Solution Manual for Mathematics: Its Power and Utility, 10th Edition
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Student's Survival and Solutions Manual for
MATHEMATICS
Its Power and Utility
TENTH EDITION
Karl J. Smith
MATHEMATICS
Its Power and Utility
TENTH EDITION
Karl J. Smith
CONTENTS
Page iii
CONTENTS
PART I:
FOUNDATIONS: THE POWER OF MATHEMATICS
1 Arithmetic, Calculators, and Problem Solving ......................... 1
2 Sets of Numbers ......................................................................... 35
3 Introduction to Algebra .............................................................. 63
4 Percents and Problem Solving ................................................... 91
5 Introduction to Geometry ......................................................... 113
6 Measurement and Problem Solving ......................................... 131
PART II:
APPLICATIONS: THE UTILITY OF MATHEMATICS
7 Applications of Percent ............................................................. 157
8 Sets and Logic ........................................................................... 195
9 Probability ................................................................................. 229
10 Statistics..................................................................................... 251
11 Graphs ....................................................................................... 279
Page iii
CONTENTS
PART I:
FOUNDATIONS: THE POWER OF MATHEMATICS
1 Arithmetic, Calculators, and Problem Solving ......................... 1
2 Sets of Numbers ......................................................................... 35
3 Introduction to Algebra .............................................................. 63
4 Percents and Problem Solving ................................................... 91
5 Introduction to Geometry ......................................................... 113
6 Measurement and Problem Solving ......................................... 131
PART II:
APPLICATIONS: THE UTILITY OF MATHEMATICS
7 Applications of Percent ............................................................. 157
8 Sets and Logic ........................................................................... 195
9 Probability ................................................................................. 229
10 Statistics..................................................................................... 251
11 Graphs ....................................................................................... 279
PREFACE
Page v
PREFACE
This manual was designed to help you bridge the gap between the textbook and a
working knowledge of mathematics. It has been said that “Mathematics is not a
Spectator Sport" and this means you cannot learn mathematics by simply attending
class, but instead you must build a body of information that will enable you to do
problem solving in the real world. I decided to entitle this supplement, Student's
Survival and Solutions Manual because I want it to be more than a Student
Solutions Manual. Thirty years of teaching experience have given me the ability to
anticipate the types of errors and difficulties you may have while taking this course.
Here I will show you some of the steps that are left out of the text, and most all of
these steps in the included problems.
There are several things you must do if you wish to be successful with
mathematics:
Attend every class.
Read the book.
Regardless of how clear and lucid your professor's lecture on a
particular topic may be, do not attempt to do the problems
without first reading the text and studying the examples. It will
serve to reinforce and clarify the concepts and procedures.
Problems, problems...Problems, Problems,
You must work problems every day; work the assigned problems;
work problems. Look over the entireESSENTIAL IDEA
problem set (even those problems which are not assigned).
Ask questions when you are stuck (and you will get stuck that
is part of the process).
Keep asking questions until receiving answers that are
understandable to you.
Today's calculators are good at obtaining answers, and if all that
is desired is an answer, then you have relegated yourself to the
level of a machine. Do not work problems to obtain answers. It
is the that are important. Even though a solutionsconcepts
manual is basically a “how to" document, always ask awhy
particular approach was used, and understand the concept the
problem is illustrating.
Page v
PREFACE
This manual was designed to help you bridge the gap between the textbook and a
working knowledge of mathematics. It has been said that “Mathematics is not a
Spectator Sport" and this means you cannot learn mathematics by simply attending
class, but instead you must build a body of information that will enable you to do
problem solving in the real world. I decided to entitle this supplement, Student's
Survival and Solutions Manual because I want it to be more than a Student
Solutions Manual. Thirty years of teaching experience have given me the ability to
anticipate the types of errors and difficulties you may have while taking this course.
Here I will show you some of the steps that are left out of the text, and most all of
these steps in the included problems.
There are several things you must do if you wish to be successful with
mathematics:
Attend every class.
Read the book.
Regardless of how clear and lucid your professor's lecture on a
particular topic may be, do not attempt to do the problems
without first reading the text and studying the examples. It will
serve to reinforce and clarify the concepts and procedures.
Problems, problems...Problems, Problems,
You must work problems every day; work the assigned problems;
work problems. Look over the entireESSENTIAL IDEA
problem set (even those problems which are not assigned).
Ask questions when you are stuck (and you will get stuck that
is part of the process).
Keep asking questions until receiving answers that are
understandable to you.
Today's calculators are good at obtaining answers, and if all that
is desired is an answer, then you have relegated yourself to the
level of a machine. Do not work problems to obtain answers. It
is the that are important. Even though a solutionsconcepts
manual is basically a “how to" document, always ask awhy
particular approach was used, and understand the concept the
problem is illustrating.
Loading page 4...
PREFACE STUDENT'S SURVIVAL AND SOLUTION MANUAL FOR Mathematics Its Power and Utility, Ninth Edition
Page vi
For the most part, this manual includes complete solutions to the odd-numbered
problems. Notice that the complete solutions for the self-tests are given at the end
of the review sections in the textbook.
Page vi
For the most part, this manual includes complete solutions to the odd-numbered
problems. Notice that the complete solutions for the self-tests are given at the end
of the review sections in the textbook.
Loading page 5...
STUDENT'S SURVIVAL AND SOLUTION MANUAL FOR SECTION 1.1Mathematics Its Power and Utility, Ninth Edition
Page 1
CHAPTER 1
Arithmetic, Calculators, and Problem Solving
SURVIVAL HINT: If your instructor does not begin with Chapter 1, you might wish to take some time
looking over this chapter anyway. Please read page 1 of the textbook because it tells you what this
part of the book is all about. As you look through this first chapter you will notice that we cover
fractions, decimals, and numeration systems. We also introduce you to using a calculator. If you have
a manual for your calculator, do not attempt to read it; part of what this book will do is to teach you how
to use a calculator to do the type of problems you are considering at the time. You may remember
some things about fractions, decimals, and numeration systems, or you may have forgotten more than
you remember. That is ok; --- it is our job.... the book and your instructor have been assigned therelax
task of making sure you understand the topic.
Find out from your instructor what is expected of you. You will probably need a copy of this
textbook, pencils and paper, a straight-edge, and a calculator. (By the way, the cover for your
calculator makes a good straight-edge.) Put your name and phone number inside the cover of your
calculator, so if you lose it, it is, at least, possible that it be returned.
As you begin on your mathematics journey, Bon Voyage!
1.1 Math Anxiety, page 4
SURVIVAL HINT: There are, of course, no right or wrong answers to Problems 1-25. In the book from
which these myths were obtained there is a whole chapter justifying that these are( )Mind Over Math
myths, and the authors cite specific studies to support their conclusions. As you do these problems,
remember that a is a popular belief or tradition that has grown up around something which is anmyth
unfounded or false notion. Don't be afraid to express your own ideas when answering these questions.
Problem Set page 91.1,
1. Write a couple of paragraphs about your math history. You are writing this for yourself,
not for your teacher.
2. Read the “Math Anxiety Bill of Rights” on page 6 and then make up your own list.
3. When you see this stop signal, stop and study the material because it will be used later.
4. When you see this caution sign, make a special note of the material next to the caution
sign, because it will be used throughout the rest of the book.
5. The yield sign means that you will need to remember the result, but it is not necessary for
you to understand the derivation.
6. The bump sign means that some unexpected or difficult material follows, and you will
need to slow down to understand the discussion.
7-36. We will generally not give answers to those questions marked IN YOUR OWN WORDS.
This problems do not have “right” or “wrong” answers, but rather are included to help you
advance your success in this course. Do you best to answer the question.
Page 1
CHAPTER 1
Arithmetic, Calculators, and Problem Solving
SURVIVAL HINT: If your instructor does not begin with Chapter 1, you might wish to take some time
looking over this chapter anyway. Please read page 1 of the textbook because it tells you what this
part of the book is all about. As you look through this first chapter you will notice that we cover
fractions, decimals, and numeration systems. We also introduce you to using a calculator. If you have
a manual for your calculator, do not attempt to read it; part of what this book will do is to teach you how
to use a calculator to do the type of problems you are considering at the time. You may remember
some things about fractions, decimals, and numeration systems, or you may have forgotten more than
you remember. That is ok; --- it is our job.... the book and your instructor have been assigned therelax
task of making sure you understand the topic.
Find out from your instructor what is expected of you. You will probably need a copy of this
textbook, pencils and paper, a straight-edge, and a calculator. (By the way, the cover for your
calculator makes a good straight-edge.) Put your name and phone number inside the cover of your
calculator, so if you lose it, it is, at least, possible that it be returned.
As you begin on your mathematics journey, Bon Voyage!
