1. Preface 2. Acknowledgements 3. Addition and Subtraction of Whole Numbers 1. Objectives . Whole Numbers . Reading and Writing Whole Numbers . Rounding Whole Numbers . Addition of Whole Numbers . Subtraction of Whole Numbers . Properties of Addition . Summary of Key Concepts . Exercise Supplement 10. Proficiency Exam 4. Multiplication and Division of Whole Numbers 1. Objectives 2. Multiplication of Whole Numbers 3. Concepts of Division of Whole Numbers 4. Division of Whole Numbers 5. Some Interesting Facts about Division 6 7. 8
WON DU BW WN
. Properties of Multiplication . Summary of Key Concepts . Exercise Supplement 9. Proficiency Exam 5. Exponents, Roots, and Factorization of Whole Numbers 1. Objectives . Exponents and Roots . Prime Factorization of Natural Numbers . The Greatest Common Factor . The Least Common Multiple . Summary of Key Concepts
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8. Exercise Supplement 9. Proficiency Exam 6. Introduction to Fractions and Multiplication and Division of Fractions 1. Objectives . Fractions of Whole Numbers . Equivalent Fractions, Reducing Fractions to Lowest Terms, and Raising Fractions to Higher Terms 5. Multiplication of Fractions 6. Division of Fractions 7. Applications Involving Fractions 8. Summary of Key Concepts 9. Exercise Supplement 10. Proficiency Exam 7. Addition and Subtraction of Fractions, Comparing Fractions, and Complex Fractions 1. Objectives 2. Addition and Subtraction of Fractions with Like Denominators 3. Addition and Subtraction of Fractions with Unlike Denominators . Addition and Subtraction of Mixed Numbers . Comparing Fractions . Complex Fractions . Combinations of Operations with Fractions . Summary of Key Concepts . Exercise Supplement 10. Proficiency Exam 8. Decimals 1. Objectives 2. Reading and Writing Decimals
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. Converting a Decimal to a Fraction
. Rounding Decimals
. Addition and Subtraction of Decimals
. Multiplication of Decimals
. Division of Decimals
. Nonterminating Divisions
. Converting a Fraction to a Decimal
. Combinations of Operations with Decimals and Fractions . Summary of Key Concepts
. Exercise Supplement
13.
Proficiency Exam
9. Ratios and Rates
. Objectives
. Ratios and Rates
. Proportions
. Applications of Proportions . Percent
. Fractions of One Percent
. Applications of Percents
. Summary of Key Concepts . Exercise Supplement
10.
Proficiency Exam
10. Techniques of Estimation
1. . Estimation by Rounding
. Estimation by Clustering
. Mental Arithmetic-Using the Distributive Property . Estimation by Rounding Fractions
. Summary of Key Concepts
. Exercise Supplement
8.
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Objectives
Proficiency Exam
11. Measurement and Geometry
CON DU BWN FE
. Objectives
. Measurement and the United States System
. The Metric System of Measurement
. Simplification of Denominate Numbers
. Perimeter and Circumference of Geometric Figures . Area and Volume of Geometric Figures and Objects . Summary of Key Concepts
. Exercise Supplement
9.
Proficiency Exam
12. Signed Numbers
., . Variables, Constants, and Real Numbers
. Signed Numbers
. Absolute Value
. Addition of Signed Numbers
. Subtraction of Signed Numbers
. Multiplication and Division of Signed Numbers . Summary of Key Concepts
. Exercise Supplement
10.
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Objectives
Proficiency Exam
13. Algebraic Expressions and Equations
. Objectives
. Algebraic Expressions
. Combining Like Terms Using Addition and Subtraction . Solving Equations of the Form x+a=b and x-a=b
. Solving Equations of the Form ax=b and x/a=b
. Applications I: Translating Words to Mathematical
Symbols
. Applications Il: Solving Problems . Summary of Key Concepts
. Exercise Supplement
. Proficiency Exam
Preface This module contains the preface for Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr.
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:
e Section Overview
¢ Sample Sets
e Practice Sets
e Section Exercises
e Exercises for Review
e 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.
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 8a when they start learning algebra. The chapter also includes the commutative and associative properties of multiplication.
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.
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]).
Acknowledgements This module contains the authors' acknowledgments and dedication of the book, Fundamentals of Mathematics by Denny Burzynski and Wade Ellis.
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.
e Barbara Conway, Berkshire Community College
e Bill Hajdukiewicz, Miami-Dade Community College e Virginia Hamilton, Shawnee State University
e David Hares, El Centro College
e Norman Lee, Ball State University
e Ginger Y. Manchester, Hinds Junior College
e John R. Martin, Tarrant County Junior College
e Shelba Mormon, Northlake College
e Lou Ann Pate, Pima Community College
e Gus Pekara, Oklahoma City Community College
e David Price, Tarrant County Junior College
¢ David Schultz, Virginia Western Community College e Sue S. Watkins, Lorain County Community College e Elizabeth M. Wayt, Tennessee State University
e 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.
Objectives
This module contains the learning objectives for the chapter "Addition and Subtraction of Whole Numbers" from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, jr.
After completing this chapter, you should Whole Numbers ({link])
e know the difference between numbers and numerals
e know why our number system is called the Hindu-Arabic numeration system
e understand the base ten positional number system
e be able to identify and graph whole numbers
Reading and Writing Whole Numbers ([{link]) e be able to read and write a whole number Rounding Whole Numbers ((link])
¢ understand that rounding is a method of approximation e be able to round a whole number to a specified position
Addition of Whole Numbers ({link])
e understand the addition process e be able to add whole numbers e be able to use the calculator to add one whole number to another
Subtraction of Whole Numbers ({link])
e understand the subtraction process
e be able to subtract whole numbers
e be able to use a calculator to subtract one whole number from another whole number
Properties of Addition ({link])
e understand the commutative and associative properties of addition
¢ understand why 0 is the additive identity
Whole Numbers
This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses many of aspects of whole numbers, including the Hindu-Arabic numeration system, the base ten positional number system, and the graphing of whole numbers. By the end of this module students should be able to: 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, and identify and graph whole numbers.
Section Overview
e Numbers and Numerals
e The Hindu-Arabic Numeration System
e The Base Ten Positional Number System ¢ Whole Numbers
¢ Graphing Whole Numbers
Numbers and Numerals
We begin our study of introductory mathematics by examining its most basic building block, the number.
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.
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.
e Hindu-Arabic numerals 4, 123, 1005
e Roman numerals IV, CX XIII, MV
e Egyptian numerals
li dd> ONNI11> biti
Strokes Coiled rope, Lotus flower heel bones, and strokes and strokes
Practice Set A Exercise:
Problem:
Wo
Do the phrases "four," "one hundred twenty-three," and "one thousand five" qualify as numerals? Yes or no?
Solution:
Yes. Letters are symbols. Taken as a collection (a written word), they represent a number.
The Hindu-Arabic Numeration System 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
Leonardo 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
Digits
The Hindu-Arabic numerals 01234567 89 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."
Base Ten Positional Systems It is for this reason that our number system is called a positional number system with base ten.
Commas When numbers are composed of more than three digits, commas are sometimes used to separate the digits into groups of three.
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.
ae | eee | ee ee) ee | ee ee eee ee Oe | Trillions Billions Millions Thousands Units
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.)
In our positional number system, the value of a digit is determined by its position in the number.
Sample Set B
Example:
Find the value of 6 in the number 7,261.
Since 6 is in the tens position of the units period, its value is 6 tens. 6 tens = 60
Example:
Find the value of 9 in the number 86,932,106,005.
Since 9 is in the hundreds position of the millions period, its value is 9 hundred millions.
9 hundred millions = 9 hundred million
Example:
Find the value of 2 in the number 102,001.
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 Exercise: Problem: Find the value of 5 in the number 65,000.
Solution:
five thousand
Exercise:
Problem: Find the value of 4 in the number 439,997,007,010.
Solution:
four hundred billion
Exercise:
Problem: Find the value of 0 in the number 108. Solution:
zero tens, Or Zero
Whole Numbers
Whole Numbers
Numbers that are formed using only the digits 0123456789
are called whole numbers. They are
On 4, 25354) 5,G,-750 79, 10.11. 120138. 14. Toya
The three dots at the end mean "and so on in this same pattern."
Graphing Whole Numbers
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.
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.
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
Example: Graph the following whole numbers: 3, 5, 9.
Example:
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.
0 106 107 872 873 874
The numbers that have been graphed are 0, 106, 873, 874
Practice Set C
Exercise:
Problem: Graph the following whole numbers: 46, 47, 48, 325, 327.
“2 “"/ ps —-
Solution: 0 46 47 48 325 326 327 Exercise: Problem:
Specify the whole numbers that are graphed on the following number line.
0 123 4 5 6 112 113 978 979 Solution: 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:
Problem: What is a number? Solution: concept
Exercise:
Problem: What is a numeral?
Exercise:
Problem: Does the word "eleven" qualify as a numeral?
Solution:
Yes, since it is a symbol that represents a number.
Exercise:
Problem: How many different digits are there? Exercise: Problem:
Our number system, the Hindu-Arabic number system, is a number system with base .
Solution:
positional; 10 Exercise: Problem:
Numbers composed of more than three digits are sometimes separated into groups of three by commas. These groups of three are called .
Exercise:
Problem:
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?
Solution:
ones, tens, hundreds Exercise:
Problem:
Each period has its own particular name. From right to left, what are the names of the first four?
Exercise:
Problem: In the number 841, how many tens are there?
Solution: 4
Exercise:
Problem: In the number 3,392, how many ones are there?
Exercise:
Problem: In the number 10,046, how many thousands are there?
Solution:
0
Exercise:
Problem:
In the number 779,844,205, how many ten millions are there? Exercise:
Problem: In the number 65,021, how many hundred thousands are there? Solution:
0
For following problems, give the value of the indicated digit in the given number. Exercise:
Problem: 5 in 599
Exercise:
Problem: 1 in 310,406
Solution:
ten thousand
Exercise:
Problem: 9 in 29,827
Exercise:
Problem: 6 in 52,561,001,100 Solution:
6 ten millions = 60 million
Exercise:
Problem
Write a two-digit number that has an eight in the tens position.
Exercise:
Problem
Write a four-digit number that has a one in the thousands position and
a zero in
the ones position.
Solution:
1,340 (answers may vary)
Exercise:
Problem
Exercise:
Problem
: How many two-digit whole numbers are there?
: How many three-digit whole numbers are there?
Solution:
900
Exercise:
Problem
Exercise:
Problem
: How many four-digit whole numbers are there?
: Is there a smallest whole number? If so, what is it?
Solution:
yes; Zero
Exercise:
Problem: Is there a largest whole number? If so, what is it?
Exercise:
Problem: Another term for "visually displaying" is .
Solution:
graphing
Exercise:
Problem: The whole numbers can be visually displayed ona. Exercise:
Problem:
Graph (visually display) the following whole numbers on the number line below: 0, 1, 31, 34.
0 12 8 4 #‘\%Y 29 30 31 32 33 34
Solution:
0 1 2 83 4 2 30 31 32 33 34
Exercise:
Problem:
Construct a number line in the space provided below and graph
(visually display) the following whole numbers: 84, 85, 901, 1006, 1007.
Exercise:
Problem:
Specify, if any, the whole numbers that are graphed on the following number line.
tJ tt MN ot
0 61 62 63 64 99 100 101 102
Solution:
61, 99, 100, 102 Exercise: Problem:
Specify, if any, the whole numbers that are graphed on the following number line.
+ —+-A\— ++ + + “| -— + 4+ A +
01 8 9 10 11 73 74 85 86 87
Reading and Writing Whole Numbers
This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses how to read and write whole numbers. By the end of this module, students should be able to read and write whole numbers.
Section Overview
e Reading Whole Numbers e Writing Whole Numbers
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.
Example: Read 42958.
1. Beginning at the right, we can separate this number into distinct periods by inserting a comma between the 2 and 9. 42,958
2. Beginning at the left, we read each period individually:
Lowa alt es , —— Forty-two thousand
Le cceaclnciceninsimeniianl Thousands period
[a 5) 8, —— nine hundred fifty-eight
Forty-two thousand, nine hundred fifty-eight.
Example: Read 307991343.
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
2. Beginning at the left, we read each period individually.
| ‘aes L_jLojJt 1, ——» Three hundred seven million, eel
Millions period
bat ear hat » —— nine hundred ninety-one thousand, Li eenacnemainadianl Thousands period
3, el 3, —— three hundred forty-three Lf Units period
Three hundred seven million, nine hundred ninety-one thousand, three hundred forty-three.
Example: Read 36000000000001.
1. Beginning at the right, we can separate this number into distinct periods by placing commas. 36,000,000,001 2. Beginning at the left, we read each period individually.
a | 3 8, , —— Thirty-six trillion, LS Trillions period 0D: @ 6 aon L_jLUJL_J,—— zero billion, Le Billions period 0,0,,0 _ LJLLjLOJ,—- zero million, LS Millions period 0 ,0,,0 L-JL JL J,-—— zero thousand, —————: | Thousands period "al el | ek LE Units period
Thirty-six trillion, one.
Practice Set A
Write each number in words. Exercise:
Problem: 12,542
Solution: Twelve thousand, five hundred forty-two Exercise:
Problem: 101,074,003
Solution: One hundred one million, seventy-four thousand, three Exercise:
Problem: 1,000,008
Solution:
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.
3. Using commas to separate periods, combine the periods to form one number.
Sample Set B
Write each number using digits.
Example: Seven thousand, ninety-two. Using the comma as a period separator, we have
Seven thousand , —— 7, nearereete eae
ninety-two, ——> 092
7,092
Example: Fifty billion, one million, two hundred thousand, fourteen. Using the commas as period separators, we have
Fifty billion,, —— 50, one million ,—— 001,
two hundred thousand , —— 200,
fourteen, —— 014
50,001,200,014
Example: Ten million, five hundred twelve. The comma sets off the periods. We notice that there is no thousands
period. We'll have to insert this ourselves.
Ten million ,—— 10,
zero thousand , —— 000,
five hundred twelve, —— 512 ee
10,000,512
Practice Set B
Express each number using digits. Exercise:
Problem: One hundred three thousand, twenty-five.
Solution:
103,025
Exercise: Problem: Six million, forty thousand, seven.
Solution:
6,040,007
Exercise:
Problem:
Twenty trillion, three billion, eighty million, one hundred nine thousand, four hundred two.
Solution:
20,003,080, 109,402
Exercise:
Problem: Eighty billion, thirty-five. Solution:
80,000,000,035
Exercises
For the following problems, write all numbers in words. Exercise:
Problem: 912 Solution:
nine hundred twelve
Exercise:
Problem: 84
Exercise:
Problem: 1491
Solution:
one thousand, four hundred ninety-one
Exercise:
Problem: 8601
Exercise:
Problem: 35,223
Solution:
thirty-five thousand, two hundred twenty-three
Exercise:
Problem: 71,006
Exercise:
Problem: 437,105
Solution:
four hundred thirty-seven thousand, one hundred five
Exercise:
Problem: 201,040
Exercise:
Problem: 8,001,001 Solution:
eight million, one thousand, one
Exercise:
Problem: 16,000,053 Exercise: Problem: 770,311,101
Solution:
seven hundred seventy million, three hundred eleven thousand, one hundred one
Exercise:
Problem: 83,000,000,007
Exercise:
Problem: 106,100,001,010 Solution:
one hundred six billion, one hundred million, one thousand ten
Exercise:
Problem: 3,333,444,777 Exercise:
Problem: 800,000,800,000
Solution:
eight hundred billion, eight hundred thousand
Exercise:
Problem:
A particular community college has 12,471 students enrolled. Exercise:
Problem:
A person who watches 4 hours of television a day spends 1460 hours a
year watching T.V. Solution:
four; one thousand, four hundred sixty Exercise: Problem: Astronomers believe that the age of the earth is about 4,500,000,000 years. Exercise: Problem:
Astronomers believe that the age of the universe is about 20,000,000,000 years.
Solution:
twenty billion Exercise:
Problem:
There are 9690 ways to choose four objects from a collection of 20.
Exercise:
Problem:
If a 412 page book has about 52 sentences per page, it will contain about 21,424 sentences.
Solution: four hundred twelve; fifty-two; twenty-one thousand, four hundred twenty-four Exercise: Problem: In 1980, in the United States, there was $1,761,000,000,000 invested in life insurance. Exercise: Problem:
In 1979, there were 85,000 telephones in Alaska and 2,905,000 telephones in Indiana.
