeros by one. 2. Draw a line through each zero in the group and make it a 9, except the rightmost zero, make it 10. 3. Proceed to subtract as usual.  Sample Set F Perform each subtraction. 40, 000 − 125 –––––––– Solution The number 40,000 contains a group of zeros. 1. The number to the immediate left of the group is 4. Decrease 4 by 1. 4-1=3 2. Make each 0, except the rightmost one, 9. Make the rightmost 0 a 10. 3. Subtract as usual.  Example 1.5.1 8, 000, 006 − 41, 107 ––––––––––– Solution The number 8,000,006 contains a group of zeros. 1. The number to the immediate left of the group is 8. Decrease 8 by 1. 8-1=7 1.5.10 https://math.libretexts.org/@go/page/48778  2. Make each zero, except the rightmost one, 9. Make the rightmost 0 a 10. 3. To perform the subtraction, we’ll need to borrow from the ten. 1 ten = 10 ones 10 ones + 6 ones = 16 ones Practice Set F Perform each subtraction. 21, 007 − 4, 873 ––––––––– Answer 16,134 Practice Set F 10, 004 − 5, 165 ––––––––– Answer 4,839 Practice Set F 16, 000, 000 − 201, 060 ––––––––––––– Answer 15,789,940 Calculators In practice, calculators are used to find the difference between two whole numbers.  Sample Set G Find the difference between 1006 and 284. Display Reads Type 1006 1006 Press −− 1006 Type 284 284 Press = 722 The difference between 1006 and 284 is 722. (What happens if you type 284 first and then 1006? We'll study such numbers in Chapter 10.) 1.5.11 https://math.libretexts.org/@go/page/48778  Practice Set G Use a calculator to find the difference between 7338 and 2809. Answer 4,529 Practice Set G Use a calculator to find the difference between 31,060,001 and 8,591,774. Answer 22,468,227 Exercises For the following problems, perform the subtractions. You may check each difference with a calculator. Exercise 1.5.1 15 − 8 –––– Answer 7 Exercise 1.5.2 19 − 8 –––– Exercise 1.5.3 11 − 5 –––– Answer 6 Exercise 1.5.4 14 − 6 –––– Exercise 1.5.5 12 − 9 –––– Answer 3 1.5.12 https://math.libretexts.org/@go/page/48778 Exercise 1.5.6 56 −12 –––– Exercise 1.5.7 74 −33 –––– Answer 41 Exercise 1.5.8 80 −61 –––– Exercise 1.5.9 350 −141 ––––– Answer 209 Exercise 1.5.10 800 −650 ––––– Exercise 1.5.11 35, 002 −14, 001 ––––––––– Answer 21,001 Exercise 1.5.12 5, 000, 566 −2, 441, 326 –––––––––––– Exercise 1.5.13 400, 605 −121, 352 –––––––––– Answer 279,253 1.5.13 https://math.libretexts.org/@go/page/48778 Exercise 1.5.14 46, 400 − 2, 012 ––––––––– Exercise 1.5.15 77, 893 − 421 –––––––– Answer 77,472 Exercise 1.5.16 42 −18 –––– Exercise 1.5.17 51 −27 –––– Answer 24 Exercise 1.5.18 622 − 88 ––––– Exercise 1.5.19 261 − 73 ––––– Answer 188 Exercise 1.5.20 242 −158 ––––– Exercise 1.5.21 3, 422 −1, 045 –––––––– Answer 2,377 1.5.14 https://math.libretexts.org/@go/page/48778 Exercise 1.5.22 5, 565 −3, 985 –––––––– Exercise 1.5.23 42, 041 −15, 355 ––––––––– Answer 26,686 Exercise 1.5.24 304, 056 − 20, 008 –––––––––– Exercise 1.5.25 64, 000, 002 − 856, 743 ––––––––––––– Answer 63,143,259 Exercise 1.5.26 4, 109 − 856 ––––––– Exercise 1.5.27 10, 113 − 2, 079 ––––––––– Answer 8,034 Exercise 1.5.28 605 − 77 ––––– Exercise 1.5.27 59 −26 –––– Answer 33 1.5.15 https://math.libretexts.org/@go/page/48778  Exercise 1.5.28 36, 107 − 8, 314 ––––––––– Exercise 1.5.29 92, 526, 441, 820 −59, 914, 805, 253 ––––––––––––––––– Answer 32,611,636,567 Exercise 1.5.30 1, 605 − 881 ––––––– Exercise 1.5.31 30, 000 −26, 062 ––––––––– Answer 3,938 Exercise 1.5.32 600 −216 ––––– Exercise 1.5.33 90, 000, 003 − 726, 048 ––––––––––– Answer 8,273,955 For the following problems, perform each subtraction. Exercise 1.5.34 Subtract 63 from 92.  Hint: The word "from" means "beginning at." Thus, 63 from 92 means beginning at 92, or 92 - 63. 1.5.16 https://math.libretexts.org/@go/page/48778 Exercise 1.5.35 Subtract 35 from 86. Answer 51 Exercise 1.5.34 Subtract 382 from 541. Exercise 1.5.35 Subtract 1,841 from 5,246. Answer 3,405 Exercise 1.5.36 Subtract 26,082 from 35,040. Exercise 1.5.37 Find the difference between 47 and 21. Answer 26 Exercise 1.5.38 Find the difference between 1,005 and 314. Exercise 1.5.39 Find the difference between 72,085 and 16. Answer 72,069 Exercise 1.5.40 Find the difference between 7,214 and 2,049. Exercise 1.5.41 Find the difference between 56,108 and 52,911. Answer 3,197 1.5.17 https://math.libretexts.org/@go/page/48778 Exercise 1.5.42 How much bigger is 92 than 47? Exercise 1.5.43 How much bigger is 114 than 85? Answer 29 Exercise 1.5.44 How much bigger is 3,006 than 1,918? Exercise 1.5.45 How much bigger is 11,201 than 816? Answer 10,385 Exercise 