1.1 Math Anxiety, page 4
SURVIVAL HINT: There are, of course, no right or wrong answers to Problems 1-25. In the book from
which these myths were obtained there is a whole chapter justifying that these are( )Mind Over Math
myths, and the authors cite specific studies to support their conclusions. As you do these problems,
remember that a is a popular belief or tradition that has grown up around something which is anmyth
unfounded or false notion. Don't be afraid to express your own ideas when answering these questions.
Problem Set page 91.1,
1. Write a couple of paragraphs about your math history. You are writing this for yourself,
not for your teacher.
2. Read the “Math Anxiety Bill of Rights” on page 6 and then make up your own list.
3. When you see this stop signal, stop and study the material because it will be used later.
4. When you see this caution sign, make a special note of the material next to the caution
sign, because it will be used throughout the rest of the book.
5. The yield sign means that you will need to remember the result, but it is not necessary for
you to understand the derivation.
6. The bump sign means that some unexpected or difficult material follows, and you will
need to slow down to understand the discussion.
7-36. We will generally not give answers to those questions marked IN YOUR OWN WORDS.
This problems do not have “right” or “wrong” answers, but rather are included to help you
advance your success in this course. Do you best to answer the question.
Loading page 6...
CHAPTER 1 STUDENT'S SURVIVAL AND SOLUTION MANUAL FOR Mathematics Its Power and Utility, Ninth Edition
Page 2
37. This sign means that a stop sign is ahead.
39. This sign means that there is a roundabout ahead.
41. This sign is used to mean a kangaroo crossing.
43. This sign is used to designate a woman's rest room.
45. This sign is used to designate a currency exchange service.
47. This sign is used to designate a restaurant.
49. Choice B, employment
51. Choice A, announcements
53. Choice E, recreation
1.2 Formulating the Problem, page 11
SURVIVAL HINT: It is important that you pay attention to the agreement. Thisorder of operations
procedure is used almost every day in this course. There are four main algebraic processes; these are
simplify evaluate, solve, factor., and In this section, we state, “To a numerical expression meanssimplify
to carry out all the operations, according to the order of operations, and to write the answer as a single
number.”
New Terms Introduced in this Section
Addition Counting number Difference
Distributive property Elementary operations Estimation
Juxtaposition Natural number Product
Quotient Simplify a numerical expression Numerical expression
Sum Translation Whole number
Problem Set page 181.2,
1. a. b.
Display:
3. (1) Parentheses first
(2) Multiplication and division, reading from left to right
(3) Addition and subtraction, reading from left to right
4. Distributive Property: ( + ) = +a b c ab ac
SURVIVAL HINT: Take a look at the essential ideas of this section. There are two:
1. State the order of operations.
2. Explain the distributive property in your own words.
Because these items are essential to your understanding of the material, we give the answer to all of
the essential ideas in this . Your instructor may or may not assign these problems, butSurvival Manual
it would be a good idea to designate a section of your notebook and then put intoESSENTIAL IDEAS
this section of the essential ideas.all
Page 2
37. This sign means that a stop sign is ahead.
39. This sign means that there is a roundabout ahead.
41. This sign is used to mean a kangaroo crossing.
43. This sign is used to designate a woman's rest room.
45. This sign is used to designate a currency exchange service.
47. This sign is used to designate a restaurant.
49. Choice B, employment
51. Choice A, announcements
53. Choice E, recreation
1.2 Formulating the Problem, page 11
SURVIVAL HINT: It is important that you pay attention to the agreement. Thisorder of operations
procedure is used almost every day in this course. There are four main algebraic processes; these are
simplify evaluate, solve, factor., and In this section, we state, “To a numerical expression meanssimplify
to carry out all the operations, according to the order of operations, and to write the answer as a single
number.”
New Terms Introduced in this Section
Addition Counting number Difference
Distributive property Elementary operations Estimation
Juxtaposition Natural number Product
Quotient Simplify a numerical expression Numerical expression
Sum Translation Whole number
Problem Set page 181.2,
1. a. b.
Display:
3. (1) Parentheses first
(2) Multiplication and division, reading from left to right
(3) Addition and subtraction, reading from left to right
4. Distributive Property: ( + ) = +a b c ab ac
SURVIVAL HINT: Take a look at the essential ideas of this section. There are two:
1. State the order of operations.
2. Explain the distributive property in your own words.
Because these items are essential to your understanding of the material, we give the answer to all of
the essential ideas in this . Your instructor may or may not assign these problems, butSurvival Manual
it would be a good idea to designate a section of your notebook and then put intoESSENTIAL IDEAS
this section of the essential ideas.all
Loading page 7...
STUDENT'S SURVIVAL AND SOLUTION MANUAL FOR SECTION 1.2Mathematics Its Power and Utility, Ninth Edition
Page 3
5.
False; the error was to add first.
7. 2 (3 + 4) 2 3 + 2 4 is an example of the distributive property; what is shown here is
simplification, so the statement is wrong.
9. False; it is a whole number.
11. a.
Since the last operation was addition, we call this a .sum
b.
Since the last operation was addition, we call this a .sum
13. a. 12 6 + 3 2 + 3
5
Since the last operation was addition, we call this a .sum
b.
5
Since the last operation was division, we call this a quotient.
15 a.. 15 3 +
Since the last operation was addition, we call this a .sum
b.
3
Since the last operation was subtraction, we call this a .difference
17 a..
32
Since the last operation was addition, we call this a .sum
b. 8 2 8 6 4
56
Since the last operation was addition, we call this a .sum
19 a.. 5 5 5
Since the last operation was addition, we call this a .sum
Page 3
5.
False; the error was to add first.
7. 2 (3 + 4) 2 3 + 2 4 is an example of the distributive property; what is shown here is
simplification, so the statement is wrong.
9. False; it is a whole number.
11. a.
Since the last operation was addition, we call this a .sum
b.
Since the last operation was addition, we call this a .sum
13. a. 12 6 + 3 2 + 3
5
Since the last operation was addition, we call this a .sum
b.
5
Since the last operation was division, we call this a quotient.
15 a.. 15 3 +
Since the last operation was addition, we call this a .sum
b.
3
Since the last operation was subtraction, we call this a .difference
17 a..
32
Since the last operation was addition, we call this a .sum
b. 8 2 8 6 4
56
Since the last operation was addition, we call this a .sum
19 a.. 5 5 5
Since the last operation was addition, we call this a .sum
Loading page 8...
CHAPTER 1 STUDENT'S SURVIVAL AND SOLUTION MANUAL FOR Mathematics Its Power and Utility, Ninth Edition
Page 4
b.
13
Since the last operation was addition, we call this a .sum
21 a.
Since the last operation was addition, we call this a .sum
b.
19
Since the last operation was addition, we call this a .sum
23 a.. 2 18 + 9 3 5 2 36 + 3 10
Since the last operation was subtraction, we call this a .difference
b. 2 (18 + 9) 3 5 2 2 27 3 5 2
54 3
Since the last operation was subtraction, we call this a .difference
25 a. b.. 3 (4 + 8) 3 4 + 3 8 7 (9 + 4) 7 9 + 7 4
27 a.. 4 (300 + 20 + 7) = 4 300 + 4 20 + 4 7
b. 6 (500 + 30 + 3) = 6 500 + 6 30 + 6 3
29 “The sum” means “ ” and “the product” means “ ”: 3 + 2 4. Parentheses are not
necessary because the order of operations requires that the product be done before the addition.
31. “Times” means “ ” and “sum” means “ ”: 10(5 + 6)
The parentheses are used to indicate the order of operations and they are also used to
indicate multiplication.
33. 8 5 + 10; if we wanted 8 (5 + 10) we would need to say “Eight times the quantity
five plus ten.”
35. 8(11 9); the difference requires parentheses for the correct order of operations.
37. Press: :716 Display 261
Since the last operation was subtraction, we call this a .difference
39. Press: : Display 800
Page 4
b.
13
Since the last operation was addition, we call this a .sum
21 a.
Since the last operation was addition, we call this a .sum
b.
19
Since the last operation was addition, we call this a .sum
23 a.. 2 18 + 9 3 5 2 36 + 3 10
Since the last operation was subtraction, we call this a .difference
b. 2 (18 + 9) 3 5 2 2 27 3 5 2
54 3
Since the last operation was subtraction, we call this a .difference
25 a. b.. 3 (4 + 8) 3 4 + 3 8 7 (9 + 4) 7 9 + 7 4
27 a.. 4 (300 + 20 + 7) = 4 300 + 4 20 + 4 7
b. 6 (500 + 30 + 3) = 6 500 + 6 30 + 6 3
29 “The sum” means “ ” and “the product” means “ ”: 3 + 2 4. Parentheses are not
necessary because the order of operations requires that the product be done before the addition.
31. “Times” means “ ” and “sum” means “ ”: 10(5 + 6)
The parentheses are used to indicate the order of operations and they are also used to
indicate multiplication.
33. 8 5 + 10; if we wanted 8 (5 + 10) we would need to say “Eight times the quantity
five plus ten.”