Solution: one thousand, nine hundred seventy-nine; eighty-five thousand; two million, nine hundred five thousand Exercise: Problem: In 1975, in the United States, it is estimated that 52,294,000 people drove to work alone. Exercise: Problem:
In 1980, there were 217 prisoners under death sentence that were divorced.
Solution:
one thousand, nine hundred eighty; two hundred seventeen Exercise: Problem: In 1979, the amount of money spent in the United States for regular- session college education was $50,721,000,000,000. Exercise: Problem:
In 1981, there were 1,956,000 students majoring in business in U.S. colleges.
Solution: one thousand, nine hundred eighty one; one million, nine hundred fifty-six thousand Exercise: Problem: In 1980, the average fee for initial and follow up visits to a medical doctors office was about $34. Exercise: Problem:
In 1980, there were approximately 13,100 smugglers of aliens apprehended by the Immigration border patrol.
Solution:
one thousand, nine hundred eighty; thirteen thousand, one hundred
Exercise:
Problem:
In 1980, the state of West Virginia pumped 2,000,000 barrels of crude oil, whereas Texas pumped 975,000,000 barrels.
Exercise:
Problem: The 1981 population of Uganda was 12,630,000 people.
Solution:
twelve million, six hundred thirty thousand Exercise:
Problem:
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:
Problem: Six hundred eighty-one
Solution:
681
Exercise:
Problem: Four hundred ninety
Exercise:
Problem: Seven thousand, two hundred one
Solution:
F201
Exercise:
Problem: Nineteen thousand, sixty-five
Exercise:
Problem: Five hundred twelve thousand, three
Solution:
512,003 Exercise:
Problem:
Two million, one hundred thirty-three thousand, eight hundred fifty- nine
Exercise: Problem: Thirty-five million, seven thousand, one hundred one
Solution:
35,007,101
Exercise:
Problem: One hundred million, one thousand
Exercise:
Problem: Sixteen billion, fifty-nine thousand, four
Solution:
16,000,059,004
Exercise:
Problem:
Nine hundred twenty billion, four hundred seventeen million, twenty- one thousand
Exercise: Problem: Twenty-three billion
Solution:
23,000,000,000 Exercise:
Problem:
Fifteen trillion, four billion, nineteen thousand, three hundred five
Exercise:
Problem: One hundred trillion, one
Solution:
100,000,000,000,001
Exercises for Review
Exercise:
Problem: ({link]) How many digits are there?
Exercise:
Problem: ({link]) In the number 6,641, how many tens are there?
Solution: 4
Exercise:
Problem: ({link]) What is the value of 7 in 44,763?
Exercise: Problem: ({link]) Is there a smallest whole number? If so, what is it?
Solution:
yes, Zero Exercise:
Problem:
({link]) Write a four-digit number with a 9 in the tens position.
Rounding Whole Numbers
This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses how to round whole numbers. By the end of the module students should be able to understand that rounding is a method of approximation and round a whole number to a specified position.
Section Overview
e Rounding as an Approximation e The Method of Rounding Numbers
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.
a Pi a x Pi Pi a a
x, eF &
The precise number of symbols in the above collection is 18.
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:
Round 67 to the nearest ten. 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.
6 — - tens
0 60 67 38670
67 is closer to 7 tens than it is to 6 tens.
Thus, 67, rounded to the nearest ten, is 70.
Example:
Round 4,329 to the nearest hundred.
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.
3 ices | ie hundreds 0 4,300 4,329 4,400 4,329 is closer to 43 hundreds than it is to 44 hundreds.
Thus, 4,329, rounded to the nearest hundred is 4,300.
Example: Round 16,500 to the nearest thousand. On the number line, 16,500 is exactly halfway from 16,000 to 17,000.
6 thousands 7% is 7 thousands
0 16,000 16,500 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:
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.
Example: Round 3,426 to the nearest ten.
1. We are rounding to the tens position. Mark the digit in the tens position
3,426 f
tens position
2. 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= 3.
3,430
Thus, 3,426 rounded to the nearest ten is 3,430.
Example: Round 9,614,018,007 to the nearest ten million.
1. We are rounding to the nearest ten million. 9,614,018,007 t ten millions position
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.
Example: Round 148,422 to the nearest million.
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.
0,148,422 millions position
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.
Example: Round 397,000 to the nearest ten thousand.
1. We are rounding to the nearest ten thousand. 397,000
ten thousand position
2. 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 = 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. Exercise:
Problem: Round 3387 to the nearest hundred.
Solution: 3400
Exercise:
Problem: Round 26,515 to the nearest thousand.
Solution: 27,000
Exercise:
Problem: Round 30,852,900 to the nearest million.
Solution: 31,000,000
Exercise:
Problem: Round 39 to the nearest hundred.
Solution:
0
Exercise:
Problem: Round 59,600 to the nearest thousand.
Solution:
60,000
Exercises For the following problems, complete the table by rounding each number to the indicated
positions. Exercise:
Problem: 1,642
hundred thousand ten thousand million Solution: hundred thousand ten thousand million 1,600 2000 0 0 Exercise:
Problem: 5,221
hundred thousand
Exercise:
Problem: 91,803
Hundred thousand Solution: Hundred thousand 91,800 92,000 Exercise:
Problem: 106,007
hundred thousand
Exercise:
ten thousand
ten thousand
ten thousand
90,000
ten thousand
million
million
million
0
million
Problem: 208
hundred thousand ten thousand Solution: hundred thousand ten thousand 200 0 0 Exercise:
Problem: 199
hundred thousand ten thousand
Exercise:
Problem: 863
million
million
0
million
hundred thousand
Solution: hundred thousand 900 1,000 Exercise:
Problem: 794
hundred thousand
Exercise:
Problem: 925
hundred thousand
Solution:
ten thousand
ten thousand
0
ten thousand
ten thousand
million
million
0
million
million
hundred
900
Exercise:
Problem: 909
hundred
Exercise:
Problem: 981
hundred
Solution:
hundred
1,000
Exercise:
thousand
1,000
thousand
thousand
thousand
1,000
ten thousand
0
ten thousand
ten thousand
ten thousand
0
million
0
million
million
million
0
Problem: 965
hundred thousand
Exercise:
Problem: 551,061,285
hundred thousand Solution: hundred thousand 551,061,300 551,061,000 Exercise:
Problem: 23,047,991,521
ten thousand
ten thousand
ten thousand
951,060,000
million
million
million
551,000,000
hundred thousand ten thousand
Exercise:
Problem: 106,999,413,206
Hundred thousand ten thousand Solution: hundred thousand ten thousand
106,999,413,200 106,999,413,000 106,999,410,000
Exercise:
Problem: 5,000,000
hundred thousand ten thousand
Exercise:
million
million
million
106,999,000,000
million
Problem: 8,006,001
hundred thousand ten thousand Solution: Hundred Thousand ten thousand 8,006,000 8,006,000 8,010,000 Exercise:
Problem: 94,312
hundred thousand ten thousand
Exercise:
Problem: 33,486
million
Million
8,000,000
million
hundred thousand
Solution: hundred thousand 33,500 33,000 Exercise:
Problem: 560,669
hundred thousand
Exercise:
Problem: 388,551
hundred thousand
Solution:
ten thousand
ten thousand
30,000
ten thousand
ten thousand
million
million
0
million
million
hundred
388,600
Exercise:
Problem: 4,752
hundred
Exercise:
Problem: 8,209
hundred
Solution:
hundred
8,200
Exercise:
thousand
389,000
thousand
thousand
thousand
8,000
ten thousand
390,000
ten thousand
ten thousand
ten thousand
10,000
million
0
million
million
million
0
Problem: In 1950, there were 5,796 cases of diphtheria reported in the United States. Round to the nearest hundred. Exercise: Problem:
In 1979, 19,309,000 people in the United States received federal food stamps. Round to the nearest ten thousand.
Solution:
19,310,000 Exercise: Problem: In 1980, there were 1,105,000 people between 30 and 34 years old enrolled in school. Round to the nearest million. Exercise: Problem:
In 1980, there were 29,100,000 reports of aggravated assaults in the United States. Round to the nearest million.
Solution:
29,000,000
For the following problems, round the numbers to the position you think is most reasonable for the situation. Exercise:
Problem: In 1980, for a city of one million or more, the average annual salary of police and firefighters was $16,096. Exercise: Problem:
The average percentage of possible sunshine in San Francisco, California, in June is 73%.
Solution:
70% or 75% Exercise:
Problem:
In 1980, in the state of Connecticut, $3,777,000,000 in defense contract payroll was awarded.
Exercise:
Problem:
In 1980, the federal government paid $5,463,000,000 to Viet Nam veterans and dependants.
Solution: $5,500,000,000
Exercise:
Problem: In 1980, there were 3,377,000 salespeople employed in the United States. Exercise:
Problem:
In 1948, in New Hampshire, 231,000 popular votes were cast for the president. Solution:
230,000
Exercise:
Problem: In 1970, the world production of cigarettes was 2,688,000,000,000. Exercise:
Problem: In 1979, the total number of motor vehicle registrations in Florida was 5,395,000. Solution:
5,400,000
Exercise:
Problem: In 1980, there were 1,302,000 registered nurses the United States.
Exercises for Review
Exercise:
Problem:
([link]) There is a term that describes the visual displaying of a number. What is the term?
Solution: graphing
Exercise:
Problem: ((link]) What is the value of 5 in 26,518,206?
Exercise:
Problem: ((link]) Write 42,109 as you would read it. Solution:
Forty-two thousand, one hundred nine
Exercise:
Problem: ([link]) Write "six hundred twelve" using digits.
Exercise:
Problem: ([link]) Write "four billion eight" using digits. Solution:
4,000,000,008
Addition of Whole Numbers
This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses how to add whole numbers. By the end of this module, students should be able to understand the addition process, add whole numbers, and use the calculator to add one whole number to another.
Section Overview
e Addition
e Addition Visualized on the Number Line e The Addition Process
e Addition Involving Carrying
e Calculators
Addition
Suppose we have two collections of objects that we combine together to form a third collection. For example,
: : ' aE P BEES is combined with = to yield Tr
a @ a G We are combining a collection of four objects with a collection of three objects to obtain a collection of seven objects.
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.
25 443 means
2 tens + 5 ones +4 tens + 3 ones 6 tens + 8 ones
We write this as 68.
We can suggest the following procedure for adding whole numbers using this example.
Example:
The Process of Adding Whole Numbers To add whole numbers,
The process:
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
Note: 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
Example: Add 276 and 103. 276 64+3=49.
uate Oke 379 2+1=3.
Example:
Add 1459 and 130 1459 d ay ‘ > ‘
+130 | ae 1589 ex
jee (es ib
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.
Exercise:
Problem: Add 63 and 25.
Solution:
88
6 tens + 3 ones
+2 tens + 5 ones
8 tens + 8 ones
Exercise:
Problem: Add 4,026 and 1,501.
Solution:
D.o27
4 thousands + 0 hundreds + 2 tens + 6 ones +1 thousand +5 hundreds + 0 tens + 1 one 5 thousands + 5 hundreds + 2 tens + 7 ones
Exercise:
Problem: Add 231,045 and 36,121. Solution:
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.
a This sum exceeds 9.
18=1ten + 8ones 12 ones +34=3tens+ 4 ones a , 4 tens + 12 ones = 4 tens + 1 ten + 2 ones \ ee tte
= 5 tens + 2 ones = 52
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.
Example: Add 1875 and 358.
111 1875 + 358 2233
oo Write 3, carry 1 ten. 14+7+5=13 °&#£2Write 3, carry 1 hundred. 14+8+3=12 #£zWrite 2, carry 1 thousand. ie al eae
The sum is 2233.
Example: Add 89,208 and 4,946.
11 1 89,208 + 4,946 94,154
8+-6=14 Write 4, carry 1 ten. 14+0+4=5 Write the 5 (nothing to carry).
2+9=11 Write 1, carry one thousand. 14+9+4=14 °#£2Write 4, carry one ten thousand. i 3= 9
The sum is 94,154.
Example: Add 38 and 95.
11 38 + 95 133
chap is) Write 3, carry 1 ten.
14+3+9=13 °&#®4Write 3, carry 1 hundred.
eel
As you proceed with the addition, it is a good idea to keep in mind what is actually happening.
38 means 3 tens + 8 ones +96 + 9tens + 5 ones —_ 12 tens +13 ones
= 12 tens +1 ten+ 3 ones = 13 tens + 3 ones = 1 hundred+ 3 tena + 3 ones = 133
The sum is 133.
Example: Find the sum 2648, 1359, and 861.
111 2648 1359 + 861 4868
Ore) lao Write 8, carry 1 ten. 14+4+5+6=16 }#£2Write 6, carry 1 hundred. 14+6+3+8=18 }#£2Write 8, carry 1 thousand.
1+2+4+1=4 The sum is 4,868.
Numbers other than 1 can be carried as illustrated in [link].
Example: Find the sum of the following numbers.
132 1 878016 9905 38951 + 56817 983689
OO le et Write 9, carry the 1. Boe Ors ool Write 8.
ee) ola ae Write 6, carry the 2. 2+8+9+8+6=33 #£Write 3, carry the 3. Soop hae ea By lis) Write 8, carry the 1. fois Write 9.
The sum is 983,689.
Example:
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?
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.
1 ll 9,289 10,108 +11,412 30,809
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. Exercise:
Problem: Add 58 and 29. Solution:
87
5 tens + 8 ones +2tens+ 9 ones 7 tens + 17 ones
— Ttens + 1ten + 7ones = 8tens + 7ones ae oi
Exercise:
Problem: Add 476 and 85. Solution: 561
4 hundreds + 7tens+ 6 ones + 8tens+ 5 ones 4 hundreds + 15 tens + 11 ones
= 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
Exercise:
Problem: Add 27 and 88.
Solution: 115
2tens+ 7 ones + 8tens+ 8 ones 10 tens + 15 ones
= 10 tens + 1 ten + 5 ones
= 11 tens + 5 ones
= 1 hundred + 1 ten + 5 ones = TED
Exercise:
Problem: Add 67,898 and 85,627. Solution:
153,020
For the next three problems, find the sums. Exercise:
57 Problem: 26 84 Solution: 167
Exercise:
847 Problem: 825
796 Solution: 2,468 Exercise: 16,945 8,472 Problem: 387,721 21,059 629 Solution: 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.
Example:
34 + 21 Type Press Type
Press
The sum is 55.
Example:
34
106 + 85 4
Type
Press
Type
322 + 406
106
85
Display Reads 34 34 21
eye)
Display Reads
The calculator keeps a
Pee running subtotal
106
85
Press = 191 - 106+ 85
Type B22 B22
Press 45 513 -— 191 + 322
Type 406 406
Press = 919 - 513 + 406 The sum is 919.
