# Solution Manual for Pathways to College Mathematics, 2nd Edition

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INSTRUCTORโS
SOLUTIONS MANUAL
DANIEL S. MILLER
Niagara County Community College
P ATHWAYS
TO C OLLEGE M ATHEMATICS
SECOND EDITION
Robert Blitzer
Miami Dade College
The author and publisher of this book have used their best efforts in preparing this book. These efforts include the
development, research, and testing of the theories and programs to determine their effectiveness. The author and
publisher make no warranty of any kind, expressed or implied, with regard to these programs or the documentation
contained in this book. The author and publisher shall not be liable in any event for incidental or consequential
damages in connection with, or arising out of, the furnishing, performance, or use of these programs.
Reproduced by Pearson from electronic files supplied by the author.
Copyright ยฉ 2020, 2016 by Pearson Education, Inc. 221 River Street, Hoboken, NJ 07030. All rights reserved.
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any
form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written
permission of the publisher. Printed in the United States of America.
ISBN-13: 978-0-13-527708-9
ISBN-10: 0-13-527708-6
Instructor’s Solutions Manual
Pathways to College Mathematics 2e
Table of Contents
Chapter P
Prealgebra Pathways ………………………………………………………………………………….1
Chapter 1
Numerical Pathways ………………………………………………………………………………..77
Chapter 2
Algebraic Pathways:
Equations and Inequalities ………………………………………………………………………143
Chapter 3
Algebraic Pathways:
Graphs, Functions, Linear Functions, and Linear Systems ………………………….215
Chapter 4
Algebraic Pathways:
Polynomials, Quadratic Equations, and Quadratic Functions ………………………295
Chapter 5
Geometric Pathways:
Measurement …………………………………………………………………………………………373
Chapter 6
Geometric Pathways ………………………………………………………………………………393
Chapter 7
Pathways to Probability:
Counting Methods and Probability Theory………………………………………………..435
Chapter 8
Statistical Pathways ……………………………………………………………………………….491
Copyright ยฉ 2020 Pearson Education Inc.
i
Chapter P
Prealgebra Pathways
Check Points P.1
1. a. millions place
b. ones place
c. hundreds place
d. millions place
2. Work from left to right. Write the name of the three-digit number in each period, followed by the name of the period and
a comma. Do not write the name of the last period, โones.โ
a. twenty-seven thousand, one hundred forty-three
b. five hundred twenty-one million, six hundred thirty thousand, fifty-seven
3. a. Begin by noting how to write the number within each period.
53 ๏ธ
406
๏ถ๏ด๏ท๏ด
๏ถ๏ด๏ด๏ท๏ด๏ด๏ธ
fifty-three thousand, four hundred six
Write the digits for the number in each period, followed by a comma.
The standard form for the number is 53,406.
b. Begin by noting how to write the number within each period.
016
๏ถ๏ด๏ด
๏ด204
๏ท๏ด๏ด๏ด
๏ธ
๏ถ๏ด๏ด๏ด๏ด932
๏ท๏ด๏ด๏ด๏ด
๏ธ
๏ถ๏ท๏ธ
two hundred four million, nine hundred thirty-two thousand, sixteen
Write the digits for the number in each period, followed by a comma.
The standard form for the number is 204,932,016.
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Chapter P Prealgebra Pathways
4. a. The place value chart shows that 704,663 contains 7 hundred-thousands, 0 ten-thousands, 4 thousands, 6 hundreds, 6
tens, and 3 ones. Thus 704,663 is written in expanded form as follows: 704,663 ๏ฝ 700, 000 ๏ซ 4000 ๏ซ 600 ๏ซ 60 ๏ซ 3.
b. The place value chart shows that 49,063,400 contains 4 ten-millions, 9 millions, 0 hundred-thousands, 6 tenthousands, 3 thousands, 4 hundreds, 0 tens, and 0 ones. Thus 49,063,400 is written in expanded form as follows:
49, 063, 400 ๏ฝ 40, 000, 000 ๏ซ 9, 000, 000 ๏ซ 60, 000 ๏ซ 3000 ๏ซ 400.
5. a. 14 ๏พ 5 because 14 is to the right of 5 on the number line.
b. 0 ๏ผ 16 because 0 is to the left of 16 on the number line.
6. a. The digit to the right of the thousands digit is 4, which is less than 5. This indicates to leave the thousands digit the
same. Replace all digits to the right with zeros.
57, 498 ๏ป 57, 000
b. The digit to the right of the hundred-thousands digit is 5. This indicates to add one to the hundred-thousands digit.
Replace all digits to the right with zeros.
4,856,902 ๏ป 4,900, 000
c. The digit to the right of the thousands digit is 6, which is greater than 5. This implies to add one to the thousands
digit. Replace all digits to the right with zeros.
9602 ๏ป 10, 000
d. The digit to the right of the millions digit is 2, which is less than 5. This implies to leave the millions digit the same.
Replace all digits to the right with zeros.
684, 236, 042 ๏ป 684, 000, 000
7. a. The digit to the right of the billions digit is 5. This implies to add one to the billions digit. Replace all digits to the
right with zeros.
7,599, 445,183 ๏ป 8, 000, 000, 000
b. The digit to the right of the ten-thousands digit is 5. This implies to add one to the ten-thousands digit. Replace all
digits to the right with zeros.
7,599, 445,183 ๏ป 7,599, 450, 000
c. The digit to the right of the thousands digit is 1, which is less than 5. This implies to leave the thousands digit the
same. Replace all digits to the right with zeros.
7,599, 445,183 ๏ป 7,599, 445, 000
8. a. The cost of a coronary bypass in the United States is $67,583
b. The country with the least amount in the CT scan column is India.
The average cost for this procedure in India is $43.
c. The charge for an appendectomy in Chile is $5509. The countries in which an appendectomy costs more than in
Chile are Canada, Switzerland, and United States.
9. a. We begin with the number of marriages between an African-American husband and a white wife in 2010. Look at
the bars labeled with the year 2010. The yellow bar to the right represents the number of marriages between an
African-American husband and a white wife. The number above this bar is 390, representing 390 thousand. Thus, in
2010, there were 390,000 marriages between an African-American husband and a white wife.
b. Look for the red bar labeled 61 (for 61 thousand, or 61,000). This is the bar to the left for the year labeled 1990.
Thus, in 1990, there were 61,000 marriages between a white husband and an African-American wife.
2
Copyright ยฉ 2020 Pearson Education, Inc.
Section P.1 Whole Numbers
Concept and Vocabulary Check P.1
1. whole; 0
11. hundred-millions
2. standard
12. hundred-millions
3. periods
13. two hundred fifty-eight
4. millions; hundred-thousands; thousands; tens
14. three hundred twenty-four
5. millions; forty-two; nine
15. eight thousand, three hundred seventy-six
6. expanded; 5000; 60; 8
16. six thousand, two hundred twenty-six
7. number line
17. thirty-six thousand, eight hundred eighty
8.
19. seven million, five hundred sixty-six thousand
10. 8; 5; add 1; 9,000,000
20. four million, three hundred two thousand
11. 2; 3; do not change; 8,542,000
21. thirty-five million, two hundred sixty thousand,
three hundred seventy-five
Exercise Set P.1
22. fifty-seven million, forty-four thousand, two
hundred eight
Note that exercises #1 – 22 use the following table:
23. The standard form is 3468.
24. The standard form is 5283.
25. The standard form is 86,500.
26. The standard form is 58,004.
27. The standard form is 16,402,012.
28. The standard form is 14,204,015.
1. hundreds
2. hundreds
3. ones
4. ones
5. hundred-thousands
6. hundred-thousands
7. millions
8. millions
9. ten-millions
10. ten-millions
29. The standard form is 9,000,009.
30. The standard form is 5,000,005.
31. The standard form is 26,034,203.
32. The standard form is 52,028,706.
33. The standard form is 620,595.
34. The standard form is 430,696.
35. The expanded form is 600 ๏ซ 40 ๏ซ 3.
36. The expanded form is 500 ๏ซ 70 ๏ซ 2.
37. The expanded form is 5000 ๏ซ 40 ๏ซ 6.
38. The expanded form is 3000 ๏ซ 50 ๏ซ 7.
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Chapter P Prealgebra Pathways
39. The expanded form is 80, 000 ๏ซ 1000 ๏ซ 300 ๏ซ 60 ๏ซ 4.
40. The expanded form is 70, 000 ๏ซ 2000 ๏ซ 500 ๏ซ 40 ๏ซ 6.
41. The expanded form is 50, 000 ๏ซ 5000 ๏ซ 30 ๏ซ 8.
42. The expanded form is 40, 000 ๏ซ 4000 ๏ซ 20 ๏ซ 9.
43. The expanded form is 20, 000, 000 ๏ซ 8, 000, 000 ๏ซ 600, 000 ๏ซ 40, 000.
44. The expanded form is 50, 000, 000 ๏ซ 6, 000, 000 ๏ซ 300, 000 ๏ซ 7000 ๏ซ 30 ๏ซ 2.
45. 9 ๏พ 3 because 9 is to the right of 3 on the number line.