35. 8(11 9); the difference requires parentheses for the correct order of operations.
37. Press: :716 Display 261
Since the last operation was subtraction, we call this a .difference
39. Press: : Display 800
Loading page 9...
STUDENT'S SURVIVAL AND SOLUTION MANUAL FOR SECTION 1.3Mathematics Its Power and Utility, Ninth Edition
Page 5
Since the last operation was subtraction, we call this a .sum
41. Press: : Display 1,080
Since the last operation was subtraction, we call this a .sum
43. Press:
Don't forget the times sign for the indicated multiplication. :Display 1,600
Since the last operation was multiplication, we call this a .product
45. Press: : Display 59
Since the last operation was subtraction, we call this a .difference
47. Press: : Display 2,700
Since the last operation was subtraction, we call this a .difference
49. Press: : Display 285,197
Since the last operation was subtraction, we call this a .sum
51. Press:
: since the last operation was subtraction, we call this a .Display sum2,323;
Estimates for Problems 53-60 may vary.
53. Each day has 24 hours so we estimate
8,000
We estimate this time to be 8,000 hours. The exact answer is 8,640 hours.
55. We estimate
, ,
We estimate the deduction to be $ . The exact answer is $ .
57. We estimate by assuming a 40 hour work week for 50 weeks a year:
,000
36,000
We estimate the annual salary to be $36,000. The exact answer is $37,440.
59. We estimate
00
We estimate we can drive 300 miles. The exact answer is 345 miles.
1.3 Fractions and Decimals, page 19
SURVIVAL HINT: A fraction is an indicated division. For example, is one unit (the proverbial pie)1
2
divided into two parts, so it is This section is devoted to developing your number sense regarding .
fractions. You may or may not have developed this number sense in previous arithmetic classes, but
now is a good time for a refresher.
Page 5
Since the last operation was subtraction, we call this a .sum
41. Press: : Display 1,080
Since the last operation was subtraction, we call this a .sum
43. Press:
Don't forget the times sign for the indicated multiplication. :Display 1,600
Since the last operation was multiplication, we call this a .product
45. Press: : Display 59
Since the last operation was subtraction, we call this a .difference
47. Press: : Display 2,700
Since the last operation was subtraction, we call this a .difference
49. Press: : Display 285,197
Since the last operation was subtraction, we call this a .sum
51. Press:
: since the last operation was subtraction, we call this a .Display sum2,323;
Estimates for Problems 53-60 may vary.
53. Each day has 24 hours so we estimate
8,000
We estimate this time to be 8,000 hours. The exact answer is 8,640 hours.
55. We estimate
, ,
We estimate the deduction to be $ . The exact answer is $ .
57. We estimate by assuming a 40 hour work week for 50 weeks a year:
,000
36,000
We estimate the annual salary to be $36,000. The exact answer is $37,440.
59. We estimate
00
We estimate we can drive 300 miles. The exact answer is 345 miles.
1.3 Fractions and Decimals, page 19
SURVIVAL HINT: A fraction is an indicated division. For example, is one unit (the proverbial pie)1
2
divided into two parts, so it is This section is devoted to developing your number sense regarding .
fractions. You may or may not have developed this number sense in previous arithmetic classes, but
now is a good time for a refresher.
Loading page 10...
CHAPTER 1 STUDENT'S SURVIVAL AND SOLUTION MANUAL FOR Mathematics Its Power and Utility, Ninth Edition
Page 6
New Terms Introduced in this Section
Approximately equal to symbol Column names Common fraction
Decimal Decimal fraction Decimal point
Denominator Division by zero Fraction
Hundred Improper fraction Mixed number
Numerator Place-value names Proper fraction
Remainder Repeating decimal Ten
Rational number Terminating decimal Trailing zeros
Problem Set page 261.3,
3. A fraction is a quotient of a whole number divided by a counting number.
4 a. b. c. d. e.. 1 10 100 1000 10000
The answers are getting larger. Answers vary; it looks like the closer the divisor is to 0 the
larger the answer.
5. The place value names (in decreasing order) are: trillions, hundred billions, ten billions,
billions, hundred millions, ten millions, millions, hundred thousands, ten thousands,
thousands, hundreds, tens, units, decimal point, tenths, hundredths, thousandths, ten-
thousandths, hundred-thousandths, and millionths.
SURVIVAL HINT: Did you remember to look at the Even if they were not assigned,ESSENTIAL IDEAS?
you should add these terms to your notebook:
1. Fraction
2. Common fraction/decimal fraction
3. Division by 0 is not permitted.
4. Place value names
7. False; can't divide by 0.
9. True; we can affix any number of trailing zeros.
11. False; 0.
13. False; by calculator, the answer is appoximately 27.96709753.
15. a b. proper; . proper;8 13 17 21
17 a b. . .4 2
19 a.. = 516 1
3 3 Divide into to obtain , with a remainder of .
b. = 625 1
4 4 Divide into to obtain , with a remainder of .
c. = 14141 1
10 10 Divide into to obtain , with a remainder of .
d. =
3 Divide into to obtain , with a remainder of .
Page 6
New Terms Introduced in this Section
Approximately equal to symbol Column names Common fraction
Decimal Decimal fraction Decimal point
Denominator Division by zero Fraction
Hundred Improper fraction Mixed number
Numerator Place-value names Proper fraction
Remainder Repeating decimal Ten
Rational number Terminating decimal Trailing zeros
Problem Set page 261.3,
3. A fraction is a quotient of a whole number divided by a counting number.
4 a. b. c. d. e.. 1 10 100 1000 10000
The answers are getting larger. Answers vary; it looks like the closer the divisor is to 0 the
larger the answer.
5. The place value names (in decreasing order) are: trillions, hundred billions, ten billions,
billions, hundred millions, ten millions, millions, hundred thousands, ten thousands,
thousands, hundreds, tens, units, decimal point, tenths, hundredths, thousandths, ten-
thousandths, hundred-thousandths, and millionths.
SURVIVAL HINT: Did you remember to look at the Even if they were not assigned,ESSENTIAL IDEAS?
you should add these terms to your notebook:
1. Fraction
2. Common fraction/decimal fraction
3. Division by 0 is not permitted.
4. Place value names
7. False; can't divide by 0.
9. True; we can affix any number of trailing zeros.
11. False; 0.
13. False; by calculator, the answer is appoximately 27.96709753.
15. a b. proper; . proper;8 13 17 21
17 a b. . .4 2
19 a.. = 516 1
3 3 Divide into to obtain , with a remainder of .
b. = 625 1
4 4 Divide into to obtain , with a remainder of .
c. = 14141 1
10 10 Divide into to obtain , with a remainder of .
d. =
3 Divide into to obtain , with a remainder of .
Loading page 11...
STUDENT'S SURVIVAL AND SOLUTION MANUAL FOR SECTION 1.3Mathematics Its Power and Utility, Ninth Edition
Page 7
21 a.. = 1
1 Divide into to obtain , with a remainder of .
b. =
10 10 Divide into to obtain , with a remainder of .
c. 33
1
= Divide into to obtain , with a remainder of .
d. =
Divide into to obtain , with a remainder of .
23 a.. =
Divide into to obtain , with a remainder of .
b. =
Divide into to obtain
c. =18
Divide into to obtain
d. =
Divide into to obtain , with a remainder of .
25 a..
b.
27 a..
b. 5 1 2
29 a. . =
=b. 2
5 5
31 a.. =
b. =
33 a.. 1 =
b. 2 =
1
35 a.. =
b. =
7
SURVIVAL HINT: 36-47In Problems , do not round the decimal representations. For example,
1
3 = 0.3
_
Note the bar over the three; this shows that the three repeats. It is NOT correct to write
1 1
3 3
= 0.33333 or = 0.33
or any other such representation. The three repeats and does not terminate. That is, in doing the long
division, there is never a remainder of zero.
37 a.. = Long division; by calculator: 5
Continue with the division until you see the digits repeat. There is one repeating digit.
b. = Long division; by calculator:
Continue with the division until you see the digits repeat. There is one repeating digit.
Page 7
21 a.. = 1
1 Divide into to obtain , with a remainder of .
b. =
10 10 Divide into to obtain , with a remainder of .
c. 33
1
= Divide into to obtain , with a remainder of .
d. =
Divide into to obtain , with a remainder of .
23 a.. =
Divide into to obtain , with a remainder of .
b. =
Divide into to obtain
c. =18
Divide into to obtain
d. =
Divide into to obtain , with a remainder of .
25 a..
b.
27 a..
b. 5 1 2
29 a. . =
=b. 2
5 5
31 a.. =
b. =
33 a.. 1 =
b. 2 =
1
35 a.. =
b. =
7
SURVIVAL HINT: 36-47In Problems , do not round the decimal representations. For example,
1
3 = 0.3
_
Note the bar over the three; this shows that the three repeats. It is NOT correct to write
1 1
3 3
= 0.33333 or = 0.33
or any other such representation. The three repeats and does not terminate. That is, in doing the long
division, there is never a remainder of zero.