1.5.46 How much bigger is 3,080,020 than 1,814,161? Exercise 1.5.47 In Wichita, Kansas, the sun shines about 74% of the time in July and about 59% of the time in November. How much more of the time (in percent) does the sun shine in July than in November? Answer 15% Exercise 1.5.48 The lowest temperature on record in Concord, New Hampshire in May is 21°F, and in July it is 35°F. What is the difference in these lowest temperatures? Exercise 1.5.49 In 1980, there were 83,000 people arrested for prostitution and commercialized vice and 11,330,000 people arrested for driving while intoxicated. How many more people were arrested for drunk driving than for prostitution? Answer 11,247,000 Exercise 1.5.50 In 1980, a person with a bachelor's degree in accounting received a monthly salary offer of $1,293, and a person with a marketing degree a monthly salary offer of $1,145. How much more was offered to the person with an accounting degree than the person with a marketing degree? 1.5.18 https://math.libretexts.org/@go/page/48778  Exercise 1.5.51 In 1970, there were about 793 people per square mile living in Puerto Rico, and 357 people per square mile living in Guam. How many more people per square mile were there in Puerto Rico than Guam? Answer 436 Exercise 1.5.52 The 1980 population of Singapore was 2,414,000 and the 1980 population of Sri Lanka was 14,850,000. How many more people lived in Sri Lanka than in Singapore in 1980? Exercise 1.5.53 In 1977, there were 7,234,000 hospitals in the United States and 64,421,000 in Mainland China. How many more hospitals were there in Mainland China than in the United States in 1977? Answer 57,187,000 Exercise 1.5.54 In 1978, there were 3,095,000 telephones in use in Poland and 4,292,000 in Switzerland. How many more telephones were in use in Switzerland than in Poland in 1978? For the following problems, use the corresponding graphs to solve the problems. Exercise 1.5.55 How many more life scientists were there in 1974 than mathematicians? Answer 165,000 Exercise 1.5.56 How many more social, psychological, mathematical, and environmental scientists were there than life, physical, and computer scientists? 1.5.19 https://math.libretexts.org/@go/page/48778 Exercise 1.5.57 How many more prosecutions were there in 1978 than in 1974? Answer 74 Exercise 1.5.58 How many more prosecutions were there in 1976-1980 than in 1970-1975? Exercise 1.5.59 How many more dry holes were drilled in 1960 than in 1975? Answer 4,547 Exercise 1.5.60 How many more dry holes were drilled in 1960, 1965, and 1970 than in 1975, 1978 and 1979? 1.5.20 https://math.libretexts.org/@go/page/48778 For the following problems, replace the ☐ with the whole number that will make the subtraction true. Exercise 1.5.61 14 − ☐ ––– 3 Answer 11 Exercise 1.5.62 21 − ☐ ––– 14 Exercise 1.5.63 35 − ☐ ––– 25 Answer 10 Exercise 1.5.64 16 − ☐ ––– 9 Exercise 1.5.65 28 − ☐ ––– 16 1.5.21 https://math.libretexts.org/@go/page/48778  Answer 12 For the following problems, find the solutions. Exercise 1.5.66 Subtract 42 from the sum of 16 and 56. Exercise 1.5.67 Subtract 105 from the sum of 92 and 89. Answer 76 Exercise 1.5.68 Subtract 1,127 from the sum of 2,161 and 387. Exercise 1.5.69 Subtract 37 from the difference between 263 and 175. Answer 51 Exercise 1.5.70 Subtract 1,109 from the difference between 3,046 and 920. Exercise 1.5.71 Add the difference between 63 and 47 to the difference between 55 and 11. Answer 60 Exercise 1.5.72 Add the difference between 815 and 298 to the difference between 2,204 and 1,016. Exercise 1.5.73 Subtract the difference between 78 and 43 from the sum of 111 and 89. Answer 165 1.5.22 https://math.libretexts.org/@go/page/48778  Exercise 1.5.74 Subtract the difference between 18 and 7 from the sum of the differences between 42 and 13, and 81 and 16. Exercise 1.5.75 Find the difference between the differences of 343 and 96, and 521 and 488. Answer 214 Exercises for Review Exercise 1.5.76 In the number 21,206, how many hundreds are there? Exercise 1.5.77 Write a three-digit number that has a zero in the ones position. Answer 330 (answers may vary) Exercise 1.5.78 How many three-digit whole numbers are there? Exercise 1.5.79 Round 26,524,016 to the nearest million. Answer 27,000,000 Exercise 1.5.80 Find the sum of 846 + 221 + 116. This page titled 1.5: Subtraction of Whole Numbers is shared under a CC BY license and was authored, remixed, and/or curated by Denny Burzynski & Wade Ellis, Jr. (OpenStax CNX) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit histo