46. 7 ๏พ 2 because 7 is to the right of 2 on the number line.
47. 0 ๏ผ 14 because 0 is to the left of 14 on the number line.
48. 0 ๏ผ 16 because 0 is to the left of 16 on the number line.
49. 3600 ๏ผ 4500 because 36000 is to the left of 4500 on the number line.
50. 2300 ๏ผ 3200 because 2300 is to the left of 3200 on the number line.
51. 200, 000 ๏พ 20, 000 because 200,000 is to the right of 20,000 on the number line.
52. 300, 000 ๏พ 30, 000 because 300,000 is to the right of 30,000 on the number line.
53. 624 rounded to the nearest ten is 620.
54. 372 rounded to the nearest ten is 370.
55. 627 rounded to the nearest ten is 630.
56. 378 rounded to the nearest ten is 380.
57. 4891 rounded to the nearest hundred is 4900.
58. 5482 rounded to the nearest hundred is 5500.
59. 4831 rounded to the nearest hundred is 4800.
60. 5432 rounded to the nearest hundred is 5400.
61. 61,529 rounded to the nearest thousand is 62,000.
62. 72,503 rounded to the nearest thousand is 73,000.
63. 61,129 rounded to the nearest thousand is 61,000.
64. 72,103 rounded to the nearest thousand is 72,000.
65. 24,628 rounded to the nearest ten-thousand is 20,000.
66. 34,628 rounded to the nearest ten-thousand is 30,000.
67. 345,207 rounded to the nearest ten-thousand is 350,000.
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Section P.1 Whole Numbers
68. 645,308 rounded to the nearest ten-thousand is 650,000.
69. 86,609,100 rounded to the nearest million is 87,000,000.
70. 75,809,100 rounded to the nearest million is 76,000,000.
71. 86,409,100 rounded to the nearest million is 86,000,000.
72. 75,309,100 rounded to the nearest million is 75,000,000.
73. 86,609,100 rounded to the nearest ten-million is 90,000,000.
74. 75,809,100 rounded to the nearest million is 80,000,000.
75. ninety-two quadrillion, two hundred thirty-three trillion, seven hundred twenty billion, three hundred sixty-eight million,
five hundred forty-seven thousand, eight hundred.
76. ten-quadrillions
77. 700, 000, 000, 000 ๏ซ 20, 000, 000, 000
78. 90, 000, 000, 000, 000, 000 ๏ซ 2, 000, 000, 000, 000, 000
79. 92,233,720,368,547,800 rounded to the nearest ten-quadrillion is 90,000,000,000,000,000.
The word name is ninety quadrillion.
80. 92,233,720,368,547,800 rounded to the nearest quadrillion is 92,000,000,000,000,000.
The word name is ninety-two quadrillion.
81. The greatest yearly earnings are for males with a Doctorate which is $131,569. In word form this is one hundred thirtyone thousand, five hundred sixty-nine dollars.
82. The least yearly earnings are for females with a 9th to 12th grade, non-graduate which is $16,812. In word form this is
sixteen thousand, eight hundred twelve dollars.
83. Women with bachelorโs degree earns $50,856. In word form this is fifty thousand, eight hundred fifty-six dollars.
84. Men with associate degree earns $51,865. In word form this is fifty-one thousand, eight hundred sixty-five dollars.
85. Men with bachelorโs degree will earns $79,927 which would round to $80,000. In word form this is seventy-nine
thousand, nine hundred twenty-seven dollars.
86. Women with masterโs degree will earns $64,834 which would round to $60,000. In word form this is sixty-four
thousand, eight hundred thirty-four dollars.
87. In 1960 there were only 9,700,000 immigrants.
88. In 2016 there were 43,700,000 immigrants.
89. In 1900 there were only 10,300,000 immigrants.
In 1940 there were only 11,600,000 immigrants.
In 1960 there were only 9,700,000 immigrants.
90. In 1980 there were 14,100,000 immigrants.
In 2000 there were 31,100,000 immigrants.
In 2016 there were 43,700,000 immigrants.
91. In 1980 there were 14,100,000 immigrants.
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Chapter P Prealgebra Pathways
92. In 2000 there were 31,100,000 immigrants.
93. 2,376,206; two million, three hundred seventy-six thousand, two hundred six
94. 1,857,160; one million; eight hundred fifty-seven thousand, one hundred sixty
95. Williams
96. Brown and Jones
97. 1,857,160 rounded to the nearest hundred-thousand is 1,900,000.
98. 2,376,206 rounded to the nearest hundred-thousand is 2,400,000.
99. The 3 is in the ten-thousands place.
100. The 8 is in the ten-thousands place.
101. 1,380,145 1,380,145
103. two thousand, four hundred fifty-three
104. two hundred two thousand, twenty-two
105. 102,063
106. 12,042
107. โ 117. Answers will vary.
118. does not make sense; Explanations will vary. Sample explanation: Adding one to this number would create a bigger
number.
119. makes sense
120. makes sense
121. makes sense
122. true
123. false; Changes to make the statement true will vary. A sample change is: The number 32,864 is written in standard form.
124. false; Changes to make the statement true will vary. A sample change is: When rounding whole numbers, the digit to be
rounded either stays the same or increases by 1.
125. false; Changes to make the statement true will vary. A sample change is: When comparing numbers of various items,
tables are just as effective as bar graphs.
126. The whole numbers from 10 to 40 would be rounded to 10 or 20 or 30 or 40. So there are four different rounded
numbers.
6
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Section P.2 Fractions and Mixed Numbers
Check Points P.2
1. a.
3
Numerator is 3. Denominator is 10.
10
b.
10
Numerator is 10. Denominator is 3.
3
c.
๏ฐ
๏ฏ
180
Numerator is ๏ฐ . Denominator is 180๏ฏ.
2. a. There are 7 parts shaded out of a total 10 equal parts. Thus, the fraction
7
(seven-tenths) represents the shaded
10
portion of the figure.
b. There are 2 parts shaded out of a total 5 equal parts. Thus, the fraction
2
(two-fifths) represents the shaded portion
5
of the figure.
c. There are 13 parts shaded out of a total 16 equal parts. Thus, the fraction
13
(thirteen-sixteenths) represents the
16
shaded portion of the figure.
number of fatal gun accidents with
children between 15 and 34
3. Fraction of fatal gun accidents with children between 15 and 34 is
.
total number of fatal gun accidents
145 ๏ซ 107
606
252
๏ฝ
606
๏ฝ
4. a.
3
3
is a proper fraction.
Because the numerator is less than the denominator,
10
10
b.
10
10
is an improper fraction.
Because the numerator and denominator are equal,
10
10
c.
10
10
Because the numerator is greater than the denominator,
is an improper fraction.
3
3
5. a. Improper Fraction: Each whole object is divided into 3 equal parts, or thirds. A total of 5 of the thirds are shaded.
5
The improper fraction represents the shaded portion of the group of figures.
3
2
2
of a second object. The mixed number 1
Mixed Number: The shaded portions include 1 whole object and
3
3
represents the shaded portion of the group of figures.
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Chapter P Prealgebra Pathways
b. Improper Fraction: Each whole object is divided into 4 equal parts, or fourths. A total of 9 of the fourths are shaded.
9
represents the shaded portion of the group of figures.
The improper fraction
4
1
1
of a second object. The mixed number 2
Mixed Number: The shaded portions include 2 whole object and
4
4
represents the shaded portion of the group of figures.
6. a. 2
5 8 ๏ 2 ๏ซ 5 16 ๏ซ 5 21
๏ฝ
๏ฝ
๏ฝ
8
8
8
8
b. 12
7. a.
7 16 ๏12 ๏ซ 7 192 ๏ซ 7 199
๏ฝ
๏ฝ
๏ฝ
16
16
16
16
1
35
3
2
5
3
Write the mixed number using quotient
b.
12
7 86
7
16
14
22
86
7
Write the mixed number using quotient
c.
5
2
remainder
. Thus, ๏ฝ 1 .
3
3
original denominator
513
19
86
2
remainder
๏ฝ 12 .
. Thus,
7
7
original denominator
27
19 513
38
133
133
200
Write the mixed number using quotient
513
remainder
๏ฝ 27.
. Thus,
19
original denominator
Concept and Vocabulary Check P.2
1. numerator; denominator
2. 3; 8;
3
8
3. proper; improper
4. mixed
5. 5; 3; 2; 5
8
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Section P.2 Fractions and Mixed Numbers
6. 2; 1
Exercise Set P.2
1.
5
Numerator is 5. Denominator is 13.
13
2.
7
Numerator is 7. Denominator is 19.
19
3.
13
Numerator is 13. Denominator is 5.
5
4.
19
Numerator is 19. Denominator is 7.
7
5.
6.
๏ฐ
60
๏ฐ
180
Numerator is ๏ฐ . Denominator is 60.
Numerator is ๏ฐ . Denominator is 180.
7. There are 3 parts shaded out of a total 5 equal parts. Thus, the fraction
3
represents the shaded portion of the figure.