37 a.. = Long division; by calculator: 5
Continue with the division until you see the digits repeat. There is one repeating digit.
b. = Long division; by calculator:
Continue with the division until you see the digits repeat. There is one repeating digit.
Loading page 12...
CHAPTER 1 STUDENT'S SURVIVAL AND SOLUTION MANUAL FOR Mathematics Its Power and Utility, Ninth Edition
Page 8
39 a.. =
Long division; by calculator:
Continue with the division until the remainder is zero.
b. =
Long division; by calculator:
Continue with the division until you see the digits repeat. There is one repeating digit.
41 a.. =
Long division; by calculator:
Continue with the division until you see the digits repeat. There is one repeating digit.
b. =
Long division; by calculator:
Continue with the division until you see the digits repeat. There is one repeating digit.
43 a.. = Long division; by calculator: 5
Continue with the division until you see the digits repeat. There is one repeating digit.
b. =
Long division; by calculator:
Continue with the division until you see the digits repeat. There is one repeating digit.
45 a.. =
Long division; by calculator:
Continue with the division until you see the digits repeat. There is one repeating digit.
b. = Long division; by calculator:
Continue with the division until you see the digits repeat. There is one repeating digit.
47 a.. = Long division; by calculator:
Continue with the division until the remainder is zero.
b. 3 = 3 1 9
9 Long division; by calculator: 9
Continue with the division until you see the digits repeat. There is one repeating digit.
49 a. . The square is divided into one hundred small parts, and we see 82 of them are shaded,
so or of the square is shaded.
. The circle is divided into 8 parts and 5 parts are shaded; .b 5
8
SURVIVAL HINT: Your answers to Problems 50-51 may vary. Estimating is not an easy skill to learn,so
relax and do the best you can.
51 a. . The square is divided into one hundred small parts, and we see 63 of them are shaded,
so 0.63 of the square is shaded.
b. The square is divide into one hundred small parts and there are four small triangles,
each pair of which can form a square so there is a total of of those squares
are not shared. Thus, the shaded portion is or 0.82 of the large square.
SURVIVAL HINT: Do not round the answers to Problem 53, but use the overbar to indicate the
repeating decimals. If your calculator does not show a pattern, you must approximate the answer.
Page 8
39 a.. =
Long division; by calculator:
Continue with the division until the remainder is zero.
b. =
Long division; by calculator:
Continue with the division until you see the digits repeat. There is one repeating digit.
41 a.. =
Long division; by calculator:
Continue with the division until you see the digits repeat. There is one repeating digit.
b. =
Long division; by calculator:
Continue with the division until you see the digits repeat. There is one repeating digit.
43 a.. = Long division; by calculator: 5
Continue with the division until you see the digits repeat. There is one repeating digit.
b. =
Long division; by calculator:
Continue with the division until you see the digits repeat. There is one repeating digit.
45 a.. =
Long division; by calculator:
Continue with the division until you see the digits repeat. There is one repeating digit.
b. = Long division; by calculator:
Continue with the division until you see the digits repeat. There is one repeating digit.
47 a.. = Long division; by calculator:
Continue with the division until the remainder is zero.
b. 3 = 3 1 9
9 Long division; by calculator: 9
Continue with the division until you see the digits repeat. There is one repeating digit.
49 a. . The square is divided into one hundred small parts, and we see 82 of them are shaded,
so or of the square is shaded.
. The circle is divided into 8 parts and 5 parts are shaded; .b 5
8
SURVIVAL HINT: Your answers to Problems 50-51 may vary. Estimating is not an easy skill to learn,so
relax and do the best you can.
51 a. . The square is divided into one hundred small parts, and we see 63 of them are shaded,
so 0.63 of the square is shaded.
b. The square is divide into one hundred small parts and there are four small triangles,
each pair of which can form a square so there is a total of of those squares
are not shared. Thus, the shaded portion is or 0.82 of the large square.
SURVIVAL HINT: Do not round the answers to Problem 53, but use the overbar to indicate the
repeating decimals. If your calculator does not show a pattern, you must approximate the answer.
Loading page 13...
STUDENT'S SURVIVAL AND SOLUTION MANUAL FOR SECTION 1.4Mathematics Its Power and Utility, Ninth Edition
Page 9
53 a. . Press: : Answer: 0.7 15 Display .466666666666
. Press: : Answer: 0.318b 7 22 Display .3181818181818
55. First, note that = 0.5: $52 $52.5
$5,250
1
2
To multiply by 100, move the decimal point two places to the right.
57. First, note that = 0.25: $ 2 $ 2. 5
$42.25
$ 4
$8,450
1
4
To multiply by 100, move the decimal point two places to the right.
59. First, note that = 0.125: $63 $63.125
8
$6,312.50
1
8
To multiply by 100, move the decimal point two places to the right.
1.4 Rounding and Estimation, page 28
SURVIVAL HINT: Take a look at the essential ideas. Add these ideas to your journal.
New Terms Introduced in this Section
Rounding money Rounding numbers
Problem Set page 311.4,
3. The rounding place digit is the name of the positional column to which the number is being
rounded.
4. Estimation is making a reasonably accurate guess. It is important in everyday life in many
situations in which an exact answers is not needed. For example, when making a purchase,
deciding on which route to take, how long a task may take to complete, or tipping at a
restaurant.
5. 30. 0 5 rounded to the nearest tenth is 30.1, so the statement is false. The error is that the
number was rounded to the nearest unit, rather than the nearest tenth.
7. 3,6 8 4,999 rounded to the nearest ten thousand is 3,680,000, so the statement is false. The
error is that one was added to the rounding place digit and it should not have been.
Page 9
53 a. . Press: : Answer: 0.7 15 Display .466666666666
. Press: : Answer: 0.318b 7 22 Display .3181818181818
55. First, note that = 0.5: $52 $52.5
$5,250
1
2
To multiply by 100, move the decimal point two places to the right.
57. First, note that = 0.25: $ 2 $ 2. 5
$42.25
$ 4
$8,450
1
4
To multiply by 100, move the decimal point two places to the right.
59. First, note that = 0.125: $63 $63.125
8
$6,312.50
1
8
To multiply by 100, move the decimal point two places to the right.
1.4 Rounding and Estimation, page 28
SURVIVAL HINT: Take a look at the essential ideas. Add these ideas to your journal.
New Terms Introduced in this Section
Rounding money Rounding numbers
Problem Set page 311.4,
3. The rounding place digit is the name of the positional column to which the number is being
rounded.
4. Estimation is making a reasonably accurate guess. It is important in everyday life in many
situations in which an exact answers is not needed. For example, when making a purchase,
deciding on which route to take, how long a task may take to complete, or tipping at a
restaurant.
5. 30. 0 5 rounded to the nearest tenth is 30.1, so the statement is false. The error is that the
number was rounded to the nearest unit, rather than the nearest tenth.
7. 3,6 8 4,999 rounded to the nearest ten thousand is 3,680,000, so the statement is false. The
error is that one was added to the rounding place digit and it should not have been.
Loading page 14...
CHAPTER 1 STUDENT'S SURVIVAL AND SOLUTION MANUAL FOR Mathematics Its Power and Utility, Ninth Edition
Page 10
9. 23 12 25 10 = 250; estimate is correct.
11. 2. 3 12 Rounding place digit is marked.
2.3 Digit to the right is a 1, so delete the “12”.
13. 6,287.4 5 13 Rounding place digit is marked.
6,287.45 Digit to the right is a 1, so delete the “13”.
15. 5. 2 91 Digit to the right is a 9, so add one
5.3 to the rounding place digit and delete the “91”.
17. 6, 2 87.4513 Digit to the right is an 8, so add
6,300 one to the rounding place digit and change the digits to the right
to zeros. Drop the zeros to the right of the decimal point.
19. 12.8 1 97 Digit to the right is a 9, so
12.82 add one to the rounding place digit and delete the “97”.
21. 4.81 7 92 Digit to the right is a 9, so
4.818 add one to the rounding place digit and delete the “92”.
23. 4 .8199 Digit to the right is a 8, so
5 add one to the rounding place digit and delete the “.8199”.
25. $12.9 9 3 Digit to the right is a 3, so
$12.99 delete the “3”.
27. $14.9 9 8 Digit to the right is a 8, so
$15.00 add one to the rounding place digit and delete the “8”. Note that
adding 1 to 9 causes a carry into the next column. That is,
$14.99 + $0.01 = $15.00.
29. 6 9 4.3814 Digit to the right is an 4, so
690 change the numerals to the right of the rounding place digit to zeros; then
delete those to the right of the decimal point.