Practice Set C
Use a calculator to find the following sums. Exercise:
Problem: 62 + 81+ 12 Solution:
155
Exercise:
Problem: 9,261 + 8,543 + 884 + 1,062
Solution:
19,750
Exercise:
Problem: 10,221 + 9,016 + 11,445
Solution:
30,682
Exercises For the following problems, perform the additions. If you can, check each
sum with a calculator. Exercise:
Problem:14 + 5
Solution:
19
Exercise:
Problem: 12 + 7
Exercise:
Problem: 46 + 2
Solution:
48
Exercise:
Problem: 83 + 16
Exercise:
Problem: 77 + 21
Solution:
98
Exercise:
Problem
Exercise:
321 “SAD
916
Problem:
“62
Solution:
978
Exercise:
Problem
Exercise:
Problem
104 "4561
265 "4103
Solution:
368
Exercise:
Problem
Exercise:
Problem
> 502 + 237
: 8,521 + 4,256
Solution:
12777 Exercise: oe 16,408 roblem: + 3,101 Exercise: pebi 16,515 roblem: 442,223 Solution: 58,738 Exercise:
Problem: 616,702 + 101,161
Exercise:
Problem: 43,156,219 + 2,013,520 Solution:
45,169,739
Exercise:
Problem: 17 + 6
Exercise:
Problem: 25 + 8
Solution
33
Exercise:
Problem
Exercise:
Problem
84 ae ak
+ 6
Solution:
81
Exercise:
Problem
Exercise:
Problem
: 36+ 48
:74+17
Solution:
a1
Exercise:
Problem
Exercise:
Problem
: 486 + 58
: 743 + 66
Solution:
809
Exercise:
Problem
Exercise:
Problem
: 381 + 88
687 p75
Solution:
862
Exercise:
Problem
Exercise:
Problem
931 "4853
: 1,428 + 893
Solution:
2,321
Exercise:
Problem
Exercise:
: 12,898 + 11,925
631,464 Problem:
+509,740 Solution: 1,141,204 Exercise: Beeb 805,996 ro om) 3 98,516 Exercise: 38,428,106 Problem:
+522,936,005
Solution:
961,364,111
Exercise:
Problem: 5,288,423,100 + 16,934,785 ,995
Exercise:
Problem: 98,876,678,521,402 + 843,425,685,685,658
Solution:
942 ,302,364,207,060
Exercise:
Problem: 41 + 61+ 85 + 62
Exercise:
Problem: 21 + 85+ 104+9-+415
Solution: 234 Exercise: 116 27 Problem: 110 110 + 8 Exercise: 75,206 Problem: 4,152 +16,007 Solution: 95,365 Exercise: 8,226 143 92,015 Problem: 8 487,553
5,218
Exercise:
50,006 1,005 100,300 20,008 1,000,009 800,800
Problem:
Solution:
1,972,128
Exercise:
616
42,018
1,687
225
Problem: 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:
problem: 1488 robiem: 2,183
Solution:
3,700
Exercise:
Problem:
Exercise:
Problem:
Solution:
3,101,500
Exercise:
Problem:
Exercise:
Problem:
Solution:
100
Exercise:
Problem:
928,725 15,685
82,006 3,019,528
18,621 5,059
92 48
16 37
Exercise:
Pal Problem: 16 Solution: 0 Exercise: 11172 Problem: 22,749 12,248 Exercise: 240 280 Problem: 210 310 Solution: 1,000 Exercise: 9,573 Problem: 101,279 122,581
For the next five problems, replace the letter m with the whole number that will make the addition true. Exercise:
Problem:
Solution:
is)
Exercise:
Problem:
Exercise:
Problem:
Solution:
19
Exercise:
Problem:
Exercise:
62 + om 67
1,893 Problem: ++ m 1,981
Solution:
88 Exercise: Problem: 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: Problem: 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?
Solution:
60,511,000 Exercise: Problem: 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:
Problem:
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?
Solution:
5,682,651 square miles
Exercise: Problem: 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 returns 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:
Problem: Find the total number of scientists employed in 1974.
EMPLOYMENT STATUS OF MATHEMATICAL SCIENTISTS — 1974
Solution:
1,190,000 Exercise:
Problem:
Find the total number of sales for space vehicle systems for the years 1965-1980.
SALES FOR SPACE VEHICLE SYSTEMS, 1965-1980
4 1,750,000,000}- ——— 3 1,400,000,000
i
1965 1970 1971 1972 1973 1974 1975 1976 Year
Exercise:
Problem: Find the total baseball attendance for the years 1960-1980.
BASEBALL ATTENDANCE 1960-1980
1965 1970 1975 1977
Year
Solution:
271,564,000 Exercise:
Problem: Find the number of prosecutions of federal officials for 1970-1980.
PROSECUTIONS OF FEDERAL OFFICIALS 1970-1980
Number of prosecutions g 8
1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 Year
For the following problems, try to add the numbers mentally. Exercise:
Problem:
N W Oot Ol
Solution:
20
Exercise:
Problem:
Exercise:
Problem:
Solution:
23
Exercise:
Problem:
Exercise:
= & bo CO
bo Oot CO FR ©
“Iw OC Ot ND Ol
Problem:
Solution:
40
Exercise:
Problem:
Exercise:
Problem:
Solution:
50
Exercise:
Problem:
Exercise:
Pont nrewo er OD
20 30
15 39
16 14
| Problem:
27 Solution: 50 Exercise: 82 Problem: 18 Exercise: 36 Problem: 14 Solution: 50
Exercises for Review
Exercise: Problem: ({link]) Each period of numbers has its own name. From right to left, what is the name of the fourth period?
Exercise:
Problem:
({link]) In the number 610,467, how many thousands are there?
Solution:
0
Exercise:
Problem
Exercise:
Problem
: (Llink]) Write 8,840 as you would read it.
: (Llink]) Round 6,842 to the nearest hundred.
Solution:
6,800
Exercise:
Problem
: (Llink]) Round 431,046 to the nearest million.
Subtraction of Whole Numbers
This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses how to subtract whole numbers. By the end of this module, students should be able to understand the subtraction process, subtract whole numbers, and use a calculator to subtract one whole number from another whole number.
Section Overview
e Subtraction
e Subtraction as the Opposite of Addition e The Subtraction Process
e Subtraction Involving Borrowing
e Borrowing From Zero
e Calculators
Subtraction
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.
The Minus Symbol The minus symbol (-) is used to indicate subtraction. For example, 11 — 4 indicates that 4 is to be subtracted from 11.
Minuend
The number immediately in front of or the minus symbol is called the minuend, and it represents the original number of units.
Subtrahend The number immediately following or below the minus symbol is called the subtrahend, and it represents the number of units to be removed.
Difference
The result of the subtraction is called the difference of the two numbers. For example, in 11 — 4 = 7, 11 is the minuend, 4 is the subtrahend, and 7 is the difference.
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
Example: 8—5=3sincee3+5=8.
Example: 9—3=6since6+3=9.
Practice Set A
Complete the following statements. Exercise:
Problem:7 — 5 = since +5 = 7.
Solution:
7—5=2since2+5=—7
Exercise:
Problem:9 — 1 = since +1 = 9.
Solution:
9—1= 8since8+1=9 Exercise:
Problem:17 — 8 = since +8 = 17.
Solution:
17 — 8= 9since 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.
48 means 4 tens + 8 ones — 35 —3 tens — 5 ones lten +3 ones
which we write as 13.
Example:
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 —30 13
Sample Set B
Perform the following subtractions.
Example: 215 —142 133 5-2=3. 7-—4=8. 2 esd,
Example:
46,042
— 1,031 45,011 2—1=1. 4—3=1. O= 0:0: 6 A= 5. 4—0=4., Example:
Find the difference between 977 and 235. Write the numbers vertically, placing the larger number on top. Line up the
columns properly. 977
—2395
5: The difference between 977 and 235 is 742.
Example:
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.
Exercise: Probl 534 m: Pawar caja Solution: 331 Exercise: eT 857 r m: oblem: — 13 Solution: 814 Exercise: —s 95,628 roblem: 34,510 Solution: 61,118
Exercise:
— 11,005 ro em: 1,005
Solution:
10,000
Exercise:
Problem: Find the difference between 88,526 and 26,412. Solution:
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
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:
84 = 8 tens + 4 ones —~ 27 = 2 tens + 7 ones
7 tens + 1 ten + 4 ones 2 tens + 7 ones
7 tens + 10 ones + 4 ones 2 tens + 7 ones
Our new name for 84 is 7 tens + 14 ones.
7 tens + 14 ones 2tens+ 7 ones 5 tens + 7 ones
=D1
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.
Borrowing The process of borrowing (converting) is illustrated in the problems of Sample Set C.
Sample Set C
Example:
714
i =21 57
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.
Example:
517
$72 — 91 581
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.
Exercise:
Problem: roblem _ 95
Solution:
18, 5 tens +3 ones — 3 tens + 5 ones
4 tens + 1 ten + 3 ones - 3 tens + 5 ones
4 tens + 13 ones — 8tens+ 5 ones
lten + 8ones =18
Exercise:
Problem: _98
Solution:
48, 7 tens + 6 ones 2 tens + 8 ones
6 tens + 1 ten + 6 ones — 2tens + 8 ones
6 tens + 16 ones — 2tens+ 8ones
4tens+ 8ones = 48
Exercise:
Probl isle roblem: _ o.
Solution:
307, 8 hundreds + 7 tens + 2 ones — 5 hundreds + 6 tens + 5 ones
8 hundreds + 6 tens + 1 ten + 2 ones - 5 hundreds + 6 tens + 5 ones 8 hundreds + 6 tens + 12 ones _ 5 hundreds + 6 tens + 5 ones 3 hundreds + 0 tens + 7 ones
= 307 Exercise: Probl 441 roblem: 356 Solution: 85 Exercise: Baa 775 m: roblem: _ 66 Solution: 709 Exercise: er 5,663 roblem: 2,559 Solution:
3,104
Borrowing More Than Once Sometimes it is necessary to borrow more than once. This is shown in the problems in [link].
Sample Set D
Perform the Subtractions. Borrowing more than once if necessary
Example: 513 $11 ¢41 — 358 283 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.
Example:
12
4914 B34 —__85 449 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.
Example: 71529
- 6952 After borrowing, we have
10 14 69412 71529 — 6952 64577
Practice Set D
Perform the following subtractions. Exercise:
peeuieni 526 roblem: 358
Solution: 168
Exercise:
63,419
Problem: _ 7,779
Solution: 55,640
Exercise:
4,312
Problem: 3,123
Solution:
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
i 503 '— 37
and
5000
We'll examine each case.
Example:
Borrowing from a single zero.
Consider the problem 37
Since we do not have a name for 3 — 7, we must borrow from 0.
503 = 5 hundreds + 0 tens + 3 ones — 37 3 tens + 7 ones
Since there are no tens to borrow, we must borrow 1 hundred. One hundred = 10 tens.
4 hundreds + 10 tens + 3 ones 3 tens + 7 ones
We can now borrow 1 ten from 10 tens (leaving 9 tens). One ten = 10 ones and 10 ones + 3 ones = 13 ones.
4 hundreds + 9 tens + 13 ones 38tens+ 7 ones 4 hundreds + 6 tens + 6 ones = 466
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.
Sample Set E
Example: Perform this subtraction. 503
=o 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
410
$93
— oF
2. Draw a line through the zero and make it a 10.
3. Borrow from the 10 and proceed.
9 41013
303 — $7 466
1 ten + 10 ones
10 ones + 3 ones = 13 ones
Practice Set E
Perform each subtraction. Exercise:
Probl se roblem: _ 18 Solution: 888
Exercise: poi 5102 roblem: _ 559 Solution: 4,543
Exercise: Bi 9055 roblem: _ 336 Solution: 8,669
Example:
Borrowing from a group of zeros 5000
Site In this case, we have a group of zeros.
Consider the problem
5000 = 5 thousands + 0 hundred + 0 tens + 0 ones — 37= 3 tens + 7 ones
Since we cannot borrow any tens or hundreds, we must borrow 1 thousand. One thousand = 10 hundreds.
4 thousands + 10 hundreds + 0 tens + 0 ones 3 tens + 7 ones
We can now borrow 1 hundred from 10 hundreds. One hundred = 10 tens.
4 thousands + 9 hundreds + 10 tens + 0 ones 3 tens + 7 ones
We can now borrow 1 ten from 10 tens. One ten = 10 ones.
4 thousands + 9 hundreds + 9 tens + 10 ones 3tens+ 7 ones 4 thousands + 9 hundreds + 6 tens + 3 ones = 4,963
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.
Example: 40,000
Zo 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.
39 9910
49,000
—- 125
3. Subtract as usual.
39 9910
49,000
—- 125 39,875
Example: 8,000,006
SAH
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.
eh =i if
2. Make each zero, except the rightmost one, 9. Make the rightmost 0 a 10.
7999 910
$,900,006
— 41,107
3. To perform the subtraction, we’ll need to borrow from the ten.
9 7 999 91016
$,909,006
— 41,107 7,958,899
1 ten = 10 ones 10 ones + 6 ones = 16 ones
Practice Set F
Perform each subtraction.
Exercise: Problem: eo — 4,873 Solution: 16,134
Exercise:
10,004
Problem: — 5,165
Solution:
4,839
Exercise:
ar 16,000,000 Tr a, ASOIED
Solution:
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 = Joe
The difference between 1006 and 284 is 722.
(What happens if you type 284 first and then 1006? We'll study such numbers in [link]Chapter 10.)
Practice Set G
Exercise:
Problem:
Use a calculator to find the difference between 7338 and 2809.
Solution:
4,529 Exercise:
Problem:
Use a calculator to find the difference between 31,060,001 and 8,591,774.
Solution:
22,468,227
Exercises
For the following problems, perform the subtractions. You may check each difference with a calculator.
Exercise: 15 Problem: — 8 Solution: 7 Exercise: 19 Problem: — 8 Exercise: 11 Problem: — 5 Solution: 6 Exercise: 14 Problem: — 6 Exercise: 12 Problem: 9
Solution:
3
Exercise:
56
Problem:
Exercise:
—12
74
Problem:
—33
Solution:
41
Exercise:
80
Problem:
Exercise:
Problem
—61
_ 390 *—141
Solution:
209
Exercise:
Problem
Exercise:
_ 800 * —650
Problem
35,002 "14,001
Solution:
21,001
Exercise:
Problem
Exercise:
Problem
5,000,566 * 2 441 326
400,605 * 121,352
Solution:
279,293
Exercise:
46,400
Problem:
Exercise:
91D
77,893
Problem:
A21
Solution:
77,472
Exercise:
A2
Problem:
Exercise:
—18
dl
Problem:
=—2%
Solution:
24
Exercise:
622
Problem:
Exercise:
— 88
261
Problem:
— 73
Solution:
188
Exercise:
Problem
Exercise:
Problem
242 " —158
3,422 1045
Solution:
2377 Exercise: Probl 5,965 roblem: 3,985 Exercise: — 42,041 ro em: 15 355 Solution: 26,686 Exercise: Probl 304,056 roblem: _ 20,008 Exercise: Probl 64,000,002 Pen oss. “BRGOTAS Solution: 63,143,259 Exercise: 4,109 Problem:
856
Exercise:
10,113
Problem:
2) 9.079
Solution:
8,034
Exercise:
605
Problem:
Exercise:
a as
59
Problem:
—26
Solution:
33
Exercise:
36,107
Problem:
Exercise:
Problem
V8 814
92.526,441,820 ” 59,914,805,253
Solution:
32,611,636,567
Exercise:
peanienL 1,605 roblem: 981 Exercise: aoe 30,000 roblem: 26,062 Solution: 3,938 Exercise: Bsa 600 roblem: 916 Exercise: pei 9,000,003 Fromme 796.048 Solution: 6:273,955
For the following problems, perform each subtraction. Exercise:
Problem: Subtract 63 from 92.
Note: The word "from" means "beginning at." Thus, 63 from 92 means beginning at 92, or 92 — 63.
Exercise:
Problem: Subtract 35 from 86. Solution: 51
Exercise:
Problem: Subtract 382 from 541.
Exercise:
Problem: Subtract 1,841 from 5,246.
Solution: 3,405
Exercise:
Problem: Subtract 26,082 from 35,040. Exercise:
Problem: Find the difference between 47 and 21.
Solution:
26
Exercise:
Problem:
Exercise:
Problem:
Solution:
72,069
Exercise:
Problem:
Exercise:
Problem:
Solution:
B17
Exercise:
Problem:
Exercise:
Problem:
Solution:
29
Exercise:
Problem:
Find the difference between 1,005 and 314.
Find the difference between 72,085 and 16.
Find the difference between 7,214 and 2,049.
Find the difference between 56,108 and 52,911.
How much bigger is 92 than 47?
How much bigger is 114 than 85?
How much bigger is 3,006 than 1,918?
Exercise:
Problem: How much bigger is 11,201 than 816?
Solution:
10,385
Exercise:
Problem: How much bigger is 3,080,020 than 1,814,161? Exercise: Problem: 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?
Solution:
15% Exercise: Problem: 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: Problem: 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?
Solution:
11,247,000
Exercise: Problem: 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?
Exercise: Problem: 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?
Solution:
436 Exercise: Problem: 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: Problem: 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?
Solution:
57,187,000 Exercise: Problem: 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:
Problem:
How many more life scientists were there in 1974 than mathematicians? ({link])
Solution:
165,000 Exercise: Problem: How many more social, psychological, mathematical, and
environmental scientists were there than life, physical, and computer scientists? ([link])
EMPLOYMENT STATUS OF SCIENTISTS— 1974
Exercise:
Problem:
How many more prosecutions were there in 1978 than in 1974? ({link])
Solution:
74 Exercise:
Problem:
How many more prosecutions were there in 1976-1980 than in 1970- 1975? ([link])
PROSECUTIONS OF FEDERAL OFFICIALS 1970-1980
1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 Year
Exercise:
Problem:
How many more dry holes were drilled in 1960 than in 1975? ([link])
Solution:
4,547 Exercise:
Problem:
How many more dry holes were drilled in 1960, 1965, and 1970 than in 1975, 1978 and 1979? ([link])
OIL WELLS—DRY HOLES DRILLED 1960-1979
Year
For the following problems, replace the L] with the whole number that will make the subtraction true.