5
8. There are 2 parts shaded out of a total 5 equal parts. Thus, the fraction
2
represents the shaded portion of the figure.
5
9. There is 1 part shaded out of a total 6 equal parts. Thus, the fraction
1
represents the shaded portion of the figure.
6
10. There are 2 parts shaded out of a total 6 equal parts. Thus, the fraction
2
represents the shaded portion of the figure.
6
11. There are 7 parts shaded out of a total 16 equal parts. Thus, the fraction
7
represents the shaded portion of the figure.
16
12. There are 11 parts shaded out of a total 16 equal parts. Thus, the fraction
13. a. Probability of landing on a red region ๏ฝ
11
represents the shaded portion of the figure.
16
red region
3
๏ฝ .
total regions 10
b. Probability of landing on a blue region ๏ฝ
blue region
2
๏ฝ .
total regions 10
c. Red is more likely.
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Chapter P Prealgebra Pathways
14. a. Probability of landing on a brown region ๏ฝ
brown region 3
๏ฝ .
total regions 10
b. Probability of landing on a yellow region ๏ฝ
yellow region 2
๏ฝ .
total regions 10
c. Brown is more likely.
15.
9
9
Because the numerator is less than the denominator,
is a proper fraction.
11
11
16.
8
8
is a proper fraction.
Because the numerator is less than the denominator,
13
13
17. c.
11
11
is an improper fraction.
Because the numerator is greater than the denominator,
9
9
18. c.
13
13
Because the numerator is greater than the denominator,
is an improper fraction.
8
8
19. 1
2
is a mixed number.
9
20. 1
5
is a mixed number.
8
21.
6
6
Because the numerator and denominator are equal,
is an improper fraction.
6
6
22.
8
8
Because the numerator and denominator are equal, is an improper fraction.
8
8
23. Improper Fraction:
4
.
3
1
Mixed Number: 1 .
3
24. Improper Fraction:
5
.
3
2
Mixed Number: 1 .
3
11
.
4
3
Mixed Number: 2 .
4
25. Improper Fraction:
10
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Section P.2 Fractions and Mixed Numbers
26. Improper Fraction:
5
.
2
1
Mixed Number: 2 .
2
17
.
5
2
Mixed Number: 3 .
5
27. Improper Fraction:
28. Improper Fraction:
18
.
5
3
Mixed Number: 3 .
5
29. Improper Fraction:
25
.
7
4
Mixed Number: 3 .
7
30. Improper Fraction:
26
.
7
5
Mixed Number: 3 .
7
3 8 ๏ 2 ๏ซ 3 19
31. 2 ๏ฝ
๏ฝ
8
8
8
32. 2
7 9 ๏ 2 ๏ซ 7 25
๏ฝ
๏ฝ
9
9
9
33. 7
3 5 ๏ 7 ๏ซ 3 38
๏ฝ
๏ฝ
5
5
5
34. 6
2 5 ๏ 6 ๏ซ 2 32
๏ฝ
๏ฝ
5
5
5
35. 8
7 16 ๏ 8 ๏ซ 7 135
๏ฝ
๏ฝ
16
16
16
36. 9
5 16 ๏ 9 ๏ซ 5 149
๏ฝ
๏ฝ
16
16
16
37. 12
18 25 ๏12 ๏ซ 18 318
๏ฝ
๏ฝ
25
25
25
38. 15
21 25 ๏15 ๏ซ 21 396
๏ฝ
๏ฝ
25
25
25
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Chapter P Prealgebra Pathways
39.
23
3
๏ฝ4
5
5
40.
47
7
๏ฝ5
8
8
41.
76
4
๏ฝ8
9
9
42.
59
5
๏ฝ6
9
9
43.
711
11
๏ฝ 35
20
20
44.
788
13
๏ฝ 31
25
25
45.
1247
๏ฝ 29
43
46.
1665
๏ฝ 45
37
47.
48.
49.
50.
51. a. Shark attacks ๏ฝ 24 ๏ซ 14 ๏ซ 3 ๏ซ 2 ๏ซ 1 ๏ซ 1 ๏ซ 1 ๏ฝ 46
b. Florida had the greatest number of attacks, 24. Thus, the fraction of attacks that occurred in Florida is
24
.
46
c. Alabama, North Carolina, and Texas had one shark attack each. Thus, the fraction of attacks that occurred in these
3
.
three states is
46
52. a. Shark attacks ๏ฝ 24 ๏ซ 14 ๏ซ 3 ๏ซ 2 ๏ซ 1 ๏ซ 1 ๏ซ 1 ๏ฝ 46
12
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Section P.3 Simplifying Fractions; Operations with Fractions
b. Hawaii had the second greatest number of attacks, 14. Thus, the fraction of attacks that occurred in Hawaii is
14
.
46
c. South Carolina and California had a combined 5 shark attacks. Thus, the fraction of attacks that occurred in these
5
.
two states is
46
53. โ 58. Answers will vary.
59. does not make sense; Explanations will vary. Sample explanation: Since
3
is an improper fraction the price would be
2
higher than the original price.
60. makes sense
61. does not make sense; Explanations will vary. Sample explanation: There are 8 total people, so the denominator would be
8.
62. true
63. true
64. false; Changes to make the statement true will vary. A sample change is: A mixed number has a whole number and a
5
is not a proper fraction.
proper fraction.
4
65. false; Changes to make the statement true will vary. A sample change is:
17
2
๏ฝ3 .
5
5
Check Points P.3
1. a. 300 ๏ฝ 3 ๏100
๏ฝ 3 ๏10 ๏10
๏ฝ 3๏ 2 ๏5๏ 2๏5
๏ฝ 2 ๏ 2 ๏3๏5๏5
b. 36 ๏ฝ 6 ๏ 6
๏ฝ 2 ๏3๏ 2 ๏3
๏ฝ 2 ๏ 2 ๏3๏3
2. a.
10 2 ๏ 5 2
๏ฝ
๏ฝ
15 3 ๏ 5 3
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Chapter P Prealgebra Pathways
b.
c.
13
13
; Because 13 and 15 share no common factors (other than 1),
is already reduced to its lowest terms.
15
15
d.
9 1๏ 9 1
๏ฝ
๏ฝ
45 5 ๏ 9 5
3. a.
b.
c.
d.
4. a.
4 2 4๏2
8
๏ ๏ฝ
๏ฝ
11 3 11 ๏ 3 33
6๏
3 6 3 18
3
๏ฝ ๏ ๏ฝ
๏ฝ3
5 1 5 5
5
3 2 3๏ 2 6 2 ๏ 3 2
๏ ๏ฝ
๏ฝ
๏ฝ
๏ฝ
7 3 7 ๏ 3 21 7 ๏ 3 7
Remember that you can divide numerators and denominators by common factors before performing multiplication.
3 2 3 2 2
๏ ๏ฝ ๏ ๏ฝ
7 3 7 3 7
1
๏ฆ 2 ๏ถ ๏ฆ 1 ๏ถ 17 3 51
๏ง 3 5 ๏ท ๏ง1 2 ๏ท ๏ฝ 5 ๏ 2 ๏ฝ 10 ๏ฝ 5 10
๏จ
๏ธ๏จ ๏ธ
5 3 5 8 5 4 ๏ 2 10
1
๏ฝ
๏ฝ3
๏ธ ๏ฝ ๏ ๏ฝ ๏
4 8 4 3 4
3
3
3
b.
2
2 3 2 1 2
๏ธ3 ๏ฝ ๏ธ ๏ฝ ๏ ๏ฝ
3
3 1 3 3 9
c.
3
1 27 9 27 4 9 ๏ 3 4 3
1
3 ๏ธ2 ๏ฝ
๏ธ ๏ฝ
๏ ๏ฝ
๏ ๏ฝ ๏ฝ1
8
4 8 4 8 9 4 ๏2 9 2
2
5. a.
2 3
5
๏ซ ๏ฝ
11 11 11
b.
5 1 4 2
๏ญ ๏ฝ ๏ฝ
6 6 6 3
c.
3
1 27 9 18 9
1
3 ๏ญ1 ๏ฝ
๏ญ ๏ฝ
๏ฝ ๏ฝ2
8
8 8 8 8 4
4
6.
14
42
2 ๏ 3 ๏7
7
๏ฝ
๏ฝ
24 2 ๏ 2 ๏ 2 ๏ 3 4
When reducing fractions, it may not be necessary to write prime factorizations.
We can use the greatest common factor to reduce this fraction.
42 7 ๏ 6 7
๏ฝ
๏ฝ
24 4 ๏ 6 4
2 2 ๏ 7 14
๏ฝ
๏ฝ
3 3 ๏ 7 21
Copyright ยฉ 2020 Pearson Education, Inc.
Section P.3 Simplifying Fractions; Operations with Fractions
7. a.
b.
c.
8. a.