31. $8 6 ,125 Digit to the right is an 1, so
$86,000 change the digits to the right of the rounding place digit to zeros.
33. From the calculator display, we see 2/3 is represented as 0.66 6 6666667. The digit to the
right is a 6, so add 1 to the rounding place digit and drop the digits to the right to obtain:
0.667.
35. From the calculator display, we see 2/17 is represented as 0.11 7 6470588. The digit to the
right is a 6, so add 1 to the rounding place digit and drop the digits to the right to obtain:
0.118.
37. From the calculator display, we see 7/51 is represented as 0.13 7 254902. The digit to the
right is a 2, so do not change the rounding place digit but drop the digits to the right to
obtain: 0.137.
39. 2
6 0.3333333333; rounded answer: 0.333; by calculator: 2 6
Page 10
9. 23 12 25 10 = 250; estimate is correct.
11. 2. 3 12 Rounding place digit is marked.
2.3 Digit to the right is a 1, so delete the “12”.
13. 6,287.4 5 13 Rounding place digit is marked.
6,287.45 Digit to the right is a 1, so delete the “13”.
15. 5. 2 91 Digit to the right is a 9, so add one
5.3 to the rounding place digit and delete the “91”.
17. 6, 2 87.4513 Digit to the right is an 8, so add
6,300 one to the rounding place digit and change the digits to the right
to zeros. Drop the zeros to the right of the decimal point.
19. 12.8 1 97 Digit to the right is a 9, so
12.82 add one to the rounding place digit and delete the “97”.
21. 4.81 7 92 Digit to the right is a 9, so
4.818 add one to the rounding place digit and delete the “92”.
23. 4 .8199 Digit to the right is a 8, so
5 add one to the rounding place digit and delete the “.8199”.
25. $12.9 9 3 Digit to the right is a 3, so
$12.99 delete the “3”.
27. $14.9 9 8 Digit to the right is a 8, so
$15.00 add one to the rounding place digit and delete the “8”. Note that
adding 1 to 9 causes a carry into the next column. That is,
$14.99 + $0.01 = $15.00.
29. 6 9 4.3814 Digit to the right is an 4, so
690 change the numerals to the right of the rounding place digit to zeros; then
delete those to the right of the decimal point.
31. $8 6 ,125 Digit to the right is an 1, so
$86,000 change the digits to the right of the rounding place digit to zeros.
33. From the calculator display, we see 2/3 is represented as 0.66 6 6666667. The digit to the
right is a 6, so add 1 to the rounding place digit and drop the digits to the right to obtain:
0.667.
35. From the calculator display, we see 2/17 is represented as 0.11 7 6470588. The digit to the
right is a 6, so add 1 to the rounding place digit and drop the digits to the right to obtain:
0.118.
37. From the calculator display, we see 7/51 is represented as 0.13 7 254902. The digit to the
right is a 2, so do not change the rounding place digit but drop the digits to the right to
obtain: 0.137.
39. 2
6 0.3333333333; rounded answer: 0.333; by calculator: 2 6
Loading page 15...
STUDENT'S SURVIVAL AND SOLUTION MANUAL FOR SECTION 1.5Mathematics Its Power and Utility, Ninth Edition
Page 11
41. 0.41666666666; rounded answer: 0.417; by calculator:5
12 5 12
43. 0.3179916318; rounded answer: 0.318; by calculator:152
478 152 478
45. B : 4.82 5 mi, so it is about 10 miles round trip; 5 10 = 50.Think
47. C : $35,000 $36,000 and $36,000 12 = $3,000.Think
49. B : $1,000 $1,200 and $1,200 12 = $100.Think
51. $15,000
12 = $1,250 ; by calculator: 15000 12
53. = $12.125 ; rounded answer: $12.13; by calculator:
$1,818.75
150 1818.75 150
55. $70.8333333 ; rounded answer: $70.83 by calculator:
$850
12 850 12
57. $112.333333 ; rounded answer $112.33; by calculator:
$674
6 674 6
1.5 Exponents and Prime Factorization, page 33
SURVIVAL HINT: There are many ideas in this section which you will use daily in your future math work.
Be sure you understand the concept of an exponent, and that of a prime factorization.
New Terms Introduced in this Section
Base Composite Cubed Exponent
Exponential notation Factor Factoring Factorization
Factor tree Googol Power Powers of ten
Prime factorization Prime number Scientific notation Squared
Problem Set page 391.5,
3. An exponent is a number used to indicate repeated multiplication. In other words, it is the
number of repeated factors. In it is the number .b nn
4. A number is in scientific notation when that number is written as a power of 10 or as a
decimal number between 1 and 10 times a power of 10.
5. An excellent way to find the prime factorization is to use factor trees.
6. The is an exponent key and EE is used for scientific notation.yx
SURVIVAL HINT: Take a look at the four preceding essential ideas. Add these ideas to your journal.
7. a b. c. d.. one million 10 6 10 10 10 10 10 10
8. a b. c. d.. one thousand 10 3 10 10 10
9. a. b. c. d.one-tenth 10 1 0.1
10. a b. c. d.. one-hundredth 10
Page 11
41. 0.41666666666; rounded answer: 0.417; by calculator:5
12 5 12
43. 0.3179916318; rounded answer: 0.318; by calculator:152
478 152 478
45. B : 4.82 5 mi, so it is about 10 miles round trip; 5 10 = 50.Think
47. C : $35,000 $36,000 and $36,000 12 = $3,000.Think
49. B : $1,000 $1,200 and $1,200 12 = $100.Think
51. $15,000
12 = $1,250 ; by calculator: 15000 12
53. = $12.125 ; rounded answer: $12.13; by calculator:
$1,818.75
150 1818.75 150
55. $70.8333333 ; rounded answer: $70.83 by calculator:
$850
12 850 12
57. $112.333333 ; rounded answer $112.33; by calculator:
$674
6 674 6
1.5 Exponents and Prime Factorization, page 33
SURVIVAL HINT: There are many ideas in this section which you will use daily in your future math work.
Be sure you understand the concept of an exponent, and that of a prime factorization.
New Terms Introduced in this Section
Base Composite Cubed Exponent
Exponential notation Factor Factoring Factorization
Factor tree Googol Power Powers of ten
Prime factorization Prime number Scientific notation Squared
Problem Set page 391.5,
3. An exponent is a number used to indicate repeated multiplication. In other words, it is the
number of repeated factors. In it is the number .b nn
4. A number is in scientific notation when that number is written as a power of 10 or as a
decimal number between 1 and 10 times a power of 10.
5. An excellent way to find the prime factorization is to use factor trees.
6. The is an exponent key and EE is used for scientific notation.yx
SURVIVAL HINT: Take a look at the four preceding essential ideas. Add these ideas to your journal.
7. a b. c. d.. one million 10 6 10 10 10 10 10 10
8. a b. c. d.. one thousand 10 3 10 10 10
9. a. b. c. d.one-tenth 10 1 0.1
10. a b. c. d.. one-hundredth 10
Loading page 16...
CHAPTER 1 STUDENT'S SURVIVAL AND SOLUTION MANUAL FOR Mathematics Its Power and Utility, Ninth Edition
Page 12
11. False; means .
13. False; there are two multiplications (not three).
15. a. In scientific notation, the decimal point goes after the first nonzero digit:
3.2 10 = 3,200
Decide on the exponent by counting the number of decimal places necessary to
write this number, 3.2, as the given number. We see it is three places.
Move the decimal point 3 places to the right.
b. 2.5 10 = 25,000
Move the decimal point 4 places to the right.
c. 1.8 10 = 18,000,000
Move the decimal point 7 places to the right.
d. 6.4 10 = 640
Move the decimal point 2 places to the right.
17. a. 4.21 10 = 0.0000
Move the decimal point 6 places to the left.
b. 9.2 10 = 92,000,000
Move the decimal point 7 places to the right.
c. 1 10 =
Don't move the decimal point.
d. 1.5 10 =
Don't move the decimal point.
19 a. . “E9” means “ 10 ”, so 6.34E9 = 6.34 10
b. 5.2019E11 = 5.2019 10
c. “08” means “ 10 ”, so 4.093745 08 = 4.093745 10
d. 8.291029292 12 8.291029292 10
21 a. . 7.2 10 = 72,000,000,000
Move the decimal point places to the right.10
b. 4.5 10 = 4,500
Move the decimal point places to the right.3
23 a. . 2.1 10 = 0.
Move the decimal point places to the left.3
b. 4.6 10 = 0.
Move the decimal point places to the left.7
25 a. . . 10 =
Move the decimal point places.
b. . 10 =
Move the decimal point places to the left.4
Page 12
11. False; means .
13. False; there are two multiplications (not three).
15. a. In scientific notation, the decimal point goes after the first nonzero digit:
3.2 10 = 3,200
Decide on the exponent by counting the number of decimal places necessary to
write this number, 3.2, as the given number. We see it is three places.