Exercise: 14 Problem: — |] 3 Solution: sla Exercise: Zi. Problem: — 1] 14
Exercise:
35
Problem: — | 25 Solution: 10 Exercise: 16 Problem: — |] 9 Exercise: 28 Problem: — |] 16 Solution: 12
For the following problems, find the solutions. Exercise:
Problem: Subtract 42 from the sum of 16 and 56.
Exercise:
Problem: Subtract 105 from the sum of 92 and 89.
Solution:
76
Exercise:
Problem: Subtract 1,127 from the sum of 2,161 and 387.
Exercise:
Problem: Subtract 37 from the difference between 263 and 175.
Solution:
pi
Exercise:
Problem: Subtract 1,109 from the difference between 3,046 and 920. Exercise:
Problem:
Add the difference between 63 and 47 to the difference between 55 and 11.
Solution:
60 Exercise:
Problem:
Add the difference between 815 and 298 to the difference between 2,204 and 1,016.
Exercise:
Problem:
Subtract the difference between 78 and 43 from the sum of 111 and 89.
Solution:
165 Exercise:
Problem:
Subtract the difference between 18 and 7 from the sum of the differences between 42 and 13, and 81 and 16.
Exercise:
Problem:
Find the difference between the differences of 343 and 96, and 521 and 488.
Solution:
214
Exercises for Review
Exercise:
Problem:
({link]) In the number 21,206, how many hundreds are there? Exercise:
Problem: ({link]) Write a three-digit number that has a zero in the ones position. Solution:
330 (answers may vary)
Exercise:
Problem: ({link]) How many three-digit whole numbers are there?
Exercise:
Problem: ({link]) Round 26,524,016 to the nearest million.
Solution:
27,000,000
Exercise:
Problem: ((link]) Find the sum of 846 + 221 + 116.
Properties of Addition
This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses properties of addition. By the end of the module students should be able to understand the commutative and associative properties of addition and understand why 0 is the additive identity.
Section Overview e The Commutative Property of Addition e The Associative Property of Addition e The Additive Identity
We now consider three simple but very important properties of addition.
The Commutative Property of Addition
Commutative Property of Addition If two whole numbers are added in any order, the sum will not change.
Sample Set A
Example: Add the whole numbers
The numbers 8 and 5 can be added in any order. Regardless of the order they are added, the sum is 13.
Practice Set A
Exercise:
Problem:
Use the commutative property of addition to find the sum of 12 and 41 in two different ways.
Solution:
12 + 41 = 53 and 41 + 12 = 53
Exercise:
Problem: Add the whole numbers
837 1,958
Solution:
837 + 1,958 = 2,795 and 1,958 + 837 = 2,795
The Associative Property of Addition
Associative Property of Addition
If three whole numbers are to be added, the sum will be the same if the first two are added first, then that sum is added to the third, or, the second two are added first, and that sum is added to the first.
Using Parentheses
It is a common mathematical practice to use parentheses to show which pair of numbers we wish to combine first.
Sample Set B
Example: Add the whole numbers. 43 and 16 are associated. 43 (43 + 16) + 27 = 59 + 27 = 86. 16 43 + (16 + 27) = 43 + 43 = 86. 27 {16 and 27 are associated.
Practice Set B
Exercise:
Problem:
Use the associative property of addition to add the following whole numbers two different ways.
Solution:
(17 + 32) + 25 = 49 + 25 = 74 and 17 + (32 +25) =17+457= 74
Exercise:
Problem:
1,629 806 429
Solution: (1,629 + 806) + 429 = 2,435 + 429 = 2,864
1,629 + (806 + 429) = 1,629 + 1,235 = 2,864
The Additive Identity 0 Is the Additive Identity
The whole number 0 is called the additive identity, since when it is added to any whole number, the sum is identical to that whole number.
Sample Set C
Example: Add the whole numbers.
Za NaeAI f—s 748) Oo 29 — 29 Zero added to 29 does not change the identity of 29.
Practice Set C
Add the following whole numbers. Exercise:
Problem:
Solution:
8 Exercise:
Problem:
Solution:
3)
Suppose we let the letter x represent a choice for some whole number. For the first two problems, find the sums. For the third problem, find the sum provided we now know that x represents the whole number 17.
Exercise:
Problem:
Solution:
x Exercise:
Problem:
0 x
Solution:
x Exercise:
Problem:
Solution:
L7
Exercises
For the following problems, add the numbers in two ways. Exercise:
Problem:
Solution:
a7 Exercise:
Problem:
a6 12
Exercise:
Problem:
Solution:
45 Exercise:
Problem:
lit
Exercise:
Problem:
Solution:
568 Exercise:
Problem:
Exercise:
Problem:
Td, 205 49,118
Solution:
122.323 Exercise:
Problem:
Exercise:
Problem:
Solution:
45 Exercise:
Problem:
Exercise:
Problem:
Solution:
100 Exercise:
Problem:
Exercise:
Problem:
Solution:
556 Exercise: Problem: 1019
11 5a
Exercise:
Problem:
Solution:
43,461
For the following problems, show that the pairs of quantities yield the same sum. Exercise:
Problem: (11 + 27) + 9 and 11 + (27+ 9)
Exercise:
Problem: (80 + 52) + 6 and 80 + (52 + 6)
Solution:
132 + 6 =80 + 58 = 138
Exercise:
Problem: (114 + 226) + 108 and 114 + (226 + 108)
Exercise:
Problem: (731 + 256) + 171 and 731 + (256 + 171)
Solution:
987 + 171 =731 + 427 = 1,158
Exercise: Problem: The fact that (a first number + a second number) + third number = a
first number + (a second number + a third number) is an example of the property of addition.
Exercise: Problem:
The fact that 0 + any number = that particular number is an example of the property of addition.
Solution:
Identity Exercise: Problem: The fact that a first number + a second number = a second number + a first number is an example of the property of addition. Exercise: Problem:
Use the numbers 15 and 8 to illustrate the commutative property of addition.
Solution:
15+8=8+4+15 = 23 Exercise:
Problem:
Use the numbers 6, 5, and 11 to illustrate the associative property of addition.
Exercise:
Problem:
The number zero is called the additive identity. Why is the term identity so appropriate?
Solution:
...because its partner in addition remains identically the same after that addition
Exercises for Review Exercise: Problem: ({link]) How many hundreds in 46,581? Exercise: Problem: ({link]) Write 2,218 as you would read it. Solution:
Two thousand, two hundred eighteen.
Exercise:
Problem: ({link]) Round 506,207 to the nearest thousand.
Exercise:
482
Problem: ([link]) Find the sum of + 68
Solution:
550
Exercise:
3,318
Problem: ({link]) Find the difference: 499
Summary of Key Concepts
This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module reviews the key concepts from the chapter Addition and Subtraction of Whole Numbers.
Summary of Key Concepts
Number / Numeral ({link])
A number is a concept. It exists only in the mind. A numeral is a symbol that represents a number. It is customary not to distinguish between the two (but we should remain aware of the difference).
Hindu-Arabic Numeration System ([link])
In our society, we use the Hindu-Arabic numeration system. It was invented by the Hindus shortly before the third century and popularized by the Arabs about a thousand years later.
Digits ([link]) The numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 are called digits.
Base Ten Positional System ((link])
The Hindu-Arabic numeration system is a positional number system with base ten. Each position has value that is ten times the value of the position to its right.
Commas / Periods ((link])
Commas are used to separate digits into groups of three. Each group of three is called a period. Each period has a name. From right to left, they are ones, thousands, millions, billions, etc.
Whole Numbers ((link]) A whole number is any number that is formed using only the digits (0, 1, 255, 4,0; 0, 7505 9).
Number Line ([link]) The number line allows us to visually display the whole numbers.
Graphing ([link]) Graphing a whole number is a term used for visually displaying the whole number. The graph of 4 appears below.
Reading Whole Numbers ([link]) To express a whole number as a verbal phrase:
1. Begin at the right and, working right to left, separate the number into distinct periods by inserting commas every three digits. 2. Begin at the left, and read each period individually.
Writing Whole Numbers ([link]) To rename a number that is expressed in words to a number expressed in digits:
1. Notice that a number expressed as a verbal phrase will have its periods set off by commas.
2. Start at the beginning of the sentence, and write each period of numbers individually.
3. Use commas to separate periods, and combine the periods to form one number.
Rounding ((link])
Rounding is the process of approximating the number of a group of objects by mentally "seeing" the collection as occurring in groups of tens, hundreds, thousands, etc.
Addition ({link]) Addition is the process of combining two or more objects (real or intuitive) to form a new, third object, the total, or sum.
Addends / Sum ([link]) In addition, the numbers being added are called addends and the result, or total, the sum.
Subtraction ({link]) Subtraction is the process of determining the remainder when part of the total is removed.
Minuend / Subtrahend Difference (({link])
18—11=7
Be Jes
minuend subtrahend difference
Commutative Property of Addition ({link])
If two whole numbers are added in either of two orders, the sum will not change.
34+5=5+4+3
Associative Property of Addition ([link])
If three whole numbers are to be added, the sum will be the same if the first two are added and that sum is then added to the third, or if the second two are added and the first is added to that sum.
(3+5)+2=3+4+(542)
Parentheses in Addition ({link]) Parentheses in addition indicate which numbers are to be added first.
Additive Identity ({link])
The whole number 0 is called the additive identity since, when it is added to any particular whole number, the sum is identical to that whole number. 0+7=7
7+0=7
Exercise Supplement
This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module is an exercise supplement for the chapter Addition and Subtraction of Whole Numbers and contains many exercise problems. Odd problems are accompanied by solutions.
Exercise Supplement
For problems 1-35, find the sums and differences.
Exercise:
Bub 908 r : oblem 4 99
Solution:
937
Exercise:
Brabiway: 529
roblem: 4161 Exercise:
TT 549 r : oblem + 16
Solution:
565
Exercise: 726
Problem
* 4899
Exercise:
Problem
390 "+169
Solution:
aoe
Exercise:
Problem
Exercise:
Problem
166 * +660
391 "4951
Solution:
1,342
Exercise:
48
Problem:
Exercise:
+36
1,103
Problem:
"4 898
Solution:
2,001
Exercise:
1,642
Problem:
Exercise:
Problem
"4 899
807 "41,156
Solution:
1,963
Exercise:
Problem
Exercise:
Problem
80,349 "+ 2,679
70,070 "+ 9.386
Solution:
79,456
Exercise:
Problem
Exercise:
90,874 "4 2.945
Problem
A5,292 "451,661
Solution:
96,953
Exercise:
Problem
Exercise:
Problem
1,617 "454,923
702,607 “+ 89,217
Solution:
791,824
Exercise:
6,670,006
Problem:
Exercise:
Problem
+ 2.495
267 "+8 034
Solution:
8,301
Exercise:
Problem:
Exercise:
Problem:
Solution:
140,381
Exercise:
Problem:
Exercise:
Problem:
Solution:
76,224
Exercise:
Problem:
Exercise:
Problem:
7,007 +11,938
131,294
+ 9,087
5,292 + 161
17,260 +58,964
7,006 —5,382
7,973 ~3,018
Solution:
4,955
Exercise:
Problem
Exercise:
16,608 "— 1,660
209,527
Problem:
23,916
Solution:
185,611
Exercise:
Problem
Exercise:
Problem
_ 584 " —226
S316 "1.075
Solution:
2,238
Exercise:
Problem
458 " —122
Exercise:
1,007
Problem:
“ats B31
Solution:
1,336
Exercise:
Problem
Exercise:
16,082 "8.018
926
Problem:
— 48
Solution:
878
Exercise:
Problem
Exercise:
Problem
736 ” 45,869
676,504 = BROTT
Solution:
618,227
For problems 36-39, add the numbers. Exercise:
769 Problem: a roblem: 598 746 Exercise: 554 Problem: 184 883 Solution: 1,621 Exercise: 30,188 79,731 pa 16,600 roblem: 66,085 39,169 95,170
Exercise:
2.129
6,190 17,044 Problem: 30,447 292 Al 428,458 Solution: 484,601
For problems 40-50, combine the numbers as indicated. Exercise:
Problem: 2,957 + 9,006
Exercise:
Problem: 19,040 + 813 Solution:
Ae Metso.
Exercise:
Problem: 350,212 + 14,533
Exercise:
Problem: 970 + 702 + 22+ 8
Solution:
1,702
Exercise:
Problem:3,704 + 2,344 + 429 + 10,374 + 74
Exercise:
Problem: 874 + 845 + 295 — 900 Solution: 1,114
Exercise:
Problem: 904 + 910 — 881
Exercise:
Problem: 521 + 453 — 334 + 600 Solution: 1,300
Exercise:
Problem: 892 — 820 — 9
Exercise:
Problem: 159 + 4,085 — 918 — 608 Solution:
25718
Exercise:
Problem: 2,562 + 8,754 — 393 — 385 — 910
For problems 51-63, add and subtract as indicated. Exercise:
Problem: Subtract 671 from 8,027. Solution:
7300
Exercise:
Problem: Subtract 387 from 6,342.
Exercise:
Problem: Subtract 2,926 from 6,341.
Solution: 3,415
Exercise:
Problem: Subtract 4,355 from the sum of 74 and 7,319.
Exercise:
Problem: Subtract 325 from the sum of 7,188 and 4,964.
Solution:
11,827
Exercise:
Problem: Subtract 496 from the difference of 60,321 and 99.
Exercise:
Problem: Subtract 20,663 from the difference of 523,150 and 95,225.
Solution:
407,262 Exercise:
Problem:
Add the difference of 843 and 139 to the difference of 4,450 and 839. Exercise:
Problem:
Add the difference of 997,468 and 292,513 to the difference of 22,140 and 8,617.
Solution:
718,478 Exercise: Problem: Subtract the difference of 8,412 and 576 from the sum of 22,140 and 8,617. Exercise: Problem:
Add the sum of 2,273, 3,304, 847, and 16 to the difference of 4,365 and 864.
Solution:
9,941 Exercise:
Problem:
Add the sum of 19,161, 201, 166,127, and 44 to the difference of the sums of 161, 2,455, and 85, and 21, 26, 48, and 187.
Exercise:
Problem:
Is the sum of 626 and 1,242 the same as the sum of 1,242 and 626? Justify your claim.
Solution:
626 + 1,242 = 1,242 + 626 = 1,868
Proficiency Exam
This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module is a proficiency exam to the chapter Addition and Subtraction of Whole Numbers. Each problem is accompanied with a reference link pointing back to the module that discusses the type of problem demonstrated in the question. The problems in this exam are accompanied by solutions.
Proficiency Exam
Exercise:
Problem: ({link]) What is the largest digit?
Solution:
9 Exercise:
Problem:
({link]) In the Hindu-Arabic 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?
Solution:
ones, tens, hundreds Exercise:
Problem: ({link]) In the number 42,826, how many hundreds are there? Solution:
8
Exercise:
Problem: ({link]) Is there a largest whole number? If so, what is it? Solution:
no Exercise:
Problem:
({link]) Graph the following whole numbers on the number line: 2, 3,
Exercise:
Problem: ({link]) Write the number 63,425 as you would read it aloud. Solution:
Sixty-three thousand, four hundred twenty-five Exercise: Problem:
({link]) Write the number eighteen million, three hundred fifty-nine thousand, seventy-two.
Solution: 18,359,072
Exercise:
Problem: ({link]) Round 427 to the nearest hundred.
Solution: 400
Exercise:
Problem: ({link]) Round 18,995 to the nearest ten.
Solution:
19,000 Exercise:
Problem:
({link]) Round to the most reasonable digit: During a semester, a mathematics instructor uses 487 pieces of chalk.
Solution:
500
For problems 11-17, find the sums and differences. Exercise:
Probl link im robiem.: ae) 4. A8
Solution:
675
Exercise:
Problem: ([link]) 3106 + 921 Solution:
4,027
Exercise:
152 Problem: ({link}) :
36 Solution: 188 Exercise: 5,189 Problem: ({link]) 4.122 +8,001 Solution: 23,501 Exercise:
Problem: (({link]) 21+ 16+ 42+ 11 Solution:
90
Exercise:
Problem: ((link]) 520 — 216
Solution:
304
Exercise:
Problem: ({link])
Solution:
70,125
Exercise:
80,001 — 9,878
Problem: ({link]) Subtract 425 from 816.
Solution:
hell
Exercise:
Problem: ({link]) Subtract 712 from the sum of 507 and 387.
Solution:
182
Exercise:
Problem:
({link]) Is the sum of 219 and 412 the same as the sum of 412 and 219°? If so, what makes it so?