1
1 3 1๏ 5 3 ๏ 2 5
6 11
or 1
๏ซ ๏ฝ
๏ซ
๏ฝ
๏ซ
๏ฝ
2 5 2 ๏ 5 5 ๏ 2 10 10 10
10
4 3 4 ๏ 4 3 ๏ 3 16 9
7
๏ญ ๏ฝ
๏ญ
๏ฝ
๏ญ
๏ฝ
3 4 3 ๏ 4 4 ๏ 3 12 12 12
1 11 19 23 19 ๏ 2 23
๏ญ
๏ฝ
๏ญ
3 ๏ญ1 ๏ฝ
6 12 6 12 6 ๏ 2 12
38 23 15
๏ฝ
๏ญ
๏ฝ
12 12 12
5
1
๏ฝ or 1
4
4
3
7
2
๏ซ10
7
17
27
b.
3
16
7
๏ซ20
16
15
35
c.
d.
5
10
2 ๏5
๏ฝ 35
๏ฝ 35
8
16
2 ๏8
7
12
10
๏ซ2
12
17
5
5
18 ๏ฝ 18 ๏ซ 1 ๏ฝ 19
12
12
12
16
19
23
5
๏ญ16
23
24
8
e.
5
7
14
23
19
20
4
๏ญ 2
20
18
16
3
15
5 ๏3
๏ฝ 16
๏ฝ 16
4
20
5 ๏4
Copyright ยฉ 2020 Pearson Education, Inc.
15
Chapter P Prealgebra Pathways
9. To subtract the fraction parts, we need to borrow from the number 8 because
This is done as follows: 8
5
7
is less than .
9
9
5
5
14
14
๏ฝ 7๏ป
๏ซ1 ๏ฝ 7 ๏ซ
๏ฝ7 .
9
9
9
8 9
5
14
๏ฝ 7
9
9
7
7
๏ญ3 ๏ฝ๏ญ3
9
9
8
4
10. a.
7
9
Using inspection, the least common denominator for the fraction parts is 35.
2
2๏5
10
4 ๏ฝ 4
๏ฝ 4
7
7 ๏5
35
11
11
11
๏ซ 9
๏ฝ๏ซ9
๏ฝ๏ซ9
35
35
35
13
21
7 ๏3
3
๏ฝ 13
๏ฝ 13
35
5
7 ๏5
b. Using inspection, the least common denominator for the fraction parts is 10.
1
1๏ 5
5
26 ๏ฝ 26
๏ฝ 26
2
2๏5
10
3
3๏ 2
6
๏ซ 10 ๏ฝ ๏ซ 10
๏ฝ ๏ซ 10
5
5๏ 2
10
36
c.
11
1
1
๏ฝ 36 ๏ซ 1 ๏ฝ 37
10
10
10
Using inspection, the least common denominator for the fraction parts is 12.
2
2๏4
8
32 ๏ฝ 32
๏ฝ 32
3
3๏ 4
12
5
5
5
๏ญ 9 ๏ฝ๏ญ9
๏ฝ๏ญ9
12
12
12
23
3
3 ๏1
1
๏ฝ 23
๏ฝ 23
12
4
3 ๏4
d. The least common denominator for the fraction is 35.
2
2๏5
10
14 ๏ฝ 14
๏ฝ 14
7
7๏5
35
3
3๏7
21
๏ญ 6 ๏ฝ๏ญ6
๏ฝ๏ญ6
5
5๏7
35
We will need to borrow from the number 14 in order to do the subtraction.
14
16
10
10
45
45
๏ฝ 13 ๏ซ 1 ๏ฝ 13 ๏ซ
๏ฝ 13 .
35
35
35
35
Copyright ยฉ 2020 Pearson Education, Inc.
Section P.3 Simplifying Fractions; Operations with Fractions
Now perform the subtraction.
10
45
๏ฝ 13
35
35
21
21
๏ญ6
๏ฝ๏ญ6
35
35
14
7
e.
24
35
We need to borrow 1 from the whole number 23 and create a fraction to perform the subtraction. We will choose 1
11
to be
.
11
11
11
4
4
๏ญ 17 ๏ฝ ๏ญ 17
11
11
7
5
11
23
11. a.
๏ฝ 22
We are interested in the fraction of jobs that will not require college education. This includes the fraction that will
require less than high school education plus the fraction that will require a high school graduate education.
11
6
11
6๏4
๏ซ
๏ฝ
๏ซ
100 25 100 25 ๏ 4
11
24
๏ฝ
๏ซ
100 100
35
๏ฝ
100
7
๏ฝ
20
7
of all jobs will not require any college education.
By 2020,
20
b. We have seen that the phrase โhow much greaterโ implies subtraction. To determine how much greater the fraction
that will require a college degree is than the fraction that will require some college, we find the difference of these
fractions.
7
3
7
3๏ 2
๏ญ
๏ฝ
๏ญ
20 10 20 10 ๏ 2
7
6
๏ฝ
๏ญ
20 20
1
๏ฝ
20
1
By 2020, the fraction of jobs that will require a college degree will be
greater than the fraction that will require
20
some college.
Copyright ยฉ 2020 Pearson Education, Inc.
17
Chapter P Prealgebra Pathways
Concept and Vocabulary Check P.3
9. composite; 140 ๏ฝ 10 ๏14
๏ฝ 2๏5๏ 2๏7
๏ฝ 2๏ 2๏5๏7
1. natural
2. prime
3. factors; product
4.
5.
10. composite; 110 ๏ฝ 10 ๏11
๏ฝ 2 ๏ 5 ๏11
a
b
11. 79 has no factors other than 1 and 79, so 79 is
prime.
a๏c
12. 83 has no factors other than 1 and 83, so 83 is
prime.
b๏d
6. reciprocals
7.
8.
13. composite; 81 ๏ฝ 9 ๏ 9
๏ฝ 3๏3๏3๏3
d
c
14. composite; 64 ๏ฝ 8 ๏ 8 ๏ฝ 2 ๏ 4 ๏ 2 ๏ 4
๏ฝ 2๏2๏2๏2๏2๏2
a๏ซc
b
15. composite; 240 ๏ฝ 10 ๏ 24
๏ฝ 2 ๏ 5 ๏ 2 ๏ 12
๏ฝ 2๏5๏ 2๏3๏ 4
๏ฝ 2๏5๏ 2๏3๏ 2๏ 2
๏ฝ 2๏ 2๏ 2๏ 2๏3๏5
9. least common denominator
Exercise Set P.3
1. composite; 22 ๏ฝ 2 ๏11
16. composite; 360 ๏ฝ 10 ๏ 36
๏ฝ 2๏5๏6๏6
๏ฝ 2 ๏5๏ 2 ๏3๏ 2 ๏3
๏ฝ 2 ๏ 2 ๏ 2 ๏3๏3๏5
2. composite; 15 ๏ฝ 3 ๏ 5
3. composite; 20 ๏ฝ 4 ๏ 5
๏ฝ 2๏2๏5
18
4. composite; 75 ๏ฝ 3 ๏ 25
๏ฝ 3๏5๏5
17.
10 2 ๏ 5 5
๏ฝ
๏ฝ
16 2 ๏ 8 8
5. 37 has no factors other than 1 and 37, so 37 is
prime.
18.
8
2 ๏4 4
๏ฝ
๏ฝ
14 2 ๏ 7 7
6. 23 has no factors other than 1 and 23, so 23 is
prime.
19.
15 3 ๏ 5 5
๏ฝ
๏ฝ
18 3 ๏ 6 6
7. composite; 36 ๏ฝ 4 ๏ 9
๏ฝ 2๏ 2 ๏3๏3
20.
18 9 ๏ 2 2
๏ฝ
๏ฝ
45 9 ๏ 5 5
8. composite; 200 ๏ฝ 10 ๏ 20
๏ฝ 2 ๏ 5 ๏ 2 ๏10
๏ฝ 2๏5๏ 2๏ 2๏5
21.
35
5 ๏7
7
๏ฝ
๏ฝ
50 5 ๏10 10
22.
45
5 ๏9
9
๏ฝ
๏ฝ
50 5 ๏10 10
Copyright ยฉ 2020 Pearson Education, Inc.
Section P.3 Simplifying Fractions; Operations with Fractions
23.
32 16 ๏ 2 2
๏ฝ
๏ฝ
80 16 ๏ 5 5
๏ฆ 3 ๏ถ ๏ฆ 3 ๏ถ 15 8 120 20 ๏ 6
39. ๏ง 3 ๏ท ๏ง1 ๏ท ๏ฝ ๏ ๏ฝ
๏ฝ
๏ฝ6
20 ๏1
๏จ 4 ๏ธ ๏จ 5 ๏ธ 4 5 20
24.
75 5 ๏15 15
๏ฝ
๏ฝ
80 5 ๏16 16
40.
๏ฆ 4 ๏ถ ๏ฆ 1 ๏ถ 14 5 70 10 ๏ 7
๏ฝ
๏ง๏จ 2 ๏ท๏ธ ๏ง๏จ1 ๏ท๏ธ ๏ฝ ๏ ๏ฝ
5
4
5 4 20 10 ๏ 2
๏ฝ
44 2 ๏ 22 22
25.
๏ฝ
๏ฝ
50 2 ๏ 25 25
7
1
or 3
2
2
38 2 ๏19 19
26.