Move the decimal point 3 places to the right.
b. 2.5 10 = 25,000
Move the decimal point 4 places to the right.
c. 1.8 10 = 18,000,000
Move the decimal point 7 places to the right.
d. 6.4 10 = 640
Move the decimal point 2 places to the right.
17. a. 4.21 10 = 0.0000
Move the decimal point 6 places to the left.
b. 9.2 10 = 92,000,000
Move the decimal point 7 places to the right.
c. 1 10 =
Don't move the decimal point.
d. 1.5 10 =
Don't move the decimal point.
19 a. . “E9” means “ 10 ”, so 6.34E9 = 6.34 10
b. 5.2019E11 = 5.2019 10
c. “08” means “ 10 ”, so 4.093745 08 = 4.093745 10
d. 8.291029292 12 8.291029292 10
21 a. . 7.2 10 = 72,000,000,000
Move the decimal point places to the right.10
b. 4.5 10 = 4,500
Move the decimal point places to the right.3
23 a. . 2.1 10 = 0.
Move the decimal point places to the left.3
b. 4.6 10 = 0.
Move the decimal point places to the left.7
25 a. . . 10 =
Move the decimal point places.
b. . 10 =
Move the decimal point places to the left.4
Loading page 17...
STUDENT'S SURVIVAL AND SOLUTION MANUAL FOR SECTION 1.5Mathematics Its Power and Utility, Ninth Edition
Page 13
27 a. b. .
29 a. . 2.18928271 10
21,892,827,100
b. 0.0000329 07 0.0000329 10
329
31.
33.
35. 7
37 a b. . .
(2 2) 5
2 5
39 a b. . .
2 2 64
2 2 2 32
2 2 2 2 16
2 2 2 2 2 8
2 2 2 2 2 2 4
2 2 2 2 2 2 2 2
2
41 a b. . . ,
2 2 2 5 2 5 2 5 2 5 2 5
2 5
2 5 10 10
2 5
43 a. . See which prime divides evenly into the given number. If it does not divide evenly,
move to the next prime. If it does divide evenly, try that same number again.
Page 13
27 a. b. .
29 a. . 2.18928271 10
21,892,827,100
b. 0.0000329 07 0.0000329 10
329
31.
33.
35. 7
37 a b. . .
(2 2) 5
2 5
39 a b. . .
2 2 64
2 2 2 32
2 2 2 2 16
2 2 2 2 2 8
2 2 2 2 2 2 4
2 2 2 2 2 2 2 2
2
41 a b. . . ,
2 2 2 5 2 5 2 5 2 5 2 5
2 5
2 5 10 10
2 5
43 a. . See which prime divides evenly into the given number. If it does not divide evenly,
move to the next prime. If it does divide evenly, try that same number again.
Loading page 18...
CHAPTER 1 STUDENT'S SURVIVAL AND SOLUTION MANUAL FOR Mathematics Its Power and Utility, Ninth Edition
Page 14
, 7 637
7 7 91
7 7 7 13
7 13
b. , 13 17,641
13 13 1,357
13 13 23 59
13 23 59
45 a. . Use your calculator and the list of primes to see which prime divides evenly into the
given number. If it does not divide evenly, move to the next prime. If it does divide
evenly, try that same number again.
, 19 2,407
19 29 83
b. , 31 961
31 31 31
31
47. First estimate the number of seconds in a year:
60 60 365 24 6 10 6 10 3.65 10
24 6 6 3.65 10
Now multiply this number by your age (in years) to see that the most reasonable exponent
is the one shown in choice A.
49. Choice A is a penny, choice B is a dime, and choice C is a dollar; the best choice is C.
51. Choice B is a very small number (and not even a possibility), and choice C is too large, so
the best choice must be A.
53. 41,840,000 = 4.184 10
55. 3.33 10 = 333,000
57 a. . , , , , , , ,
By calculator,b.
59. NOTE: no calculator; this is an estimation problem.
$ $ $ $ $ $ $ $ $ $
$ $ $ $ + $ + $ + $ + $1 + $ $ $ $ $ $1
$ $2 $20 (Actual, with calculator is $
Page 14
, 7 637
7 7 91
7 7 7 13
7 13
b. , 13 17,641
13 13 1,357
13 13 23 59
13 23 59
45 a. . Use your calculator and the list of primes to see which prime divides evenly into the
given number. If it does not divide evenly, move to the next prime. If it does divide
evenly, try that same number again.
, 19 2,407
19 29 83
b. , 31 961
31 31 31
31
47. First estimate the number of seconds in a year:
60 60 365 24 6 10 6 10 3.65 10
24 6 6 3.65 10
Now multiply this number by your age (in years) to see that the most reasonable exponent
is the one shown in choice A.
49. Choice A is a penny, choice B is a dime, and choice C is a dollar; the best choice is C.
51. Choice B is a very small number (and not even a possibility), and choice C is too large, so
the best choice must be A.
53. 41,840,000 = 4.184 10
55. 3.33 10 = 333,000
57 a. . , , , , , , ,
By calculator,b.
59. NOTE: no calculator; this is an estimation problem.
$ $ $ $ $ $ $ $ $ $
$ $ $ $ + $ + $ + $ + $1 + $ $ $ $ $ $1
$ $2 $20 (Actual, with calculator is $
Loading page 19...
STUDENT'S SURVIVAL AND SOLUTION MANUAL FOR SECTION 1.6Mathematics Its Power and Utility, Ninth Edition
Page 15
1.6 Common Fractions, page 42
SURVIVAL HINT: You may have had problems with fractions in the past, but the emphasis here is on
understanding fractions and learning how to use a calculator to help you with calculation involving
fractions. If you are feeling anxious about this topic, it may be time to take a break with some of the
anxiety reducing techniques introduced in the first section of this book. Relax.... ask questions, and you
will succeed!
New Terms Introduced in this Section
Canceling Completely reduce fraction Complex decimal
Divide fractions Fundamental property of fractions Invert
Multiply fractions Reciprocal Reduced fraction
Reducing fractions Simplify a fractional expression
Problem Set page 481.6,
3. Use the fundamental property of fractions (see the answer to Problem 1) to eliminate all
common factors (other than 1 or 1).
4. A fraction is reduced when there is no number (except 1) that divides into both the
numerator and denominator.
5. To multiply fractions, multiply numerators and multiply denominators.
6. To divide fractions, invert and multiply.
7. A terminating decimal is a fraction whose decimal representation ends.
8. (1) Multiply the given number without its decimal point by the decimal name of the last digit.
(2) By the decimal name of the last digit, we mean: one place is tenth, or ; two places is1
10
hundredth, or ; three places is thousandth, or ; .1 1
100 1,000
9. A reduced fraction is one in which there is no common factor (other than and )
between the numerator and denominator; this is true.
11. F;
13. but These results are different, so the statement is
false.
15. a b c d. . . .
Page 15
1.6 Common Fractions, page 42
SURVIVAL HINT: You may have had problems with fractions in the past, but the emphasis here is on
understanding fractions and learning how to use a calculator to help you with calculation involving
fractions. If you are feeling anxious about this topic, it may be time to take a break with some of the
anxiety reducing techniques introduced in the first section of this book. Relax.... ask questions, and you
will succeed!
New Terms Introduced in this Section
Canceling Completely reduce fraction Complex decimal
Divide fractions Fundamental property of fractions Invert
Multiply fractions Reciprocal Reduced fraction
Reducing fractions Simplify a fractional expression
Problem Set page 481.6,
3. Use the fundamental property of fractions (see the answer to Problem 1) to eliminate all
common factors (other than 1 or 1).
4. A fraction is reduced when there is no number (except 1) that divides into both the
numerator and denominator.
5. To multiply fractions, multiply numerators and multiply denominators.
6. To divide fractions, invert and multiply.
7. A terminating decimal is a fraction whose decimal representation ends.
8. (1) Multiply the given number without its decimal point by the decimal name of the last digit.
(2) By the decimal name of the last digit, we mean: one place is tenth, or ; two places is1
10
hundredth, or ; three places is thousandth, or ; .1 1
100 1,000
9. A reduced fraction is one in which there is no common factor (other than and )
between the numerator and denominator; this is true.
11. F;
13. but These results are different, so the statement is
false.
15. a b c d. . . .
Loading page 20...
CHAPTER 1 STUDENT'S SURVIVAL AND SOLUTION MANUAL FOR Mathematics Its Power and Utility, Ninth Edition
Page 16
17 a b c d. . . . .
3 42 3 14 16 2 8
5 3 14 14 24 3 8
24
5
2
3
19 a b c d. . . . .
40 5 48 5,670
672 14 48 ,150
5
1
21. Shade two of the five rows and three of the four columns. There are 6 of the 20 squares
that are double shaded. This can also be stated as three-tenths of the large square.
23. Shade four of the five rows and one of the three columns. There are 4 of the 15 squares
that are double shaded.