Solution:
Yes, commutative property of addition
Objectives This module contains Chapter 2 of Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr.
After completing this chapter, you should Multiplication of Whole Numbers (({link])
e understand the process of multiplication
e be able to multiply whole numbers
e be able to simplify multiplications with numbers ending in zero
¢ be able to use a calculator to multiply one whole number by another
Concepts of Division of Whole Numbers ({link])
e understand the process of division
¢ understand division of a nonzero number into zero
¢ understand why division by zero is undefined
¢ be able to use a calculator to divide one whole number by another
Division of Whole Numbers ({link])
e be able to divide a whole number by a single or multiple digit divisor e be able to interpret a calculator statement that a division results in a remainder
Some Interesting Facts about Division ({link])
e be able to recognize a whole number that is divisible by 2, 3, 4, 5, 6, 8, 9, or 10
Properties of Multiplication ({link]) e understand and appreciate the commutative and associative properties
of multiplication e understand why 1 is the multiplicative identity
Multiplication of Whole Numbers
This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses how to multiply whole numbers. By the end of the module students should be able to understand the process of multiplication, multiply whole numbers, simplify multiplications with numbers ending in zero, and use a calculator to multiply one whole number by another.
Section Overview
e¢ Multiplication
The Multiplication Process With a Single Digit Multiplier The Multiplication Process With a Multiple Digit Multiplier e Multiplication With Numbers Ending in Zero
e Calculators
Multiplication
Multiplication is a description of repeated addition. In the addition of
5+5+5
the number 5 is repeated 3 times. Therefore, we say we have three times five and describe it by writing
3x5
Thus,
3xX5=5+5+4+5
Multiplicand
In a multiplication, the repeated addend (number being added) is called the multiplicand. In 3 x 5, the 5 is the multiplicand.
Multiplier
Also, in a multiplication, the number that records the number of times the multiplicand is used is called the multiplier. In 3 x 5, the 3 is the multiplier.
Sample Set A
Express each repeated addition as a multiplication. In each case, specify the multiplier and the multiplicand.
Example: (eee pel tie pur We et Akai 6 x 7. Multiplier is 6. Multiplicand is 7.
Example: 18+18+18 3 xX 18. Multiplier is 3. Multiplicand is 18.
Practice Set A
Express each repeated addition as a multiplication. In each case, specify the multiplier and the multiplicand. Exercise:
Problem: 12 + 12+ 12+ 12
. Multiplier is . Multiplicand is .
Solution:
4 x 12. Multiplier is 4. Multiplicand is 12. Exercise:
Problem: 36 + 36 + 36 + 36 + 36 + 36 + 36 + 36
. Multiplier is . Multiplicand is .
Solution:
8 x 36. Multiplier is 8. Multiplicand is 36. Exercise:
Problem: 0+0+0+0+0
. Multiplier is . Multiplicand is .
Solution:
5 x 0. Multiplier is 5. Multiplicand is 0.
Exercise:
1847 + 1847 4+ ... + 1847 Problem: 12,000 times . Multiplier is . Multiplicand is .
Solution:
12,000 x 1,847. Multiplier is 12,000. Multiplicand is 1,847.
Factors In a multiplication, the numbers being multiplied are also called factors.
Products The result of a multiplication is called the product. In 3 x 5 = 15, the 3 and 5 are not only called the multiplier and multiplicand, but they are also called factors. The product is 15.
Indicators of Multiplication <,-,() The multiplication symbol (x) is not the only symbol used to indicate multiplication. Other symbols include the dot ( - ) and pairs of parentheses ( ). The expressions
3 x 5, 3-5, 3(5), (3)5, (3)(5)
all represent the same product.
The Multiplication Process With a Single Digit Multiplier
Since multiplication is repeated addition, we should not be surprised to notice that carrying can occur. Carrying occurs when we find the product of 38 and 7:
First, we compute 7 x 8 = 56. Write the 6 in the ones column. Carry the 5. Then take 7 x 3 = 21. Add to 21 the 5 that was carried: 21 + 5 = 26. The product is 266.
Sample Set B
Find the following products.
Example:
o 7
x_3 192 eet iy) Beco=6
Write the 2, carry the 1. Add to 18 the 1 that was carried: 18 + 1 = 19.
The product is 192.
oo 25
Write the 0, carry the 3. Add to 10 the 3 that was carried: 10 + 3 = 13. Write the 3, carry the 1. Add to 25 the 1 that was carried: 25 + 1= 6.
The product is 2,630.
Example:
Write the 6, carry the 3.
Add to the 0 the 3 that was carried:0 + 3 = 3. Write the 3.
Write the 2, carry the 7.
Add to the 9 the 7 that was carried: 9 + 7 = 16.
Since there are no more multiplications to perform,write both the 1 and 6.
The product is 16,236.
Practice Set B
Find the following products.
Exercise:
37
Problem:
x 5
Solution:
185
Exercise:
78
Problem:
“x 8
Solution:
624
Exercise:
536
Problem:
Solution:
3,792
Exercise:
Problem:
40,019 x 8
Solution:
320,152
Exercise:
Problem
_ 301,599 “y 3
Solution:
904,797
The Multiplication Process With a Multiple Digit Multiplier
In a multiplication in which the multiplier is composed of two or more digits, the multiplication must take place in parts. The process is as follows:
¢ Part 1 First Partial Product Multiply the multiplicand by the ones digit of the multiplier. This product is called the first partial product.
¢ Part 2 Second Partial Product Multiply the multiplicand by the tens digit of the multiplier. This product is called the second partial product. Since the tens digit is used as a factor, the second partial product is written below the first partial product so that its rightmost digit appears in the tens column.
e Part 3 If necessary, continue this way finding partial products. Write each one below the previous one so that the rightmost digit appears in the column directly below the digit that was used as a factor.
e Part 4 Total Product Add the partial products to obtain the total product.
Note:It may be necessary to carry when finding each partial product.
Sample Set C
Example: Multiply 326 by 48.
e Part 1
24 326 x 48 2608 <— First partial product.
e Part 2
12 24 326 X_48 2608 1304 <— Second partial product.
e Part 3This step is unnecessary since all of the digits in the multiplier have been used. e Part 4Add the partial products to obtain the total product.
12
24 326 x_48 2608
+1304 15648
<— Total product.
e The product is 15,648.
Example:
Multiply 5,369 by 842.
e Part 1
11 5369 X 842 10738
e Part 2
123
11 5369 X_842 10738
21476
e Part 3
257
123
11 5369 X_842 10738
21476
42952
4520698
<— First partial product.
«— Second partial product.
<— Third partial product. <— Total product (Part 4).
e The product is 4,520,698.
Example: Multiply 1,508 by 206.
e Part 1
34 1508 X 206 9048 #<— First partial product (in first column from the right).
e Part 2
364
1508 X_206
9048
Since 0 times 1508 is 0, the partial product will not change the identity of the total product (which is obtained by addition). Go to the next partial product. e Part 3
11
3.4
1508 x 206
9048 3016 == «— Third partial product (in third column from the right). 310648 <— Total product (Part 4).
e The product is 310,648
Practice Set C Exercise: Problem: Multiply 73 by 14.
Solution:
1,022
Exercise:
Problem: Multiply 86 by 52.
Solution:
4,472
Exercise:
Problem: Multiply 419 by 85. Solution:
35,615
Exercise:
Problem: Multiply 2,376 by 613. Solution:
1,456,488
Exercise:
Problem: Multiply 8,107 by 304. Solution:
2,464,528
Exercise:
Problem: Multiply 66,260 by 1,008. Solution:
66,790,080
Exercise:
Problem: Multiply 209 by 501. Solution:
104,709
Exercise:
Problem: Multiply 24 by 10.
Solution:
240 Exercise: Problem: Multiply 3,809 by 1,000. Solution: 3,809,000 Exercise: Problem: Multiply 813 by 10,000.
Solution:
8,130,000
Multiplications With Numbers Ending in Zero
Often, when performing a multiplication, one or both of the factors will end in zeros. Such multiplications can be done quickly by aligning the numbers so that the rightmost nonzero digits are in the same column.
Sample Set D Perform the multiplication (49,000) (1,200).
(49,000)(1,200) = 49000 x 1200
Since 9 and 2 are the rightmost nonzero digits, put them in the same column.
49000 1200
Draw (perhaps mentally) a vertical line to separate the zeros from the nonzeros.
49/000 X 12/00
Multiply the numbers to the left of the vertical line as usual, then attach to the right end of this product the total number of zeros.
Attach these 5 zeros to 588.
The product is 58,800,000
Practice Set D
Exercise: Problem: Multiply 1,800 by 90.
Solution:
162,000
Exercise:
Problem: Multiply 420,000 by 300.
Solution:
126,000,000
Exercise:
Problem: Multiply 20,500,000 by 140,000. Solution:
2,870,000,000,000
Calculators
Most multiplications are performed using a calculator.
Sample Set E
Example: Multiply 75,891 by 263.
Display Reads
Type 75891 75891 Press x 75891 Type 263 263 Press - 19959333
The product is 19,959,333.
Example: Multiply 4,510,000,000,000 by 1,700.
Display Reads Type 451 451 Press x 451 Type 17 17 Press - 7667
The display now reads 7667. We'll have to add the zeros ourselves. There are a total of 12 zeros. Attaching 12 zeros to 7667, we get 7,667,000,000,000,000. The product is 7,667,000,000,000,000.
Example: Multiply 57,847,298 by 38,976.
Display Reads
Type 57847298 57847298
Press x 57847298 Type 38976 38976 Press = 2.2546563 12
The display now reads 2.2546563 12. What kind of number is this? This is an example of a whole number written in scientific notation. We'll study this concept when we get to decimal numbers.
Practice Set E
Use a calculator to perform each multiplication. Exercise:
Problem: 52 x 27 Solution: 1,404 Exercise: Problem: 1,448 x 6,155 Solution: 8,912,440 Exercise: Problem: 8,940,000 x 205,000 Solution:
1,832,700,000,000
Exercises
For the following problems, perform the multiplications. You may check each product with a calculator. Exercise:
Problem: x3 Solution: 24 Exercise: 3 Problem: x5 Exercise: 8 Problem: x6 Solution: 48 Exercise: 5 Problem: x7 Exercise:
Problem:6 x 1
Solution: 6
Exercise:
Problem:4 x 5 Exercise:
Problem:75 x 3
Solution:
225
Exercise:
Problem:35 x 5
Exercise: 45 Problem: x 6 Solution: 270 Exercise: 31 Problem: x 7 Exercise: 97 Problem: x 6 Solution: 582 Exercise: Probl 2 roblem: x57 Exercise: 64 Problem: x15 Solution: 960 Exercise: 73 Problem: x15
Exercise:
81 Problem: x95
Solution:
7,695 Exercise: Problem: x
Exercise:
Problem:57 x 64 Solution: 3,648
Exercise:
Problem:76 x 42 Exercise:
Problem:894 x 52
Solution:
46,488
Exercise:
Problem:684 x 38
Exercise:
115 Problem: x. 22 Solution: 2,530
Exercise:
706
Problem:
x 8&1 Exercise: 328
Problem:
x
Solution:
6,888
Exercise:
Probl on Tr m: as x 94
Exercise:
Problem:930 x 26 Solution:
24,180
Exercise:
Problem:318 x 63
Exercise:
Beebe 582 ro em: 127 Solution: 73,914 Exercise: 24 Problem: f x116 Exercise: Problem: =P
x 225
Solution:
68,625 Exercise: Probl 782 m: OR Sapa Exercise: 1 Problem: ie x 663 Solution: 511,173 Exercise: Problem: age x516 Exercise:
Problem:1,905 x 710 Solution: 1,352,550
Exercise:
Problem:5,757 x 5,010
Exercise:
Se 3,106 Pere TSO Solution: 5,441,712
Exercise:
bl 9,300 Problem: «1,130
Exercise:
bl 7,057 Problem: «5,229
Solution:
36,901,053
Exercise: ; 8,051 Problem: x 5,580
Exercise:
bl 5,804 Pro em: 4.300
Solution:
24,957,200
Exercise:
Problem: roblem ale
Exercise:
724
Problem: ro em: | 0
Solution:
0
Exercise:
problem: 2°049 Tr em: 7 ye AL
Exercise:
_ 5,173
Problem: x 8 Solution: 41,384 Exercise: Problem: nee x 0 Exercise: Problem: £2008 x 0 Solution: 0 Exercise: 1 Problem: le x 142 Exercise: Problem: ape x 190 Solution: 73,530 Exercise: Probl 3,400 Tr m: oble 70 Exercise: pean 460,000 roore™ 14,000
Solution:
6,440,000,000
Exercise: , 558,000,000 Problem: | 81,000 Exercise: 37,000 Problem: : Bese AG Solution: 4,440,000 Exercise: 498,000 Problem: x 0 Exercise: rer 4,585,000 ro em: 140 Solution: 641,900,000 Exercise: Beanie 30,700,000 ro em: 180 Exercise: Probl 8,000 m: ciara x 10 Solution: 80,000
Exercise:
Problem: Suppose a theater holds 426 people. If the theater charges $4 per ticket and sells every seat, how much money would they take in? Exercise: Problem: In an English class, a student is expected to read 12 novels during the semester and prepare a
report on each one of them. If there are 32 students in the class, how many reports will be prepared?
Solution:
384 reports Exercise: Problem: In a mathematics class, a final exam consists of 65 problems. If this exam is given to 28 people, how many problems must the instructor grade? Exercise: Problem:
A business law instructor gives a 45 problem exam to two of her classes. If each class has 37 people in it, how many problems will the instructor have to grade?
Solution:
3,330 problems Exercise: Problem: An algebra instructor gives an exam that consists of 43 problems to four of his classes. If the
classes have 25, 28, 31, and 35 students in them, how many problems will the instructor have to grade?
Exercise: Problem: In statistics, the term "standard deviation" refers to a number that is calculated from certain data. If the data indicate that one standard deviation is 38 units, how many units is three standard deviations?
Solution:
114 units
Exercise: Problem: Soft drinks come in cases of 24 cans. If a supermarket sells 857 cases during one week, how many individual cans were sold? Exercise: Problem:
There are 60 seconds in 1 minute and 60 minutes in 1 hour. How many seconds are there in 1 hour?
Solution:
3,600 seconds Exercise: Problem: There are 60 seconds in 1 minute, 60 minutes in one hour, 24 hours in one day, and 365 days in one year. How many seconds are there in 1 year? Exercise: Problem:
Light travels 186,000 miles in one second. How many miles does light travel in one year? (Hint: Can you use the result of the previous problem?)
Solution:
5,865,696,000,000 miles per year Exercise:
Problem:
An elementary school cafeteria sells 328 lunches every day. Each lunch costs $1. How much money does the cafeteria bring in in 2 weeks?
Exercise:
Problem:
A computer company is selling stock for $23 a share. If 87 people each buy 55 shares, how much money would be brought in?
Solution:
$110,055
Exercises for Review Exercise:
Problem: ({link]) In the number 421,998, how may ten thousands are there? Exercise:
Problem: (({link]) Round 448,062,187 to the nearest hundred thousand.
Solution:
448,100,000
Exercise:
Problem: (({link]) Find the sum. 22,451 + 18,976.
Exercise: Problem: (({link]) Subtract 2,289 from 3,001.
Solution:
712 Exercise:
Problem:
([link]) Specify which property of addition justifies the fact that (a first whole number + a second whole number) = (the second whole number + the first whole number)
Concepts of Division of Whole Numbers
This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses how to divide whole numbers. By the end of the module students should be able to understand the process of division, understand division of a nonzero number into zero, understand why division by zero is undefined, and use a calculator to divide one whole number by another.
Section Overview e Division e Division into Zero (Zero As a Dividend: a a #0) e Division by Zero (Zero As a Divisor: . a#0) e Division by and into Zero (Zero As a Dividend and Divisor: ) e Calculators
Division
Division is a description of repeated subtraction.
In the process of division, the concern is how many times one number is contained in another number. For example, we might be interested in how many 5's are contained in 15. The word times is significant because it
implies a relationship between division and multiplication.
There are several notations used to indicate division. Suppose Q records the number of times 5 is contained in 15. We can indicate this by writing
5)15 5 ; 15 divided by 5 5 into 15
15/5=Q 15+5=Q 15 divided by 5 15 divided by 5
Each of these division notations describes the same number, represented here by the symbol @. Each notation also converts to the same multiplication form. It is 15 = 5 x Q
In division,
Dividend the number being divided into is called the dividend.
Divisor the number dividing into the dividend is the divisor.