๏ฝ
๏ฝ
50 2 ๏ 25 25
41.
5 4 5 3 5 ๏ 3 15
๏ธ ๏ฝ ๏ ๏ฝ
๏ฝ
4 3 4 4 4 ๏ 4 16
120 2 ๏ 60 60
27.
๏ฝ
๏ฝ
86
2 ๏ 43 43
42.
7 2 7 3 7 ๏ 3 21
5
๏ธ ๏ฝ ๏ ๏ฝ
๏ฝ
or 1
8 3 8 2 8 ๏ 2 16
16
116 2 ๏ 58 58
28.
๏ฝ
๏ฝ
86
2 ๏ 43 43
43.
29.
2 1 2 ๏1 2
๏ ๏ฝ
๏ฝ
5 3 5 ๏ 3 15
44.
18
18 1
๏ธ2 ๏ฝ ๏
5
5 2
18 ๏1 18 2 ๏ 9 9
4
๏ฝ
๏ฝ
๏ฝ
๏ฝ or 1
5 ๏ 2 10 2 ๏ 5 5
5
12
12 1
๏ธ3 ๏ฝ ๏
7
7 3
12 ๏1 12 3๏ฏ ๏ 4 4
๏ฝ
๏ฝ
๏ฝ
๏ฝ
7 ๏ 3 21 3๏ฏ ๏ 7 7
30.
3 1 3 ๏1
3
๏ ๏ฝ
๏ฝ
7 4 7 ๏ 4 28
31.
3 7
3 ๏ 7 21
๏ ๏ฝ
๏ฝ
8 11 8 ๏11 88
45. 2 ๏ธ
18 2 5 10 2 ๏ 5 5
๏ฝ ๏ ๏ฝ
๏ฝ
๏ฝ
5 1 18 18 2 ๏ 9 9
32.
5 3
5 ๏ 3 15
๏ ๏ฝ
๏ฝ
8 11 8 ๏11 88
46. 3 ๏ธ
12 3 7 21 3๏ฏ ๏ 7 7
3
๏ฝ ๏ ๏ฝ
๏ฝ
๏ฝ or 1
7 1 12 12 3๏ฏ ๏ 4 4
4
33. 9 ๏
4 9 4 9 ๏ 4 36
1
or 5
๏ฝ ๏ ๏ฝ
๏ฝ
7 1 7 1๏ 7
7
7
47.
3 1 3 4 3 ๏ 4 12
๏ธ ๏ฝ ๏ ๏ฝ
๏ฝ
๏ฝ3
4 4 4 1 4 ๏1 4
34. 8 ๏
3 8 3 8 ๏ 3 24
3
๏ฝ ๏ ๏ฝ
๏ฝ
or 3
7 1 7 1๏ 7
7
7
48.
3 1 3 7 3 ๏ 7 21
๏ธ ๏ฝ ๏ ๏ฝ
๏ฝ
๏ฝ3
7 7 7 1 7 ๏1 7
35.
1 5 1๏ 5
5
5 ๏1
1
๏ ๏ฝ
๏ฝ
๏ฝ
๏ฝ
10 6 10 ๏ 6 60 5 ๏12 12
49.
7 5 7 3 7 ๏ 3 21
3 ๏7
7
๏ธ ๏ฝ ๏ ๏ฝ
๏ฝ
๏ฝ
๏ฝ
6 3 6 5 6 ๏ 5 30 3 ๏10 10
36.
1 2 1๏ 2
2
2 ๏1
1
๏ฝ
๏ ๏ฝ
๏ฝ
๏ฝ
8 3 8 ๏ 3 24 2 ๏12 12
50.
37.
5 6 5 ๏ 6 30 2 ๏15 15
1
๏ ๏ฝ
๏ฝ
๏ฝ
๏ฝ
or 1
4 7 4 ๏ 7 28 2 ๏14 14
14
7 6
7 ๏ 6 42 2 ๏ 21 21
38.
๏ ๏ฝ
๏ฝ
๏ฝ
๏ฝ
4 11 4 ๏11 44 2 ๏ 22 22
7 3 7 8 7 ๏ 8 56 4๏ฏ ๏14
๏ธ ๏ฝ ๏ ๏ฝ
๏ฝ
๏ฝ
4 8 4 3 4 ๏ 3 12
4๏ฏ ๏ 3
14
2
๏ฝ
or 4
3
3
51.
1 1 1 7 7
7 ๏1 1
๏ธ ๏ฝ ๏ ๏ฝ
๏ฝ
๏ฝ
14 7 14 1 14 7 ๏ 2 2
52.
1 1 1 4 4 4๏ฏ ๏1 1
๏ธ ๏ฝ ๏ ๏ฝ ๏ฝ
๏ฝ
8 4 8 1 8 4๏ฏ ๏ 2 2
Copyright ยฉ 2020 Pearson Education, Inc.
19
Chapter P Prealgebra Pathways
3
1 33 11
๏ธ
53. 6 ๏ธ 1 ๏ฝ
5 10 5 10
33 10 11 ๏ 3 5 ๏ 2 6
๏ฝ
๏ ๏ฝ
๏
๏ฝ ๏ฝ6
5 11
1
5
11
3
5 7 21
54. 1 ๏ธ 2 ๏ฝ ๏ธ
4
8 4 8
7 8 56 28 ๏ 2 2
๏ฝ ๏ ๏ฝ
๏ฝ
๏ฝ
4 21 84 28 ๏ 3 3
55.
56.
57.
66.
3
67.
2 4 2๏ซ4 6
๏ซ ๏ฝ
๏ฝ
11 11
11
11
5 2 5๏ซ2 7
๏ซ ๏ฝ
๏ฝ
13 13
13
13
68.
5 1
6
2 ๏3 3
๏ซ
๏ฝ
๏ฝ
๏ฝ
58.
16 16 16 2 ๏ 8 8
5 5 10 2 ๏ 5 5
1
๏ซ ๏ฝ
๏ฝ
๏ฝ
or 1
8 8 8
2 ๏4 4
4
60.
3 3 6 2 ๏3 3
๏ซ ๏ฝ ๏ฝ
๏ฝ
8 8 8 2 ๏4 4
61.
7 5
2
2 ๏1 1
๏ญ
๏ฝ
๏ฝ
๏ฝ
12 12 12 2 ๏ 6 6
62.
13 5
8
2 ๏4 4
๏ญ ๏ฝ
๏ฝ
๏ฝ
18 18 18 2 ๏ 9 9
63.
16 2 14 7 ๏ 2
๏ญ ๏ฝ
๏ฝ
๏ฝ2
7 7 7
7 ๏1
64.
17 2 15
๏ญ ๏ฝ
๏ฝ3
5 5 5
65.
2
5
1
๏ซ2
5
20
70.
71.
72.
3
5
3
28
or
5
5
3
8
1
๏ญ2
8
73.
2
1
13
๏ฝ 3 or
8
4
4
3
10
1
๏ญ1
10
4
3
69.
6
3
15
๏ฝ 3 or
8
4
4
5
3
7 1
8
4 ๏2 2
๏ซ
๏ฝ
๏ฝ
๏ฝ
12 12 12 4 ๏ 3 3
59.
5
8
1
๏ซ1
8
2
2
1
16
๏ฝ 3 or
10
5
5
1 1 1 5 1 2
๏ซ ๏ฝ ๏ ๏ซ ๏
2 5 2 5 5 2
5 2 5๏ซ2 7
๏ฝ
๏ซ
๏ฝ
๏ฝ
10 10
10
10
1 1 1 5 1 3
๏ซ ๏ฝ ๏ ๏ซ ๏
3 5 3 5 5 3
5 3 5๏ซ3 8
๏ฝ ๏ซ
๏ฝ
๏ฝ
15 15
15
15
3 3 3 5 3
๏ซ
๏ฝ ๏ ๏ซ
4 20 4 5 20
15 3
๏ฝ
๏ซ
20 20
18
2 ๏9
9
๏ฝ
๏ฝ
๏ฝ
20 2 ๏10 10
2 2 2 3 2
๏ซ
๏ฝ ๏ ๏ซ
5 15 5 3 15
6 2
8
๏ฝ ๏ซ
๏ฝ
15 15 15
3 5 3 3 5 2
๏ซ
๏ฝ ๏ ๏ซ ๏
8 12 8 3 12 2
9 10 19
๏ฝ
๏ซ
๏ฝ
24 24 24
Copyright ยฉ 2020 Pearson Education, Inc.
Section P.3 Simplifying Fractions; Operations with Fractions
74.
75.
2
1 11 5
82. 3 ๏ญ 2 ๏ฝ ๏ญ
3
2 3 2
11 2 5 3
๏ฝ ๏ ๏ญ ๏
3 2 2 3
22 15 7
1
๏ฝ
๏ญ
๏ฝ
or 1
6
6 6
6
3 2
3 3 2 2
๏ซ
๏ฝ ๏ ๏ซ ๏
10 15 10 3 15 2
9
4 13
๏ฝ
๏ซ
๏ฝ
30 30 30
11 2 11 2 2 11 4
7
๏ญ ๏ฝ ๏ญ ๏ ๏ฝ ๏ญ ๏ฝ
18 9 18 9 2 18 18 18
17 4 17 4 2 17 8
9
๏ญ ๏ฝ
๏ญ ๏ ๏ฝ
๏ญ
๏ฝ
76.