25. Divide the side into six parts and the top into three parts. Then, shade one of the six rows
and two of the three columns. There are 2 of the 18 squares that are double shaded. This
can also be stated as one-ninth of the large square.
27. In this problem, you are verifying for yourself that division gives the same result as “invert
and multiply.”
a. The divisor is 5, and the answer is 3. Also, 15
b. The divisor is 3, the answer is 2. Also,
29. In this problem, you are verifying for yourself that division gives the same result as “invert
and multiply.”
a. The divisor is , and the answer is . Also,
The divisor is , the answer is . Also,b.
31 a. b..
c. d.
7 100
20 14
SURVIVAL HINT: Note is reduced. You do not need to write this answer as the mixed number
2
e. f.
Page 16
17 a b c d. . . . .
3 42 3 14 16 2 8
5 3 14 14 24 3 8
24
5
2
3
19 a b c d. . . . .
40 5 48 5,670
672 14 48 ,150
5
1
21. Shade two of the five rows and three of the four columns. There are 6 of the 20 squares
that are double shaded. This can also be stated as three-tenths of the large square.
23. Shade four of the five rows and one of the three columns. There are 4 of the 15 squares
that are double shaded.
25. Divide the side into six parts and the top into three parts. Then, shade one of the six rows
and two of the three columns. There are 2 of the 18 squares that are double shaded. This
can also be stated as one-ninth of the large square.
27. In this problem, you are verifying for yourself that division gives the same result as “invert
and multiply.”
a. The divisor is 5, and the answer is 3. Also, 15
b. The divisor is 3, the answer is 2. Also,
29. In this problem, you are verifying for yourself that division gives the same result as “invert
and multiply.”
a. The divisor is , and the answer is . Also,
The divisor is , the answer is . Also,b.
31 a. b..
c. d.
7 100
20 14
SURVIVAL HINT: Note is reduced. You do not need to write this answer as the mixed number
2
e. f.
Loading page 21...
STUDENT'S SURVIVAL AND SOLUTION MANUAL FOR SECTION 1.6Mathematics Its Power and Utility, Ninth Edition
Page 17
33 a. b. c..
d. e. f.
35 a. b. c..
d. e. f.
SURVIVAL HINT: You may be able to do these mentally, any number divided by itself is .
37 a. b. c..
d. e. f.
39 a. b..
c. d.
e. f.
Page 17
33 a. b. c..
d. e. f.
35 a. b. c..
d. e. f.
SURVIVAL HINT: You may be able to do these mentally, any number divided by itself is .
37 a. b. c..
d. e. f.
39 a. b..
c. d.
e. f.
Loading page 22...
CHAPTER 1 STUDENT'S SURVIVAL AND SOLUTION MANUAL FOR Mathematics Its Power and Utility, Ninth Edition
Page 18
41 a. b.. 1 8
2 9
c. d.
e. f.
43. a b c. 0. . .
,
45. a. b c
. . ,
47. a. b. c.
16 1
7
7
50
1
6
SURVIVAL HINT: You may know these answers from memory.
49. a. b. c.
1
51.
$
$
1 1
10 10 $ , AGE SALARY
Page 18
41 a. b.. 1 8
2 9
c. d.
e. f.
43. a b c. 0. . .
,
45. a. b c
. . ,
47. a. b. c.
16 1
7
7
50
1
6
SURVIVAL HINT: You may know these answers from memory.
49. a. b. c.
1
51.
$
$
1 1
10 10 $ , AGE SALARY
Loading page 23...
STUDENT'S SURVIVAL AND SOLUTION MANUAL FOR SECTION 1.7Mathematics Its Power and Utility, Ninth Edition
Page 19
53.
$ ,
$ ,
1 1
10 10 $ , AGE SALARY
SURVIVAL HINT: Translate the word “of to mean multiply.”
55. 3
16 $12,512 = $2,346
By calculator: 3
57. $227.00 $
By calculator:
59. $ , $44,000 22
1 5
$44,000 5
1 22
$2,000 22 5
22
$10,000
1.7 Adding and Subtracting Fractions, page 50
SURVIVAL HINT: The in the first two problems summarize what you need toESSENTIAL IDEAS
remember from this section.
New Terms Introduced in this Section
Common denominator Extended order of operations LCD
Lowest common denominator Subtraction of fractions
Problem Set page 561.7,
1. Step 1 Find the LCD.
Step 2 Change the forms of the fractions to obtain forms with common denominators.
Step 3 Add the numerators of the fractions with common denominators.
3. This job would require at least two cuts, so the total is:
2 + 5 + 7 + 5 + 5 + 7 + 5
1 1 1 3 2 4 8 3
16 4 2 16 16 16 16 16
17 17
16
18 1
16
Page 19
53.
$ ,
$ ,
1 1
10 10 $ , AGE SALARY
SURVIVAL HINT: Translate the word “of to mean multiply.”
55. 3
16 $12,512 = $2,346
By calculator: 3
57. $227.00 $
By calculator:
59. $ , $44,000 22
1 5
$44,000 5
1 22
$2,000 22 5
22
$10,000
1.7 Adding and Subtracting Fractions, page 50
SURVIVAL HINT: The in the first two problems summarize what you need toESSENTIAL IDEAS
remember from this section.
New Terms Introduced in this Section
Common denominator Extended order of operations LCD
Lowest common denominator Subtraction of fractions
Problem Set page 561.7,
1. Step 1 Find the LCD.
Step 2 Change the forms of the fractions to obtain forms with common denominators.
Step 3 Add the numerators of the fractions with common denominators.
3. This job would require at least two cuts, so the total is:
2 + 5 + 7 + 5 + 5 + 7 + 5
1 1 1 3 2 4 8 3
16 4 2 16 16 16 16 16
17 17
16
18 1
16
Loading page 24...
CHAPTER 1 STUDENT'S SURVIVAL AND SOLUTION MANUAL FOR Mathematics Its Power and Utility, Ninth Edition
Page 20
The smallest single length is 18 inches.
5. (1) First, perform any operations enclosed in parentheses.
(2) Next, perform any operations that involve raising to a power.
(3) Perform multiplication and division, reading from left to right.
(4) Do addition and subtraction, reading from left to right.
6. Step 1 Factor all given denominators into prime factors; write this factorization using
exponents.
Step 2 List each different prime factor you found in the prime factorization of the
denominators.
Step 3 On each prime in the list from Step 2, place the largest exponent that appears on
that prime factor anywhere in the factorization of the denominators.
Step 4 The LCD is the product of the prime factors with the exponents found in Step 3.
7. False; do not add the denominators; the correct answer is + = .3 4 7
8 8 8
9. This problem and solution have no errors.
11. This problem and solution have no errors.
13. 19 20 1
40 40 2
= The correct response is A.
15. 19 3 20 12
40 10 40 40
The correct response is D.
17. If I multiply a number by a number larger than 1, the result is larger, but if I multiply a
number by a number smaller than 1, the result is smaller. The correct response is B.
19. If I divide a number by a number larger than 1 the result is smaller, but if I divide a number
by a number smaller than 1, then the result is larger. The correct response is A.
21 a b c. . + . + . +
2 1 3 3 5 8 5 3 8
5 5 5 7 7 7 11 11 11
SURVIVAL HINT: A number is if there is no number other than 1 that divides into both thereduced
numerator and denominator. This means, for example, the result in Problem 21b is reduced.
d. e f
3 5 5 1
2 9 9
. . +
Page 20
The smallest single length is 18 inches.
5. (1) First, perform any operations enclosed in parentheses.
(2) Next, perform any operations that involve raising to a power.
(3) Perform multiplication and division, reading from left to right.
(4) Do addition and subtraction, reading from left to right.
6. Step 1 Factor all given denominators into prime factors; write this factorization using
exponents.
Step 2 List each different prime factor you found in the prime factorization of the
denominators.
Step 3 On each prime in the list from Step 2, place the largest exponent that appears on
that prime factor anywhere in the factorization of the denominators.
Step 4 The LCD is the product of the prime factors with the exponents found in Step 3.
7. False; do not add the denominators; the correct answer is + = .3 4 7
8 8 8
9. This problem and solution have no errors.
11. This problem and solution have no errors.
13. 19 20 1
40 40 2
= The correct response is A.
15. 19 3 20 12
40 10 40 40
The correct response is D.
17. If I multiply a number by a number larger than 1, the result is larger, but if I multiply a
number by a number smaller than 1, the result is smaller. The correct response is B.
19. If I divide a number by a number larger than 1 the result is smaller, but if I divide a number
by a number smaller than 1, then the result is larger. The correct response is A.
21 a b c. . + . + . +
2 1 3 3 5 8 5 3 8
5 5 5 7 7 7 11 11 11
SURVIVAL HINT: A number is if there is no number other than 1 that divides into both thereduced
numerator and denominator. This means, for example, the result in Problem 21b is reduced.
d. e f
3 5 5 1
2 9 9
. . +
Loading page 25...