Quotient the result of the division is called the quotient.
quotient divisor \dividend
dividend
divisor = quotient
dividend/divisor = quotient dividend + divisor = quotient
Sample Set A
Find the following quotients using multiplication facts.
Example:
18 = 6
Sivice: 6X3. — ES: 18 +6=3 Notice also that
—6 Repeated subtraction
Thus, 6 is contained in 18 three times.
Example: 24
3 Since 3 x 8 = 24,
pas 3} =
Notice also that 3 could be subtracted exactly 8 times from 24. This implies that 3 is contained in 24 eight times.
Example: 36 6
Since 6 x 6 = 36,
36) = 6 = 6
Thus, there are 6 sixes in 36.
Example:
9)72 Since 9 x 8 = 72, 8
9)72 Thus, there are 8 nines in 72.
Practice Set A
Use multiplication facts to determine the following quotients. Exercise:
Problem: 32 — 8 Solution:
4
Exercise:
Problem: 18 — 9 Solution:
2
Exercise:
Problem:
Solution:
D
Exercise:
Problem:
Solution:
6
Exercise:
Problem:
Solution:
4
Exercise:
Problem: 4)36
Solution:
9
Division into Zero (Zero as a Dividend: o yas 0)
Let's look at what happens when the dividend (the number being divided into) is zero, and the divisor (the number doing the dividing) is any whole number except zero. The question is
: : 5 What number, if any, 18 any nonzero whole number *
Let's represent this unknown quotient by Q. Then,
aapromao trom = any nonzero whole number ~~
Converting this division problem to its corresponding multiplication problem, we get
0 = Q x (any nonzero whole number)
From our knowledge of multiplication, we can understand that if the product of two whole numbers is zero, then one or both of the whole numbers must be zero. Since any nonzero whole number is certainly not zero, @ must represent zero. Then,
= (0
any nonzero whole number
Zero Divided By Any Nonzero Whole Number Is Zero Zero divided any nonzero whole number is zero.
Division by Zero (Zero as a Divisor: = , a # 0)
Now we ask,
any nonzero whole number 5
What number, if any, is 7
Letting Q represent a possible quotient, we get
any nonzero whole number __ Q age
Converting to the corresponding multiplication form, we have (any nonzero whole number) = Q x 0
Since Q x 0 = 0, (any nonzero whole number) = 0. But this is absurd. This would mean that 6 = 0, or 37 = 0. A nonzero whole number cannot equal 0! Thus,
any nonzero whole number
0 does not name a number
Division by Zero is Undefined Division by zero does not name a number. It is, therefore, undefined.
Division by and Into Zero (Zero as a Dividend and Divisor: a)
We are now curious about zero divided by zero (2). If we let Q represent a potential quotient, we get
0 9 7@ Converting to the multiplication form,
0=Qx0
This results in 0=—0
This is a statement that is true regardless of the number used in place of Q. For example,
o = 5, since 0 = 5 x 0. o = 31, since 0 = 31 x 0. Se = 286, since 0 = 286 x 0.
A unique quotient cannot be determined.
Indeterminant Since the result of the division is inconclusive, we say that - is
indeterminant.
- is Indeterminant The division ~ is indeterminant.
Sample Set B
Perform, if possible, each division.
Example: 2. Since division by 0 does not name a whole number, no quotient exists, and we state = is undefined
Example:
0)14. Since division by 0 does not name a defined number, no quotient exists, and we state 0)14. is undefined
Example: 9)0. Since division into 0 by any nonzero whole number results in 0, we
i 0 ave 9)0
Example: q. Since division into 0 by any nonzero whole number results in 0, we
have 2 ==a(i)
Practice Set B
Perform, if possible, the following divisions. Exercise:
olor
Problem:
Solution:
undefined
Exercise:
=)
Problem:
Solution:
0
Exercise:
Problem: 0)0
Solution:
indeterminant
Exercise:
Problem: 0s
Solution:
undefined
Exercise:
Problem: a
Solution:
undefined
Exercise:
Problem: A.
Solution:
0
Calculators
Divisions can also be performed using a calculator.
Sample Set C
Example:
Divide 24 by 3. Display Reads Type 24 24 Press + 24 Type 3 3 Press = 8
The display now reads 8, and we conclude that 24 + 3 = 8.
Example:
Divide 0 by 7.
Display Reads
Type 0 0
Press = 0 Type 7 v7
Press = 0
The display now reads 0, and we conclude that 0 + 7 = 0.
Example:
Divide 7 by 0.
Since division by zero is undefined, the calculator should register some kind of error message.
Display Reads
Type i 7 Press cr ih Type 0 0 Press = Error
The error message indicates an undefined operation was attempted, in this case, division by zero.
Practice Set C
Use a calculator to perform each division. Exercise:
Problem: 35 — 7
Solution: 5
Exercise:
Problem: 56 ~ 8
Solution: 7
Exercise:
Problem: 0 — 6
Solution: 0
Exercise:
Problem: 3 — 0
Solution:
An error message tells us that this operation is undefined. The particular message depends on the calculator.
Exercise:
Problem: 0 — 0
Solution:
An error message tells us that this operation cannot be performed. Some calculators actually set 0 + 0 equal to 1. We know better! 0 + 0 is indeterminant.
Exercises
For the following problems, determine the quotients (if possible). You may use a Calculator to check the result. Exercise:
Problem: 4)32
Solution:
8
Exercise:
Problem: 7)42
Exercise:
Problem: 6)18
Solution:
3
Exercise:
Problem: 2)14
Exercise:
Problem:
Solution:
9
Exercise:
Problem:
Exercise:
Problem:
Solution:
7
Exercise:
Problem:
Exercise:
Problem:
Solution:
4
Exercise:
Problem:
Exercise:
Problem:
24+ 8
10+ 2
Solution:
5
Exercise:
Problem
Exercise:
Problem
:21+7
:21+3
Solution:
fi
Exercise:
Problem:
Exercise:
Problem:
Solution:
not defined
Exercise:
Problem:
Exercise:
Problem:
12+4
3)9
Solution:
3
Exercise:
Problem:
Exercise:
Problem:
Solution:
0
Exercise:
Problem:
Exercise:
Problem:
Solution:
fs)
Exercise:
Problem:
Exercise:
Problem:
Solution:
8
Exercise:
7)0
DO
co|o
Problem:
Exercise:
Problem: 72 — 8
Solution:
J
Exercise:
Problem: Write =~ = 8 using three different notations.
Exercise:
Problem: Write 22 = 3 using three different notations.
9 Solution: 27+9=3;9)07 =3; 2% =3 Exercise:
4 Problem: [n the statement 6)24
6 is called the . 24 is called the . 4 is called the .
Exercise:
Problem: In the statement 56 + 8 = 7,
7 is called the . 8 is called the . 56 is called the . Solution:
7 is quotient; 8 is divisor; 56 is dividend
Exercises for Review Exercise:
Problem: ({link]) What is the largest digit?
Exercise:
8,006
Problem: ([link]) Find the sum. +4118
Solution:
12,124
Exercise:
631
Problem: ({link]) Find the difference. 599
Exercise:
Problem:
({link]) Use the numbers 2, 3, and 7 to illustrate the associative property of addition.
Solution:
(24+3)4+7=24(3+7)=12 5+7=2+10=12
Exercise:
Problem: (({link]) Find the product
86 12
Division of Whole Numbers
This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses how to divide whole numbers. By the end of the module students should be able to be able to divide a whole number by a single or multiple digit divisor and interpret a calculator statement that a division results in a remainder.
Section Overview
e Division with a Single Digit Divisor
e Division with a Multiple Digit Divisor e Division with a Remainder Calculators
Division with a Single Digit Divisor
Our experience with multiplication of whole numbers allows us to perform such divisions as 75 + 5. We perform the division by performing the corresponding multiplication, 5 x Q = 75. Each division we considered in [link] had a one-digit quotient. Now we will consider divisions in which the quotient may consist of two or more digits. For example, 75 + 5.
Let's examine the division 75 + 5. We are asked to determine how many 5's are contained in 75. We'll approach the problem in the following way.
1. Make an educated guess based on experience with multiplication.
2. Find how close the estimate is by multiplying the estimate by 5.
3. If the product obtained in step 2 is less than 75, find out how much less by subtracting it from 75.
4. If the product obtained in step 2 is greater than 75, decrease the estimate until the product is less than 75. Decreasing the estimate makes sense because we do not wish to exceed 75.
We can suggest from this discussion that the process of division consists of The Four Steps in Division
1. an educated guess
2. a multiplication 3. a subtraction 4. bringing down the next digit (if necessary)
The educated guess can be made by determining how many times the divisor is contained in the dividend by using only one or two digits of the dividend.
Sample Set A
Example: Find 75 + 5. 5)75 Rewrite the problem using a division bracket.
10
5)75
Make an educated guess by noting that one 5 is contained in 75 at most 10
times.
Since 7 is the tens digit, we estimate that 5 goes into 75 at most 10 times. 10
5)75
=50
25
Now determine how close the estimate is.
10 fives is 10 x 5 = 50. Subtract 50 from 75.
Estimate the number of 5's in 25.
There are exactly 5 fives in 25. 5 10 fives + 5 fives = 15 fives. 10
5)75 =o) 25 —25 0
There are 15 fives contained in 75.
Check:
75215X5 75 ~ 75
Thus, 75 +5 = 15. The notation in this division can be shortened by writing.
15 5)75 ah 25 —25 0 Divide: 5 goes into 7 at most 1 time. Multiply: 1 x 5=5. Write 5 below 7.
Subtract: 7-5 = 2. Bring down the 5. Divide: 5 goes into 25 exactly 5 times.
Multiply: 5 x 5 = 25. Write 25 below 25. Subtract? 25-25 =;
Example:
Find 4,944 = 8.
8 4944
Rewrite the problem using a division bracket. 600
8 )4944
—4800
144
8 goes into 49 at most 6 times, and 9 is in the hundreds column. We'll
guess 600.
Then, 8 x 600 = 4800.
10 600 8 \4944 —4800 144 — 80 64
8 goes into 14 at most 1 time, and 4 is in the tens column. We'll guess 10.
8 goes into 64 exactly 8 times. 600 eights + 10 eights + 8 eights = 618 eights. Check:
494428 X618 4944 ~ 4944
Thus, 4,944 + 8 = 618. As in the first problem, the notation in this division can be shortened by eliminating the subtraction signs and the zeros in each educated guess.
Divide: 8 goes into 49 at most 6 times. Multiply: 6 x 8 = 48. Write 48 below 49.
Subtract: 49 - 48 = 1. Bring down the 4. Divide: 8 goes into 14 at most 1 time.
Multiply: 1 x 8 = 8. Write 8 below 14.
| Subtract: 14-8 —=6. Bring down the 4. Divide: 8 goes into 64 exactly 8 times.
Multiply: 8 x 8 = 64. Write 64 below 64. Subtract: 64-64=0.
Note: Not all divisions end in zero. We will examine such divisions in a subsequent subsection.
Practice Set A
Perform the following divisions. Exercise:
Problem: 126 ~ 7
Solution:
18
Exercise:
Problem: 324 — 4
Solution:
81
Exercise:
Problem: 2,559 =~ 3 Solution:
853
Exercise:
Problem: 5,645 =~ 5 Solution:
1,129
Exercise:
Problem: 757,125 ~ 9
Solution:
84,125
Division with a Multiple Digit Divisor The process of division also works when the divisor consists of two or more
digits. We now make educated guesses using the first digit of the divisor and one or two digits of the dividend.
Sample Set B
Example: Find 2,232 + 36.
36) 2232
Use the first digit of the divisor and the first two digits of the dividend to make the educated guess.
3 goes into 22 at most 7 times.
Try 7: 7 X 36 = 252 which is greater than 223. Reduce the estimate. Try 6: 6 x 36 = 216 which is less than 223.
6 36) 2232 —216|
72
Multiply: 6 x 36 = 216. Write 216 below 223.
Subtract: 223 - 216 = 7. Bring down the 2. Divide 3 into 7 to estimate the number of times 36 goes into 72. The 3 goes
into 7 at most 2 times. pede OX OF = fs
62 36)2232 2164 72
-72
0
Check:
2232 2 36 X 62 2232 ~ 2232
Thus, 2,232 + 36 = 62.
Example:
Find 2,417,228 + 802.
802 )2417228
First, the educated guess: 24 + 8 = 3. Then 3 x 802 = 2406, which is less than 2417. Use 3 as the guess. Since 3 x 802 = 2406, and 2406 has
four digits, place the 3 above the fourth digit of the dividend.
3 802) 2417228 — 2406] 112
Subtract: 2417 - 2406 = 11. Bring down the 2. The divisor 802 goes into 112 at most 0 times. Use 0.
30 802) 2417228
— 2406! 112 —0y
1122
Multiply: Oe 2802 — Or Subtract: jee Orem 1 ILA
Bring down the 2. The 8 goes into 11 at most 1 time, and 1 x 802 = 802, which is less than i ry: A
Subtract 1122 — 802 = 320 Bring down the 8.
8 goes into 32 at most 4 times. A x 802 = 3208.
Use 4.
3014 802 ) 2417228
Check:
2417228 2 3014 X 802 2417228 ~ 2417228
Thus, 2,417,228 + 802 = 3,014.
Practice Set B
Perform the following divisions. Exercise:
Problem: 1,376 ~ 32
Solution:
43
Exercise:
Problem: 6,160 ~ 55
Solution:
112
Exercise:
Problem: 18,605 ~ 61 Solution:
305
Exercise:
Problem: 144,768 ~ 48 Solution:
3,016
Division with a Remainder
We might wonder how many times 4 is contained in 10. Repeated subtraction yields
10
6 —4
Since the remainder is less than 4, we stop the subtraction. Thus, 4 goes into 10 two times with 2 remaining. We can write this as a division as follows.
2 4)10 — 8 2 Divide: A goes into 10 at most 2 times. Multiply: 2 x 4=8. Write 8 below 0. Subtract: 10-8 = 2.
Since 4 does not divide into 2 (the remainder is less than the divisor) and there are no digits to bring down to continue the process, we are done. We write
2R2 4) Morl+4= eRe ) 2 with remainder 2
Sample Set C
Example: Find 85 + 3.
w AIS Sle Bly
Divide: 3 goes into 8 at most 2 times. Multiply: 2 x 3=6. Write 6 below 8. Subtract: 8-6 = 2. Bring down the 5.
Divide: 3 goes into 25 at most 8 times. Multiply: 3 x 8 = 24. Write 24 below 25.
Subtract: 25-24=1. There are no more digits to bring down to continue the process. We are done. One is the remainder. Check: Multiply 28 and 3, then add 1. 28 x_ 3 84 rel
85 Thus, 89 + 3 = 28R1.
Example: Find 726 + 23.
Check: Multiply 31 by 23, then add 13.
31 X 23 93 62_ 713 + 13 726
Thus, 726 + 23 = 31 R13.
Practice Set C
Perform the following divisions. Exercise:
Problem:75 — 4 Solution:
18 R3
Exercise:
Problem:346 — 8 Solution:
43 R2
Exercise:
Problem:489 ~ 21 Solution:
23 R6
Exercise:
Problem:5,016 + 82
Solution:
61 R14
Exercise:
Problem:41,196 ~ 67
Solution:
614 R58
Calculators
The calculator can be useful for finding quotients with single and multiple digit divisors. If, however, the division should result in a remainder, the calculator is unable to provide us with the particular value of the remainder.
Also, some calculators (most nonscientific) are unable to perform divisions in which one of the numbers has more than eight digits.
Sample Set D
Use a calculator to perform each division.
Example: 328 = 8
Type 328 Press = Type 8
Press =
The display now reads 41.
Example:
53,136 + 82 Type 593136 Press ae Type 82 Press a
The display now reads 648.
Example:
730,019,001 + 326
We first try to enter 730,019,001 but find that we can only enter 73001900. If our calculator has only an eight-digit display (as most nonscientific calculators do), we will be unable to use the calculator to perform this division.
Example: 3727 + 49
Type 277 Press ice Type 49
Press =
The display now reads 76.061224.
This number is an example of a decimal number (see [link]). When a decimal number results in a calculator division, we can conclude that the division produces a remainder.
Practice Set D
Use a calculator to perform each division. Exercise:
Problem: 3,330 + 74
Solution:
45
Exercise:
Problem: 63,365 ~ 115
Solution:
dol
Exercise:
Problem: 21,996,385,287 + 53
Solution: Since the dividend has more than eight digits, this division cannot be
performed on most nonscientific calculators. On others, the answer is 415,026,137.4
Exercise: Problem: 4,558 = 67 Solution: This division results in 68.02985075, a decimal number, and therefore, we cannot, at this time, find the value of the remainder. Later, we will discuss decimal numbers.