18 9 18 9 2 18 18 18
9 ๏1 1
๏ฝ
๏ฝ
9 ๏2 2
77.
78.
79.
80.
4 3 4 4 3 3
๏ญ ๏ฝ ๏ ๏ญ ๏
3 4 3 4 4 3
16 9
7
๏ฝ
๏ญ
๏ฝ
12 12 12
83.
84.
3 2 3 3 2 2
๏ญ ๏ฝ ๏ ๏ญ ๏
2 3 2 3 3 2
9 4 5
๏ฝ ๏ญ ๏ฝ
6 6 6
85.
3
3๏ 2
6
๏ฝ 3
๏ฝ 3
7
7๏2
14
1
1
1
๏ซ 2 ๏ฝ๏ซ 2 ๏ฝ๏ซ2
14
14
14
7
1
11
5 ๏ฝ 5 or
14
2
2
3
86.
2
4
1 ๏ฝ 1
3
6
1
3
๏ซ 4 ๏ฝ๏ซ4
2
6
5
7
1
37
๏ฝ 6 or
6
6
6
3
1 15 7
81. 3 ๏ญ 2 ๏ฝ ๏ญ
4
3 4 3
15 3 7 4
๏ฝ ๏ ๏ญ ๏
4 3 3 4
45 28 17
5
๏ฝ
๏ญ
๏ฝ
or 1
12 12 12
12
87.
2
21
7
๏ซ3
21
9
3
9 ๏ฝ9
21
7
6
3
14
4
๏ซ 14
14
7
1
23 ๏ฝ 23
14
2
9
11
14
7
๏ซ 20
14
18
4
2
25 ๏ฝ 26 ๏ฝ 26
14
14
7
5
13
14
7
๏ซ 30
14
20
6
3
37
๏ฝ 38 ๏ฝ 38
14
14
7
7
8
15
2
๏ญ 2
15
14
12
88.
6
2
๏ฝ 12
15
5
5
6
1
๏ญ 2
6
13
4
2
11 ๏ฝ 11
6
3
Copyright ยฉ 2020 Pearson Education, Inc.
21
Chapter P Prealgebra Pathways
89.
2
11
๏ฝ 6
9
9
5
5
๏ญ4 ๏ฝ๏ญ4
9
9
2
90.
2
9
๏ฝ 5
7
7
5
5
๏ญ3 ๏ฝ๏ญ3
7
7
14
1
2
15 ๏ฝ 15
3
6
1
1
๏ซ 12 ๏ฝ ๏ซ 12
6
6
3
6
๏ฝ 18
5
10
3
3
๏ซ 22 ๏ฝ ๏ซ 22
10
10
18
9
10
4
16
๏ฝ 87
5
20
3
15
๏ซ 72 ๏ฝ ๏ซ 72
4
20
87
4
16
๏ฝ 55
9
36
3
27
๏ซ 63 ๏ฝ ๏ซ 63
4
36
43
7
๏ฝ 119
36
36
5
25
๏ฝ 56
6
30
7
21
๏ญ 18 ๏ฝ ๏ญ 18
10
30
56
38
22
31
11
๏ฝ 160
20
20
55
118
95.
98.
5
6
1
2
22
๏ฝ 29
๏ฝ 28
10
20
20
1
5
5
๏ญ 5 ๏ฝ ๏ญ5
๏ฝ๏ญ5
4
20
20
29
3
1
๏ฝ 27
6
2
159
94.
1
6
1
2
8
19 ๏ฝ 19 ๏ฝ 18
3
6
6
1
3
3
๏ญ 4 ๏ฝ ๏ญ4 ๏ฝ๏ญ4
2
6
6
4
7
40
93.
25
97.
27
92.
2
4
51 ๏ฝ 51
3
6
1
3
๏ญ 26 ๏ฝ ๏ญ 26
2
6
6
2
๏ฝ2
9
3
6
2
91.
96.
7
23
17
20
๏ฆ1 1๏ถ 5 ๏ฆ3 2๏ถ 5
99. ๏ง ๏ญ ๏ท ๏ธ ๏ฝ ๏ง ๏ญ ๏ท ๏ธ
๏จ 2 3๏ธ 8 ๏จ6 6๏ธ 8
1 5
๏ฝ ๏ธ
6 8
1 8 8
2 ๏4
4
๏ฝ ๏ ๏ฝ
๏ฝ
๏ฝ
6 5 30 2 ๏15 15
๏ฆ1 1๏ถ ๏ฆ1 1๏ถ ๏ฆ2 1๏ถ ๏ฆ3 2๏ถ
100. ๏ง ๏ซ ๏ท ๏ธ ๏ง ๏ซ ๏ท ๏ฝ ๏ง ๏ซ ๏ท ๏ธ ๏ง ๏ซ ๏ท
๏จ 2 4๏ธ ๏จ 2 3๏ธ ๏จ 4 4๏ธ ๏จ 6 6๏ธ
3 5
๏ฝ ๏ธ
4 6
3 6
๏ฝ ๏
4 5
18
2 ๏9
9
๏ฝ
๏ฝ
๏ฝ
20 2 ๏10 10
๏ฆ2 4๏ถ ๏ฆ1 3๏ถ ๏ฆ2 3๏ถ ๏ฆ 2 9 ๏ถ
101. ๏ง ๏ธ ๏ท ๏ซ ๏ง ๏ซ ๏ท ๏ฝ ๏ง ๏ ๏ท ๏ซ ๏ง ๏ซ ๏ท
๏จ 3 3 ๏ธ ๏จ 6 4 ๏ธ ๏จ 3 4 ๏ธ ๏จ 12 12 ๏ธ
1 11
๏ฝ ๏ซ
2 12
6 11
๏ฝ
๏ซ
12 12
17
5
๏ฝ
๏ฝ1
12
12
4
2
๏ฝ 38
30
15
Copyright ยฉ 2020 Pearson Education, Inc.
Section P.3 Simplifying Fractions; Operations with Fractions
๏ฆ 3 9 ๏ถ ๏ฆ 7 8 ๏ถ ๏ฆ 3 10 ๏ถ ๏ฆ 21 16 ๏ถ
102. ๏ง ๏ธ ๏ท ๏ซ ๏ง ๏ญ ๏ท ๏ฝ ๏ง ๏ ๏ท ๏ซ ๏ง ๏ญ ๏ท
๏จ 5 10 ๏ธ ๏จ 10 15 ๏ธ ๏จ 5 9 ๏ธ ๏จ 30 30 ๏ธ
2 5
๏ฝ ๏ซ
3 30
2 1
๏ฝ ๏ซ
3 6
4 1
๏ฝ ๏ซ
6 6
5
๏ฝ
6
103. a.
10 6 16 8
๏ซ
๏ฝ
๏ฝ
50 50 50 25
b. The phrase โhow much greaterโ implies we should subtract fractions.
10 6 10 ๏ญ 6 4
2
๏ญ
๏ฝ
๏ฝ
๏ฝ
50 50
50
50 25
104. a.
7
5 12
6
๏ซ
๏ฝ
๏ฝ
50 50 50 25
b. The phrase โhow much greaterโ implies we should subtract fractions.
7
5 7๏ญ5 2
1
๏ญ
๏ฝ
๏ฝ
๏ฝ
50 50
50
50 25
105. a. 11
b. 11
106. a. 10
b. 10
107. a.
1
9
10
๏ซ 6 ๏ฝ 17 ๏ฝ 18 hours
10
10
10
1
9
11
9
2
1
๏ญ 6 ๏ฝ 10 ๏ญ 6 ๏ฝ 4 ๏ฝ 4 hours
10
10
10
10
10
5
4
7
11
1
๏ซ 7 ๏ฝ 17 ๏ฝ 18 hours
10
10
10
10
4
7
14
7
7
๏ญ7 ๏ฝ 9 ๏ญ7 ๏ฝ 2
10
10
10
10
10
8 8 64
๏ ๏ฝ
9 9 81
b.
3 8 24 12 ๏ 2 2
๏ ๏ฝ
๏ฝ
๏ฝ
4 9 36 12 ๏ 3 3
c.
There are black keys to the left of the keys for the notes D, E, G, A, and B.
108. The numbers are the prime numbers between 2 and 97, inclusive.
109. โ 116. Answers will vary.
117. makes sense
118. makes sense
Copyright ยฉ 2020 Pearson Education, Inc.
23
Chapter P Prealgebra Pathways
119. makes sense
120. does not make sense; Explanations will vary. Sample explanation: When adding fractions, we need to have a common
denominator before adding the fractions.
121. does not make sense; Explanations will vary. Sample explanation: Although that method will work, it is easier to leave
them as mixed numbers and perform the addition.
122. makes sense
123.