STUDENT'S SURVIVAL AND SOLUTION MANUAL FOR SECTION 1.7Mathematics Its Power and Utility, Ninth Edition
Page 21
23 a. b. c..
d. e. f.
5
25. Pick larger exponents for LCD. These are shown in boldface.
LCD LCD
a. b.
c. d.
LCD LCD 2 3 5 7 = 630
27. Pick larger exponents for LCD. These are shown in boldface.
,
LCD , LCD ,
a. b.
c. d.
LCD ,
LCD 2 3 5 ,
29 a b. . LCD is 6; . LCD is 24; + +
5 1 5 2 5 5 20 15
6 3 6 6 6 8 24 24
Page 21
23 a. b. c..
d. e. f.
5
25. Pick larger exponents for LCD. These are shown in boldface.
LCD LCD
a. b.
c. d.
LCD LCD 2 3 5 7 = 630
27. Pick larger exponents for LCD. These are shown in boldface.
,
LCD , LCD ,
a. b.
c. d.
LCD ,
LCD 2 3 5 ,
29 a b. . LCD is 6; . LCD is 24; + +
5 1 5 2 5 5 20 15
6 3 6 6 6 8 24 24
Loading page 26...
CHAPTER 1 STUDENT'S SURVIVAL AND SOLUTION MANUAL FOR Mathematics Its Power and Utility, Ninth Edition
Page 22
. LCD is 24; LCD is 12;c d.
. LCD is 30; . LCD is 12;e f
31 a b. c.. .
33 a..
LCD 2 3
LCD
b.
Page 22
. LCD is 24; LCD is 12;c d.
. LCD is 30; . LCD is 12;e f
31 a b. c.. .
33 a..
LCD 2 3
LCD
b.
Loading page 27...
STUDENT'S SURVIVAL AND SOLUTION MANUAL FOR SECTION 1.7Mathematics Its Power and Utility, Ninth Edition
Page 23
35 a..
LCD 2 3
b.
LCD
37 a..
b.
39 a. b..
Page 23
35 a..
LCD 2 3
b.
LCD
37 a..
b.
39 a. b..
Loading page 28...
CHAPTER 1 STUDENT'S SURVIVAL AND SOLUTION MANUAL FOR Mathematics Its Power and Utility, Ninth Edition
Page 24
41. Multiplication before addition: + +
4 17 4 78 4 17 78
5 95 5 95 5 95 95
1
4
5
By calculator:
Display: 0.8 Answer is 0.8 or .4
5
43. Parentheses first: + + 2 + + 2
4 2 1 12 10 1
5 3 5 15 15 5
22 5
15 1 + 2
22 6
3 3
+
28
3
By calculator:
Display: 9.333333333 Answer is 9. or .
45. By calculator:
Display: (This is an approximate answer.).4487116145
47. First find the regions labeled “B” and add up their areas: The first one at the left is of ;1 1
2 4
The second one is at the right and is of :1 1
3 2
+ +
1 1 1 1 1 1
2 4 3 2 8 6
1 3 1 4
8 3 6 4
+
3 4
24 24
+
7
24
49. For this one, we find the portion that is not C. The C portion at the left is and the C1
4
portion at the right is of or : + +
1 1 1 3 1 2
4 6 4 3 6 2
3 2
12 12
+
5
12
1 1 1
3 2 6
Page 24
41. Multiplication before addition: + +
4 17 4 78 4 17 78
5 95 5 95 5 95 95
1
4
5
By calculator:
Display: 0.8 Answer is 0.8 or .4
5
43. Parentheses first: + + 2 + + 2
4 2 1 12 10 1
5 3 5 15 15 5
22 5
15 1 + 2
22 6
3 3
+
28
3
By calculator:
Display: 9.333333333 Answer is 9. or .
45. By calculator:
Display: (This is an approximate answer.).4487116145
47. First find the regions labeled “B” and add up their areas: The first one at the left is of ;1 1
2 4
The second one is at the right and is of :1 1
3 2
+ +
1 1 1 1 1 1
2 4 3 2 8 6
1 3 1 4
8 3 6 4
+
3 4
24 24
+
7
24
49. For this one, we find the portion that is not C. The C portion at the left is and the C1
4
portion at the right is of or : + +
1 1 1 3 1 2
4 6 4 3 6 2
3 2
12 12
+
5
12
1 1 1
3 2 6
Loading page 29...
STUDENT'S SURVIVAL AND SOLUTION MANUAL FOR SECTION 1.7Mathematics Its Power and Utility, Ninth Edition
Page 25
Thus, the portion that is not C is: 1 =
5 7
12 12
51. This is the portion that is not B. From Problem 47, the portion that is B is . Thus, we7
24
find
1 =
7 17
24 24
53. We see the square is divided into three columns of equal size. We are looking for the parts
labeled K: in the first column K is of or of that column; in the second column it is1 1 1
3 2 3
of that column; there is no K in the third column. Thus,
1 1 1 1 1 1 1
6 3 3 3 3 18 9
+ + 0 +
1 2
18 18
+
3
18
1
6
55. We see the square is divided into three columns of equal size. We are looking for the parts
labeled Y: in the first column Y is of that column; in the second column it is of that1 1
2 3
column; there is no Y in the third column. Thus,
1 1 1 1 1 1 1
2 3 3 3 3 6 9
+ + 0 +
3 2
18 18
+
5
18
57. R or G or Y is everything that is not K:
1 =
1 5
6 6
The fact that K is of the square was found in Problem 53.1
6
Page 25
Thus, the portion that is not C is: 1 =
5 7
12 12
51. This is the portion that is not B. From Problem 47, the portion that is B is . Thus, we7
24
find
1 =
7 17
24 24
53. We see the square is divided into three columns of equal size. We are looking for the parts
labeled K: in the first column K is of or of that column; in the second column it is1 1 1
3 2 3
of that column; there is no K in the third column. Thus,
1 1 1 1 1 1 1
6 3 3 3 3 18 9
+ + 0 +
1 2
18 18
+
3
18
1
6
55. We see the square is divided into three columns of equal size. We are looking for the parts
labeled Y: in the first column Y is of that column; in the second column it is of that1 1
2 3
column; there is no Y in the third column. Thus,
1 1 1 1 1 1 1
2 3 3 3 3 6 9
+ + 0 +
3 2
18 18
+
5
18
57. R or G or Y is everything that is not K:
1 =
1 5
6 6
The fact that K is of the square was found in Problem 53.1
6
Loading page 30...
CHAPTER 1 STUDENT'S SURVIVAL AND SOLUTION MANUAL FOR Mathematics Its Power and Utility, Ninth Edition
Page 26
59
1 + 2 + 3 1 + 2 + 3
1 2 1 3 8 6
4 3 2 12 12 12
6 17
12
7 5
12
The total weight is 7 pounds.
1.8 Hindu-Arabic Numeration System, page 58
SURVIVAL HINT: The in the first three problems summarize what you need toESSENTIAL IDEAS
remember from this section.
New Terms Introduced in this Section
Abacus Expanded notation Hindu-Arabic numerals Number
Number Numeral Numeration system
Problem Set page 611.8,
1. A decimal numeration system refers to a numeration system with 10 symbols and rules for
combining those symbols to represent all numbers. If we refer to the decimal numeration
system we use everyday then (1) there are ten symbols, (2) larger numbers are expressed in
terms of powers of 10, and (3) it is positional.
3. A Hindu-Arabic numeral is one of the decimal numerals we use everyday:
and .
5. A is an expression of quantity whereas a is a symbol used to represent anumber numeral
number.
6. Expanded notation is writing a number by showing the meaning of each digit in that
number.
7 8 a. b.. .a b. 100 . 64 2
9.
Page 26
59
1 + 2 + 3 1 + 2 + 3
1 2 1 3 8 6
4 3 2 12 12 12
6 17
12
7 5
12
The total weight is 7 pounds.
1.8 Hindu-Arabic Numeration System, page 58
SURVIVAL HINT: The in the first three problems summarize what you need toESSENTIAL IDEAS
remember from this section.
New Terms Introduced in this Section
Abacus Expanded notation Hindu-Arabic numerals Number
Number Numeral Numeration system
Problem Set page 611.8,
1. A decimal numeration system refers to a numeration system with 10 symbols and rules for
combining those symbols to represent all numbers. If we refer to the decimal numeration
system we use everyday then (1) there are ten symbols, (2) larger numbers are expressed in
terms of powers of 10, and (3) it is positional.
3. A Hindu-Arabic numeral is one of the decimal numerals we use everyday:
and .
5. A is an expression of quantity whereas a is a symbol used to represent anumber numeral
number.
6. Expanded notation is writing a number by showing the meaning of each digit in that
number.
7 8 a. b.. .a b. 100 . 64 2
9.
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