Exercises
For the following problems, perform the divisions.
The first 38 problems can be checked with a calculator by multiplying the
divisor and quotient then adding the remainder. Exercise:
Problem: 52 — 4
Solution:
13
Exercise:
Problem: 776 — 8
Exercise:
Problem: 603 ~ 9
Solution:
67
Exercise:
Problem
Exercise:
Problem
: 240 = 8
: 208 + 4
Solution:
52
Exercise:
Problem
Exercise:
Problem
576 + 6
:21+7
Solution:
3
Exercise:
Problem:
Exercise:
Problem
: 140 = 2
Solution:
70
Exercise:
Problem:
Exercise:
Problem:
Solution:
61
Exercise:
Problem:
Exercise:
Problem:
Solution:
og
Exercise:
Problem:
Exercise:
Problem:
Solution:
67
Exercise:
Problem:
528 + 8 244 + 4 0+7
Lips 96=8 orl
896 + 56
Exercise:
Problem
Solution:
87
Exercise:
Problem
Exercise:
Problem
Solution:
04
Exercise:
Problem
Exercise:
Problem
Solution:
a2
Exercise:
Problem
Exercise:
1,044 + 12 988 + 19 : 5,238 + 97 £2,530 + 55 : 4,264 + 82 : 637 + 13
Problem:
Solution:
38
Exercise:
Problem:
Exercise:
Problem:
Solution:
45
Exercise:
Problem:
Exercise:
Problem:
Solution:
777
Exercise:
Problem:
Exercise:
Problem:
3,420 + 90
5,655 + 87
F115 47
9,328 + 22
55,167 + 71
68,356 + 92
27702 = 81
Solution: 342
Exercise:
Problem: 6,510 + 31
Exercise:
Problem: 60,536 + 94 Solution:
644
Exercise:
Problem: 31,844 ~ 38
Exercise:
Problem: 23,985 ~ 45 Solution:
see,
Exercise:
Problem: 60,606 ~ 74
Exercise:
Problem: 2,975,400 + 285 Solution:
10,440
Exercise:
Problem: 1,389,660 ~ 795
Exercise:
Problem: 7,162,060 = 879 Solution:
8,147 remainder 847
Exercise:
Problem: 7,561,060 ~ 909
Exercise:
Problem: 38 —~ 9 Solution:
4 remainder 2
Exercise:
Problem: 97 — 4
Exercise:
Problem: 199 ~ 3 Solution:
66 remainder 1
Exercise:
Problem: 573 — 6
Exercise:
Problem: 10,701 ~ 13 Solution:
823 remainder 2
Exercise:
Problem: 13,521 + 53
Exercise:
Problem: 3,628 ~ 90 Solution:
4O remainder 28
Exercise:
Problem: 10,592 ~ 43
Exercise:
Problem: 19,965 ~ 30 Solution:
665 remainder 15
Exercise:
Problem: 8,320 + 21
Exercise:
Problem: 61,282 ~ 64 Solution:
957 remainder 34
Exercise:
Problem: 1,030 + 28
Exercise:
Problem: 7,319 + 11 Solution:
665 remainder 4
Exercise:
Problem: 3,628 ~ 90
Exercise:
Problem: 35,279 ~ 77 Solution:
458 remainder 13
Exercise:
Problem: 52,196 + 55
Exercise:
Problem: 67,751 ~ 68
Solution:
996 remainder 23
For the following 5 problems, use a calculator to find the quotients. Exercise:
Problem: 4,346 ~ 53 Exercise:
Problem: 3,234 + 77
Solution:
42
Exercise:
Problem: 6,771 ~ 37 Exercise:
Problem: 4,272,320 + 520
Solution:
8,216
Exercise:
Problem: 7,558,110 + 651
Exercise:
Problem:
A mathematics instructor at a high school is paid $17,775 for 9 months. How much money does this instructor make each month?
Solution:
$1,975 per month Exercise: Problem: A couple pays $4,380 a year for a one-bedroom apartment. How much does this couple pay each month for this apartment? Exercise: Problem:
Thirty-six people invest a total of $17,460 in a particular stock. If they each invested the same amount, how much did each person invest?
Solution:
$485 each person invested Exercise:
Problem:
Each of the 28 students in a mathematics class buys a textbook. If the
bookstore sells $644 worth of books, what is the price of each book? Exercise:
Problem:
A certain brand of refrigerator has an automatic ice cube maker that
makes 336 ice cubes in one day. If the ice machine makes ice cubes at a constant rate, how many ice cubes does it make each hour?
Solution:
14 cubes per hour Exercise: Problem: A beer manufacturer bottles 52,380 ounces of beer each hour. If each bottle contains the same number of ounces of beer, and the
manufacturer fills 4,365 bottles per hour, how many ounces of beer does each bottle contain?
Exercise: Problem:
A computer program consists of 68,112 bits. 68,112 bits equals 8,514 bytes. How many bits in one byte?
Solution:
8 bits in each byte Exercise: Problem: A 26-story building in San Francisco has a total of 416 offices. If each
floor has the same number of offices, how many floors does this building have?
Exercise: Problem: A college has 67 classrooms and a total of 2,546 desks. How many
desks are in each classroom if each classroom has the same number of desks?
Solution:
38
Exercises for Review
Exercise:
Problem: ({link]) What is the value of 4 in the number 124,621?
Exercise:
Problem: ({link]) Round 604,092 to the nearest hundred thousand.
Solution: 600,000
Exercise:
Problem: ({link]) What whole number is the additive identity?
Exercise:
Problem: ({link]) Find the product. 6,256 x 100. Solution:
625,600
Exercise:
Problem: ({link]) Find the quotient. 0 + 11.
Some Interesting Facts about Division
This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses interesting facts about diving whole numbers. By the end of the module students should be able to recognize a whole number that is divisible by 2, 3, 4, 5, 6, 8, 9, or 10.
Section Overview
e Division by 2, 3, 4, and 5 e Division by 6, 8, 9, and 10
Quite often, we are able to determine if a whole number is divisible by
another whole number just by observing some simple facts about the number. Some of these facts are listed in this section.
Division by 2, 3, 4, and 5
Division by 2 A whole number is divisible by 2 if its last digit is 0, 2, 4, 6, or 8.
The numbers 80, 112, 64, 326, and 1,008 are all divisible by 2 since the last digit of each is 0, 2, 4, 6, or 8, respectively.
The numbers 85 and 731 are not divisible by 2.
Division by 3 A whole number is divisible by 3 if the sum of its digits is divisible by 3.
The number 432 is divisible by 3 since 4 + 3 + 2 = 9 and 9 is divisible by 2
432 +3 = 144
The number 25 is not divisible by 3 since 2 + 5 = 7, and 7 is not divisible by 3:
Division by 4
A whole number is divisible by 4 if its last two digits form a number that is divisible by 4.
The number 31,048 is divisible by 4 since the last two digits, 4 and 8, form a number, 48, that is divisible by 4.
31048 + 4 = 7262 The number 137 is not divisible by 4 since 37 is not divisible by 4.
Division by 5 A whole number is divisible by 5 if its /ast digit is 0 or 5.
Sample Set A
Example: The numbers 65, 110, 8,030, and 16,955 are each divisible by 5 since the last digit of each is 0 or 5.
Practice Set A
State which of the following whole numbers are divisible by 2, 3, 4, or 5. A number may be divisible by more than one number. Exercise:
Problem: 26
Solution:
2
Exercise:
Problem: 81
Solution: 3 Exercise:
Problem: 51
Solution: 3 Exercise:
Problem: 385
Solution: 5 Exercise:
Problem: 6,112
Solution: Ded Exercise:
Problem: 470
Solution:
25D
Exercise:
Problem: 113,154 Solution:
2,0
Division by 6, 8, 9, 10
Division by 6 A number is divisible by 6 if it is divisible by both 2 and 3.
The number 234 is divisible by 2 since its last digit is 4. It is also divisible by 3 since 2+ 3+ 4 = 9 and 9 is divisible by 3. Therefore, 234 is divisible by 6.
The number 6,532 is not divisible by 6. Although its last digit is 2, making it divisible by 2, the sum of its digits,6 + 5+ 3+ 2 = 16, and 16 is not divisible by 3.
Division by 8
A whole number is divisible by 8 if its last three digits form a number that is divisible by 8.
The number 4,000 is divisible by 8 since 000 is divisible by 8.
The number 13,128 is divisible by 8 since 128 is divisible by 8.
The number 1,170 is not divisible by 8 since 170 is not divisible by 8.
Division by 9
A whole number is divisible by 9 if the sum of its digits is divisible by 9. The number 702 is divisible by 9 since 7 + 0 + 2 is divisible by 9.
The number 6588 is divisible by 9 since 6 + 5 + 8 + 8 = 27 is divisible by 9.
The number 14,123 is not divisible by 9 since 1+4+1+2+3=11is not divisible by 9.
Division by 10
A Whole number is divisible by 10 if its last digit is 0.
Sample Set B
Example: The numbers 30, 170, 16,240, and 865,000 are all divisible by 10.
Practice Set B State which of the following whole numbers are divisible 6, 8, 9, or 10.
Some numbers may be divisible by more than one number. Exercise:
Problem: 900
Solution:
6, 9, 10
Exercise:
Problem: 6,402
Solution:
6
Exercise:
Problem: 6,660
Solution:
6, 9, 10 Exercise:
Problem: 55,116
Solution:
G39
Exercises
For the following 30 problems, specify if the whole number is divisible by 2, 3, 4, 5, 6, 8, 9, or 10. Write "none" if the number is not divisible by any digit other than 1. Some numbers may be divisible by more than one number.
Exercise:
Problem: 48 Solution: 23:4. 6.8
Exercise:
Problem: 85 Exercise:
Problem: 30
Solution:
2365). 0..10)
Exercise:
Problem:
Exercise:
Problem:
Solution: 2
Exercise:
Problem:
Exercise:
Problem:
Solution: 2/4
Exercise:
Problem:
Exercise:
Problem:
Solution:
3
Exercise:
Problem:
83
98
972
892
676
903
800
Exercise:
Problem: 223
Solution:
none
Exercise:
Problem: 836
Exercise:
Problem: 665
Solution: 5
Exercise:
Problem: 4,381
Exercise:
Problem: 2,195 Solution:
5
Exercise:
Problem: 2,544
Exercise:
Problem: 5,172
Solution:
2,3, 4,6
Exercise:
Problem: 1,307
Exercise:
Problem: 1,050
Solution:
Zoos Uy LO
Exercise:
Problem: 3,898
Exercise:
Problem: 1,621
Solution:
none
Exercise:
Problem: 27,808
Exercise:
Problem: 45,764
Solution:
2,4
Exercise:
Problem: 49,198
Exercise:
Problem: 296,122
Solution: 2
Exercise:
Problem: 178,656
Exercise:
Problem: 5,102,417
Solution: none
Exercise:
Problem: 16,990,792
Exercise:
Problem: 620,157,659 Solution:
none
Exercise:
Problem: 457,687,705
Exercises for Review
Exercise:
Problem: ({link]) In the number 412, how many tens are there?
Solution: 1
Exercise:
Problem: ({link]) Subtract 613 from 810.
Exercise:
Problem: ([link]) Add 35, 16, and 7 in two different ways.
Solution:
(35 +16) +7=514+7=58 35 + (16 +7) = 35 +23 =58
Exercise:
Problem: ({link]) Find the quotient 35 + 0, if it exists.
Exercise:
Problem: ({link]) Find the quotient. 3654 + 42.
Solution:
87
Properties of Multiplication
This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses properties of multiplication of whole numbers. By the end of the module students should be able to understand and appreciate the commutative and associative properties of multiplication and understand why 1 is the multiplicative identity.
Section Overview e The Commutative Property of Multiplication e The Associative Property of Multiplication e The Multiplicative Identity
We will now examine three simple but very important properties of multiplication.
The Commutative Property of Multiplication
Commutative Property of Multiplication The product of two whole numbers is the same regardless of the order of the factors.
Sample Set A
Example: Multiply the two whole numbers.
6 hal 6-7 = 42
7:6= 42 The numbers 6 and 7 can be multiplied in any order. Regardless of the order they are multiplied, the product is 42.
Practice Set A
Use the commutative property of multiplication to find the products in two ways. Exercise:
Problem:
Solution:
15-6 = 90 and 6-15 = 90 Exercise:
Problem:
432 428
Solution:
432 - 428 = 184,896 and 428 - 432 = 184,896
The Associative Property of Multiplication
Associative Property of Multiplication
If three whole numbers are multiplied, the product will be the same if the first two are multiplied first and then that product is multiplied by the third, or if the second two are multiplied first and that product is multiplied by the first. Note that the order of the factors is maintained.
It is a common mathematical practice to use parentheses to show which pair of numbers is to be combined first.
Sample Set B
Example: Multiply the whole numbers.
8 3 14
(8-3)-14= 24-14 = 336 8- (3-14) = 8-42 = 336
Practice Set B
Use the associative property of multiplication to find the products in two ways. Exercise:
Problem:
Solution:
168 Exercise: Problem: 73
18 126
Solution:
165,564 The Multiplicative Identity
The Multiplicative Identity is 1 The whole number 1 is called the multiplicative identity, since any whole number multiplied by 1 is not changed.
Sample Set C
Example: Multiply the whole numbers.
12 1 Who I= IH Iho WA = 1
Practice Set C
Multiply the whole numbers. Exercise:
Problem:
843
Solution:
843
Exercises
For the following problems, multiply the numbers. Exercise:
Problem:
Solution:
234 Exercise:
Problem:
18 41
Exercise:
Problem:
Solution:
4,032 Exercise:
Problem:
132
Exercise:
Problem:
1000 326
Solution:
326,000 Exercise:
Problem:
1400
Exercise:
Problem:
Solution:
252 Exercise: Problem:
40 16
Exercise:
Problem:
Solution:
21,340 Exercise:
Problem:
110 85 0
Exercise:
Problem:
462
18
Solution:
8,316 Exercise:
Problem:
101
For the following 4 problems, show that the quantities yield the same products by performing the multiplications. Exercise:
Problem: (4 - 8) - 2 and 4 - (8 - 2)
Solution: 32-2=—64=4-16
Exercise:
Problem: (100 - 62) - 4 and 100 - (62 - 4)
Exercise:
Problem: 23 - (11 - 106) and (23 - 11) - 106
Solution: 23 - 1,166 = 26,818 = 253 - 106
Exercise:
Problem: 1 - (5-2) and (1-5) - 2 Exercise:
Problem:
The fact that (a first number - a second number) - a third number = a first number - (a second number - a third num is an example of the property of multiplication.
Solution:
associative Exercise:
Problem:
The fact that 1 - any number = that particular numberis an example of the property of multiplication.
Exercise:
Problem: Use the numbers 7 and 9 to illustrate the commutative property of multiplication.
Solution:
7-9=63=9-7
Exercise:
Problem: Use the numbers 6, 4, and 7 to illustrate the associative property of multiplication.
Exercises for Review
Exercise:
Problem: ({link]) In the number 84,526,098,441, how many millions are there?
Solution:
6
Exercise: 85 Problem: ({link]) Replace the letter m with the whole number that makes the addition true. + m™ 97
Exercise:
Problem: ((link]) Use the numbers 4 and 15 to illustrate the commutative property of addition.
Solution:
4+15=19
15+4=19 Exercise:
Problem: ((link]) Find the product. 8,000,000 x 1,000.
Exercise:
Problem: ((link]) Specify which of the digits 2, 3, 4, 5, 6, 8,10 are divisors of the number 2,244. Solution:
2,3,4,6
Summary of Key Concepts
This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module summarizes the concepts discussed in the chapter "Multiplication and Division of Whole Numbers."
Summary of Key Concepts
Multiplication ({link]) Multiplication is a description of repeated addition. (es ont Mes ht eee ad 7 appears 4 times
This expression is described by writing 4 X 7.
Multiplicand/Multiplier/Product ({link])
In a multiplication of whole numbers, the repeated addend is called the multiplicand, and the number that records the number of times the multiplicand is used is the multiplier. The result of the multiplication is the product.
Factors ({link]) In a multiplication, the numbers being multiplied are also called factors. Thus, the multiplicand and the multiplier can be called factors.
Division ((link]) Division is a description of repeated subtraction.