1
1
1
1
1
1
1
1
1
1
๏ซ
๏ซ
๏ซ
๏ซ
๏ฝ ๏ซ
๏ซ
๏ซ 2 ๏ซ
2
1๏ 2 2 ๏ 3 3 ๏ 4 4 ๏ 5 5 ๏ 6 2 2 ๏ 3 3 ๏ 2
2 ๏5 5๏ 2๏3
1๏ 2 ๏ 3 ๏ 5
1๏ 2 ๏ 5
1๏ 5
1๏ 3
1๏ 2
๏ฝ
๏ซ
๏ซ
๏ซ 2
๏ซ
2
2 ๏ 2 ๏3๏5 2 ๏3๏ 2 ๏5 3๏ 2 ๏5 2 ๏5๏3 5๏ 2 ๏3๏ 2
30 10 5
3
2
๏ฝ
๏ซ
๏ซ
๏ซ
๏ซ
60 60 60 60 60
50
๏ฝ
60
5
๏ฝ
6
124. The notes in each measure add to
3
.
4
Check Points P.4
1. a. The underlined digit of 3.258 is in the hundredths place.
b. The underlined digit of 6.9347 is in the tenths place.
c. The underlined digit of 2056.31479 is in the thousandths place.
d. The underlined digit of 0.002576 is in the hundred-thousandths place.
e. The underlined digit of 22.005124689 is in the millionths place.
2. Work from left to right. Write the whole-number part in words, followed by the word โandโ for the decimal point. Then
write the decimal part in words as though it were a whole number and follow it by the place value of the last digit.
a. nineteen and twenty-four hundredths
b. eight hundred twenty-six and three hundred seventy-five thousandths
c. fifty-two ten-thousandths
24
Copyright ยฉ 2020 Pearson Education, Inc.
Section P.4 Decimals
3. a. Write the digits for the whole-number part, which comes before the word โand.โ Write a decimal point for the word
โand.โ Then write the digits for the decimal part as if it were a whole number.
45 ๏ธ ๏ฝ
32
๏
๏ถ๏ด๏ท๏ด
๏ถ๏ด๏ด๏ด๏ท๏ด๏ด๏ด๏ธ
forty-five and thirty-two hundredths
Make sure the last digit is in the hundredths place.
The standard form for the number is 45.32.
b. Write the digits for the whole-number part, which comes before the word โand.โ Write a decimal point for the word
โand.โ Then write the digits for the decimal part as if it were a whole number.
11
๏
๏ถ๏ด
๏ด400
๏ท๏ด๏ด
๏ธ ๏ฝ
๏ถ๏ด๏ด๏ด๏ท๏ด๏ด๏ด
๏ธ
four hundred and eleven thousandths
Make sure the last digit is in the thousandths place by inserting a 0 in the tenths place.
The standard form for the number is 400.011.
c. The word form does not contain the word โandโ thus, the whole-number part is zero.
๏ถ๏ด๏ด๏ด๏ด๏ด
๏ด411
๏ท๏ด๏ด๏ด๏ด๏ด๏ด
๏ธ
four hundred eleven thousandths
Make sure the last digit is in the thousandths place.
The standard form for the number is 0.411.
d. Write the digits for the whole-number part, which comes before the word โand.โ Write a decimal point for the word
โand.โ Then write the digits for the decimal part as if it were a whole number.
52
8
๏
๏ถ๏ด๏ด๏ด๏ด๏ท๏ด๏ด๏ด๏ด
๏ธ
๏ฝ
๏ฝ
eight and fifty-two ten-thousandths
Make sure the last digit is in the ten-thousandths place by inserting a 0 in the tenths place and the hundredths place.
The standard form for the number is 8.0052.
4. a. 0.7 has 1 decimal place so the denominator will contain 1 zero, or 10.
7
0.7 ๏ฝ
10
b. 0.59 has 2 decimal places so the denominator will contain 2 zeros, or 100.
59
0.59 ๏ฝ
100
c. 0.012 has 3 decimal places so the denominator will contain 3 zeros, or 1000.
12
4 ๏3
3
0.012 ๏ฝ
๏ฝ
๏ฝ
1000 4 ๏ 250 250
5. a. 72.6 has 1 decimal place so the denominator will contain 1 zero, or 10.
6
2 ๏3
3
72.6 ๏ฝ 72 ๏ฝ 72
๏ฝ 72
10
5
2 ๏5
b. 184.083 has 3 decimal places so the denominator will contain 3 zeros, or 1000.
83
184.083 ๏ฝ 184
1000
c. 5.0044 has 4 decimal places so the denominator will contain 4 zeros, or 10,000.
44
4 ๏11
11
5.0044 ๏ฝ 5
๏ฝ5
๏ฝ5
10, 000
2500
4 ๏ 2500
Copyright ยฉ 2020 Pearson Education, Inc.
25
Chapter P Prealgebra Pathways
6. a. The digit to the right of the tenths digit is 3, which is less than 5. Therefore do not change the digit to be rounded.
Drop all digits to the right of the tenths place.
23.436 ๏ป 23.4
b. The digit to the right of the hundredths digit is 6, which is greater than 5. Therefore we add 1 to the hundredths digit.
Drop all digits to the right of the hundredths place.
392.7869 ๏ป 392.79
c. The digit to the right of the thousandths digit is 5. Therefore we add 1 to the thousandths digit. Drop all digits to the
right of the thousandths place.
1400.61558 ๏ป 1400.616
d. The digit to the right of the ones digit is 6, which is greater than 5. Therefore we add 1 to the ones digit. Drop all
digits to the right of the ones place.
298.603 ๏ป 299
7. The digit to the right of the hundredths digit is 6, which is greater than 5. Therefore we add 1 to the hundredths digit:
9 ๏ซ 1 ๏ฝ 10. Record 0 in the hundredths place and carry the 1 to the tenths place.
38.496 ๏ป 38.50
Concept and Vocabulary Check P.4
1. whole-number; decimal
2. tens; ones; tenths; hundredths
3. and; thousandths
4. 10; 100; 1000
5. 6; 7; add 1; 12.37
Exercise Set P.4
1. hundredths
2. hundredths
3. thousandths
4. thousandths
5. tens
6. tens
7. ten-thousandths
8. ten-thousandths
9. millionths
10. millionths
11. hundred-thousandths
26
Copyright ยฉ 2020 Pearson Education, Inc.
Section P.4 Decimals
12. hundred-thousandths
39. 0.3 ๏ฝ
3
10
40. 0.9 ๏ฝ
9
10
13. seven and sixty-three hundredths
14. eight and fifty-seven hundredths
15. sixteen and six tenths
16. eighteen and eight tenths
41. 0.39 ๏ฝ
39
100
17. eight hundred sixty-five and two hundred five
thousandths
42. 0.37 ๏ฝ
37
100
18. six hundred eighty-five and three hundred seven
thousandths
43. 0.8 ๏ฝ
8
2 ๏4 4
๏ฝ
๏ฝ
10 2 ๏ 5 5
44. 0.6 ๏ฝ
6
2 ๏3 3
๏ฝ
๏ฝ
10 2 ๏ 5 5
19. twenty-three ten-thousandths
20. thirty-two ten-thousandths
21. six thousand and three thousandths
22. three thousand and six thousandths
45. 0.64 ๏ฝ
64
4 ๏16 16
๏ฝ
๏ฝ
100 4 ๏ 25 25
46. 0.75 ๏ฝ
75
25 ๏ 3 3
๏ฝ
๏ฝ
100 25 ๏ 4 4
23. two hundred seven and twenty-seven hundredthousandths
24. three hundred nine and thirty-nine hundredthousandths
25. The standard form is 5.9.
47. 3.005 ๏ฝ 3
5
5 ๏1
1
๏ฝ3
๏ฝ3
1000
200
5 ๏ 200
48. 7.005 ๏ฝ 7
5
5 ๏1
1
๏ฝ7
๏ฝ7
1000
200
5 ๏ 200
26. The standard form is 4.7.
27. The standard form is 17.37.
28. The standard form is 19.29.
29. The standard form is 800.08.
49. 876.32 ๏ฝ 876
32
4 ๏8
8
๏ฝ 876
๏ฝ 876
100
25
4 ๏ 25
50. 678.34 ๏ฝ 678
34
2 ๏17
17
๏ฝ 678
๏ฝ 678
100
50
2 ๏ 50
30. The standard form is 900.09.
31. The standard form is 306.535.
32. The standard form is 507.543.
33. The standard form is 0.292.
34. The standard form is 0.529.
51. 0.0009 ๏ฝ
9
10, 000
52. 0.0007 ๏ฝ
7
10, 000
35. The standard form is 200.092.
36. The standard form is 500.029.
53. 17.268 ๏ฝ 17
268
4 ๏ 67
67
๏ฝ 17
๏ฝ 17
1000
250
4 ๏ 250
54. 19.375 ๏ฝ 19
375
125 ๏ 3
3
๏ฝ 19
๏ฝ 19
1000
8
125 ๏ 8
37. The standard form is 0.0075.
38. The standard form is 0.0057.
Copyright ยฉ 2020 Pearson Education, Inc.
27
Chapter P Prealgebra Pathways
55. 0.4006 ๏ฝ
4006
2 ๏ 2003 2003
๏ฝ
๏ฝ
10, 000 2 ๏ 5000 5000
56. 0.6004 ๏ฝ
6004
4 ๏1501 1501
๏ฝ
๏ฝ
10, 000 4 ๏ 2500 2500
57. 0.38 rounded to the nearest tenth is 0.4.
58. 0.27 rounded to the nearest tenth is 0.3.
59. 0.8647 rounded to the nearest hundredth is 0.86.
60. 0.7843 rounded to the nearest hundredth is 0.78.
61. 24.23651 rounded to the nearest thousandth is 24.237.
62. 37.32851 rounded to the nearest thousandth is 37.329.
63. 63.498 rounded to the nearest hundredth is 63.50.
64. 71.396 rounded to the nearest hundredth is 71.40.
65. 32.98601 rounded to the nearest tenth is 33.0.
66. 43.97801 rounded to the nearest tenth is 44.0.
67. 32.98601 rounded to the nearest ten is 30.
68. 43.97801 rounded to the nearest ten is 40.
69. 0.041652 rounded to the nearest ten-thousandth is 0.0417.
70. 0.032751 rounded to the nearest ten-thousandth is 0.0328.
71. 318.489 rounded to the nearest whole number is 318.
72. 617.498 rounded to the nearest whole number is 617.
73. one thousand, five and one hundred five ten-billionths
74. two thousand, seven and two hundred seven ten-billionths
75. 11.00011101 ๏ฝ 11
11,101
100, 000, 000
76. 13.00011103 ๏ฝ 13
11,103
100, 000, 000
77. In 2014 the average age of cars on U.S. roads was at a maximum of 11.4 ๏ฝ 11
78. In 2008 the average age of cars on U.S. roads was at a minimum of 10.1 ๏ฝ 10
4
2
๏ฝ 11 years.
10
5
1
years.
10
79. In 2010 the average vehicle age was between 10.5 and 11.0 years at ten and sixth tenths years.
28
Copyright ยฉ 2020 Pearson Education, Inc.
Section P.4 Decimals
80. In 2012 the average vehicle age was between 11.0 and 11.4 years at eleven and two tenths years.
81. In 2008 the average age of vehicles on the U.S. roads was 10.1 years , which rounded to the nearest whole number
equals 10 years.
82. In 2010, 2012, and 2014 the average age of vehicles on the U.S. roads when rounded to the nearest whole number
equals 11 years.
1
2
1
83. In 2012 the average age of cars on U.S. roads was 11 years because 11.2 ๏ฝ 11 ๏ฝ 11 years.
5
10
5
84. In 2010 the average age of cars on U.S. roads was 10
3
6
3
years because 10.6 ๏ฝ 10 ๏ฝ 10 years.
5
10
5
85. Check to Safeway:
86. Check to Dr. Jill Cantor:
87. Cobra: nine thousandths
88. Krait, five tenths
89. cobra, mamba, brown snake, taipan, krait because 0.009 ๏ผ 0.02 ๏ผ 0.05 ๏ผ 0.11 ๏ผ 0.5.
90. krait, taipan, brown snake, mamba, cobra because 0.009 ๏ผ 0.02 ๏ผ 0.05 ๏ผ 0.11 ๏ผ 0.5.
91. a. $4.539 rounded to the nearest cent is $4.54.
b. $4.539 rounded to the nearest whole dollar is $5.
92. a. $4.239 rounded to the nearest cent is $4.24.
b. $4.239 rounded to the nearest whole dollar is $4.
93. 2.718281828459045 rounded to the nearest hundredth is 2.72.
94. 2.718281828459045 rounded to the nearest thousandth is 2.718.
95. 2.718281828459045 rounded to the nearest ten-thousandth is 2.7183.
Copyright ยฉ 2020 Pearson Education, Inc.
29
Chapter P Prealgebra Pathways
96. 2.718281828459045 rounded to the nearest millionth is 2.718282.
97. โ 101. Answers will vary.
102. does not make sense; Explanations will vary. Sample explanation: The place values to the immediate left of a decimal
start with ones.
103. does not make sense; Explanations will vary. Sample explanation: 0.17 in words is seventeen hundredths.
104. does not make sense; Explanations will vary. Sample explanation: Adding zeroes before the first digit in a numberโs
decimal part will change the value of the number. Adding zeroes after the last digit in a numberโs decimal part will not
change the value of the number.
105. Answers will vary; an example is 18.6001.
106. Answers will vary; an example is 23.5399.
107. Answers will vary; an example is 62.35.
108. 9
3
3๏ 4
12
๏ฝ9
๏ฝ9
๏ฝ 9.12
25
25 ๏ 4
100
Mid-Chapter P Check Point
1. eight billion, sixty-three million, five hundred sixty-one thousand, four
2. 54,302,628
3. a. nearest hundred:
64,517 ๏ป 64,500
b. nearest thousand:
64,517 ๏ป 65,000
4. 18 ๏พ 0
5. 18 ๏ผ 19
6. a. The James Bond franchise had the greatest number of movies.
The total world gross for this franchise was $5,116,147,171.
b. three billion, two hundred eighty-seven million, two hundred eighty-five thousand, five dollars
c. Star Wars had 7 movies.
$4, 279, 632, 749 ๏ป $4, 000, 000, 000
d. Shrek and Lord of the Rings each had total world gross less than $3,500,000,000.
7. a.
61 (1980)
๏ญ 28 (Ancient Greece and Rome)
33
People born in 1980 are expected to live 33 years more than people born in ancient Greece and Rome.
b. Average life expectancy was 48 years in 1950.
30
Copyright ยฉ 2020 Pearson Education, Inc.
Mid-Chapter P Check Point
c. The following life expectancies round to 30.
Stone Age: 25 ๏ป 30
Ancient Greece and Rome: 28 ๏ป 30
Middle Ages: 30 ๏ป 30
1900: 31 ๏ป 30
d. life expectancy 1950: 48
life expectancy Middle Ages: 30
2
1
14 7 14 5 14 5 2
17.
๏ธ ๏ฝ ๏ ๏ฝ
๏ ๏ฝ
15 5 15 7 15 7 3
3
1
5
2
?
2
48 ๏ฝ 2 ๏ 30 ๏ญ 12
3
1
2 25 20 25 9
25 9
15
7
18. 4 ๏ธ 2 ๏ฝ
๏ธ
๏ฝ
๏
๏ฝ
๏
๏ฝ
๏ฝ1
6
9 6
9
6 20
8
8
6 20
4
2
10 15 10 14 10 14 4
19.
๏ธ ๏ฝ ๏ ๏ฝ
๏
๏ฝ
21 14 21 15 21 15 9
?
48 ๏ฝ 60 ๏ญ 12
3
3
?
48 ๏ฝ 48 true
20.
8. There are 2 parts shaded out of a total 5 equal parts.
2
Thus, the fraction (two-fifths) represents the
5
shaded portion of the figure.
9. Improper Fraction:
7
.
3
10. 4
11.
14 7
๏ฝ
30 15
21.
1
Mixed Number: 2 .
3
13
26
๏ฝ
18
36
5
15
๏ญ
๏ฝ ๏ญ
12
36
7 10 ๏ 4 ๏ซ 7 47
๏ฝ
๏ฝ
10
10
10
11
36
85
1
๏ฝ 14
6
6
22.
12. 150 ๏ฝ 10 ๏15
๏ฝ 2 ๏5๏3๏5
13.
1
5
๏ฝ
6
30
3
9
๏ซ
๏ฝ ๏ซ
10
30
4
12 4 ๏ 3 3
๏ฝ
๏ฝ
28 4 ๏ 7 7
23.
1
1
5 8
5 8
1
14.
๏ ๏ฝ
๏
๏ฝ
24 15 24 15 9
3
2
16
1 ๏ฝ 1
5
40
7
35
๏ซ 3 ๏ฝ ๏ซ3
8
40
1
3
9
๏ฝ 5 ๏ฝ 4
2
6
6
2
4
4
๏ญ 2 ๏ฝ ๏ญ2 ๏ฝ ๏ญ2
3
6
6
5
3
2
1
15. 6 ๏
51
11
๏ฝ5
40
40
7
6 7
7
1
๏ฝ ๏
๏ฝ ๏ฝ3
12 1 12 2
2
24. 35 ๏ญ 15 ๏ฝ 20 questions left.
20 4
๏ฝ of the test left.
35 7
2
3
3
๏ฆ 2 ๏ถ๏ฆ 1 ๏ถ 5 9 5 9 15
16. ๏ง1 ๏ท๏ง 2 ๏ท ๏ฝ ๏ ๏ฝ ๏ ๏ฝ
๏ฝ3
4
4
๏จ 3 ๏ธ๏จ 4 ๏ธ 3 4 3 4
1
5
6
25. hundredths
26. twenty-three and two hundred three thousandths
27. The standard form is 3003.03.
Copyright ยฉ 2020 Pearson Education, Inc.
31

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