Dividend/Divisor/Quotient ({link]) In a division, the number divided into is called the dividend, and the number dividing into the dividend is called the divisor. The result of the division is called the quotient.
quotient
divisor ) dividend
Division into Zero ([link]) Zero divided by any nonzero whole number is zero.
Division by Zero ([link])
Division by zero does not name a whole number. It is, therefore, undefined. The quotient 4 is indeterminant.
Division by 2, 3, 4, 5, 6, 8, 9, 10 ([link]) Division by the whole numbers 2, 3, 4, 5, 6, 8, 9, and 10 can be determined by noting some certain properties of the particular whole number.
Commutative Property of Multiplication ({link]) The product of two whole numbers is the same regardless of the order of the factors. 3x 5=5 x3
Associative Property of Multiplication ({link])
If three whole numbers are to be multiplied, the product will be the same if the first two are multiplied first and then that product is multiplied by the third, or if the second two are multiplied first and then that product is multiplied by the first.
(3x5) x2=3 x (5 2}
Note that the order of the factors is maintained.
Multiplicative Identity ([link])
The whole number 1 is called the multiplicative identity since any whole number multiplied by 1 is not changed.
4x1=4
1x4=4
Exercise Supplement
This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module is an exercise supplement for the chapter "Multiplication and Division of Whole Numbers" and contains many exercise problems. Odd problems are accompanied by solutions.
Exercise Supplement
Multiplication of Whole Numbers ((link])
Exercise:
Problem:
In the multiplication 5 x 9 = 45, 5 and 9 are called and 45 is called the .
Solution:
factors; product Exercise:
Problem:
In the multiplication 4 x 8 = 32, 4 and 8 are called and 32 is called the .
Concepts of Division of Whole Numbers ([link])
Exercise:
Problem: In the division 24 ~ 6 = 4, 6 is called the , and 4 is called the .
Solution:
divisor; quotient Exercise:
Problem:
In the division 36 ~ 2 = 18, 2 is called the , and 18 is called the .
Some Interesting Facts about Division ([link])
Exercise:
Problem: A number is divisible by 2 only if its last digit is . Solution:
an even digit (0, 2, 4, 6, or 8) Exercise:
Problem:
A number is divisible by 3 only if of its digits is divisible by 3. Exercise:
Problem:
A number is divisible by 4 only if the rightmost two digits form a number that is .
Solution:
divisible by 4
Multiplication and Division of Whole Numbers ((link],[link])
Find each product or quotient.
Exercise:
24
Problem:
Exercise:
x3
14
Problem:
x 8
Solution:
112
Exercise:
Problem
Exercise:
Problem
:21+7
230+5
Solution:
7
Exercise:
36
Problem:
Exercise:
| DD
87
Problem:
" x35
Solution:
3,045 Exercise: 117 Problem: x42 Exercise:
Problem: 208 ~ 52
Solution:
4
Exercise:
521 Problem:
Exercise:
Problem:
Solution:
15,075
Exercise:
Problem: 1338 ~ 446
Exercise:
Problem: 2814 — 201
Solution:
14
Exercise:
Problem:
Exercise:
5821 x8
6016
Problem:
ya |
Solution:
42,112
Exercise:
Problem
Exercise:
Problem
> 576 + 24
: 3969 + 63
Solution:
63
Exercise:
5482
Problem:
Exercise:
x $22
9104 Problem:
x 115 Solution: 1,046,960 Exercise: 6102 Problem: x 1000 Exercise: 10101 Problem: x 10000 Solution: 101,010,000 Exercise:
Problem: 162,006 ~ 31
Exercise:
Problem: 0 — 25 Solution:
0
Exercise:
Problem: 25 — 0
Exercise:
Problem: 4280 ~ 10
Solution:
428
Exercise:
Problem: 2126000 — 100
Exercise:
Problem: 84 — 15
Solution:
5 remainder 9
Exercise:
Problem: 126 ~ 4
Exercise:
Problem: 424 — 0
Solution:
not defined
Exercise:
Problem: 1198 ~ 46
Exercise:
Problem: 995 =~ 31
Solution:
32 remainder 3
Exercise:
Problem: 0 — 18
Exercise: 2162 Problem: x 1421 Solution: 3,072,202 Exercise:
Problem: 0 x 0
Exercise:
Problem: 5 x 0
Solution:
0
Exercise:
Problem: 64 x 1
Exercise:
Problem: 1 x 0
Solution:
0
Exercise:
Problem: 0 — 3
Exercise:
Problem: 14 — 0
Solution: not defined
Exercise:
Problem: 35 — 1
Exercise:
Problem: 1 — 1
Solution:
1
Properties of Multiplication ({link])
Exercise:
Problem:
Use the commutative property of multiplication to rewrite 36 x 128.
Exercise:
Problem: Use the commutative property of multiplication to rewrite 114 x 226.
Solution:
226-114 Exercise:
Problem:
Use the associative property of multiplication to rewrite (5 - 4) - 8. Exercise:
Problem: Use the associative property of multiplication to rewrite 16 - (14 - 0).
Solution:
(16-14) -0
Multiplication and Division of Whole Numbers ((link],[link])
Exercise: Problem: A computer store is selling diskettes for $4 each. At this price, how much would 15 diskettes cost? Exercise: Problem:
Light travels 186,000 miles in one second. How far does light travel in 23 seconds?
Solution:
4,278,000 Exercise:
Problem:
A dinner bill for eight people comes to exactly $112. How much should each person pay if they all agree to split the bill equally?
Exercise:
Problem:
Each of the 33 students in a math class buys a textbook. If the bookstore sells $1089 worth of books, what is the price of each book?
Solution:
$33
Proficiency Exam
This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module is a proficiency exam to the chapter "Multiplication and Division of Whole Numbers." Each problem is accompanied with a reference link pointing back to the module that discusses the type of problem demonstrated in the question. The problems in this exam are accompanied by solutions.
Proficiency Exam
Exercise:
Problem:
({link]) In the multiplication of 8 x 7 = 56, what are the names given to the 8 and 7 and the 56?
Solution:
8 and 7 are factors; 56 is the product Exercise:
Problem: ({link]) Multiplication is a description of what repeated process? Solution:
Addition Exercise:
Problem:
({link]) In the division 12 + 3 = 4, what are the names given to the 3 and the 4?
Solution:
3 is the divisor; 4 is the quotient Exercise:
Problem:
({link]) Name the digits that a number must end in to be divisible by 2.
Solution:
0, 2, 4, 6, or 8 Exercise:
Problem:
({link]) Name the property of multiplication that states that the order of the factors in a multiplication can be changed without changing the product.
Solution:
commutative
Exercise:
Problem: ({link]) Which number is called the multiplicative identity? Solution:
1
For problems 7-17, find the product or quotient. Exercise:
Problem: ({link]) 14 x 6
Solution:
84
Exercise:
Problem: ({link]) 37 x 0 Solution:
0
Exercise:
Problem: ({link]) 352 x 1000 Solution:
352,000
Exercise:
Problem: (({link]) 5986 x 70 Solution:
419,020
Exercise:
Problem: ([link]) 12 x 12 Solution:
252
Exercise:
Problem: ((link]) 856 + 0
Solution:
not defined
Exercise:
Problem: ({link]) 0 + 8 Solution:
0
Exercise:
Problem: ({link]) 136 + 8 Solution:
17
Exercise:
Problem: ([link]) 432 + 24 Solution:
18
Exercise:
Problem: ([link]) 5286 + 37 Solution:
142 remainder 32
Exercise:
Problem: (({link]) 211 x 1
Solution:
211
For problems 18-20, use the numbers 216, 1,005, and 640. Exercise:
Problem: ({link]) Which numbers are divisible by 3? Solution: 216; 1,005
Exercise: Problem: ({link]) Which number is divisible by 4? Solution:
216; 640
Exercise: Problem: ({link]) Which number(s) is divisible by 5?
Solution:
1,005; 640
Objectives
This module contains the learning objectives for the chapter "Exponents, Roots, and Factorizations of Whole Numbers" from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, jr.
After completing this chapter, you should Exponents and Roots ({link])
e understand and be able to read exponential notation e understand the concept of root and be able to read root notation e be able to use a calculator having the y* key to determine a root
Grouping Symbols and the Order of Operations ({link])
e understand the use of grouping symbols e understand and be able to use the order of operations e use the calculator to determine the value of a numerical expression
Prime Factorization of Natural Numbers ({link])
e be able to determine the factors of a whole number
e be able to distinguish between prime and composite numbers e be familiar with the fundamental principle of arithmetic
e be able to find the prime factorization of a whole number
The Greatest Common Factor (({link])
e be able to find the greatest common factor of two or more whole numbers
The Least Common Multiple ({link])
e be able to find the least common multiple of two or more whole numbers
Exponents and Roots
This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses exponents and roots. By the end of the module students should be able to understand and be able to read exponential notation, understand the concept of root and be able to read root notation, and use a calculator having the y* key to determine a root.
Section Overview
e Exponential Notation
e Reading Exponential Notation e Roots
e Reading Root Notation
e Calculators
Exponential Notation
Exponential Notation We have noted that multiplication is a description of repeated addition. Exponential notation is a description of repeated multiplication.
Suppose we have the repeated multiplication 8-8-8-8-8
Exponent
The factor 8 is repeated 5 times. Exponential notation uses a superscript for the number of times the factor is repeated. The superscript is placed on the repeated factor, 8°, in this case. The superscript is called an exponent.
The Function of an Exponent
An exponent records the number of identical factors that are repeated in a multiplication.
Sample Set A
Write the following multiplication using exponents.
Example: 3-3. Since the factor 3 appears 2 times, we record this as 32
Example:
62 - 62-62-62 - 62-62-62 - 62 - 62. Since the factor 62 appears 9 times, we record this as
62°
Expand (write without exponents) each number.
Example:
12, ihe exponent 4 is recording 4 factors of 12 in a multiplication. Thus, Telesis PS
Example:
706°. The exponent 3 is recording 3 factors of 706 in a multiplication. Thus, 706° = 706 - 706 - 706
Practice Set A
Write the following using exponents. Exercise:
Problem: 37 - 37 Solution:
a7?
Exercise:
Problem: 16-16-16-16-16 Solution:
16°
Exercise: Problem: 9-9-9-9-9-9-9-9-9-9 Solution: 910
Write each number without exponents. Exercise:
Problem: 85°
Solution:
85-85-85
Exercise:
Problem: 4’
Solution:
4-4-4-4-4.-4.4
Exercise:
Problem: 1,739”
Solution:
1,739 - 1,739
Reading Exponential Notation In a number such as 8°,
Base 8 is called the base.
Exponent, Power 5 is called the exponent, or power. 8° is read as "eight to the fifth power," or more simply as "eight to the fifth," or "the fifth power of eight."
Squared When a whole number is raised to the second power, it is said to be squared. The number 5? can be read as
5 to the second power, or 5 to the second, or 5 squared.
Cubed When a whole number is raised to the third power, it is said to be cubed. The number 5? can be read as
5 to the third power, or 5 to the third, or 5 cubed.
When a whole number is raised to the power of 4 or higher, we simply say that that number is raised to that particular power. The number 58 can be read as
5 to the eighth power, or just 5 to the eighth.
Roots
In the English language, the word "root" can mean a source of something. In mathematical terms, the word "root" is used to indicate that one number is the source of another number through repeated multiplication.
Square Root
We know that 49 = 7”, that is, 49 = 7 - 7. Through repeated multiplication, 7 is the source of 49. Thus, 7 is a root of 49. Since two 7's must be multiplied together to produce 49, the 7 is called the second or square root of 49.
Cube Root
We know that 8 = 2°, that is, 8 = 2-2-2. Through repeated multiplication, 2 is the source of 8. Thus, 2 is a root of 8. Since three 2's must be multiplied together to produce 8, 2 is called the third or cube root of 8.
We can continue this way to see such roots as fourth roots, fifth roots, sixth roots, and so on.
Reading Root Notation
There is a symbol used to indicate roots of a number. It is called the radical sign an
The Radical Sign of The symbol is called a radical sign and indicates the nth root of a number.
We discuss particular roots using the radical sign as follows:
Square Root
number indicates the square root of the number under the radical sign. It is customary to drop the 2 in the radical sign when discussing square roots. The symbol ,/— is understood to be the square root radical sign.
/49 =7 since 7-7 = 72 = 49
Cube Root number indicates the cube root of the number under the radical sign.
4/8 = 2since2-2-2—223 —8
Fourth Root Vv number indicates the fourth root of the number under the radical sign.
V7 81 = 3since3-3-3-3=34 = 81 In an expression such as 1/32
Radical Sign /__ is called the radical sign.
Index 5 is called the index. (The index describes the indicated root.)
Radicand 32 is called the radicand.
Radical ¥ 32 is called a radical (or radical expression).
Sample Set B
Find each root.
Example:
/25 To determine the square root of 25, we ask, "What whole number squared equals 25?" From our experience with multiplication, we know this number to be 5. Thus,
/25=5 Check: 5-5 = 5% = 25
Example: 4/32 To determine the fifth root of 32, we ask, "What whole number raised to the fifth power equals 32°?" This number is 2.
4/32 = 2
CHECK 27 e220) ere oP
Practice Set B
Find the following roots using only a knowledge of multiplication. Exercise:
Problem: / 64
Solution:
8
Exercise:
Problem: / 100
Solution:
10
Exercise:
Problem: </ 64
Solution:
4
Exercise:
Problem: </ 64
Solution:
2
Calculators
Calculators with the ,/z, y*, and 1/z keys can be used to find or approximate roots.
Sample Set C
Example: —_ Use the calculator to find A {aah
Display Reads
Type 121 121 Press fx 11 Example: Find V/2187. Display Reads Type 2187 2187 Press y” 2187 Type i i Press Le 14285714 Press = 3
¥/ 2187 = 3 (Which means that 3’ = 2187 .)
Practice Set C
Use a calculator to find the following roots. Exercise:
Problem: </ 729
Solution: 9
Exercise:
Problem: */8503056
Solution: 54
Exercise:
Problem: \/53361
Solution: Zak Exercise: Problem: \/16777216
Solution:
4
Exercises
For the following problems, write the expressions using exponential notation. Exercise:
Problem: 4 - 4
Solution: 42
Exercise:
Problem: 12 - 12
Exercise: Problem: 9-9-9-9 Solution:
o4
Exercise:
Problem: 10-10-10-10-10-10
Exercise:
Problem: 826 - 826 - 826 Solution:
826°
Exercise:
Problem: 3,021 - 3,021 - 3,021 - 3,021 - 3,021
Exercise:
Problem: 6-6-:::: 6 85 factors of 6
Solution: 685
Exercise:
Problem: i 2 112 factors of 2
Exercise:
Problem: 1-1---- il 3,008 factors of 1
Solution:
1 3008
For the following problems, expand the terms. (Do not find the actual value.) Exercise:
Problem: 52 Exercise: Problem: 7*
Solution:
Clete
Exercise:
Problem: 157 Exercise: Problem: 117°
Solution: 117-117-117-117-117
Exercise:
Problem: 61° Exercise: Problem: 30°
Solution:
30 - 30
For the following problems, determine the value of each of the powers. Use a calculator to check each result. Exercise:
Problem: 32 Exercise: Problem: 42
Solution:
4-4=16
Exercise:
Problem: 12 Exercise: Problem: 107 Solution: 10-10 = 100 Exercise: Problem: 117 Exercise:
Problem: 197
Solution:
12-12 = 144
Exercise:
Problem: 137
Exercise:
Problem: 157
Solution:
15-15 = 225
Exercise:
Problem: 14
Exercise:
Problem: 34
Solution: 3°3-3-3=—81
Exercise:
Problem: 7° Exercise: Problem: 10°
Solution: 10-10-10 = 1,000
Exercise:
Problem: 1007 Exercise: Problem: 8°
Solution: 8-8-8 =—512
Exercise:
Problem: 5°
Exercise:
Problem: 92
Solution: 9-9-9 = 729
Exercise:
Problem: 62 Exercise:
Problem: 7!
Solution: 7i=7
Exercise:
Problem: 12°
Exercise:
Problem: 2°
Solution:
22024 222° 22 = 128
Exercise:
Problem: 0°
Exercise:
Problem: 84
Solution:
8-8-8-8 = 4,096
Exercise:
Problem: 5°
Exercise:
Problem: 6°
Solution:
6:6:-6-6-6-6-6-6-6 = 10,077,696
Exercise:
Problem: 25°
Exercise:
Problem: 427
Solution:
42-42 = 1,764
Exercise:
Problem: 31°
Exercise:
Problem: 15°
Solution: