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Chapter 4
Section 4.1
Practice Problems
1. In the fraction 11,2
the numerator is 11 and the
denominator is 2.
2. In the fraction 10 ,17y
the numerator is 10y and
the denominator is 17.
3. 3 out of 8 equal parts are shaded: 38
4. 1 out of 6 equal parts are shaded: 16
5. 7 out of 10 equal parts are shaded: 7
10
6. 9 out of 16 equal parts are shaded: 9
16
7. answers may vary; for example,
8. answers may vary; for example,
9. number of planets farther 5number of planets in our solar system 8
→→
58
of the planets in our solar system are farther
from the Sun than Earth.
10. Each part is 13
of a whole and there are 8 parts
shaded, or 2 wholes and 2 more parts. 8 2; 23 3
11. Each part is 14
of a whole and there are 5 parts
shaded, or 1 whole and 1 more part. 5 1; 14 4
12. a.
1∞0
b. 1s0
c. 10 46
13. a. 1 2 3h0
b. 1 2@0
c. 0 1 or 77
14. 9 19=
15. 6 16−
=−
16. 0 01=
−
17. 4 41=
18. 130−
is undefined.
19. 13 131−
= −
20. a. 2 7 5 2 35 2 3757 7 7 7
⋅ + += = =
b. 2 3 6 2 18 2 2063 3 3 3
⋅ + += = =
c. 9 10 10 9 100 9 10910
10 10 10 10⋅ + +
= = =
d. 1 5 4 1 20 1 2145 5 5 5
⋅ + += = =
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Chapter 4: Fractions and Mixed Numbers ISM: Prealgebra
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21. a. 1
5 954
9 415 5=
b. 2
9 2318
5
23 529 9=
c. 12
4 484880
48 124=
d. 4
13 625210
62 10413 13
=
e. 7
7 5149
2
51 277 7=
f. 1
20 2120
1
21 1120 20
=
Vocabulary and Readiness Check
1. The number 1731
is called a fraction. The number
31 is called its denominator and 17 is called its numerator.
2. If we simplify each fraction, 9 1,9−
=−
0 0,4=
−
and we say 4
0−
is undefined.
3. The fraction 83
is called an improper fraction,
the fraction 38
is called a proper fraction, and
3108
is called a mixed number.
4. The value of an improper fraction is always ≥1 and the value of a proper fraction is always <1.
Exercise Set 4.1
1. In the fraction 1 ,2
the numerator is 1 and the
denominator is 2. Since 1 < 2, the fraction is proper.
2. In the fraction 1 ,4
the numerator is 1 and the
denominator is 4. Since 1 < 4, the fraction is proper.
3. In the fraction 10 ,3
the numerator is 10 and the
denominator is 3. Since 10 > 3, the fraction is improper.
4. In the fraction 53 ,21
the numerator is 53 and the
denominator is 21. Since 53 > 21, the fraction is improper.
5. In the fraction 15 ,15
the numerator is 15 and the
denominator is 15. Since 15 ≥ 15, the fraction is improper.
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ISM: Prealgebra Chapter 4: Fractions and Mixed Numbers
137
6. In the fraction 26 ,26
the numerator is 26 and the
denominator is 26. Since 26 ≥ 26, the fraction is improper.
7. 1 out of 3 equal parts is shaded: 13
8. 5 out of 6 equal parts are shaded: 56
9. Each part is 14
of a whole and there are 11 parts
shaded, or 2 wholes and 3 more parts.
a. 114
b. 324
10. Each part is 14
of a whole and there are 10 parts
shaded, or 2 wholes and 2 more parts.
a. 104
b. 224
11. Each part is 16
of a whole and there are 23 parts
shaded, or 3 wholes and 5 more parts.
a. 236
b. 536
12. Each part is 13
of a whole and there are 11 parts
shaded, or 3 wholes and 2 more parts.
a. 113
b. 233
13. 7 out of 12 equal parts are shaded: 7
12
14. 5 out of 8 equal parts are shaded: 58
15. 3 out of 7 equal parts are shaded: 37
16. 5 out of 12 equal parts are shaded: 5
12
17. 4 out of 9 equal parts are shaded: 49
18. 7 out of 8 equal parts are shaded: 78
19. Each part is 13
of a whole and there are 4 parts
shaded, or 1 whole and 1 more part.
a. 43
b. 113
20. Each part is 15
of a whole and there are 6 parts
shaded, or 1 whole and 1 more part.
a. 65
b. 115
21. Each part is 12
of a whole and there are 11 parts
shaded, or 5 wholes and 1 more part.
a. 112
b. 152
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Chapter 4: Fractions and Mixed Numbers ISM: Prealgebra
138
22. Each part is 15
of a whole and there are 23 parts
shaded, or 4 wholes and 3 more parts.
a. 235
b. 345
23. 1 out of 6 equal parts are shaded: 16
24. 3 out of 8 equal parts are shaded: 38
25. 5 of 8 equal parts are shaded: 58
26. 13 out of 16 equal parts are shaded: 1316
27. answers may vary; for example,
28. answers may vary; for example,
29. answers may vary; for example,
30. answers may vary; for example,
31. answers may vary; for example,
32. answers may vary; for example,
33. freshmen 42students 131
→→
42131
of the students are freshmen.
34. men 22employees 63
→→
2263
of the employees are men.
35. a. number of students not freshmen 131 4289
= −=
b. not freshmen 89students 131
→→
89131
of the students are not freshmen.
36. a. number of women employees 63 2241
= −=
b. women 41employees 63
→→
4163
of the employees are women.
37. born in Virginia 8U.S. presidents 43
→→
843
of U.S. presidents were born in Virginia.
38. planets with longer days 4number of planets in solar system 8
→→
48
of the planets in our solar system have longer
days than Earth.
39. number turned into hurricanes 15number of tropical storms 28
→→
1528
of the tropical storms turned into hurricanes.
40. 5 of 12 inches is 5
12 of a foot.
41. 11 of 31 days of March is 1131
of the month.
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ISM: Prealgebra Chapter 4: Fractions and Mixed Numbers
139
42. 37 of 60 minutes is 3760
of an hour.
43. number of sophomores 10number of students in class 31
→→
1031
of the class is sophomores.
44. number of boys 9number on team 20
→→
920
of the team is boys.
45. There are 50 states total. 33 states contain federal Indian reservations.
a. 3350
of the states contain federal Indian reservations.
b. 50 − 33 = 17 17 states do not contain federal Indian reservations.
c. 1750
of the states do not contain federal Indian reservations.
46. There are 50 states total. Consumer fireworks are legal in 40 states.
a. Consumer fireworks are legal in 4050
of the states.
b. 50 − 40 = 10 Consumer fireworks are illegal in 10 states.
c. Consumer fireworks are illegal in 1050
of the states.
47. a. blue 21total 50
→→
2150
of the marbles are blue.
b. 50 − 21 = 29 29 of the marbles are red.
c. red 29total 50
→→
2950
of the marbles are red.
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Chapter 4: Fractions and Mixed Numbers ISM: Prealgebra
140
48. a. number of watercolors 15total number of pieces 37
→→
1537
of the pieces are watercolor paintings.
b. number of oil paintings 17total number of pieces 37
→→
1737
of the pieces are oil paintings.
c. 37 − 15 − 17 = 5 There are 5 sculptures.
d. numbers of sculptures 5total number of pieces 37
→→
537
of the pieces are sculptures.
49.
~
1 2 30
50. a
1 2 3 40
51. ¢
1 20
52. X
1 20
53.
U
1 20
54. ∫
10
55. g
1 2 3 40
56.
œ
1 20
57. 12 112
=
58. 3 13−
=−
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ISM: Prealgebra Chapter 4: Fractions and Mixed Numbers
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59. 5 5
1−
= −
60. 20 201−
= −
61. 0 02=
−
62. 0 08=
−
63. 8 18−
=−
64. 14 114−
=−
65. 9 is undefined
0−
66. 7 is undefined
0−
67. 3 31=
68. 5 15=
69. 1 3 2 1 6 1 723 3 3 3
⋅ + += = =
70. 13 17 1 13 17 13 30117 17 17 17
⋅ + += = =
71. 3 5 3 3 15 3 1835 5 5 5
⋅ + += = =
72. 5 9 2 5 18 5 2329 9 9 9
⋅ + += = =
73. 5 8 6 5 48 5 5368 8 8 8
⋅ + += = =
74. 3 8 7 3 56 3 5978 8 8 8
⋅ + += = =
75. 6 7 11 6 77 6 83117 7 7 7
⋅ + += = =
76. 2 5 12 2 60 2 62125 5 5 5
⋅ + += = =
77. 7 20 9 7 180 7 187920 20 20 20
⋅ + += = =
78. 14 27 10 14 270 14 2841027 27 27 27
⋅ + += = =
79. 2 3 166 2 498 2 5001663 3 3 3
⋅ + += = =
80. 2 7 114 2 798 2 8001147 7 7 7
⋅ + += = =
81. 3
5 17152
17 235 5=
82. 1
7 1376
13 617 7=
83. 4
8 3732
5
37 548 8=
84. 7
9 6463
1
64 179 9=
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Chapter 4: Fractions and Mixed Numbers ISM: Prealgebra
142
85. 3
15 47452
47 2315 15
=
86. 5
12 6560
5
65 5512 12
=
87. 15
15 22515
75750
225 1515
=
88. 14
14 196145656
0
196 1414
=
89. 1
175 182175
7
182 71175 175
=
90. 1
143 149143
6
149 61143 143
=
91. 6
112 737672
65
737 656112 112
=
92. 7
123 90186140
901 407123 123
=
93. 23 3 3 9= ⋅ =
94. 34 4 4 4 64= ⋅ ⋅ =
95. 35 5 5 5 125= ⋅ ⋅ =
96. 43 3 3 3 3 81= ⋅ ⋅ ⋅ =
97. 11 11 112 2 2
−− = =
−
98. 21 21 214 4 4
−− = =
−
99. 13 13 13
15 15 15−
= = −−
100. 45 45 4557 57 57
−= = −
−
101. answers may vary
102.
103.
104. 1 is not close to 8, so 18
is closer to 0 than to 1.
178
rounded to the nearest whole number is 7.
105. 11 is close to 12, so 1112
is close to 1. 11512
rounded to the nearest whole number is 6.
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ISM: Prealgebra Chapter 4: Fractions and Mixed Numbers
143
106. 1591 + 498 + 959 + 5 = 3053 959 of the 3053 stores are named Old Navy: 9593053
107. 87 + 57 + 20 + 7 = 171 57 of the 171 licensees are universities or
colleges: 57171
108. 1651 + 634 = 2285 1651 of the 2285 affiliates are located in the
United States: 16512285
109. 5 + 3 + 2 + 1 + 1 + 1 = 13 Five of the 13 USMC training centers are in
California: 5 .
13
Section 4.2
Practice Problems
1. a. 30 2 15
2 3 5
= ⋅↓ ↓⋅ ⋅
b.
3
56 2 28
2 2 14
2 2 2 7 2 7
= ⋅↓ ↓⋅ ⋅
↓ ↓ ↓
⋅ ⋅ ⋅ = ⋅
c.
3 2
72 2 36
2 2 18
2 2 2 9
2 2 2 3 3 2 3
= ⋅↓ ↓⋅ ⋅
↓ ↓ ↓⋅ ⋅ ⋅
↓ ↓ ↓ ↓
⋅ ⋅ ⋅ ⋅ = ⋅
2.
2
60 2 30
2 2 15
2 2 3 5 2 3 5
= ⋅↓ ↓⋅ ⋅
↓ ↓ ↓
⋅ ⋅ ⋅ = ⋅ ⋅
3. 11
3 333 993 297
The prime factorization of 297 is 33 11.⋅
4. 30 15 2 245 15 3 3
⋅= =
⋅
5. 39 3 13 13 1351 3 17 17 17x x x x⋅ ⋅ ⋅= = =
⋅
6. 9 3 3
50 2 5 5⋅
− = −⋅ ⋅
Since 9 and 50 have no common factors, 9
50− is
already in simplest form.
7. 49 7 7 7
112 7 16 16⋅
= =⋅
8. 64 4 16 1620 4 5 5
⋅− = − = −
⋅
9. 3
27 7
7 8 856a a a a a
a aa⋅ ⋅ ⋅
= =⋅ ⋅ ⋅
10. 79
is in simplest form.
21 3 7 727 3 9 9
⋅= =
⋅
Since 79
and 2127
both simplify to 7 ,9
they are
equivalent.
11. Not equivalent, since the cross products are not equal: 13 ⋅ 5 = 65 and 4 ⋅ 18 = 72
12. 6 parks in Washington 2 3 3
58 national parks 2 29 29⋅
= =⋅
329
of the national parks are in Washington
state.
Calculator Explorations
1. 128 4224 7
=
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Chapter 4: Fractions and Mixed Numbers ISM: Prealgebra
144
2. 231 7396 12
=
3. 340 20459 27
=
4. 999 37
1350 50=
5. 432 8810 15
=
6. 225 5315 7
=
7. 54 2243 9
=
8. 455 35689 53
=
Vocabulary and Readiness Check
1. The number 40 equals 2 ⋅ 2 ⋅ 2 ⋅ 5. Since each factor is prime, we call 2 ⋅ 2 ⋅ 2 ⋅ 5 the prime factorization of 40.
2. A natural number, other than 1, that is not prime is called a composite number.
3. A natural number that has exactly two different factors, 1 and itself, is called a prime number.
4. In 11 ,48
since 11 and 48 have no common factors
other than 1, 1148
is in simplest form.
5. Fractions that represent the same portion of a whole are called equivalent fractions.
6. In the statement 5 15 ,
12 36= 5 ⋅ 36 and 12 ⋅ 15 are
called cross products.
Exercise Set 4.2
1.
2
20 2 10
2 2 5 2 5
= ⋅↓
⋅ ⋅ = ⋅
2.
2
12 2 6
2 2 3 2 3
= ⋅↓ ↓⋅ ⋅ = ⋅
3.
4
48 2 24
2 2 12
2 2 2 6
2 2 2 2 3 2 3
= ⋅↓ ↓⋅ ⋅
↓ ↓ ↓⋅ ⋅ ⋅
↓ ↓ ↓ ↓⋅ ⋅ ⋅ ⋅ = ⋅
4.
2
75 3 25
3 5 5 3 5
= ⋅↓ ↓⋅ ⋅ = ⋅
5.
4
81 9 9
3 3 3 3 3
= ⋅
⋅ ⋅ ⋅ =
6.
6
64 8 8
2 4 2 4
2 2 2 2 2 2 2
= ⋅
⋅ ⋅ ⋅↓
⋅ ⋅ ⋅ ⋅ ⋅ =
7.
4
162 2 81
2 9 9
2 3 3 3 3 2 3
= ⋅↓
⋅ ⋅↓⋅ ⋅ ⋅ ⋅ = ⋅
8.
7
128 2 64
2 2 32
2 2 2 16
2 2 2 2 8
2 2 2 2 2 4
2 2 2 2 2 2 2 2
= ⋅↓ ↓⋅ ⋅
↓ ↓ ↓⋅ ⋅ ⋅
↓ ↓ ↓ ↓⋅ ⋅ ⋅ ⋅
↓ ↓ ↓ ↓ ↓⋅ ⋅ ⋅ ⋅ ⋅
↓ ↓ ↓ ↓ ↓ ↓
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ =
9. 110 2 55
2 5 11 2 5 11
= ⋅↓ ↓⋅ ⋅ = ⋅ ⋅
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ISM: Prealgebra Chapter 4: Fractions and Mixed Numbers
145
10. 130 2 65
2 5 13 2 5 13
= ⋅↓ ↓⋅ ⋅ = ⋅ ⋅
11. 85 = 5 ⋅ 17
12. 93 = 3 ⋅ 31
13.
4
240 2 120
2 2 60
2 2 2 30
2 2 2 2 15
2 2 2 2 3 5 2 3 5
= ⋅↓ ↓⋅ ⋅
↓ ↓ ↓⋅ ⋅ ⋅
↓ ↓ ↓ ↓⋅ ⋅ ⋅ ⋅
↓ ↓ ↓ ↓ ↓
⋅ ⋅ ⋅ ⋅ ⋅ = ⋅ ⋅
14.
2
836 2 418
2 2 209
2 2 11 19 2 11 19
= ⋅↓ ↓⋅ ⋅
↓ ↓ ↓⋅ ⋅ ⋅ = ⋅ ⋅
15.
2 2
828 2 414
2 2 207
2 2 3 69
2 2 3 3 23 2 3 23
= ⋅↓ ↓⋅ ⋅
↓ ↓ ↓⋅ ⋅ ⋅
↓ ↓ ↓ ↓⋅ ⋅ ⋅ ⋅ = ⋅ ⋅
16.
3 2
504 2 252
2 2 126
2 2 2 63
2 2 2 3 21
2 2 2 3 3 7 2 3 7
= ⋅↓ ↓⋅ ⋅
↓ ↓ ↓⋅ ⋅ ⋅
↓ ↓ ↓ ↓⋅ ⋅ ⋅ ⋅
↓ ↓ ↓ ↓ ↓
⋅ ⋅ ⋅ ⋅ ⋅ = ⋅ ⋅
17. 3 3 1 1
12 3 4 4⋅
= =⋅
18. 5 5 1 1
30 5 6 6⋅
= =⋅
19. 4 2 2 2 242 2 3 7 3 7 21x x x x⋅ ⋅ ⋅= = =
⋅ ⋅ ⋅
20. 9 3 3 3 348 3 16 16 16y y y y⋅ ⋅ ⋅= = =
⋅
21. 14 2 7 716 2 8 8
⋅= =
⋅
22. 22 2 11 1134 2 17 17
⋅= =
⋅
23. 20 2 10 230 3 10 3
⋅= =
⋅
24. 70 7 10 780 8 10 8
⋅= =
⋅
25. 35 5 7 750 5 10 10a aa a
⋅ ⋅= =
⋅ ⋅
26. 25 5 5 555 5 11 11z zz z
⋅ ⋅= =
⋅ ⋅
27. 63 9 7 781 9 9 9
⋅− = − = −
⋅
28. 21 7 3 349 7 7 7
⋅− = − = −
⋅
29. 230 5 6 5 5
36 6 6 6 6x x x x xx x
⋅ ⋅ ⋅ ⋅= = =
⋅ ⋅
30. 245 5 9 9 9
5 16 16 1680b b
b b b bb⋅ ⋅
= = =⋅ ⋅ ⋅ ⋅
31. 27 3 3 364 4 4 4
⋅ ⋅=
⋅ ⋅
Since 27 and 64 have no common factors, 2764
is
already in simplest form.
32. 32 2 2 2 2 263 3 3 7
⋅ ⋅ ⋅ ⋅=
⋅ ⋅
Since 32 and 63 have no common factors, 3263
is
already in simplest form.
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Chapter 4: Fractions and Mixed Numbers ISM: Prealgebra
146
33. 25 5 5 5 540 5 8 8 8xy x y x xy y
⋅ ⋅ ⋅ ⋅= = =
⋅ ⋅
34. 36 6 6 6 642 6 7 7 7y yyz y z z z
⋅ ⋅= = =
⋅ ⋅ ⋅ ⋅
35. 40 8 5 564 8 8 8
⋅− = − = −
⋅
36. 28 4 7 760 4 15 15
⋅− = − = −
⋅
37. 3 2 236 3 12 3 3
24 2 12 2 2x y x x x y y x x y x yxy x y
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅= = =
⋅ ⋅ ⋅
38. 2
3 260 5 12 5 5
3 12 336 3a b a a b a a
a b b b b bab b⋅ ⋅ ⋅ ⋅ ⋅
= = =⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
39. 90 30 3 3
120 30 4 4⋅
= =⋅
40. 60 30 2 2
150 30 5 5⋅
= =⋅
41. 40 5 8 5 564 8 8 8 8xy x yxyz x y z z z
⋅ ⋅ ⋅= = =
⋅ ⋅ ⋅ ⋅ ⋅
42. 28 7 4 7 760 15 4 15 15abc a b c b bac a c
⋅ ⋅ ⋅ ⋅ ⋅= = =
⋅ ⋅ ⋅
43. 66 22 3 3308 22 14 14
⋅= =
⋅
44. 65 5 13 5234 18 13 18
⋅= =
⋅
45. 55 5 11 11 11
85 5 17 17 17y y y y⋅
− = − = − = −⋅ ⋅ ⋅
46. 78 6 13 13 13
90 6 15 15 15x x x x⋅
− = − = − = −⋅ ⋅ ⋅
47. 189 7 27 7216 8 27 8z zz z
⋅ ⋅= =
⋅ ⋅
48. 144 18 8 8162 18 9 9
y yy y
⋅ ⋅= =
⋅ ⋅
49. 3 4 2
4 2
2
224 14 161 1616
141
14
a b c a a a b b b b c ca b b b b c cab c
a a
a
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅=
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅⋅ ⋅
=
=
50. 4 3 3
3 3 3270 18 15
1 151518
118
x y z x x x x y y y z z zx x x y y y z z zx y z
x
x
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅=
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅⋅
=
=
51. Equivalent, since the cross products are equal: 2 ⋅ 12 = 24 and 6 ⋅ 4 = 24
52. Equivalent, since the cross products are equal: 3 ⋅ 10 = 30 and 6 ⋅ 5 = 30
53. Not equivalent, since the cross products are not equal: 7 ⋅ 8 = 56 and 5 ⋅ 11 = 55
54. Not equivalent, since the cross products are not equal: 2 ⋅ 11 = 22 and 5 ⋅ 4 = 20
55. Equivalent, since the cross products are equal: 10 ⋅ 9 = 90 and 15 ⋅ 6 = 90
56. Equivalent, since the cross products are equal: 4 ⋅ 15 = 60 and 10 ⋅ 6 = 60
57. Equivalent, since the cross products are equal: 3 ⋅ 18 = 54 and 9 ⋅ 6 = 54
58. Equivalent, since the cross products are equal: 2 ⋅ 28 = 56 and 8 ⋅ 7 = 56
59. Not equivalent, since the cross products are not equal: 10 ⋅ 15 = 150 and 13 ⋅ 13 = 169
60. Not equivalent, since the cross products are not equal: 16 ⋅ 16 = 256 and 20 ⋅ 9 = 180
61. Not equivalent, since the cross products are not equal: 8 ⋅ 24 = 192 and 12 ⋅ 18 = 216
62. Not equivalent, since the cross products are not equal: 6 ⋅ 35 = 210 and 14 ⋅ 21 = 294
63. 2 hours 1 2 18 hours 4 2 4
⋅= =
⋅
2 hours represents 14
of a work shift.
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ISM: Prealgebra Chapter 4: Fractions and Mixed Numbers
147
64. 200 caps 200 1 12000 caps 200 10 10
⋅= =
⋅
200 caps represents 1
10 of the total caps.
65. 2640 feet 2640 1 15280 feet 2640 2 2
⋅= =
⋅
2640 feet represents 12
of a mile.
66. 20 centimeters 1 20 1
100 centimeters 5 20 5⋅
= =⋅
20 centimeters is 15
of a meter.
67. a. 15 3 5 350 10 5 10
⋅= =
⋅
310
of the states can claim at least one Ritz-
Carlton hotel.
b. 50 − 15 = 35 35 states do not have a Ritz-Carlton hotel.
c. 35 7 5 750 10 5 10
⋅= =
⋅
710
of the states do not have a Ritz-Carlton
hotel.
68. a. 10 2 5 578 2 39 39
⋅= =
⋅
539
of the national monuments can be found
in New Mexico.
b. 78 − 10 = 68 68 monuments are found outside New Mexico.
c. 68 2 34 3478 2 39 39
⋅= =
⋅
3439
of the national monuments are found
outside New Mexico.
69. 10 inches 2 5 524 inches 2 12 12
⋅= =
⋅
512
of the wall is concrete.
70. 10 5 2 235 5 7 7
⋅= =
⋅
27
of the students made an A on the first test.
71. a. 50 − 22 = 28 28 states do not have this type of Web site.
b. 28 2 14 1450 2 25 25
⋅= =
⋅
1425
of the states do not have this type of
Web site.
72. a. 22,000 − 10,000 = 12,000 $12,000 was not covered by his trade-in.
b. $12,000 2000 6 6$22,000 2000 11 11
⋅= =
⋅
611
of the purchase price was not covered
by the trade-in.
73. 291 individuals 291449 individuals 449
=
291449
of the individuals who had flown in space
were American.
74. 4500 employees 4500 1 1
18,000 employees 4500 4 4⋅
= =⋅
14
of the employees work at the Hallmark
headquarters in Kansas City, Missouri.
75. 3 3( 3) 27 3
9 9 9x − −
= = = −
76. 3 3( 5) 125 25
5 5 5y − −
= = = −
77. 2y = 2(−7) = −14
78. −5a = −5(−4) = 20
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Chapter 4: Fractions and Mixed Numbers ISM: Prealgebra
148
79. answers may vary
80. answers may vary
81. 3975 3 1325 36625 5 1325 5
⋅= =
⋅
82. 9506 7 1358 7
12,222 9 1358 9⋅
= =⋅
83. 36 blood donors have blood type A Rh-positive. 36 donors 4 9 9
100 donors 4 25 25⋅
= =⋅
925
of blood donors have type A Rh-positive
blood type.
84. 37 + 7 = 44 blood donors have an O blood type. 44 donors 4 11 11
100 donors 4 25 25⋅
= =⋅
1125
of blood donors have an O blood type.
85. 3 + 1 = 4 blood donors have an AB blood type. 4 donors 4 1 1
100 donors 4 25 25⋅
= =⋅
125
of blood donors have an AB blood type.
86. 9 + 1 = 10 blood donors have a B blood type. 10 donors 1 10 1100 donors 10 10 10
⋅= =
⋅
110
of blood donors have a B blood type.
87.
34,020 2 17,010
2 2 8505
2 2 3 2835
2 2 3 3 945
2 2 3 3 3 315
2 2 3 3 3 3 105
2 2 3 3 3 3 3 35
2 2 3 3 3 3 3 5 7
= ⋅↓ ↓⋅ ⋅
↓ ↓ ↓⋅ ⋅ ⋅
↓ ↓ ↓ ↓⋅ ⋅ ⋅ ⋅
↓ ↓ ↓ ↓ ↓⋅ ⋅ ⋅ ⋅ ⋅
↓ ↓ ↓ ↓ ↓ ↓⋅ ⋅ ⋅ ⋅ ⋅ ⋅
↓ ↓ ↓ ↓ ↓ ↓ ↓⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
2 534,020 2 3 5 7= ⋅ ⋅ ⋅
88.
131,625 3 43,875
3 3 14,625
3 3 3 4875
3 3 3 3 1625
3 3 3 3 5 325
3 3 3 3 5 5 65
3 3 3 3 5 5 5 3
= ⋅↓ ↓⋅ ⋅
↓ ↓ ↓⋅ ⋅ ⋅
↓ ↓ ↓ ↓⋅ ⋅ ⋅ ⋅
↓ ↓ ↓ ↓ ↓⋅ ⋅ ⋅ ⋅ ⋅
↓ ↓ ↓ ↓ ↓ ↓⋅ ⋅ ⋅ ⋅ ⋅ ⋅
↓ ↓ ↓ ↓ ↓ ↓ ↓⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
4 3131,625 3 5 13= ⋅ ⋅
89. answers may vary
90. answers may vary
91. no; answers may vary
92. The piece representing education is labeled 1 ,
10
so 1
10 of entering college freshmen plan to
major in education.
93. The piece representing biological sciences is
labeled 2 ,25
so 225
of entering college freshmen
plan to major in the biological sciences.
94. answers may vary
95. answers may vary
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96. 8691, 786, 2235, 105, 222, 900, and 1470 are divisible by 3 because the sum of each number’s digits is divisible by 3. 786, 22, 222, 900, and 1470 are divisible by 2 because they are even numbers. 786, 222, 900, and 1470 are divisible by both 2 and 3.
97. 1235, 2235, 85, 105, 900, and 1470 are divisible by 5 because each number ends with a 0 or 5. 8691, 786, 2235, 105, 222, 900, and 1470 are divisible by 3 because the sum of each number’s digits is divisible by 3. 2235, 105, 900, and 1470 are divisible by both 3 and 5.
98. 6; answers may vary
99. 15; answers may vary
Section 4.3
Practice Problems
1. 3 5 3 5 157 11 7 11 77
⋅⋅ = =
⋅
2. 1 1 1 1 13 9 3 9 27
⋅⋅ = =
⋅
3. 6 7 6 7 2 3 7 3 3
77 8 77 8 7 11 2 4 11 4 44⋅ ⋅ ⋅
⋅ = = = =⋅ ⋅ ⋅ ⋅ ⋅
4. 4 3 4 3 1 4 3 1 1
27 8 27 8 3 9 4 2 9 2 18⋅ ⋅ ⋅
⋅ = = = =⋅ ⋅ ⋅ ⋅ ⋅
5. 1 11 1 11 112 28 2 28 56
⋅⎛ ⎞⋅ − = − = −⎜ ⎟ ⋅⎝ ⎠
6. 4 33 4 33 4 3 11 3
11 16 11 16 11 4 4 4⋅ ⋅ ⋅⎛ ⎞⎛ ⎞− − = = =⎜ ⎟⎜ ⎟ ⋅ ⋅ ⋅⎝ ⎠⎝ ⎠
7. 1 3 25 1 3 256 10 16 6 10 16
1 3 5 52 3 2 5 16
1 52 2 165
64
⋅ ⋅⋅ ⋅ =
⋅ ⋅⋅ ⋅ ⋅
=⋅ ⋅ ⋅ ⋅⋅
=⋅ ⋅
=
8. 2 3 2 33 2 3 2 1 1
y y y y⋅ ⋅⋅ = = =
⋅ ⋅
9. 3 3
2 2 2 2a b a b a a a b a
b b a a bb a b a⋅ ⋅ ⋅ ⋅
⋅ = = =⋅ ⋅ ⋅⋅
10. a. 33 3 3 3 3 3 3 27
4 4 4 4 4 4 4 64⋅ ⋅⎛ ⎞ = ⋅ ⋅ = =⎜ ⎟ ⋅ ⋅⎝ ⎠
b. 24 4 4 4 4 16
5 5 5 5 5 25⋅⎛ ⎞ ⎛ ⎞ ⎛ ⎞− = − ⋅ − = =⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⋅⎝ ⎠ ⎝ ⎠ ⎝ ⎠
11. 8 2 8 9 8 9 2 4 9 4 9 367 9 7 2 7 2 7 2 7 7
⋅ ⋅ ⋅ ⋅÷ = ⋅ = = = =
⋅ ⋅
12. 4 1 4 2 4 2 89 2 9 1 9 1 9
⋅÷ = ⋅ = =
⋅
13. 10 2 10 94 9 4 2
10 94 22 5 94 2
5 94
454
− ÷ = − ⋅
⋅= −
⋅⋅ ⋅
= −⋅⋅
= −
= −
14.
33
3
3
2
3 3 554 4 1
3 14 53 1
4 53 1
4 53 1
4 53
20
y y yy
yy
yyyy y y
y y
y
÷ = ÷
= ⋅
⋅=
⋅⋅ ⋅
=⋅ ⋅ ⋅ ⋅
⋅=
⋅ ⋅ ⋅
=
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Chapter 4: Fractions and Mixed Numbers ISM: Prealgebra
150
15.
2 9 7 2 9 73 14 15 3 14 15
2 3 3 73 7 2 153 77 153 157 7
3 157 74549
⋅⎛ ⎞ ⎛ ⎞− ⋅ ÷ = − ÷⎜ ⎟ ⎜ ⎟⋅⎝ ⎠ ⎝ ⎠⋅ ⋅⎛ ⎞= − ÷⎜ ⎟⋅ ⋅⎝ ⎠
⎛ ⎞= − ÷⎜ ⎟⎝ ⎠⎛ ⎞= − ⋅⎜ ⎟⎝ ⎠
⋅= −
⋅
= −
16. a. 3 9 3 9 274 2 4 2 8
xy ⋅= − ⋅ = − = −
⋅
b.
3 94 23 24 93 24 91 3 2
2 2 3 31
2 316
x y÷ = − ÷
= − ⋅
⋅= −
⋅⋅ ⋅
= −⋅ ⋅ ⋅
= −⋅
= −
17.
924
9 928 4
2 9 91 8 4
2 9 91 8 4
2 9 91 2 4 4
9 9 True4 4
x = −
⎛ ⎞− −⎜ ⎟⎝ ⎠⎛ ⎞⋅ − −⎜ ⎟⎝ ⎠
⋅− −
⋅⋅
− −⋅ ⋅
− = −
Yes, 98
− is a solution of the equation.
18. 1 1 60 1 60 1 6 1060 106 6 1 6 1 6 1
⋅ ⋅ ⋅⋅ = ⋅ = = =
⋅ ⋅
Thus, there are 10 roller coasters in Hershey Park.
Vocabulary and Readiness Check
1. To multiply two fractions, we write
.a c a cb d b d
⋅⋅ =
⋅
2. Two numbers are reciprocals of each other if their product is 1.
3. The expression 32 2 2 2
7 7⋅ ⋅
= while
32 2 2 2.7 7 7 7
⎛ ⎞ = ⋅ ⋅⎜ ⎟⎝ ⎠
4. Every number has a reciprocal expect 0.
5. To divide two fractions, we write .a c a db d b c
⋅÷ =
⋅
6. The word “of” indicates multiplication.
Exercise Set 4.3
1. 6 3 6 3 18
11 7 11 7 77⋅
⋅ = =⋅
2. 5 7 5 7 359 4 9 4 36
⋅⋅ = =
⋅
3. 2 5 2 5 2 5 5 57 8 7 8 7 2 4 7 4 28
⋅ ⋅− ⋅ = − = − = − = −
⋅ ⋅ ⋅ ⋅
4. 4 1 4 1
15 20 15 204 1
15 4 51
15 5175
⋅⋅ − = −
⋅⋅
= −⋅ ⋅
= −⋅
= −
5. 1 2 1 2 12 15 2 15 15
⋅− ⋅ − = =
⋅
6. 3 11 3 11 3 11 1 1
11 12 11 12 11 3 4 4⋅ ⋅ ⋅
− ⋅ − = = =⋅ ⋅ ⋅
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ISM: Prealgebra Chapter 4: Fractions and Mixed Numbers
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7. 18 36 18 3620 99 20 99
2 9 2 182 2 5 9 11185 111855
x x
x
x
x
⋅⋅ =
⋅⋅ ⋅ ⋅ ⋅
=⋅ ⋅ ⋅ ⋅⋅
=⋅
=
8. 5 64 5 64 5 32 2
32 100 32 100 32 2 5 10 10y y y y⋅ ⋅ ⋅ ⋅
⋅ = = =⋅ ⋅ ⋅ ⋅
9. 2 2 2
2 1 3 1 3 1 334 1 4 1 4 4
a a aa ⋅⋅ = ⋅ = =
⋅
10. 3
3
3
3
3
3
2 2 663 3 1
2 63 1
2 3 23 1
2 21
4
yy
y
y
y
y
− ⋅ = − ⋅
⋅= −
⋅⋅ ⋅ ⋅
= −⋅
⋅ ⋅= −
= −
11. 3 2 3 2 2
3 3x y x y x x x y y x x x
x y y y x y yy y x⋅ ⋅ ⋅ ⋅ ⋅ ⋅
⋅ = = = =⋅ ⋅ ⋅⋅
12. 3 3 3 3
2 2
1
1
1
a b a bb a b a
a bb b b a a a
a a b b
a b
⋅⋅ =
⋅⋅ ⋅
=⋅ ⋅ ⋅ ⋅ ⋅
=⋅ ⋅ ⋅
=
13. 80 09⋅ =
14. 11 0 012
⋅ =
15. 17 4 17 420 5 20 5
17 45 4 5175 51725
y yy y
yy
⋅− ⋅ = −
⋅⋅ ⋅
= −⋅ ⋅ ⋅
= −⋅
= −
16. 13 5 13 520 6 20 6
13 54 5 6134 61324
x xx x
xx
⋅− ⋅ = −
⋅⋅ ⋅
= −⋅ ⋅ ⋅
= −⋅
= −
17. 11 1 5 11 1 520 7 22 20 7 22
11 1 55 2 2 7 11 2
12 2 7 2156
⋅ ⋅⋅ ⋅ =
⋅ ⋅⋅ ⋅
=⋅ ⋅ ⋅ ⋅ ⋅
=⋅ ⋅ ⋅
=
18. 27 10 16 27 10 1632 13 30 32 13 30
3 3 3 10 1616 2 13 10 33 32 13926
⋅ ⋅⋅ ⋅ =
⋅ ⋅⋅ ⋅ ⋅ ⋅
=⋅ ⋅ ⋅ ⋅⋅
=⋅
=
19. 31 1 1 1 1 1 1 1
5 5 5 5 5 5 5 125⋅ ⋅⎛ ⎞ ⎛ ⎞⎛ ⎞⎛ ⎞= = =⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⋅ ⋅⎝ ⎠ ⎝ ⎠⎝ ⎠⎝ ⎠
20. 28 8 8 8 8 64
9 9 9 9 9 81⋅⎛ ⎞ ⎛ ⎞⎛ ⎞= = =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⋅⎝ ⎠ ⎝ ⎠⎝ ⎠
21. 22 2 2 2 2 4
3 3 3 3 3 9⋅⎛ ⎞− = − ⋅− = =⎜ ⎟ ⋅⎝ ⎠
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Chapter 4: Fractions and Mixed Numbers ISM: Prealgebra
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22. 41 1 1 1 1
2 2 2 2 21 1 1 1
2 2 2 21
16
⎛ ⎞ ⎛ ⎞⎛ ⎞⎛ ⎞⎛ ⎞− = − − − −⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠⎝ ⎠⎝ ⎠
⋅ ⋅ ⋅=
⋅ ⋅ ⋅
=
23. 32 1 2 2 2 1
3 2 3 3 3 22 2 2 13 3 3 22 2 13 3 34
27
⎛ ⎞ ⎛ ⎞⎛ ⎞⎛ ⎞− ⋅ = − − − ⋅⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠⎝ ⎠
⋅ ⋅ ⋅= −
⋅ ⋅ ⋅⋅ ⋅
= −⋅ ⋅
= −
24. 33 1 3 3 3 1
4 3 4 4 4 33 3 3 14 4 4 33 3 14 4 4964
⎛ ⎞ ⎛ ⎞⎛ ⎞⎛ ⎞− ⋅ = − − − ⋅⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠⎝ ⎠
⋅ ⋅ ⋅= −
⋅ ⋅ ⋅⋅ ⋅
= −⋅ ⋅
= −
25. 2 5 2 6 2 6 2 2 3 2 2 43 6 3 5 3 5 3 5 5 5
⋅ ⋅ ⋅ ⋅÷ = ⋅ = = = =
⋅ ⋅
26. 5 3 5 4 5 4 5 4 5 58 4 8 3 8 3 4 2 3 2 3 6
⋅ ⋅÷ = ⋅ = = = =
⋅ ⋅ ⋅ ⋅
27. 6 12 6 5
15 5 15 126 5
15 126 5 1
5 3 6 21
3 216
− ÷ = − ⋅
⋅= −
⋅⋅ ⋅
= −⋅ ⋅ ⋅
= −⋅
= −
28. 4 8 4 3
15 3 15 84 3
15 84 3 1
5 3 4 21
5 21
10
− ÷ − = − ⋅−
⋅=
⋅⋅ ⋅
=⋅ ⋅ ⋅
=⋅
=
29. 8 8 2 8 2 169 2 9 9 9
xx x x
⋅− ÷ = − ⋅ = − = −
⋅
30. 10 4 10 511 5 11 4
10 511 42 5 511 2 25 511 22522
xx
x
x
x
x
÷ − = ⋅−
⋅= −
⋅⋅ ⋅ ⋅
= −⋅ ⋅
⋅ ⋅= −
⋅
= −
31. 11 3 11 11 11 11 12120 11 20 3 20 3 60y y y y⋅÷ = ⋅ = =
⋅
32. 9 2 9 9 9 9 8120 9 20 2 20 2 40z z z z⋅÷ = ⋅ = =
⋅
33.
2 2 443 3 1
2 13 42 13 4
2 13 2 21
3 216
− ÷ = − ÷
= − ⋅
⋅= −
⋅⋅
= −⋅ ⋅
= −⋅
= −
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ISM: Prealgebra Chapter 4: Fractions and Mixed Numbers
153
34.
5 5 10106 6 1
5 16 105 1
6 105 1
6 5 21
6 21
12
− ÷ = − ÷
= − ⋅
⋅= −
⋅⋅
= −⋅ ⋅
= −⋅
= −
35. 2
2
2
1 5 15 5 5
15 515 515 5
25
xx xx
xxx xxx
x
÷ = ⋅
⋅=
⋅⋅ ⋅
=⋅ ⋅⋅
=⋅
=
36. 3 3
2 3 2 23 9 3 3 3
9 3 3 39y y y y y y
y yy y y y⋅ ⋅ ⋅ ⋅
÷ = ⋅ = = =⋅ ⋅ ⋅⋅
37. 2 5 2 5 103 9 3 9 27
⋅⋅ = =
⋅
38. 8 5 8 5 1 8 5 1 1
15 32 15 32 5 3 8 4 3 4 12⋅ ⋅ ⋅
⋅ = = = =⋅ ⋅ ⋅ ⋅ ⋅
39. 23 5 3 6 3 6 18
7 6 7 5 7 5 35x x x x x x
x⋅
÷ = ⋅ = =⋅
40. 22 5 2 11 2 11 22
5 11 5 5 5 5 25y
y y y y y y⋅
÷ = ⋅ = =⋅
41. 16 8 16 1527 15 27 8
16 1527 88 2 3 5
3 9 82 59
109
yy y y
yy
yy
÷ = ⋅
⋅=
⋅⋅ ⋅ ⋅ ⋅
=⋅ ⋅ ⋅⋅
=
=
42. 12 4 12 721 7 21 4
12 721 44 3 7 17 3 4 11
y y yy
yyyy
÷ = ⋅
⋅=
⋅⋅ ⋅ ⋅ ⋅
=⋅ ⋅ ⋅ ⋅
=
43. 5 35 5 35 5 7 5 1 128 25 28 25 7 4 5 5 4
⋅ ⋅ ⋅ ⋅− ⋅ = − = − = −
⋅ ⋅ ⋅ ⋅
44. 24 5 24 5 8 3 5 1 145 8 45 8 5 3 3 8 3
⋅ ⋅ ⋅ ⋅⋅ − = − = − = −
⋅ ⋅ ⋅ ⋅
45. 23 3 3 3 3 9
4 4 4 4 4 16⋅⎛ ⎞− = − ⋅− = =⎜ ⎟ ⋅⎝ ⎠
46. 51 1 1 1 1 1
2 2 2 2 2 21 1 1 1 1
2 2 2 2 21
32
⎛ ⎞ ⎛ ⎞⎛ ⎞⎛ ⎞⎛ ⎞⎛ ⎞− = − − − − −⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠⎝ ⎠⎝ ⎠⎝ ⎠
⋅ ⋅ ⋅ ⋅= −
⋅ ⋅ ⋅ ⋅
= −
47. 2 3 2 3
21 1
x y x y x x y y y x y y xyy x y x y x
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅⋅ = = = =
⋅ ⋅ ⋅
48. 3 3
2 3 2 3 2b a b a b a a a a a
a a b b b b ba b a b b⋅ ⋅ ⋅ ⋅
⋅ = = = =⋅ ⋅ ⋅ ⋅ ⋅⋅
49. 2 7 2 7 11 7 11 777
11 1 11 1 2 1 2 2⋅
÷ = ÷ = ⋅ = =⋅
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154
50. 1 100 11002 1 2
100 21 1
100 21 1
200
− ÷ = − ÷
= − ⋅
⋅= −
⋅= −
51.
2 2
2
2
3312 1 12
3 121
3 1213 1213 12136
x x xx
xx
xxxx x
x
x
− ÷ = − ÷
= − ⋅
⋅= −
⋅⋅ ⋅
= −⋅ ⋅⋅
= −⋅
= −
52.
22
2
2
14 7 1473 1 3
7 31 14
7 31 147 31 2 7
31 232
x x xx
xx
xx
x xx
x
x
− ÷ = − ÷
= − ⋅
⋅= −
⋅⋅ ⋅ ⋅
= −⋅ ⋅ ⋅⋅
= −⋅
= −
53. 2 7 3 2 2 37 2 4 7 7 4
2 2 37 7 4
2 2 37 7 2 2
37 73
49
⎛ ⎞ ⎛ ⎞÷ ⋅ = ⋅ ⋅⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
⋅ ⋅=
⋅ ⋅⋅ ⋅
=⋅ ⋅ ⋅
=⋅
=
54. 1 5 1 1 5 122 6 12 2 6 1
1 5 122 6 1
1 5 6 22 6 1
1 51
5
⎛ ⎞ ⎛ ⎞⋅ ÷ = ⋅ ⋅⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
⋅ ⋅=
⋅ ⋅⋅ ⋅ ⋅
=⋅ ⋅
⋅=
=
55. 2
2
2
19 19 9963 63 1
19 963 1
19 99 7 1
197 1
197
yyy y
yyy yy
y
y
− ⋅ = − ⋅
⋅= −
⋅⋅ ⋅ ⋅
= −⋅ ⋅ ⋅⋅
= −⋅
= −
56. 2
2
2
31 16 311624 1 24
16 311 24
2 8 311 8 3
2 311 3
623
aaa a
aaa a
aa
a
⋅ − = ⋅−
⋅= −
⋅⋅ ⋅ ⋅ ⋅
= −⋅ ⋅ ⋅
⋅ ⋅= −
⋅
= −
57. 2 6 2 6 2 2 3 2 2 43 11 3 11 3 11 11 11
⋅ ⋅ ⋅ ⋅− ⋅− = = = =
⋅ ⋅
58. 1 6 1 6 65 7 5 7 35
⋅− ⋅ − = =
⋅
59. 4 3 4 16 4 16 4 8 2 4 2 88 16 8 3 8 3 8 3 3 3
⋅ ⋅ ⋅ ⋅÷ = ⋅ = = = =
⋅ ⋅
60. 9 16 9 15 9 15 1352 15 2 16 2 16 32
⋅÷ = ⋅ = =
⋅
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155
61. 2 2
2
21 14 21 2510 25 10 14
21 2510 14
3 7 5 52 5 2 7
3 52 2
154
x x x yy y y x
x yy xx x yy x
x
x
÷ = ⋅
⋅=
⋅⋅ ⋅ ⋅ ⋅ ⋅ ⋅
=⋅ ⋅ ⋅ ⋅ ⋅⋅ ⋅
=⋅
=
62. 2 2
2
17 13 17 1824 18 24 13
17 1824 13
17 6 36 4 13
17 34 13
5152
y y y xx x x y
y xx yy y xx y
y
y
÷ = ⋅
⋅=
⋅⋅ ⋅ ⋅ ⋅ ⋅
=⋅ ⋅ ⋅ ⋅⋅ ⋅
=⋅
=
63. 3 2 1 3 214 3 1 4 3
1 4 21 3 3
1 4 21 3 389
⎛ ⎞ ⎛ ⎞÷ ⋅ = ÷ ⋅⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
⎛ ⎞= ⋅ ⋅⎜ ⎟⎝ ⎠⋅ ⋅
=⋅ ⋅
=
64.
2 5 33 2 53311 9 1 11 9
33 11 51 2 9
33 11 52 9
3 11 11 52 3 3
11 11 52 3
6056
⎛ ⎞ ⎛ ⎞÷ ⋅ = ÷ ⋅⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
⎛ ⎞= ⋅ ⋅⎜ ⎟⎝ ⎠
⋅ ⋅=
⋅⋅ ⋅ ⋅
=⋅ ⋅⋅ ⋅
=⋅
=
65.
3 3 33
3
3
3
3
30302 2 1
12 30
12 30
12 30
12 30160
a a aa
aa
aa
a a aa a a
÷ = ÷
= ⋅
⋅=
⋅⋅ ⋅ ⋅
=⋅ ⋅ ⋅ ⋅
=⋅
=
66. 2 3 2
3
3
2
3
2
3 15 3155 1 5
15 51 3
15 51 3
c c cc
cc
cc
÷ = ÷
= ⋅
⋅=
⋅
5 3 51 3
5 51
25
c c cc c
c
c
⋅ ⋅ ⋅ ⋅ ⋅=
⋅ ⋅ ⋅⋅ ⋅
=
=
67. 2 2ab c ab c a b b c bc ab c ab c a b
⋅ ⋅ ⋅ ⋅⋅ = = =
⋅ ⋅ ⋅
68. 3 3 2
2 2ac b ac b a c b b b b b bb b a a c a aa c b a c
⋅ ⋅ ⋅ ⋅ ⋅ ⋅⋅ = = = =
⋅ ⋅ ⋅⋅
69.
1 2 5 1 2 52 3 6 2 3 6
1 53 61 63 51 63 51 2 3
3 51 2
525
⋅⎛ ⎞ ⎛ ⎞⋅ ÷ = ÷⎜ ⎟ ⎜ ⎟⋅⎝ ⎠ ⎝ ⎠
= ÷
= ⋅
⋅=
⋅⋅ ⋅
=⋅
⋅=
=
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Chapter 4: Fractions and Mixed Numbers ISM: Prealgebra
156
70.
3 8 2 3 8 24 9 5 4 9 5
3 4 2 24 3 3 5
2 23 52 53 22 53 253
⋅⎛ ⎞ ⎛ ⎞⋅ ÷ = ÷⎜ ⎟ ⎜ ⎟⋅⎝ ⎠ ⎝ ⎠⋅ ⋅⎛ ⎞= ÷⎜ ⎟⋅ ⋅⎝ ⎠
= ÷
= ⋅
⋅=
⋅
=
71.
4 4 3 4 4 37 5 7 7 5 7
4 127 354 357 124 357 124 7 57 4 353
⋅⎛ ⎞ ⎛ ⎞− ÷ ⋅ = − ÷⎜ ⎟ ⎜ ⎟⋅⎝ ⎠ ⎝ ⎠
= − ÷
= − ⋅
⋅= −
⋅⋅ ⋅
= −⋅ ⋅
= −
72.
5 4 5 5 4 58 7 16 8 7 16
5 4 58 7 4 45 58 285 288 5
5 288 5
5 7 44 2 572
⋅⎛ ⎞ ⎛ ⎞÷ ⋅− = ÷ −⎜ ⎟ ⎜ ⎟⋅⎝ ⎠ ⎝ ⎠⋅⎛ ⎞= ÷ −⎜ ⎟⋅ ⋅⎝ ⎠
⎛ ⎞= ÷ −⎜ ⎟⎝ ⎠⎛ ⎞= ⋅ −⎜ ⎟⎝ ⎠⋅
= −⋅⋅ ⋅
= −⋅ ⋅
= −
73. a. 2 5 2 5 2 5 15 6 5 6 5 2 3 3
xy ⋅ ⋅= ⋅ = = =
⋅ ⋅ ⋅
b. 2 5 2 6 2 6 125 6 5 5 5 5 25
x y ⋅÷ = ÷ = ⋅ = =
⋅
74. a. 8 1 8 1 2 4 1 2 1 29 4 9 4 9 4 9 9
xy ⋅ ⋅ ⋅ ⋅= ⋅ = = = =
⋅ ⋅
b. 8 1 8 4 8 4 329 4 9 1 9 1 9
x y ⋅÷ = ÷ = ⋅ = =
⋅
75. a. 4 9 4 9 365 11 5 11 55
xy ⋅= − ⋅ = − = −
⋅
b. 4 9 4 11 4 11 445 11 5 9 5 9 45
x y ⋅÷ = − ÷ = − ⋅ = − = −
⋅
76. a. 7 1 7 1 76 2 6 2 12
xy ⋅= ⋅− = − = −
⋅
b. 7 16 27 26 1
7 26 17 2
2 3 173
x y÷ = ÷−
= ⋅−
⋅= −
⋅⋅
= −⋅ ⋅
= −
77.
536
5 5318 6
3 5 51 18 6
3 5 51 18 63 5 53 6 6
5 5 True6 6
x = −
⎛ ⎞− −⎜ ⎟⎝ ⎠
⋅ − −
⋅− −
⋅⋅
− −⋅
− = −
Yes, 5
18− is a solution to the equation.
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ISM: Prealgebra Chapter 4: Fractions and Mixed Numbers
157
78.
2 63 11
2 9 63 11 112 9 63 11 11
2 611
2 3 611 11
6 6 True11 11
y =
⋅
⋅⋅
⋅ 3⋅33⋅11
⋅
=
Yes, 911
is a solution of the equation.
79. 1 12 10
1 2 12 5 101 2 12 5 10
1 1 False5 10
z− =
− ⋅
⋅−
⋅
− =
No, 25
is not a solution of the equation.
80. 153
3 155 3
5 3 11 5 35 3 11 5 3
3 1 False1 3
x =
⋅
⋅
⋅⋅
=
No, 35
is not a solution of the equation.
81. 1 1 of 200 2004 4
1 2004 11 200
4 11 4 50
4 1501
50
= ⋅
= ⋅
⋅=
⋅⋅ ⋅
=⋅
=
=
82. 1 1 of 2005 5
1
1
401
40
= ⋅200
1 200= ⋅5 1⋅200
=5⋅1⋅5⋅ 40
=5⋅1
=
=
83.
5 5 of 246 6
5
5
5
201
20
= ⋅24
5 24= ⋅6 1⋅ 24
=6⋅1⋅6 ⋅ 4
=6 ⋅1⋅ 4
=1
=
=
84.
5 5 of 248 8
5
5
5
151
15
= ⋅24
5 24= ⋅8 1⋅24
=8⋅1⋅8⋅3
=8⋅1⋅3
=1
=
=
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Chapter 4: Fractions and Mixed Numbers ISM: Prealgebra
158
85. 4 4 of 800 80025 25
4 80025 14 80025 1
4 25 3225 1
4 321
128
= ⋅
= ⋅
⋅=
⋅⋅ ⋅
=⋅
⋅=
=
128 of the students would be expected to major in business.
86. 7 7 of 7660
10 10
7
7
7
5362
= ⋅7660
7 7660= ⋅10 1⋅7660
=10 ⋅1⋅766 ⋅10
=10 ⋅1⋅766
=1
=
The money received was $5362 million.
87. 2 2 of 21705 5
2
2
2
868
= ⋅2170
2 2170= ⋅5 1⋅2170
=5⋅1⋅5⋅ 434
=5⋅1⋅ 434
=1
=
He hiked 868 miles.
88.
3 3 of 816 16
3
3
3232
= ⋅8
3 8= ⋅16 1⋅8
=16⋅1
⋅8=8⋅2 ⋅1
=⋅1
=
The screw sinks 32
inches deep after 8 turns.
89. 1 3 1 3 1 3 3 of 2 8 2 8 2 8 16
⋅= ⋅ = =
⋅
The radius of the circle is 3 inch.
16
90. 1 1 of 30005 5
1
1
600
= ⋅3000
1 3000= ⋅5 1⋅3000
=5⋅1⋅5⋅600
=5⋅1
=
The diet can contain 600 calories from fat per day.
91. 2 2 of 27573 3
2
2
2
1838
= ⋅2757
2 2757= ⋅3 1⋅2757
=3⋅1⋅3⋅919
=3⋅1⋅919
=1
=
The sale price is $1838.
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ISM: Prealgebra Chapter 4: Fractions and Mixed Numbers
159
92. 3 3 of 102,000
50 50
3
3
3
6120
= ⋅102,000
3 102,000= ⋅50 1⋅102,000
=50 ⋅1
⋅50 ⋅ 2040=
50⋅1⋅ 2040
=1
=
The Gonzales family pays $6120 to the real estate companies.
93. 1 1 of 180 1809 9
1 1809 11 180
9 11 9 20
9 1201
20
= ⋅
= ⋅
⋅=
⋅⋅ ⋅
=⋅
=
=
20 contestants participated more than once.
94. 23 23 of 854,00050 50
23
23
392,840
= ⋅854,000
23 854,000= ⋅50 1
⋅50 ⋅17,080=
50 ⋅1⋅17,080
=1
=
They might believe there were 392,840 people between the ages of 16 and 24.
95. 5 5 1area length width
14 141 ⋅1
= ⋅ = ⋅ = =5 14 ⋅5
The area is 1
14 square foot.
96. 1 1 3area length width2 16
3 ⋅3= ⋅ = ⋅ = =
8 2⋅8
The area is 3
16 square mile.
97. 8 8 12,00012,00025 25 1
8 25 48025 1
8 4801
3840
⋅ = ⋅
⋅ ⋅=
⋅⋅
=
=
The family drove 3840 miles for work.
98. 3 3 of 12,00025 25
3
3
3
1440
= ⋅12,000
3 12,000= ⋅25 1⋅12,00
=25⋅1⋅ 25⋅480
=25⋅1⋅ 480
=1
=
The family drove 1440 miles for shopping.
99. 1 1 of 12,0005 5
1
2400
= ⋅12,000
1 12,000= ⋅5 1⋅5⋅ 2400
=5⋅1
=
The family drove 2400 miles for family business.
100. 1
1001
120
1 12,000⋅12,000 = ⋅
100 1⋅120 ⋅100
=100⋅1
=
The family drove 120 miles for medical.
101. 277698
201+
102. 8114269
922+
103. 968772196
−
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Chapter 4: Fractions and Mixed Numbers ISM: Prealgebra
160
104. 882773109
−
105. answers may vary
106. answers may vary
107. 42 125 7 42 125 625 36 6 25 36 7
42 125 625 36 76 7 5 25 625 6 6 7
5
⋅ ÷ = ⋅ ⋅
⋅ ⋅=
⋅ ⋅⋅ ⋅ ⋅ ⋅
=⋅ ⋅ ⋅
=
108.
2 2
2
2
8 39 8 1 8 39 8 113 16 9 2 13 16 9 2
8 13 3 2 4 113 8 2 3 3 24 13 24 4 13 3 24 4 13 3 2
16 19 2
16 29 1
16 29 1
329
⋅ ⋅⎛ ⎞ ⎛ ⎞⋅ ⋅ ÷ = ÷⎜ ⎟ ⎜ ⎟⋅ ⋅⎝ ⎠ ⎝ ⎠⋅ ⋅ ⋅ ⋅⎛ ⎞= ÷⎜ ⎟⋅ ⋅ ⋅ ⋅⎝ ⎠
⎛ ⎞= ÷⎜ ⎟⎝ ⎠⎛ ⎞⎛ ⎞= ÷⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
⋅⎛ ⎞= ÷⎜ ⎟⋅⎝ ⎠
= ÷
= ⋅
⋅=
⋅
=
109. 3 3 of 290,810,000 290,810,00025 25
3 290,810,00025 13 290,810,000
25 13 25 11,632, 400
25 13 11,632, 400
134,897, 200
= ⋅
= ⋅
⋅=
⋅⋅ ⋅
=⋅
⋅=
=
The population of California is approximately 34,897,200.
110. 32 32 of 10,600265 265
32
32
32
1280
= ⋅10,600
32 10,600= ⋅265 1⋅10,600
=265⋅1⋅ 265⋅ 40
=265⋅1⋅ 40
=1
=
There were 1280 news/talk stations.
111. 34
of 36 on the first bus and 13
of 30 on the
second bus are girls. 3 14 3
3 1
3 1
3 101
27 1037
1 3 36 30⋅36 + ⋅30 = ⋅ + ⋅
3 4 1 1⋅36 ⋅30
= +4 ⋅1 3⋅1⋅ 4 ⋅9 ⋅3⋅10
= +4 ⋅1 3⋅1⋅9
= +1
= +=
There are 37 girls on the two buses.
112.
42 of fleet 252113
42fleet 252113
2521
252
42
6
678
=
= ÷
113= ⋅
42⋅113
=1⋅42⋅6 ⋅113
=1⋅42⋅113
=1
=
There were 678 aircraft in the FedEx Express air fleet.
The Bigger Picture
1. 2 2 163 27
8 ⋅8⋅ = =9 3⋅9
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ISM: Prealgebra Chapter 4: Fractions and Mixed Numbers
161
2. 2 8 23 9 3
2
2
34
9− ÷ = − ⋅
8⋅9
= −3⋅8⋅3⋅3
= −3⋅ 2 ⋅ 4
= −
3. 12 12 3 41 3
1 ⋅1 ⋅ 4 ⋅1⋅ = = =9 1⋅9 1⋅3⋅3
4. 9 15 92 8 2
9 82 153 3 2 4
2 3 53 45
125
x x xx
xxxx
8÷ = ⋅
15⋅
=⋅⋅ ⋅ ⋅ ⋅
=⋅ ⋅ ⋅
⋅=
=
5. 80 ÷ 20 ⋅ 2 = 4 ⋅ 2 = 8
6. 2 33 2 3 3 2 2 2 72⋅ = ⋅ ⋅ ⋅ ⋅ =
7. 11 5 4 11 5 4 11 5 4 1 1 120 8 33 20 8 33 4 5 8 3 11 8 3 24
⋅ ⋅ ⋅ ⋅ ⋅⋅ ⋅ = = = =
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
8. 1 20 1 20 2 20 2 4020 402 1 2 1 1 1 1 1
⋅÷ = ÷ = ⋅ = = =
⋅
9. 3 33 4(18 16) 3 4(2)3 4(8)3 3235
+ − = += += +=
10. 100 − 76 = 24
11. 12 7 712 7 7 7 7
12 012 012 12
0
xx
xx
x
− + =− + − = −
− =−
=− −
=
12. 9 14 6 329 6 14 6 6 32
3 14 323 14 14 32 14
3 183 183 3
6
n nn n n n
nn
nn
n
− = −− − = − −
− = −− + = − +
= −−
=
= −
Section 4.4
Practice Problems
1. 6 2 6 2 8
13 13 13 13+
+ = =
2. 5 1 5 1 6 2 3 38 8 8 8 2 4 4
+ ⋅+ = = = =
⋅
3. 20 6 7 20 6 7 33 or 311 11 11 11 11
+ ++ + = =
4. 11 6 11 6 512 12 12 12
−− = =
5. 7 2 7 2 5 1 5 1
15 15 15 15 3 5 3− ⋅
− = = = =⋅
6. 8 4 8 4 4 4 or
17 17 17 17 17− + −
− + = = −
7. 2 7 2 75 5 5
y y−− =
8. 4 6 3 4 6 3 5 5 or
11 11 11 11 11 11− − −
− − = = −
9. 10 5 10 5 5 5 or 12 12 12 12 12
x y − + −+ = − + = = −
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162
10. 3 3 3 3perimeter
20 20 20 203 3 3 3
2012203 45 435
= + + +
+ + +=
=
⋅=
⋅
=
The perimeter is 35
mile.
11. 13 11 13 11 2 1 2 14 4 4 4 2 2 2
− ⋅− = = = =
⋅
He ran 12
mile farther on Monday.
12. 16 is a multiple of 8, so the LCD of 78
and 1116
is 16.
13. 30 ⋅ 1 = 30 Not a multiple of 25. 30 ⋅ 2 = 60 Not a multiple of 25. 30 ⋅ 3 = 90 Not a multiple of 25. 30 ⋅ 4 = 120 Not a multiple of 25. 30 ⋅ 5 = 150 A multiple of 25.
The LCD of 2325
and 130
is 150.
14. 40 = 2 ⋅ 2 ⋅ 2 ⋅ 5 108 = 2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 3 LCD = 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 5 = 1080
The LCD of 340
− and 11
108 is 1080.
15. 20 = 2 ⋅ 2 ⋅ 5 24 = 2 ⋅ 2 ⋅ 2 ⋅ 3 45 = 3 ⋅ 3 ⋅ 5 LCD = 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 5 = 360
The LCD of 7 1, ,20 24
and 1345
is 360.
16. y = y 11 = 11
The LCD of 7y
and 6
11 is 11y.
17. 7 7 7 7 7 498 8 7 8 7 56
⋅= ⋅ = =
⋅
18. 1 1 5 1 5 54 4 5 4 5 20
⋅= ⋅ = =
⋅
19. 3 3 6 3 6 187 7 6 7 6 42x x x x⋅= ⋅ = =
⋅
20. 4 6 4 6 2441 6 1 6 6
⋅= ⋅ = =
⋅
21. 9 9 9 9 9 814 4 9 4 9 36x x x x
⋅= ⋅ = =
⋅
Vocabulary and Readiness Check
1. The fractions 911
and 1311
are called like
fractions while 34
and 13
are called unlike
fractions.
2. a c a cb b b
++ = and .a c a c
b b b−
− =
3. As long as b is not 0, .a a ab b b−
= = −−
4. The distance around a figure is called its perimeter.
5. The smallest positive number divisible by all the denominators of a list of fractions is called the least common denominator (LCD).
6. Fractions that represent the same portion of a whole are called equivalent fractions.
Exercise Set 4.4
1. 5 2 5 2 7
11 11 11 11+
+ = =
2. 9 2 9 2 11
17 17 17 17+
+ = =
3. 2 4 2 4 6 3 2 29 9 9 9 3 3 3
+ ⋅+ = = = =
⋅
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163
4. 3 2 3 2 5 1 5 1
10 10 10 10 2 5 2+ ⋅
+ = = = =⋅
5. 6 1 6 1 5 1 5 120 20 20 20 4 5 4
− + − ⋅− + = = = − = −
⋅
6. 3 1 3 1 2 1 2 18 8 8 8 4 2 4
− + − ⋅− + = = = − = −
⋅
7. 3 4 3 ( 4) 7 1 7 1
14 14 14 14 2 7 2− + − − ⋅⎛ ⎞− + − = = = − = −⎜ ⎟ ⋅⎝ ⎠
8. 5 7 5 ( 7) 12 1 12 124 24 24 24 2 12 2
− + − − ⋅⎛ ⎞− + − = = = − = −⎜ ⎟ ⋅⎝ ⎠
9. 2 4 2 4 6 2 3 2
9 9 9 9 3 3 3x x x x x x+ ⋅
+ = = = =⋅ ⋅
10. 3 2 3 2 5 1 5 1
10 10 10 10 2 5 2y y y y y y+ ⋅
+ = = = =⋅ ⋅
11. 7 3 2 7 3 218 18 18 18
21822 9
9
x x x x x x
x
x
x
− + +− + + =
−=
⋅= −
⋅
= −
12. 7 3 1 7 3 115 15 15 15
31533 5
5
z z z z z z
z
z
z
− + +− + + =
−=
⋅= −
⋅
= −
13. 10 4 10 4 611 11 11 11
−− = =
14. 9 5 9 5 4
13 13 13 13−
− = =
15. 7 1 7 1 6 3 2 38 8 8 8 4 2 4
− ⋅− = = = =
⋅
16. 5 1 5 1 4 2 2 26 6 6 6 3 2 3
− ⋅− = = = =
⋅
17. 1 4 1 4 3 3y y y y y
− −− = = = −
18. 4 7 4 7 3 3z z z z z
− −− = = = −
19. 27 8 27 833 33 33 33
27 833
19331933
⎛ ⎞− − − = − +⎜ ⎟⎝ ⎠
− +=
−=
= −
20. 37 18 37 1845 45 45 45
37 1845
19451945
⎛ ⎞− − − = − +⎜ ⎟⎝ ⎠
− +=
−=
= −
21. 20 10 17 20 10 1721 21 21 21
721
1 73 713
− −− − =
−=
⋅= −
⋅
= −
22. 27 5 28 27 5 2828 28 28 28
628
3 214 23
14
− −− − =
−=
⋅= −
⋅
= −
23. 7 3 7 34 4 4a a −− =
24. 18 3 18 3
5 5 5b b −− =
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Chapter 4: Fractions and Mixed Numbers ISM: Prealgebra
164
25. 9 99 9 99 90 9 10 9
100 100 100 100 10 10 10− + ⋅
− + = = = =⋅
26. 15 85 15 85 70 7 10 7200 200 200 200 20 10 20
− + ⋅− + = = = =
⋅
27. 13 13 13 1328 28 28
26282 132 14
1314
x x x x
x
x
x
− −− − =
−=
⋅ ⋅= −
⋅
= −
28. 15 15 15 15
26 26 2630
262 15
2 1315
13
y y y
y
y
y
− −− − =
−=
⋅= −
⋅ ⋅
= −
29. 9 1 9 115 15 15x x ++ =
30. 2 7 2 715 15 15x x ++ =
31. 7 15 7 15 8 816 16 16 16 8 2 2x x x x x x x− − ⋅− = = = − = −
⋅
32. 3 15 3 15 12 3 4 3
16 16 16 16 4 4 4z z z z z z− − ⋅
− = = = − = −⋅ ⋅
33. 9 7 10 9 7 10 8 2 4 2
12 12 12 12 12 3 4 3− − − ⋅
− − = = = − = −⋅
34. 1 15 2 1 15 2 12 4 3 38 8 8 8 8 4 2 2
− + − ⋅− + = = = − = −
⋅
35. 3 2 3 2 3
4 4 4 4 4 4x x x x x x x x x+ − ++ − + = =
36. 9 2 5 4 9 2 5 48 8 8 8 8
128
4 34 2
32
y y y y y y y y
y
y
y
+ + −+ + − =
=
⋅ ⋅=
⋅
=
37. 3 2 3 2 54 4 4 4
x y ++ = + = =
38. 7 9 7 9 2 1 2 18 8 8 8 4 2 4
x y − − ⋅− = − = = = − = −
⋅
39. 1 3 1 3 1 3 25 5 5 5 5 5
x y − +⎛ ⎞− = − − − = − + = =⎜ ⎟⎝ ⎠
40. 1 5 1 5 4 2 2 26 6 6 6 2 3 3
x y − + ⋅+ = − + = = = =
⋅
41. 4 7 9 4 7 9 20 120 20 20 20 20
+ ++ + = = =
The perimeter is 1 inch.
42. 3 6 4 3 2 3 6 4 3 2 18
13 13 13 13 13 13 13+ + + +
+ + + + = =
The perimeter is 1813
feet.
43. 5 7 5 7 5 7 5 7 24 2
12 12 12 12 12 12+ + +
+ + + = = =
The perimeter is 2 meters.
44. 1 1 1 1 1 1 1 1 4 2 2 26 6 6 6 6 6 2 3 3
+ + + ⋅+ + + = = = =
⋅
The perimeter is 23
centimeter.
45. North America makes up 16100
of the world’s
land area, while South America makes up 12100
of the land area. 16 12 16 12 28 4 7 7100 100 100 100 4 25 25
+ ⋅+ = = = =
⋅
725
of the world’s land area is within North
America and South America.
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165
46. Asia makes up 30
100 of the world’s land area,
while Africa makes up 20
100 of the world’s land
area. 30 20 30 20 50 1 50 1
100 100 100 100 2 50 2+ ⋅
+ = = = =⋅
12
of the world’s land area is within Asia and
Africa.
47. Antarctica makes up 9
100 of the world’s land
area, while Europe makes up 7
100 of the world’s
land area. 9 7 9 7 2 1 2 1
100 100 100 100 50 2 50− ⋅
− = = = =⋅
Antarctica’s land area is 150
greater than that of
Europe.
48. Asia makes up 30
100 of the world’s land area,
while Australia makes up 6
100 of the world’s
land area. 30 6 30 6 24 6 4 6
100 100 100 100 25 4 25− ⋅
− = = = =⋅
Asia’s land area is 625
greater than that of
Australia’s.
49. To find the remaining amount of track to be
inspected, subtract the 520
mile that has already
been inspected from the 1920
mile total that must
be inspected. 19 5 19 5 14 2 7 720 20 20 20 2 10 10
− ⋅− = = = =
⋅
710
of a mile of track remains to be inspected.
50. To find the remaining distance, subtract the 118
miles already run from the 168
total miles.
16 11 16 11 58 8 8 8
−− = =
He must run 58
mile more.
51. To find the fraction that had speed limits less
than 70 mph, subtract the 1750
that have 70 mph
speed limits from the 3750
that have speed limits
up to and including 70. 37 17 37 17 20 2 10 250 50 50 50 5 10 5
− ⋅− = = = =
⋅
25
of the states have speed limits that were less
than 70 mph.
52. To find the fraction treating pain other than
headaches, subtract the 750
for treating
headaches from the 3150
for some type of pain.
31 7 31 7 24 2 12 1250 50 50 50 2 25 25
− ⋅− = = = =
⋅
1225
of aspirin use is for treating pain other than
headaches.
53. Multiples of 9: 9, 18, 27, 36, 45, 54 Multiples of 15: 15, 30, 45, 60 LCD: 45
54. Multiples of 12: 12, 24, 36, 48, 60, 72 Multiples of 20: 20, 40, 60, 80 LCD: 60
55. Multiples of 36: 36, 72 Multiples of 24: 24, 48, 72 LCD: 72
56. Multiples of 15: 15, 30, 45, 60, 75, 90 Multiples of 90: 90, 180 LCD: 90
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Chapter 4: Fractions and Mixed Numbers ISM: Prealgebra
166
57. 6 = 2 ⋅ 3 15 = 3 ⋅ 5 25 = 5 ⋅ 5 LCD = 2 ⋅ 3 ⋅ 5 ⋅ 5 = 150
58. 4 = 2 ⋅ 2 14 = 2 ⋅ 7 20 = 2 ⋅ 2 ⋅ 5 LCD = 2 ⋅ 2 ⋅ 5 ⋅ 7 = 140
59. 24 = 2 ⋅ 2 ⋅ 2 ⋅ 3 x = x LCD = 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ x = 24x
60. y = y 70 = 2 ⋅ 5 ⋅ 7 LCD = 2 ⋅ 5 ⋅ 7 ⋅ y = 70y
61. 18 = 2 ⋅ 3 ⋅ 3 21 = 3 ⋅ 7 LCD = 2 ⋅ 3 ⋅ 3 ⋅ 7 = 126
62. 24 = 2 ⋅ 2 ⋅ 2 ⋅ 3 45 = 3 ⋅ 3 ⋅ 5 LCD = 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 5 = 360
63. 3 = 3 21 = 3 ⋅ 7 56 = 2 ⋅ 2 ⋅ 2 ⋅ 7 LCD = 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 7 = 168
64. 11 = 11 33 = 3 ⋅ 11 121 = 11 ⋅ 11 LCD = 3 ⋅ 11 ⋅ 11 = 363
65. 2 2 7 2 7 143 3 7 3 7 21
⋅= ⋅ = =
⋅
66. 5 5 4 5 4 206 6 4 6 4 24
⋅= ⋅ = =
⋅
67. 4 4 5 4 5 207 7 5 7 5 35
⋅= ⋅ = =
⋅
68. 3 3 20 3 20 605 5 20 5 20 100
⋅= ⋅ = =
⋅
69. 1 1 25 1 25 252 2 25 2 25 50
⋅= ⋅ = =
⋅
70. 1 1 10 1 10 105 5 10 5 10 50
⋅= ⋅ = =
⋅
71. 14 14 4 14 4 5617 17 4 17 4 68x x x x⋅= ⋅ = =
⋅
72. 19 19 6 19 6 11421 21 6 21 6 126z z z z⋅= ⋅ = =
⋅
73. 2 2 4 2 4 83 3 4 3 4 12y y y y⋅= ⋅ = =
⋅
74. 3 3 6 3 6 182 2 6 2 6 12x x x x⋅= ⋅ = =
⋅
75. 5 5 4 5 4 209 9 4 9 4 36
a a aa a a
⋅= ⋅ = =
⋅
76. 7 7 6 7 6 426 6 6 6 6 36
a a aa a a
⋅= ⋅ = =
⋅
77. Denmark: 89 89
100 100=
Finland: 9 9 10 90
10 10 10 100⋅
= =⋅
Israel: 24 24 4 9625 25 4 100
⋅= =
⋅
Spain: 23 23 4 9225 25 4 100
⋅= =
⋅
Japan: 17 17 4 6825 25 4 100
⋅= =
⋅
Norway: 91 91
100 100=
Singapore: 4 4 20 805 5 20 100
⋅= =
⋅
South Korea: 69 69
100 100=
India: 2 2 4 825 25 4 100
⋅= =
⋅
United States: 7 7 10 70
10 10 10 100⋅
= =⋅
78. 96
100 is the largest fraction, so Israel has the
largest fraction of cell phone users.
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79. 8
100 is the smallest fraction, so India has the
smallest fraction of cell phone users.
80. 3 3 25 754 4 25 100= ⋅ =
Find those countries whose fraction is over 75 .
100
Denmark, Finland, Israel, Spain, Norway, and
Singapore have over 34
of the population using
cell phones.
81. 23 3 3 9= ⋅ =
82. 34 4 4 4 64= ⋅ ⋅ =
83. 35 5 5 5 125= ⋅ ⋅ =
84. 43 3 3 3 3 81= ⋅ ⋅ ⋅ =
85. 27 7 7 49= ⋅ =
86. 45 5 5 5 5 625= ⋅ ⋅ ⋅ =
87. 32 3 2 2 2 3 24⋅ = ⋅ ⋅ ⋅ =
88. 24 5 4 4 5 80⋅ = ⋅ ⋅ =
89. 2 9 2 9 117 7 7 7
++ = =
90. 3 1 3 1 2 1 2 14 4 4 4 2 2 2
− ⋅− = = = =
⋅
91. answers may vary
92. answers may vary
93. 16 12 7 20 30 6 9100 100 100 100 100 100 100
16 12 7 20 30 6 9100
1001001
+ + + + + +
+ + + + + +=
=
=
answers may vary
94. 3 3 4 3 3 4 2 1 2 18 8 8 8 8 4 2 4
+ − ⋅+ − = = = =
⋅
He is 14
mile from home.
95. 37 37 22 37 22 814165 165 22 165 22 3630x x x x⋅= ⋅ = =
⋅
96. 108 108 19 108 19 2052215 215 19 215 19 4085y y y y
⋅= ⋅ = =
⋅
97. answers may vary
98. answers may vary
99. 2 2 5 103 3 5 152 2 20 403 3 20 602 2 100 2003 3 100 300
⋅= =
⋅⋅
= =⋅⋅
= =⋅
a, b, and d are equivalent to 2 .3
100. False; 7 7 4 7 4 28
12 12 4 12 4 48⋅
= ⋅ = =⋅
Section 4.5
Practice Problems
1. The LCD of 7 and 21 is 21. 2 8 2 3 8 6 8 14 2 7 27 21 7 3 21 21 21 21 3 7 3
⋅ ⋅+ = + = + = = =
⋅ ⋅
2. The LCD of 6 and 9 is 18. 5 2 5 3 2 2 15 4 196 9 6 3 9 2 18 18 18y y y y y y y⋅ ⋅+ = + = + =
⋅ ⋅
3. The LCD of 5 and 20 is 20. 1 9 1 4 95 20 5 4 20
4 920 20
5201 54 514
⋅− + = − +
⋅
= − +
=
⋅=
⋅
=
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4. The LCD of 7 and 10 is 70. 5 9 5 10 9 7 50 63 137 10 7 10 10 7 70 70 70
⋅ ⋅− = − = − = −
⋅ ⋅
5. The LCD of 8, 3, and 12 is 24. 5 1 1 5 3 1 8 1 28 3 12 8 3 3 8 12 2
15 8 224 24 24524
⋅ ⋅ ⋅− − = − −
⋅ ⋅ ⋅
= − −
=
6. Recall that 55 .1
= The LCD of 1 and 4 is 4.
5 5 4 20 201 4 1 4 4 4 4 4
y y y y⋅ −− = − = − =
⋅
7. The LCD of 8 and 20 is 40. Write each fraction as an equivalent fraction with a denominator of 40. 5 5 5 258 8 5 40
⋅= =
⋅
11 11 2 2220 20 2 40
⋅= =
⋅
Since 25 > 22, 25 22 ,40 40
> so 5 11 .8 20>
8. The LCD of 20 and 5 is 20. Write each fraction as an equivalent fraction with a denominator of 20.
1720
−
4 4 4 165 5 4 20
⋅− = − = −
⋅
Since −17 < −16, 17 16 ,20 20
− < − so 17 4 .20 5
− < −
9. The LCD of 11 and 9 is 99. 5 4 5 9 4 11 45 44 1
11 9 11 9 9 11 99 99 99x y ⋅ ⋅− = − = − = − =
⋅ ⋅
10. The LCD of 5, 10, and 15 is 30. 3 3 1 3 6 3 3 1 25 10 15 5 6 10 3 15 2
18 9 230 30 302930
⋅ ⋅ ⋅+ + = + +
⋅ ⋅ ⋅
= + +
=
The homeowner needs 2930
cubic yard of cement.
11. The LCD of 4 and 3 is 12. 3 2 3 3 2 4 9 8 14 3 4 3 3 4 12 12 12
⋅ ⋅− = − = − =
⋅ ⋅
The difference in length is 1 foot.
12
Calculator Explorations
1. 1 2 37
16 5 80+ =
2. 3 2 2320 25 100
+ =
3. 4 7 959 8 72+ =
4. 9 5 16311 12 132
+ =
5. 10 12 39417 19 323
+ =
6. 14 15 25331 21 217
+ =
Vocabulary and Readiness Check
1. To add or subtract unlike fractions, we first write the fractions as equivalent fractions with a common denominator. The common denominator we use is called the least common denominator.
2. The LCD for 16
and 58
is 24.
3. 1 5 1 4 5 3 4 15 19.6 8 6 4 8 3 24 24 24+ = ⋅ + ⋅ = + =
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4. 1 5 1 4 5 3 4 15 11.6 8 6 4 8 3 24 24 24− = ⋅ − ⋅ = − = −
5. x − y is an expression while 135
x = is an
equation.
6. Since −10 < −1, we know that 10 1 .13 13
− < −
Exercise Set 4.5
1. The LCD of 3 and 6 is 6. 2 1 2 2 1 4 1 53 6 3 2 6 6 6 6
⋅+ = + = + =
⋅
2. The LCD of 6 and 12 is 12. 5 1 5 2 1 10 1 116 12 6 2 12 12 12 12
⋅+ = + = + =
⋅
3. The LCD of 2 and 3 is 6. 1 1 1 3 1 2 3 2 12 3 2 3 3 2 6 6 6
⋅ ⋅− = − = − =
⋅ ⋅
4. The LCD of 3 and 4 is 12. 2 1 2 4 1 3 8 3 53 4 3 4 4 3 12 12 12
⋅ ⋅− = − = − =
⋅ ⋅
5. The LCD of 11 and 33 is 33. 2 2 2 3 2 6 2 4
11 33 11 3 33 33 33 33⋅
− + = − + = − + = −⋅
6. The LCD of 9 and 3 is 9. 5 1 5 1 3 5 3 29 3 9 3 3 9 9 9
⋅− + = − + = − + = −
⋅
7. The LCD of 14 and 7 is 14. 3 3 3 3 2 3 6 3
14 7 14 7 2 14 14 14⋅
− = − = − = −⋅
8. The LCD of 15 and 5 is 15. 2 2 2 2 3 2 6 4
15 5 15 5 3 15 15 15⋅
− = − = − = −⋅
9. The LCD of 35 and 7 is 35. 11 2 11 2 535 7 35 7 5
11 1035 352135
3 75 7
35
x x x x
x x
x
x
x
⋅+ = +
⋅
= +
=
⋅ ⋅=
⋅
=
10. The LCD of 5 and 25 is 25. 2 3 2 5 3 10 3 135 25 5 5 25 25 25 25y y y y y y y⋅+ = + = + =
⋅
11. The LCD of 1 and 12 is 12. 2 2 12 24 242
12 1 12 1 12 12 12 12 12y y y y y⋅ −
− = − = − = − =⋅
12. The LCD of 1 and 20 is 20. 55
20 1 205 201 20 2010020 20
10020
y y
y
y
y
− = −
⋅= −
⋅
= −
−=
13. The LCD of 12 and 9 is 36. 5 1 5 3 1 4 15 4 11
12 9 12 3 9 4 36 36 36⋅ ⋅
− = − = − =⋅ ⋅
14. The LCD of 12 and 18 is 36. 7 5 7 3 5 2 21 10 11
12 18 12 3 18 2 36 36 36⋅ ⋅
− = − = − =⋅ ⋅
15. The LCD of 1 and 7 is 7. 5 7 5 7 7 5 49 5 4477 1 7 1 7 7 7 7 7
⋅− + = − + = − + = − + = −
⋅
16. The LCD of 1 and 10 is 10. 7 10 710
10 1 1010 10 71 10 10
100 710 109310
− + = − +
⋅= − +
⋅
= − +
= −
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17. The LCD of 11 and 9 is 99. 5 4 5 9 4 1111 9 11 9 9 11
45 4499 99
8999
a a a a
a a
a
⋅ ⋅+ = +
⋅ ⋅
= +
=
18. The LCD of 18 and 9 is 18. 7 2 7 2 2 7 4 1118 9 18 9 2 18 18 18x x x x x x x⋅+ = + = + =
⋅
19. The LCD of 3 and 6 is 6. 2 1 2 2 1 4 1 4 13 6 3 2 6 6 6 6y y y y⋅ −− = − = − =
⋅
20. The LCD of 6 and 12 is 12. 5 1 5 2 1 10 1 10 16 12 6 2 12 12 12 12z z z z⋅ −− = − = − =
⋅
21. The LCD of 2 and x is 2x. 1 3 1 3 2 6 62 2 2 2 2 2
x x xx x x x x x
⋅ ⋅ ++ = + = + =
⋅ ⋅
22. The LCD of 5 and x is 5x. 2 3 2 3 5 2 15 2 155 5 5 5 5 5
x x xx x x x x x
⋅ ⋅ ++ = + = + =
⋅ ⋅
23. The LCD of 11 and 33 is 33. 2 2 2 3 2 6 2 8
11 33 11 3 33 33 33 33⋅
− − = − − = − − = −⋅
24. The LCD of 9 and 3 is 9. 5 1 5 1 3 5 3 89 3 9 3 3 9 9 9
⋅− − = − − = − − = −
⋅
25. The LCD of 14 and 7 is 14. 9 3 9 3 2 9 6 3
14 7 14 7 2 14 14 14⋅
− = − = − =⋅
26. The LCD of 5 and 15 is 15. 4 2 4 3 2 12 2 10 2 5 25 15 5 3 15 15 15 15 3 5 3
⋅ ⋅− = − = − = = =
⋅ ⋅
27. The LCD of 35 and 7 is 35. 11 2 11 2 5 11 10 11 1035 7 35 7 5 35 35 35y y y y⋅ −− = − = − =
⋅
28. The LCD of 5 and 25 is 25. 2 3 2 5 3 10 3 10 35 25 5 5 25 25 25 25b b b b⋅ −− = − = − =
⋅
29. The LCD of 9 and 12 is 36. 1 5 1 4 5 3 4 15 119 12 9 4 12 3 36 36 36
⋅ ⋅− = − = − = −
⋅ ⋅
30. The LCD of 18 and 12 is 36. 5 7 5 2 7 3 10 21 11
18 12 18 2 12 3 36 36 36⋅ ⋅
− = − = − = −⋅ ⋅
31. The LCD of 15 and 12 is 60. 7 5 7 4 5 5
15 12 15 4 12 528 2560 603
601 320 3120
⋅ ⋅− = −
⋅ ⋅
= −
=
⋅=
⋅
=
32. The LCD of 8 and 20 is 40. 5 3 5 5 3 2 25 6 198 20 8 5 20 2 40 40 40
⋅ ⋅− = − = − =
⋅ ⋅
33. The LCD of 7 and 8 is 56. 5 1 5 8 1 7 40 7 337 8 7 8 8 7 56 56 56
⋅ ⋅− = − = − =
⋅ ⋅
34. The LCD of 13 and 10 is 130. 10 7 10 10 7 13 100 91 913 10 13 10 10 13 130 130 130
⋅ ⋅− = − = − =
⋅ ⋅
35. The LCD of 8 and 16 is 16. 7 3 7 2 3 14 3 178 16 8 2 16 16 16 16
⋅+ = + = + =
⋅
36. The LCD of 18 and 9 is 18. 7 2 7 2 2 7 4 11
18 9 18 9 2 18 18 18⋅
+ = + = + =⋅
37. 3 5 3 5 29 9 9 9
−− = = −
38. 1 4 1 4 3
13 13 13 13−
− = = −
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39. The LCD of 5, 3, and 10 is 30. 2 1 3 2 6 1 10 3 35 3 10 5 6 3 10 10 3
12 10 930 30 301130
⋅ ⋅ ⋅− + − = − + −
⋅ ⋅ ⋅
= − + −
= −
40. The LCD of 3, 4, and 5 is 60. 1 1 2 1 20 1 15 2 123 4 5 3 20 4 15 5 12
20 15 2460 60 601160
⋅ ⋅ ⋅− − + = − − +
⋅ ⋅ ⋅
= − − +
= −
41. The LCD of 11 and 3 is 33. 5 5 3 11
11 3 11 3 3 1115 1133 3315 11
33
y y
y
y
⋅ ⋅+ = +
⋅ ⋅
= +
+=
42. The LCD of 13 and 26 is 26. 5 3 5 2 3 10 3 10 313 26 13 2 26 26 26 26z z z z⋅ ++ = + = + =
⋅
43. The LCD of 6 and 7 is 42. 5 3 5 7 3 6 35 18 536 7 6 7 7 6 42 42 42
⋅ ⋅− − = − − = − − = −
⋅ ⋅
44. The LCD of 2 and 29 is 58. 1 3 1 29 3 2 29 6 352 29 2 29 29 2 58 58 58
⋅ ⋅− − = − − = − − = −
⋅ ⋅
45. The LCD of 2, 4, and 16 is 16. 2 8 4 2
2 4 16 2 8 4 4 168 4 216 16 1614162 72 8
78
x x x x x x
x x x
x
x
x
⋅ ⋅+ + = + +
⋅ ⋅
= + +
=
⋅=
⋅
=
46. The LCD of 4, 8, and 16 is 16. 2 4 2 2
4 8 16 4 4 8 2 164 2 216 16 1681688 2
2
z z z z z z
z z z
z
z
z
⋅ ⋅+ + = + +
⋅ ⋅
= + +
=
⋅=
⋅
=
47. The LCD of 9 and 6 is 18. 7 1 7 2 1 3 14 3 119 6 9 2 6 3 18 18 18
⋅ ⋅− = − = − =
⋅ ⋅
48. The LCD of 16 and 8 is 16. 9 3 9 3 2 9 6 3
16 8 16 8 2 16 16 16⋅
− = − = − =⋅
49. The LCD of 3 and 13 is 39. 2 6 2 13 6 33 13 3 13 13 3
26 1839 39
4439
a a a a
a a
a
⋅ ⋅+ = +
⋅ ⋅
= +
=
50. The LCD of 4 and 7 is 28. 3 3 7 4 21 4 254 7 4 7 7 4 28 28 28y y y y y y y⋅ ⋅+ = + = + =
⋅ ⋅
51. The LCD of 30 and 12 is 60. 7 5 7 2 5 5 14 25 1130 12 30 2 12 5 60 60 60
⋅ ⋅− = − = − = −
⋅ ⋅
52. The LCD of 30 and 20 is 60. 7 3 7 2 3 330 20 30 2 20 3
14 960 605
601 5
12 51
12
⋅ ⋅− = −
⋅ ⋅
= −
=
⋅=
⋅
=
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53. The LCD of 9 and y is 9y. 5 1 5 1 9 5 9 5 99 9 9 9 9 9
y y yy y y y y y
⋅ ⋅ ++ = + = + =
⋅ ⋅
54. The LCD of 12 and x is 12x. 1 5 1 5 12
12 12 1260
12 1260
12
xx x x
xx x
xx
⋅ ⋅− = −
⋅ ⋅
= −
−=
55. The LCD of 5, 4, and 2 is 20. 6 3 1 6 4 3 5 1 105 4 2 5 4 4 5 2 10
24 15 1020 20 201920
⋅ ⋅ ⋅− + = − +
⋅ ⋅ ⋅
= − +
=
56. The LCD of 5, 4, and 2 is 20. 6 3 1 6 4 3 5 1 105 4 2 5 4 4 5 2 10
24 15 1020 20 202920
⋅ ⋅ ⋅+ − = + −
⋅ ⋅ ⋅
= + −
=
57. The LCD of 5 and 9 is 45. 4 4 4 9 4 5 36 20 565 9 5 9 9 5 45 45 45
⋅ ⋅+ = + = + =
⋅ ⋅
58. The LCD of 12 and 24 is 24. 11 7 11 2 7 22 7 15 3 5 512 24 12 2 24 24 24 24 3 8 8
⋅ ⋅− = − = − = = =
⋅ ⋅
59. The LCD of 9x and 8 is 72x. 5 1 5 8 1 9
9 8 9 8 8 940 9
72 7240 9
72
xx x x
xx xxx
⋅ ⋅+ = +
⋅ ⋅
= +
+=
60. The LCD of 8 and 12x is 24x. 3 5 3 3 5 28 12 8 3 12 2
9 1024 249 10
24
xx x x
xx xxx
⋅ ⋅+ = +
⋅ ⋅
= +
+=
61. The LCD of 12, 24, and 6 is 24. 9 17 1 9 2 17 1 4
12 24 6 12 2 24 6 418 17 424 24 24524
⋅ ⋅− + − = − + −
⋅ ⋅
= − + −
= −
62. The LCD of 14, 7, and 2 is 14. 5 3 1 5 3 2 1 7
14 7 2 14 7 2 2 75 6 7
14 14 146
143 27 237
⋅ ⋅− + − = − + −
⋅ ⋅
= − + −
= −
⋅= −
⋅
= −
63. The LCD of 8, 7, and 14 is 56. 3 2 5 3 7 2 8 5 48 7 14 8 7 7 8 14 4
21 16 2056 56 5637 20
56
x x x x
x x
x
⋅ ⋅ ⋅+ − = + −
⋅ ⋅ ⋅
= + −
−=
64. The LCD of 10, 2, and 5 is 10. 9 1 9 1 5 210 2 5 10 2 5 5 2
9 5 210 10 1011 5
10
x x x x
x x
x
⋅ ⋅− + = − +
⋅ ⋅
= − +
−=
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65. The LCD of 7 and 10 is 70. Write each fraction as an equivalent fraction with a denominator of 70. 2 2 10 207 7 10 70
⋅= =
⋅
3 3 7 2110 10 7 70
⋅= =
⋅
Since 20 < 21, 20 21 ,70 70
< so 2 3 .7 10<
66. The LCD of 9 and 11 is 99. Write each fraction as an equivalent fraction with a denominator of 99. 5 5 11 559 9 11 99
⋅= =
⋅
6 6 9 5411 11 9 99
⋅= =
⋅
Since 55 > 54, 55 54 ,99 99
> so 5 6 .9 11>
67. A positive fraction is greater than a negative fraction. 5 136 15> −
68. The LCD of 8 and 6 is 48. Write each fraction as an equivalent fraction with a denominator of 48.
7 7 6 428 8 6 48
⋅− = − = −
⋅
5 5 8 406 6 8 48
⋅− = − = −
⋅
Since −42 < −40, 42 40 ,48 48
− < − so 7 5 .8 6
− < −
69. The LCD of 4 and 14 is 28. Write each fraction as an equivalent fraction with a denominator of 28.
3 3 7 214 4 7 28
⋅− = − = −
⋅
11 11 2 2214 14 2 28
⋅− = − = −
⋅
Since −21 > −22, 21 22 ,28 28
− > − so 3 11.4 14
− > −
70. The LCD of 9 and 13 is 117. Write each fraction as an equivalent fraction with a denominator of 117.
2 2 13 269 9 13 117
⋅− = − = −
⋅
3 3 9 2713 13 9 117
⋅− = − = −
⋅
Since −26 > −27, 26 27 ,
117 117− > − so
2 3 .9 13
− > −
71. The LCD of 3 and 4 is 12. 1 3 1 4 3 3 4 9 133 4 3 4 4 3 12 12 12
x y ⋅ ⋅+ = + = + = + =
⋅ ⋅
72. The LCD of 3 and 4 is 12. 1 3 1 4 3 3 4 9 53 4 3 4 4 3 12 12 12
x y ⋅ ⋅− = − = − = − = −
⋅ ⋅
73. 1 3 1 3 13 4 3 4 4
xy ⋅= ⋅ = =
⋅
74. 1 3 1 4 1 4 43 4 3 3 3 3 9
x y ⋅÷ = ÷ = ⋅ = =
⋅
75.
3 12 24 3
6 14 32 3 12 2 33 12 33 3 1 22 3 3 29 26 6116
y x ⎛ ⎞+ = +⎜ ⎟⎝ ⎠
= +
⋅= +
⋅
= +
⋅ ⋅= +
⋅ ⋅
= +
=
76. 1 32 23 4
2 33 42 4 3 33 4 4 38 9
12 121712
x y ⎛ ⎞+ = +⎜ ⎟⎝ ⎠
= +
⋅ ⋅= +
⋅ ⋅
= +
=
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174
77. The LCD of 3 and 5 is 15. 4 1 4 1 4 3 1 5 4 3 1 55 3 5 3 5 3 3 5 5 3 3 5
12 5 12 515 15 15 153415
+ + + = ⋅ + ⋅ + ⋅ + ⋅
= + + +
=
The perimeter is 3415
or 42
15 centimeters.
78. The LCD of 8 and 2 is 8. 3 5 1 3 5 1 48 8 2 8 8 2 4
3 5 48 8 81284 34 23 1 or 12 2
⋅+ + = + +
⋅
= + +
=
⋅=
⋅
=
The perimeter is 32
miles or 112
miles.
79. The LCD of 4, 5, and 2 is 20. 1 1 1 3 1 5 1 4 1 10 3 54 5 2 4 4 5 5 4 2 10 4 5
5 4 10 1520 20 20 20342017 210 217 7 or 110 10
⋅ ⋅ ⋅ ⋅+ + + = + + +
⋅ ⋅ ⋅ ⋅
= + + +
=
⋅=
⋅
=
The perimeter is 1710
meters or 71
10 meters.
80. The LCD of 21 and 7 is 21. 10 1 10 1 10 1 3 10 1 321 7 21 7 21 7 3 21 7 3
10 3 10 321 21 21 2126 5 or 121 21
⋅ ⋅+ + + = + + +
⋅ ⋅
= + + +
=
The perimeter is 2621
yards or 5121
yards.
81. The sum of a number and 12
translates as 1 .2
x +
82. A number increased by 25
− translates as
2 .5
x ⎛ ⎞+ −⎜ ⎟⎝ ⎠
83. A number subtracted from 38
− translates as
3 .8x− −
84. The difference of a number and 720
translates as
7 .20
x −
85. 3 3 16 3 3 10 51
16 16 16 16 16 16 8− − = − − = =
The inner diameter is 5 inch.8
86. 3 1 1 3 2 1 14 8 8 4 2 8 8
6 1 18 8 84812
⋅− − = − −
⋅
= − −
=
=
The inner diameter is 12
inch.
87. The LCD of 10 and 100 is 100. 17 1 17 1 10 17 10 7100 10 100 10 10 100 100 100
⋅− = − = − =
⋅
The sloth can travel 7
100 mph faster in trees.
88. The LCD of 4 and 264 is 264. 1 5 1 66 5 66 5 614 264 4 66 264 264 264 264
⋅− = − = − =
⋅
A killer bee will chase a person 61
264 mile
farther than a domestic honeybee.
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89. The LCD of 2, 16, and 32 is 32. 1 11 9 1 16 11 2 92 16 32 2 16 16 2 32
16 22 932 32 324732
+ + = ⋅ + ⋅ +
= + +
=
The total length is 47 inches.32
90. The LCD of 16 and 8 is 16. 11 5 11 11 5 2 1116 8 16 16 8 2 16
11 10 1116 16 1632162
+ + = + ⋅ +
= + +
=
=
The total width is 2 inches.
91. The LCD of 20 and 25 is 100. 13 4 13 5 4 4 65 16 4920 25 20 5 25 4 100 100 100
⋅ ⋅− = − = − =
⋅ ⋅
49100
of the American students ages 10 to 17
name math or science as their favorite subject in school.
92. The LCD of 100 and 50 is 100. 49 3 49 3 2 49 6 43
100 50 100 50 2 100 100 100⋅
− = − = − =⋅
43100
of the world’s mail is handled by the postal
service of the United States.
93. The LCD of 50 and 2 is 50. 13 1 13 1 2550 2 50 2 25
13 2550 5038502 192 251925
+ = + ⋅
= +
=
⋅=
⋅
=
The Pacific and Atlantic Oceans account for 1925
of the world’s water surface.
94. The LCD of 25 and 5 is 25. 1 1 1 1 5 1 5 625 5 25 5 5 25 25 25
+ = + ⋅ = + =
The Arctic and Indian Oceans account for 625
of
the world’s water surface.
95. The LCD of 25 and 10 is 50. 3 1 3 2 1 5 6 5 1125 10 25 2 10 5 50 50 50
⋅ ⋅+ = + = + =
⋅ ⋅
1150
of the beginning freshmen plan to major in
arts and humanities or education.
96. The LCD of 100 and 20 is 100. 3 3 3 3 5
100 20 100 20 53 15
100 10018
1002 92 50950
⋅+ = +
⋅
= +
=
⋅=
⋅
=
950
of the beginning freshmen plan to major in a
technical or professional field.
97. 4 25 4 21125 25 25 25
− = − =
2125
of the beginning freshmen do not plan to
major in business.
98. 1 10 1 91
10 10 10 10− = − =
910
of the beginning freshmen do not plan to
major in engineering.
99. −50 ÷ 5 ⋅ 2 = −10 ⋅ 2 = −20
100. 8 − 6 ⋅ 4 − 7 = 8 − 24 − 7 = −16 − 7 = −23
101. (8 − 6) ⋅ (4 − 7) = 2 ⋅ (−3) = −6
102. 50 ÷ (5 ⋅ 2) = 50 ÷ 10 = 5
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Chapter 4: Fractions and Mixed Numbers ISM: Prealgebra
176
103. a.
b. There seems to be an error.
c. 3 4 3 4 75 5 5 5
++ = =
14 7or not 10 10
⎛ ⎞⎜ ⎟⎝ ⎠
104. a.
b. There seems to be an error.
c. The LCD of 8 and 4 is 8. 5 3 5 3 2 5 6 18 4 8 4 2 8 8 8
⋅− = − = − = −
⋅
2not 4
⎛ ⎞⎜ ⎟⎝ ⎠
105. The LCD of 3, 4, and 540 is 540. 2 1 2 2 180 1 135 23 4 540 3 180 4 135 540
360 135 2540 540 540225 2540 540223540
− − = ⋅ − ⋅ −
= − −
= −
=
106. The LCD of 10, 200, and 3 is 600. 9 7 1 9 60 7 3 1 200
10 200 3 10 60 200 3 3 200540 21 200600 600 600519 200600 600319600
− − = ⋅ − ⋅ − ⋅
= − −
= −
=
107. The LCD of 55 and 1760 is 1760. 30 1000 30 32 100055 1760 55 32 1760
960 10001760 17601960176049 4044 404944
⋅+ = +
⋅
= +
=
⋅=
⋅
=
108. The LCD of 26 and 1352 is 1352. 19 968 19 52 96826 1352 26 52 1352
988 9681352 1352
201352
5 4338 4
5338
⋅− = −
⋅
= −
=
⋅=
⋅
=
109. answers may vary
110. 1; answers may vary
111. The LCD of 53 and 106 is 106. Write each fraction as an equivalent fraction with a denominator of 106. 24 24 2 4853 53 2 106
⋅= =
⋅
51106
Since 51 > 48, 51 48 ,
106 106> so
51 24 .106 53
>
Standard mail accounted for the greater portion of the mail handled by volume.
Integrated Review
1. 3 out of 7 equal parts are shaded: 37
2. Each part is 14
of a whole and there are 5 parts
shaded, or 1 whole and 1 more part: 5 1or 14 4
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ISM: Prealgebra Chapter 4: Fractions and Mixed Numbers
177
3. number that get fewer than 8 73total number of people 85
→→
7385
of people get fewer than 8 hours of sleep
each night.
4.
5. 11 111
= −−
6. 17 171=
7. 0 03=
−
8. 7 is undefined0
9. 65 = 5 ⋅ 13
10. 70 2 35
2 5 7
= ⋅↓ ↓⋅ ⋅
70 = 2 ⋅ 5 ⋅ 7
11. 315 3 105
3 3 35
3 3 5 7
= ⋅↓ ↓⋅ ⋅
↓ ↓ ↓⋅ ⋅ ⋅
2315 3 5 7= ⋅ ⋅
12. 441 3 147
3 3 49
3 3 7 7
= ⋅↓ ↓⋅ ⋅
↓ ↓ ↓⋅ ⋅ ⋅
2 2441 3 7= ⋅
13. 2 2 1 1
14 2 7 7⋅
= =⋅
14. 24 6 4 620 5 4 5
⋅= =
⋅
15. 56 14 4 1460 15 4 15
⋅− = − = −
⋅
16. 72 8 9 980 8 10 10
⋅− = − = −
⋅
17. 54 27 2 2135 27 5 5x x x⋅ ⋅= =
⋅
18. 90 30 3 3
240 30 8 8y y y⋅
= =⋅ ⋅
19. 3 2165 15 11 11 11
210 15 14 14 14z z z z z z zz z
⋅ ⋅ ⋅ ⋅ ⋅ ⋅= = =
⋅ ⋅
20. 2 3
2
245 35 735 11385
711
711
ab a ba a b b ba b
a b b
ab
⋅ ⋅ ⋅=
⋅ ⋅ ⋅ ⋅ ⋅ ⋅
=⋅ ⋅ ⋅
=
21. Not equivalent, since the cross products are not equal: 7 ⋅ 10 = 70, 8 ⋅ 9 = 72
22. Equivalent, since the cross products are equal: 10 ⋅ 18 = 180, 12 ⋅ 15 = 180
23. a. number not adjacent 2total number 50
→→
2 1 2 150 25 2 25
⋅= =
⋅ of the states are not
adjacent to any other states.
b. 50 − 2 = 48; 48 states are adjacent to other states.
c. 48 2 24 2450 2 25 25
⋅= =
⋅ of the states are adjacent
to other states.
24. a. number rated R 92total number 460
→→
92 92 1 1460 92 5 5
⋅= =
⋅ of the new films were
rated R.
b. 460 − 92 = 368; 368 of the new films were rated other than R.
c. 368 4 92 4460 5 92 5
⋅= =
⋅ of the new films were
rated other than R.
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Chapter 4: Fractions and Mixed Numbers ISM: Prealgebra
178
25. 5 = 5 6 = 2 ⋅ 3 LCM = 2 ⋅ 3 ⋅ 5 = 30
26. 2 = 2 14 = 2 ⋅ 7 LCM = 2 ⋅ 7 = 14
27. 6 = 2 ⋅ 3 18 = 2 ⋅ 3 ⋅ 3 30 = 2 ⋅ 3 ⋅ 3 ⋅ 5 LCM = 2 ⋅ 3 ⋅ 3 ⋅ 5 = 90
28. 7 7 4 7 4 289 9 4 9 4 36
⋅= ⋅ = =
⋅
29. 11 11 5 11 5 5515 15 5 15 5 75
⋅= ⋅ = =
⋅
30. 5 5 8 5 8 406 6 8 6 8 48
⋅= ⋅ = =
⋅
31. 1 3 1 3 45 5 5 5
++ = =
32. 1 3 1 3 2 25 5 5 5 5
− −− = = = −
33. 1 3 1 3 35 5 5 5 25
⋅⋅ = =
⋅
34. 1 3 1 5 1 5 15 5 5 3 5 3 3
⋅÷ = ⋅ = =
⋅
35. 2 5 2 6 2 6 2 2 3 2 2 43 6 3 5 3 5 3 5 5 5
⋅ ⋅ ⋅ ⋅÷ = ⋅ = = = =
⋅ ⋅
36. 2 5 2 5 2 5 5 53 6 3 6 3 2 3 3 3 9a a a
a a a⋅ ⋅ ⋅
⋅ = = = =⋅ ⋅ ⋅ ⋅ ⋅
37. The LCD of 3y and 6y is 6y. 2 5 2 2 5
3 6 3 2 62 2 5
3 2 64 5
6 61
6
y y y y
y y
y y
y
− = ⋅ −
⋅= −
⋅
= −
= −
38. The LCD of 3 and 6 is 6. 2 5 2 2 53 6 3 2 6
2 2 53 2 6
4 56 6
96
3 33 2
32
x x x x
x x
x x
x
x
x
+ = ⋅ +
⋅= +
⋅
= +
=
⋅ ⋅=
⋅
=
39. 1 7 1 7 17 18 7 18 18
⋅− ⋅ − = =
⋅
40. 4 3 4 3 4 3 4 49 7 9 7 3 3 7 3 7 21
⋅ ⋅− ⋅ − = = = =
⋅ ⋅ ⋅ ⋅
41. 2
2
2
2
7 7 668 8 1
7 18 67 1
8 67 1
8 67
48
z z zz
zz
zzzz z
z
− ÷ = − ÷
= − ⋅
⋅= −
⋅⋅ ⋅
= −⋅ ⋅ ⋅
= −
42. 9 9 5 9 1 9 1 95
10 10 1 10 5 10 5 50⋅
− ÷ = − ÷ = − ⋅ = − = −⋅
43. The LCD of 8 and 20 is 40. 7 1 7 5 1 2 35 2 378 20 8 5 20 2 40 40 40
⋅ ⋅+ = + = + =
⋅ ⋅
44. The LCD of 12 and 9 is 36. 5 1 5 3 1 4 15 4 11
12 9 12 3 9 4 36 36 36⋅ ⋅
− = − = − =⋅ ⋅
45. The LCD of 9, 18, and 3 is 18. 2 1 1 2 2 1 1 69 18 3 9 2 18 3 6
4 1 618 18 181118
⋅ ⋅+ + = + +
⋅ ⋅
= + +
=
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ISM: Prealgebra Chapter 4: Fractions and Mixed Numbers
179
46. The LCD of 10, 5, and 25 is 50. 3 6 3 5 10 6 210 5 25 10 5 5 10 25 2
15 10 1250 50 5025 1250 50
25 1250
y y y y
y y
y
y
⋅ ⋅ ⋅+ + = + +
⋅ ⋅ ⋅
= + +
= +
+=
47. 23
of a number translates as 23x⋅ or
2 .3x
48. The quotient of a number and 15
− translates as
1 .5
x ⎛ ⎞÷ −⎜ ⎟⎝ ⎠
49. A number subtracted from 89
− translates as
8 .9x− −
50. 6
11 increased by a number translates as
6 .11
x+
51. 2 2 153015303 3 1
2 15303 1
2 3 5103 1
2 5101
1020
⋅ = ⋅
⋅=
⋅⋅ ⋅
=⋅
⋅=
=
23
of 1530 is 1020.
52. 3 18 3184 1 4
18 41 3
18 41 33 6 4
1 36 41
24
÷ = ÷
= ⋅
⋅=
⋅⋅ ⋅
=⋅⋅
=
=
He can sell 24 lots.
53. The LCD of 8 and 16 is 16. 7 1 1 7 2 1 18 16 16 8 2 16 16
14 1 116 16 1612164 34 434
⋅− − = − −
⋅
= − −
=
⋅=
⋅
=
The inner diameter is 34
foot.
Section 4.6
Practice Problems
1. 71015
7 1 7 5 7 5 7 5 710 5 10 1 10 1 2 5 1 2
yy y y y y⋅ ⋅ ⋅
= ÷ = ⋅ = = =⋅ ⋅ ⋅
2.
131 1 12 6 2 3 63 3 32 2 44 3 4 3 3 4
3 16 69 8
12 12461
124 16 124 126 14 2 6
6 18 or 81
⋅⋅⋅ ⋅⋅ ⋅
+ +=
− −
+=
−
=
= ÷
= ⋅
⋅ ⋅=
⋅
=
3. The LCD of 2, 6, 4, and 3 is 12.
( )( )
1 11 12 62 6
3 2 3 24 3 4 3
1 12 63 24 3
12
12
12 12
12 126 29 88 or 81
++=
− −
⋅ + ⋅=
⋅ − ⋅+
=−
=
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Chapter 4: Fractions and Mixed Numbers ISM: Prealgebra
180
4. The LCD of 4, 5, and 1 is 20.
( )( )
33 344 4
5 55
20 20 154 201 20 20 120 1x xx x
⋅= = =
−− ⋅ − ⋅−
5. 32 8 8 54 462 2
3 27 27 27 27⎛ ⎞ − = − = − = −⎜ ⎟⎝ ⎠
6. 1 1 7 1 1 5 1 2 7 12 5 8 8 2 5 5 2 8 8
5 2 7 110 10 8 83 8
10 83 (1)
103
10
⋅ ⋅⎛ ⎞⎛ ⎞ ⎛ ⎞⎛ ⎞− + + = − + +⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⋅ ⋅⎝ ⎠⎝ ⎠ ⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞= − + +⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞= −⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠⎛ ⎞= −⎜ ⎟⎝ ⎠
= −
7. 3 3 3 25 5 10 3
3 3 25 5 2 33 15 545
xy− − = − − ⋅
⋅= − −
⋅ ⋅
= − −
= −
Vocabulary and Readiness Check
1. A fraction whose numerator or denominator or both numerator and denominator contain fractions is called a complex fraction.
2. To simplify 1 2 7 ,2 3 8
− + ⋅ which operation do we
perform first? multiplication
3. To simplify 1 2 7 ,2 3 8
− ÷ ⋅ which operation do we
perform first? division
4. To simplify 7 1 2 ,8 2 3⎛ ⎞⋅ −⎜ ⎟⎝ ⎠
which operation do we
perform first? subtraction
5. To simplify 21 1 9 3 ,
3 4 11 8⎛ ⎞÷ ⋅ +⎜ ⎟⎝ ⎠
which operation
do we perform first? addition
6. To simplify 239 ,
4⎛ ⎞− −⎜ ⎟⎝ ⎠
which operation do we
perform first? evaluate the exponential expression
Exercise Set 4.6
1. 1834
1 3 1 4 1 4 1 4 18 4 8 3 8 3 2 4 3 6
⋅ ⋅= ÷ = ⋅ = = =
⋅ ⋅ ⋅
2. 5
121512
5 15 5 12 5 12 1 5 12 112 12 12 15 12 15 12 5 3 3
⋅ ⋅ ⋅= ÷ = ⋅ = = =
⋅ ⋅ ⋅
3. 2327
2 2 2 7 2 7 73 7 3 2 3 2 3
⋅= ÷ = ⋅ = =
⋅
4. 925625
9 625 259 2525 69 2525 63 3 2525 2 332
= ÷
= ⋅
⋅=
⋅⋅ ⋅
=⋅ ⋅
=
5. 22749
2 4 2 9 2 9 2 927 9 27 4 27 4 3 9 2 2 6
x x x x x x⋅ ⋅ ⋅= ÷ = ⋅ = = =
⋅ ⋅ ⋅ ⋅
6. 31112
3 1 3 2 3 2 611 2 11 1 11 1 11
yy y y y⋅
= ÷ = ⋅ = =⋅
7. The LCD of 4, 5, and 2 is 20.
( )( )
3 23 24 54 5
31 312 5 2 5
3 24 5
312 5
20
20
20 20
20 2015 810 122322
⋅ ++=
+ ⋅ +
⋅ + ⋅=
⋅ + ⋅+
=+
=
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ISM: Prealgebra Chapter 4: Fractions and Mixed Numbers
181
8. The LCD of 6, 3, 2, and 9 is 18.
( )( )
7 27 26 36 3
3 8 3 82 9 2 9
7 26 33 82 9
18
18
18 18
18 1821 1227 1633113
⋅ ++=
− ⋅ −
⋅ + ⋅=
⋅ − ⋅+
=−
=
=
9. The LCD of 4, 1, and 8 is 8.
( )( )
3344
1 18 8
18
8
5 8 56
8 5 86
40 16392 33 13
213
xx
x
x
x
x
x
⋅=
− ⋅ −
=⋅ − ⋅
=−
=
⋅ ⋅=
⋅
=
10. The LCD of 10, 1, and 5y is 10y.
( )331010
2 25 5
310
10 22
10
10 10 2
43 20
423
4
y yy
y
y
y y
y y
y
⋅ ++=
⋅
⋅ + ⋅=
+=
=
11. 1 1 1 1 1 15 3 4 5 3 4
1 15 121 12 1 55 12 12 512 560 601760
⋅+ ⋅ = +
⋅
= +
⋅ ⋅= +
⋅ ⋅
= +
=
12.
1 1 1 1 1 12 6 3 2 6 3
1 12 181 9 12 9 189 1
18 1810182 52 959
⋅+ ⋅ = +
⋅
= +
⋅= +
⋅
= +
=
⋅=
⋅
=
13. 5 1 1 5 3 1 5 3 1 5 56 3 4 6 1 4 2 3 1 2 2 2 2 2 8
⋅ ⋅÷ ⋅ = ⋅ ⋅ = = =
⋅ ⋅ ⋅ ⋅ ⋅ ⋅
14. 7 1 1 7 4 1 7 4 1 18 4 7 8 1 7 4 2 1 7 2
⋅ ⋅÷ ⋅ = ⋅ ⋅ = =
⋅ ⋅ ⋅
15. 2
2 1 12 43 9
4 11 94 9 11 9 936 19 9
359
⎛ ⎞− = −⎜ ⎟⎝ ⎠
= −
⋅= −
⋅
= −
=
16. 2
2 1 13 92 4
9 11 49 4 11 4 436 14 4
354
⎛ ⎞− = −⎜ ⎟⎝ ⎠
= −
⋅= −
⋅
= −
=
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Chapter 4: Fractions and Mixed Numbers ISM: Prealgebra
182
17.
2 4 1 9 2 4 1 10 9 39 9 3 10 9 3 10 10 3
6 10 279 30 306 179 306 179 30
6 173 3 5 6
173 3 51745
+⎛ ⎞⎛ ⎞ ⎛ ⎞⎛ ⎞+ − = ⋅ − ⋅⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠⎝ ⎠
⎛ ⎞⎛ ⎞= −⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
−⎛ ⎞⎛ ⎞= ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
⋅= −
⋅⋅
= −⋅ ⋅ ⋅
= −⋅ ⋅
= −
18. 1 1 1 1 1 2 1 1 2 15 10 5 10 5 2 10 5 2 10
2 1 2 110 10 10 101 3
10 101 3
10 103
100
⎛ ⎞⎛ ⎞ ⎛ ⎞⎛ ⎞− + = ⋅ − ⋅ +⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠⎝ ⎠
⎛ ⎞⎛ ⎞= − +⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞= ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠⋅
=⋅
=
19.
7 1 3 7 1 4 38 2 11 8 2 4 11
7 4 38 8 11
3 38 113 118 33 118 3
118
⎛ ⎞ ⎛ ⎞− ÷ = − ⋅ ÷⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
⎛ ⎞= − ÷⎜ ⎟⎝ ⎠
= ÷
= ⋅
⋅=
⋅
=
20. 2 7 4 9 43 3 9 3 9
3 41 93 91 43 91 4274
⎛ ⎞− − ÷ = − ÷⎜ ⎟⎝ ⎠
= − ÷
= − ⋅
⋅= −
⋅
= −
21.
1 1 1 5 1 42 2 2 24 5 4 5 5 4
5 42 220 2092 220
2 9 21 20
2 9 21 2 109 2 10
10 1 109 20
10 102910
⋅ ⋅⎛ ⎞ ⎛ ⎞⋅ + + = ⋅ + +⎜ ⎟ ⎜ ⎟⋅ ⋅⎝ ⎠ ⎝ ⎠⎛ ⎞= + +⎜ ⎟⎝ ⎠⎛ ⎞= +⎜ ⎟⎝ ⎠
= ⋅ +
⋅= +
⋅ ⋅⋅
= +⋅
= +
=
22.
2 1 2 5 2 15 1 15 2 5 1 2 2
2 10 1 15 2 22 9 15 22 9 15 29 55 545
⋅⎛ ⎞ ⎛ ⎞⋅ − − = ⋅ − −⎜ ⎟ ⎜ ⎟⋅⎝ ⎠ ⎝ ⎠⎛ ⎞= ⋅ − −⎜ ⎟⎝ ⎠⎛ ⎞= ⋅ −⎜ ⎟⎝ ⎠⋅
= −⋅
= −
=
23.
2 2
2
2
3 3 1 3 3 3 14 4 12 4 4 3 12
3 9 14 12 123 84 12
9 816 129 12
16 89 3 44 4 82732
⋅⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞÷ − = ÷ −⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⋅⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
⎛ ⎞ ⎛ ⎞= ÷ −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
⎛ ⎞= ÷⎜ ⎟⎝ ⎠
= ÷
= ⋅
⋅ ⋅=
⋅ ⋅
=
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ISM: Prealgebra Chapter 4: Fractions and Mixed Numbers
183
24.
2 2
2
2
8 2 8 2 3 229 3 9 1 3 3
8 6 29 3 38 49 3
64 481 364 381 416 4 327 3 41627
⋅⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞÷ − = ÷ −⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⋅⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
⎛ ⎞ ⎛ ⎞= ÷ −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
⎛ ⎞= ÷⎜ ⎟⎝ ⎠
= ÷
= ⋅
⋅ ⋅=
⋅ ⋅
=
25. 2 2
2
2
2 3 2 2 35 10 5 2 10
4 310 101
101 1
10 101
100
⎛ ⎞ ⎛ ⎞− = ⋅ −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
⎛ ⎞= −⎜ ⎟⎝ ⎠
⎛ ⎞= ⎜ ⎟⎝ ⎠
= ⋅
=
26. 3 3
3
3
3 4 3 3 4 22 3 2 3 3 2
9 86 6161 1 16 6 61
216
⋅ ⋅⎛ ⎞ ⎛ ⎞− = −⎜ ⎟ ⎜ ⎟⋅ ⋅⎝ ⎠ ⎝ ⎠
⎛ ⎞= −⎜ ⎟⎝ ⎠
⎛ ⎞= ⎜ ⎟⎝ ⎠⎛ ⎞ ⎛ ⎞ ⎛ ⎞= ⋅ ⋅⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠
=
27.
2 2
2
2
3 1 1 1 3 2 1 1 4 14 8 2 8 4 2 8 2 4 8
6 1 4 18 8 8 87 58 8
49 564 849 5 864 8 849 4064 64964
⋅ ⋅⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ − + = + − +⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⋅ ⋅⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
⎛ ⎞ ⎛ ⎞= + − +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
⎛ ⎞= −⎜ ⎟⎝ ⎠
= −
⋅= −
⋅
= −
=
28.
3 2 3 2
3 2
3 2
3 2
1 1 2 3 1 1 2 2 36 3 5 4 6 3 2 5 4
1 2 66 6 203 66 201 32 10
1 98 1001 25 9 28 25 100 225 18
200 20043
200
⋅ ⋅⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ + ⋅ = + +⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⋅ ⋅⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
⎛ ⎞ ⎛ ⎞= + +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
⎛ ⎞ ⎛ ⎞= +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
⎛ ⎞ ⎛ ⎞= +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
= +
⋅ ⋅= +
⋅ ⋅
= +
=
29. 2 55 55 6526
2 6 51 6 612 56 676
y z ⎛ ⎞− = −⎜ ⎟⎝ ⎠
= −
= ⋅ −
= −
=
30. 5 1 5 1 62 2 26 3 3 3 3
z x ⎛ ⎞ ⎛ ⎞− = − − = + = =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
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Chapter 4: Fractions and Mixed Numbers ISM: Prealgebra
184
31. 13
56
1 5 1 6 1 3 2 23 6 3 5 3 5 5
xz
− ⋅ ⋅= = − ÷ = − ⋅ = − = −
⋅
32. ( )
( )( )
2 15 3
562 15 3
56
30
3012 ( 10)
25225
y xz
+ −+=
⎡ ⎤+ −⎣ ⎦=
+ −=
=
33.
22 1 2 5
3 5 61 1 2 53 3 5 2 3
1 19 31 1 39 3 31 39 9
2929
x yz ⎛ ⎞ ⎛ ⎞⎛ ⎞− = − −⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠
⋅⎛ ⎞⎛ ⎞= − − −⎜ ⎟⎜ ⎟ ⋅ ⋅⎝ ⎠⎝ ⎠
= −
= − ⋅
= −
−=
= −
34. 1 5(1 )(1 ) 1 13 6
3 1 6 53 3 6 62 113 62 11
3 2 3119
x z ⎡ ⎤⎛ ⎞ ⎡ ⎤+ + = + − +⎜ ⎟⎢ ⎥ ⎢ ⎥⎝ ⎠ ⎣ ⎦⎣ ⎦⎛ ⎞⎛ ⎞= − +⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞= ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠⋅
=⋅ ⋅
=
35. 5241
12
5 1 5 12 5 12 524 12 24 1 12 2 1 2
a a a a a⋅ ⋅= ÷ = ⋅ = =
⋅ ⋅
36. 7
101425
7 1410 257 25
10 147 5 5
5 2 7 25
2 254
zz
z
z
z
z
= ÷
= ⋅
⋅ ⋅=
⋅ ⋅ ⋅ ⋅
=⋅ ⋅
=
37. 3 33 1 27 1 28 4 7 7
2 2 8 8 8 4 2 2⋅⎛ ⎞ ⎛ ⎞+ = + = = =⎜ ⎟ ⎜ ⎟ ⋅⎝ ⎠ ⎝ ⎠
38. 5 1 1 1 5 2 1 121 2 7 3 21 1 7 3
5 2 1 121 1 7 310 121 211121
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞÷ + ⋅ = ⋅ + ⋅⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
⋅ ⋅= +
⋅ ⋅
= +
=
39. 21 1 1 1 1 5 1 4 5 4 9
2 5 4 5 4 5 5 4 20 20 20⋅ ⋅⎛ ⎞− + = + = + = + =⎜ ⎟ ⋅ ⋅⎝ ⎠
40. 23 3 9 3 9 3 2 9 6 15
4 8 16 8 16 8 2 16 16 16⋅⎛ ⎞− + = + = + = + =⎜ ⎟ ⋅⎝ ⎠
41. ( )( )
1166
4 43 3
1643
6 22
1 6 1
6 2 6
6 1 612 16 813
2132
++=
− −
⋅ + ⋅=
⋅ − ⋅+
=−
=−
= −
42. ( )( )
11 122 2
1 115 55
10 33 10 3 10 30 5 2540 2 424 10 4 1010 4
−− ⋅ − ⋅ −= = = =
++ ⋅ + ⋅+
43. 2 2 22 5 2 3 3 3 91
5 5 5 5 5 5 25⎛ ⎞ ⎛ ⎞ ⎛ ⎞− = − = = ⋅ =⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠
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ISM: Prealgebra Chapter 4: Fractions and Mixed Numbers
185
44. 2 21 3 1 1 3 3
2 4 2 2 4 41 94 161 4 94 4 164 9
16 165
16
⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞⎛ ⎞− − = − − −⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠⎝ ⎠
= −
⋅= −
⋅
= −
= −
45. 3 1 1 3 4 1 414 8 2 4 4 8 8
1 54 8
1 54 85
32
⎛ ⎞⎛ ⎞ ⎛ ⎞⎛ ⎞− + = − +⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠⎝ ⎠
⎛ ⎞⎛ ⎞= −⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
⋅= −
⋅
= −
46. 1 3 1 1 2 3 1 51
10 20 5 10 2 20 5 52 3 1 5
20 20 5 55 420 5
5 45 4 515
⋅⎛ ⎞⎛ ⎞ ⎛ ⎞⎛ ⎞+ − = + −⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⋅⎝ ⎠⎝ ⎠ ⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞= + −⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞= −⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
⋅= −
⋅ ⋅
= −
47. 4 4
4
2 7 99 9 9
( 1)( 1)( 1)( 1)( 1)1
⎛ ⎞ ⎛ ⎞− − = −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
= −= − − − −=
48. 2 2
2
2
5 2 5 2 39 3 9 3 3
5 69 9
191 19 9
181
⋅⎛ ⎞ ⎛ ⎞− = −⎜ ⎟ ⎜ ⎟⋅⎝ ⎠ ⎝ ⎠
⎛ ⎞= −⎜ ⎟⎝ ⎠
⎛ ⎞= −⎜ ⎟⎝ ⎠⎛ ⎞⎛ ⎞= − −⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
=
49.
( )( )
51513 63 6
3 1 3 14 2 4 2
513 63 14 2
12
12
12 12
12 124 109 66
152 33 525
−−=
+ +
⋅ − ⋅=
⋅ + ⋅−
=+
−=
⋅= −
⋅
= −
50. ( )( )
7 17 110 210 2
34 345 4 5 4
7 110 2
345 4
20
20
20 20
20 2014 1016 152431
++=
+ +
⋅ + ⋅=
⋅ + ⋅+
=+
=
51.
3 6 3 6 3 5 3 64 5 4 5 4 6 4 5
3 5 3 64 6 4 53 5 3 2 3
4 2 3 2 2 55 98 105 5 9 48 5 10 425 3640 40
1140
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞÷ − ⋅ = ⋅ − ⋅⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
⋅ ⋅= −
⋅ ⋅⋅ ⋅ ⋅
= −⋅ ⋅ ⋅ ⋅
= −
⋅ ⋅= −
⋅ ⋅
= −
= −
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Chapter 4: Fractions and Mixed Numbers ISM: Prealgebra
186
52.
1 2 1 2 1 2 1 72 7 2 7 2 7 2 2
1 2 1 72 7 2 22 7
14 42 2 7 7
14 2 4 74 4928 28
4528
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⋅ − ÷ = ⋅ − ⋅⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
⋅ ⋅= −
⋅ ⋅
= −
⋅ ⋅= −
⋅ ⋅
= −
= −
53. ( )( )
331 13 3
313
3 22
5 3 5
3 3 2
3 5 36
15 16
16
xx
x
x
x
++=
+ +
⋅ + ⋅=
⋅ + ⋅+
=++
=
54. ( )( )
443 38 8
438
8 11
2 8 2
8 1 8
8 2 88 216 38 2
19
xx
x
x
x
⋅ −−=
+ ⋅ +
⋅ − ⋅=
⋅ + ⋅−
=+−
=
55. 1 3 2 1 6 1 7 13 or 32 1 2 2 2 2 2 2
+ = ⋅ + = + =
56. 2 2 3 2 6 2 8 22 or 23 1 3 3 3 3 3 3
+ = ⋅ + = + =
57. 5 9 6 5 54 5 49 19 or 86 1 6 6 6 6 6 6
− = ⋅ − = − =
58. 1 4 5 1 20 1 19 44 or 35 1 5 5 5 5 5 5
− = ⋅ − = − =
59. no; answers may vary
60. yes; answers may vary
61. ( )313 31 1
2 42 4 2 44 4 4 2 3 5
2 4(2) 4 2 8 8
++ ⋅ + ⋅ += = = =
⋅
62.
( )3 93 95 105 10
3 95 10
10
2 10(2)10 10
10 26 920
15205 35 434
++=
⋅ + ⋅=
⋅+
=
=
⋅=
⋅
=
63. ( )1 21 2 1 2
4 144 14 4 1428 28 28
2 28(2) 28 27 456
1156
++ ⋅ + ⋅+ =
⋅+
=
=
64. ( )5 75 7
6 96 9
5 76 9
18
2 18(2)18 18
18 215 14
362936
++=
⋅ + ⋅=
⋅+
=
=
65. The average of a and b should be halfway between a and b.
66. False; the average cannot be greater than the greatest number.
67. False; the average cannot be less than the smallest number.
68. True
69. False, if the absolute value of the negative fraction is greater than the absolute value of the positive fraction, the sum is negative.
70. True
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ISM: Prealgebra Chapter 4: Fractions and Mixed Numbers
187
71. True
72. addition; answers may vary
73. No; answers may vary.
74. Subtraction, multiplication, addition, division
75. Addition, multiplication, subtraction, division
76. Division, multiplication, subtraction, addition
77. Subtraction, multiplication, division, addition
78.
( )( )
( )
34
47
34
47
34
47
22
28 2
28
28 2 28
2856 21
167716
xy
++=
−
+=
−
⋅ + ⋅=
⋅ −
+=
−
= −
79.
3 44 44 7
4 3 41 4 712 44 7
12 7 4 44 7 7 484 1628 2868284 174 7
177
x y ⎛ ⎞+ = ⋅ + −⎜ ⎟⎝ ⎠⎛ ⎞= ⋅ + −⎜ ⎟⎝ ⎠
= −
⋅ ⋅= −
⋅ ⋅
= −
=
⋅=
⋅
=
80. 2
2 3 47 74 7
9 416 19 4 16
16 1 169 64
16 165516
x y ⎛ ⎞ ⎛ ⎞+ = + −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
= −
⋅= −
⋅
= −
= −
81.
( )( )
( )
9 914 14
3 44 7
914
3 44 7
3 44 7
28
2818
28 2818
21 16185
x y=
+ + −
=−
=⋅ − ⋅
=−
=
Section 4.7
Practice Problems
1.
112345 2 3 4 50
4q 2!Ωq
2. 2 11 5 11 5 11 11 21 or 13 15 3 15 3 5 3 9 9
⋅⋅ = ⋅ = =
⋅ ⋅
3. 5 5 18 5 18 5 6 3 1518 or 156 6 1 6 1 6 1 1
⋅ ⋅ ⋅⋅ = ⋅ = = =
⋅ ⋅
4. 1 3 16 11 16 11 4 4 11 44 43 2 or 85 4 5 4 5 4 5 4 5 5
⋅ ⋅ ⋅⋅ = ⋅ = = =
⋅ ⋅
Estimate: 135
rounds to 3, 324
rounds to 3.
3 ⋅ 3 = 9, so the answer is reasonable.
5. 7 3 97 3 97 97 23 6 or 19
15 1 15 1 5 3 5 5⋅
⋅ = ⋅ = =⋅ ⋅
Estimate: 76
15 rounds to 6 and 3 ⋅ 6 = 18, so the
answer is reasonable.
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Chapter 4: Fractions and Mixed Numbers ISM: Prealgebra
188
6. 4 4 7 4 1 4 1 479 9 1 9 7 9 7 63
⋅÷ = ÷ = ⋅ = =
⋅
7. 8 4 8 193
15 5 15 58 5
15 198 5
15 198 5
5 3 198
57
÷ = ÷
= ⋅
⋅=
⋅⋅
=⋅ ⋅
=
8. 2 3 23 313 27 14 7 14
23 147 3123 7 2
7 3146 15 or 131 31
÷ = ÷
= ⋅
⋅ ⋅=
⋅
=
9. 126245
+
52
301243017630
+
Estimate: 126
rounds to 2, 245
rounds to 4, and
2 + 4 = 6, so the answer is reasonable.
10. 53
14627
+
53
141221417514
+
3 35 1 614 14
= + =
11. 12637125
+
12303357235371735
+
2 217 1 1835 35
= + =
12. 7329516
18−
143218516
18916
18
−
1162
=
13. 79
15345
−
79
1594
15−
2281594
1513415
−
14. 252109
−
924921097149
−
15. 1234519
12−
323
12519
12−
152212519
1210312
−
536
=
The girth of the largest known American beech
tree is 536
feet larger than the girth of the largest
known sugar maple tree.
16. 1 2244 3 447 7
44 71 222 22 71 22
14 or 141
÷ = ÷
= ⋅
⋅ ⋅=
⋅
=
Therefore, 14 dresses can be made from 44 yards.
17. 3 7 9 3 6697 7 7
⋅ +− = − = −
18. 10 11 5 10 65511 11 11
⋅ +− = − = −
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ISM: Prealgebra Chapter 4: Fractions and Mixed Numbers
189
19. 37 548 8
− = − 4
8 3732
5
20. 46 195 5
− = − 9
5 4645
1
21. 3 3 11 182 34 5 4 5
11 184 5
11 2 92 2 599 9 or 910 10
⎛ ⎞ ⎛ ⎞⋅ − = ⋅ −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
⋅= −
⋅⋅ ⋅
= −⋅ ⋅
= − −
22. 2 1 30 54 17 4 7 4
30 47 5
30 47 5
5 6 47 5
24 3 or 37 7
− ÷ = − ÷
= − ⋅
⋅= −
⋅⋅ ⋅
= −⋅
= − −
23. 3124263
−
912
1286
1216
12
−
2 3 16 12 63 4 12− = −
24. 297
113014
+
49
14113014153914
+
14014
=
2 11 19 30 407 14 14
− − = −
Calculator Explorations
1. 5 28025
11 11=
2. 14 10196715 15
=
3. 31 377610735 35
=
4. 17 392318621 21
=
5. 365 12614 14
=
6. 290 42213 13
=
7. 2769 39230 10
=
8. 3941 1423117 17
=
Vocabulary and Readiness Check
1. The number 354
is called a mixed number.
2. For 35 ,4
the 5 is called the whole number part
and 34
is called the fraction part.
3. To estimate operations on mixed numbers, we round mixed numbers to the nearest whole number.
4. The mixed number 528
written as an improper
fraction is 21.8
Exercise Set 4.7
1.
2s a √
112345 2 3 4 50
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Chapter 4: Fractions and Mixed Numbers ISM: Prealgebra
190
2. 1~ 3~~
112345 2 3 4 50
3. 3R 1aa
112345 2 3 4 50
4. 4a ≈Ï
112345 2 3 4 50
5. 11212
rounds to 3.
114
rounds to 1.
3 ⋅ 1 = 3 The best estimate is b.
6. 156
rounds to 5.
537
rounds to 4.
5 ⋅ 4 = 20 The best estimate is d.
7. 212
11 rounds to 12.
9310
rounds to 4.
12 ÷ 4 = 3 The best estimate is a.
8. 320
14 rounds to 20.
8411
rounds to 5.
20 ÷ 5 = 4 The best estimate is c.
9. 2 1 8 1 823 7 3 7 21⋅ = ⋅ =
10. 5 1 5 21 5 3 7 7 14 or 29 5 9 5 3 3 5 3 3
⋅ ⋅⋅ = ⋅ = =
⋅ ⋅
11. 3 7 8 7 5 7 5 35 37 1 or 45 1 5 1 8 1 8 8 8
⋅÷ = ÷ = ⋅ = =
⋅
12. 2 9 5 9 3 9 3 27 29 1 or 53 1 3 1 5 1 5 5 5
⋅÷ = ÷ = ⋅ = =
⋅
13. Exact: 1 1 11 7 77 72 3 or 75 2 5 2 10 10⋅ = ⋅ =
Estimate: 125
rounds to 2, 132
rounds to 4.
2 ⋅ 4 = 8 so the answer is reasonable.
14. Exact: 1 1 9 57 513 12 7 or 164 8 4 8 32 32⋅ = ⋅ =
Estimate: 124
rounds to 2, 178
rounds to 7.
2 ⋅ 7 = 14
15. Exact: 4 2 19 44 836 313 6 or 235 7 5 7 35 35⋅ = ⋅ =
Estimate: 435
rounds to 4, 267
rounds to 6.
4 ⋅ 6 = 24
16. Exact: 5 3 35 38 7 5 2 19 133 15 7 or 446 5 6 5 2 3 5 3 3
⋅ ⋅ ⋅⋅ = ⋅ = =
⋅ ⋅
Estimate: 556
rounds to 6, 375
rounds to 8.
6 ⋅ 8 = 48
17. 1 5 5 25 15 2 or 122 1 2 2 2
⋅ = ⋅ =
18. 1 6 10 2 3 10 206 3 203 1 3 1 3 1
⋅ ⋅⋅ = ⋅ = = =
⋅
19. 2 1 11 3 11 3 11 13 1 or 53 2 3 2 3 2 2 2
⋅⋅ = ⋅ = =
⋅
20. 4 5 14 21 2 7 21 147 72 2 or 75 8 5 8 5 2 4 20 20
⋅ ⋅⋅ = ⋅ = =
⋅ ⋅
21. 2 1 8 7 56 22 or 183 7 3 1 3 3÷ = ⋅ =
22. 5 1 5 21 5 5 5 5 2549 5 9 5 9 21 9 21 189
⋅÷ = ÷ = ⋅ = =
⋅
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ISM: Prealgebra Chapter 4: Fractions and Mixed Numbers
191
23. 738
rounds to 4.
125
rounds to 2.
4 + 2 = 6 The best estimate is a.
24. 738
rounds to 4.
125
rounds to 2.
4 − 2 = 2 The best estimate is d.
25. 183
rounds to 8.
112
rounds to 2.
8 + 2 = 10 The best estimate is b.
26. 183
rounds to 8.
112
rounds to 2.
8 − 2 = 6 The best estimate is c.
27. Exact: 74
1212
128 26 6
12 3
+
=
Estimate: 74
12 rounds to 5,
1212
rounds to 2.
5 + 2 = 7 so the answer is reasonable.
28. Exact: 47
1123
11610
11
+
Estimate: 47
11 rounds to 7,
2311
rounds to 3.
7 + 3 = 10 so the answer is reasonable.
29. Exact: 310
14437
+
310
1483
14111314
+
Estimate: 310
14 rounds to 10,
437
rounds to 4.
10 + 4 = 14 so the answer is reasonable.
30. Exact: 512
12146
+
512
1224
12716
12
+
Estimate: 512
12 rounds to 12,
146
rounds to 4.
12 + 4 = 16 so the answer is reasonable.
31. 1952825
+
5925282571725
+
32. 26
137826
+
46
267826111426
+
33. 312
1410
52512
+
181284
10352584534784
+
34. 289
3210921
+
14863
3230963444963
+
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Chapter 4: Fractions and Mixed Numbers ISM: Prealgebra
192
35. 4157
11914
+
815
141191419 5 524 24 1 2514 14 14
+
= + =
36. 323588
15+
923
1588
1517 2 231 31 1 3215 15 15
+
= + =
37. 538126374
+
1532442
241872437 13 1312 12 1 1324 24 24
+
= + =
38. 143295136
+
104301293053
3027 916 1630 10
+
=
39. Exact: 74
1012
10−
6 32 2
10 5=
Estimate: 74
10 rounds to 5,
1210
rounds to 2.
5 − 2 = 3 so the answer is reasonable.
40. Exact: 479239249
−
Estimate: 479
rounds to 7, 239
rounds to 3.
7 − 3 = 4 so the answer is reasonable.
41. Exact: 131014437
−
13101483
14−
57
14
Estimate: 131014
rounds to 11, 437
rounds to 4.
11 − 4 = 7 so the answer is reasonable.
42. Exact: 512
12146
−
512
1224
12−
3 18 8
12 4=
Estimate: 512
12 rounds to 12,
146
rounds to 4.
12 − 4 = 8 so the answer is reasonable.
43. 1956825
−
59256825
−
308256825
−
2425
44. 25
137426
−
45267426
−
3042674262326
−
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ISM: Prealgebra Chapter 4: Fractions and Mixed Numbers
193
45. 6429
−
959429539
−
46. 871
10−
1071071
1036
10
−
47. 1636547
12−
263
12547
12−
146212547
129 315 15
12 4
−
=
48. 286
15327
10−
486
30927
30−
3485309273025 558 5830 6
−
=
49. 32411443 3 1 44
+
= + =
50. 55832887 7 1 88
+
= + =
51. 4157
11914
−
815
1411914
−
2214141191411514
−
52. 323588
15−
923
1588
15115
15
−
53. 1 28 2 56 23 2 69 9 1 9 9⋅ = ⋅ = =
54. 1 9 3 27 14 3 132 2 1 2 2⋅ = ⋅ = =
55. 2 1 5 11 5 5 251 23 5 3 5 3 11 33÷ = ÷ = ⋅ =
56.
1 1 26 135 35 4 5 4
26 45 132 13 45 13
2 45
85315
÷ = ÷
= ⋅
⋅ ⋅=
⋅⋅
=
=
=
57. 4 5 8 5 1322 13 22 13 359 18 18 18 18+ = + =
58. 3 2 3 10 28 10 1815 5 15 5 14 5 925 5 25 25 25 25 25
− = − = − =
59. 2 1 4 1 3 15 3 5 3 2 23 6 6 6 6 2− = − = =
60. 3 13 6 13 22 13 95 2 5 2 4 2 28 16 16 16 16 16 16− = − = − =
61. 1155320
10237
15+
615
3092030437
30197230
+
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Chapter 4: Fractions and Mixed Numbers ISM: Prealgebra
194
62. 377
151202
+
67
1415
72014134214
+
63. 467
11514
−
86
1411514
−
22514115141114
−
64. 547
12192324
−
104724192324
−
34462419232415 523 2324 8
−
=
65. 2 3 30 134 17 10 7 10
30 137 103 10 13
7 10397457
⋅ = ⋅
⋅=
⋅⋅ ⋅
=⋅
=
=
66. 2 3 20 11 20 11 4 5 11 55 16 2 183 4 3 4 3 4 3 4 3 3
⋅ ⋅ ⋅⋅ = ⋅ = = = =
⋅ ⋅
67. 26
11310433
+
6633
310433161333
+
68. 73
16162398
+
73
1686
1669
1621 5 518 18 1 1916 16 16
+
= + =
69. 257
− decreased by a number translates as
25 .7x− −
70. The sum of 384
and a number translates as
38 .4x+
71. Multiply 91
10 by a number translates as
91 .10
x⋅
72. Divide a number by 1611
− translates as
16 .11
x ⎛ ⎞÷ −⎜ ⎟⎝ ⎠
73. Subtract the standard gauge in the U.S. from the standard gauge in Spain.
965101562
−
965
10556
104 29 9
10 5
−
=
Spain’s standard gauge is 295
inches wider than
the U.S. standard gauge.
74. Subtract the standard gauge in Portugal from the standard gauge in Spain.
96510116520
−
186520116520720
−
Spain’s standard gauge is 720
inch wider than
that of Portugal.
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195
75. 1114335
−
5112012320
−
2510201232013720
−
Tucson gets an average of 13720
inches more
rain than Yuma.
76. 435418
−
84285418318
−
The short crutch should be lengthened 318
inches.
77. 3 2 7 2 7 2 7 72 14 1 4 1 4 1 2 2 2
⋅ ⋅⋅ = ⋅ = = =
⋅ ⋅ ⋅ or
132
The area is 72
or 132
square yards.
78. 1 5 7 5 7 35 15 3 or 172 1 2 1 2 2 2
⋅⋅ = ⋅ = =
⋅
The area is 352
or 1172
square inches.
79. 3 1 3 5 3 5 1514 4 4 4 4 4 16
⋅⋅ = ⋅ = =
⋅
The area is 1516
square inch.
80. 1 1 9 19 3 3 19 57 14 6 or 282 3 2 3 2 3 2 2
⋅ ⋅⋅ = ⋅ = =
⋅
The area is 572
or 1282
square yards.
81. 153
5778
3+
8524
521724
329 5 520 20 1 2124 24 24
+
= + =
The perimeter is 52124
meters.
82. 1 13 5 65 13 5 164 4 1 4 4⋅ = ⋅ = =
The perimeter is 1164
yards.
83. 2 1 1 2 1 215 3 2 15 3 23 4 2 3 4 4
2 315 53 4
⎛ ⎞ ⎛ ⎞− + = − +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
= −
2153354
−
815
1295
12−
20141295
1211912
−
No; the remaining board is 11912
feet which is
112
foot short.
84. 6338
−
858338528
−
The remaining piece is 528
feet.
85. 3 51 4 51 1 5112 44 4 1 4 4 16÷ = ÷ = ⋅ =
The patient walked 51 3316 16
= miles per day.
86. 3 1 111 1 111 4 11127 1114 4 4 4 4 1 1÷ = ÷ = ⋅ = =
This will make 111 quarter-pound hamburgers.
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196
87. 4 12 18 12 7 6 2 7 14 212 2 47 1 7 1 18 1 6 3 3 3
⋅ ⋅÷ = ÷ = ⋅ = = =
⋅ ⋅
The length is 243
meters.
88. 1 47 4 47 1 47 1 47 723 4 52 2 1 2 4 2 4 8 8
⋅÷ = ÷ = ⋅ = = =
⋅
The length of each side is 758
feet.
89. 922022374
15+
27260402602846095 19 7 78 8 8 1 960 12 12 12
+
= = + =
The total duration of the eclipses is 79 minutes.
12
90. 922074
15+
272602846055 116 660 12
+
=
The total duration for the eclipses in even-
numbered years is 11612
minutes.
91. 74
15223
−
74
1510215
−
223151021512 41 115 5
−
=
It will be 415
minutes longer.
92. 2239220
−
40260272601360
−
It will be 1360
minute longer.
93. 2 3 22 234 25 10 5 10
2 11 235 2 5
25325
31025
− ⋅ = − ⋅
⋅ ⋅= −
⋅ ⋅
= −
= −
94. 5 2 23 113 36 3 6 3
23 36 1123 3
2 3 112322
1122
⎛ ⎞− ÷ − = ÷⎜ ⎟⎝ ⎠
= ⋅
⋅=
⋅ ⋅
=
=
95. 1 3 1 35 19 5 198 4 8 4
1 65 198 8
7248
7248
⎛ ⎞− − = − +⎜ ⎟⎝ ⎠⎛ ⎞= − +⎜ ⎟⎝ ⎠⎛ ⎞= −⎜ ⎟⎝ ⎠
= −
96. 5 2 5 217 14 17 149 3 9 3
5 617 149 914 616 149 9
829
⎛ ⎞+ − = −⎜ ⎟⎝ ⎠
= −
= −
=
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ISM: Prealgebra Chapter 4: Fractions and Mixed Numbers
197
97. 2 3 8 931 17 31 17
15 20 60 6068 930 1760 60591360
− + = − +
= − +
= −
98. 5 1 12 5 6 2 5 30 21 2 47 2 7 2 7 2 7 7
⋅ ⋅⎛ ⎞− ⋅ − = ⋅ = = =⎜ ⎟ ⋅⎝ ⎠
99. 3 1 7 71 34 2 4 2
7 24 7
7 24 72412
⎛ ⎞ ⎛ ⎞÷ − = ÷ −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
⎛ ⎞= ⋅ −⎜ ⎟⎝ ⎠⋅
= −⋅
= −
= −
100. 7 5 7 531 26 31 268 12 8 12
21 1031 2624 24
11524
⎛ ⎞− − − = − +⎜ ⎟⎝ ⎠
= − +
= −
101. 7 5 21 2011 13 11 138 6 24 24
20 2113 1124 2444 2112 1124 24
23124
− = −
⎛ ⎞= − −⎜ ⎟⎝ ⎠⎛ ⎞= − −⎜ ⎟⎝ ⎠
= −
102. 2 3 4 3 720 30 20 30 505 10 10 10 10
⎛ ⎞ ⎛ ⎞− + − = − + − = −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
103. 3 73 1007 ( 100)
10 10 173 110 10073 1
10 10073
1000
⎛ ⎞− ÷ − = − ÷ −⎜ ⎟⎝ ⎠⎛ ⎞= − ⋅ −⎜ ⎟⎝ ⎠
⋅=
⋅
=
104. 1 3 17 194 24 8 4 8
17 84 19
17 4 24 19
341915119
− ÷ = − ÷
= − ⋅
⋅ ⋅= −
⋅
= −
= −
105. 1 1(3 ) 3 13 3x x x x⎛ ⎞= ⋅ = ⋅ =⎜ ⎟
⎝ ⎠
106. 1 1(5 ) 5 15 5
y y y y⎛ ⎞= ⋅ = ⋅ =⎜ ⎟⎝ ⎠
107. 2 3 2 3 13 2 3 2
a a a a⎛ ⎞ ⎛ ⎞= ⋅ = ⋅ =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
108. 9 10 9 10 1
10 9 10 9m m m m⎛ ⎞ ⎛ ⎞− − = ⋅ = ⋅ =⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
109. a. 59 9 1 105= + =
b. 1009 9 1 10100
= + =
c. 446 6 4 1011
= + =
d. 138 8 1 913
= + =
a, b, and c are equivalent to 10.
110. a. 7 3 36 6 1 74 4 4= + =
b. 11 3 35 5 2 74 4 4= + =
c. 12 37 716 4
=
The answer is d, all of them.
111. Incorrect, to divide mixed number, first write each mixed number as an improper fraction.
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Chapter 4: Fractions and Mixed Numbers ISM: Prealgebra
198
112. answers may vary
113. answers may vary
114. answers may vary
115. Incorrect; 2 1 11 8 88 43 1 or 43 7 3 7 21 21⋅ = ⋅ =
116. answers may vary
117. answers may vary
The Bigger Picture
1. 3 2 3 2 5
17 17 17 17+
+ = =
2. 9 1 9 1 8 4
10 10 10 10 5−
− = = =
3. 2 3 2 10 3 3 20 9 293 10 3 10 10 3 30 30 30
⋅ ⋅+ = + = + =
⋅ ⋅
4. 23 11 23 11 2 23 22 124 12 24 12 2 24 24 24
⋅− = − = − =
⋅
5. 7 19 7 5 19 2 35 38 73 3318 20 8 5 20 2 40 40 40 40
⋅ ⋅+ = + = + = =
⋅ ⋅
6. 3
33 3 3 3 27
4 4 4 644⋅ ⋅
= =⋅ ⋅
7. 16437
−
71574373127
−
8. 528116354
+
1522441
241852437 13 138 8 1 924 24 24
+
= + =
9. 6 8 6 8 2 3 8 2 8 16
11 9 11 9 11 3 3 11 3 33⋅ ⋅ ⋅ ⋅
⋅ = = = =⋅ ⋅ ⋅ ⋅
10. 4 4 34 9 34 5 34 5 34 72 1 1
15 5 15 5 15 9 3 5 9 27 27⋅
÷ = ÷ = ⋅ = = =⋅ ⋅
Section 4.8
Practice Problems
1. 2 53 12
2 2 5 23 3 12 3
5 2 412 3 45 8
12 121312
y
y
y
y
y
− =
− + = +
⋅= +
⋅
= +
=
Check: 2 53 12
13 2 512 3 12
13 2 4 512 3 4 12
13 8 512 12 12
5 5 True12 12
y − =
−
⋅−
⋅
−
=
The solution is 13 .12
2. 1 2515 5 25
10
y
y
y
=
⋅ = ⋅
=
Check: 1 25
1 10 25
2 2 True
y =
⋅
=
The solution is 10.
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ISM: Prealgebra Chapter 4: Fractions and Mixed Numbers
199
3. 5 257
7 5 7 255 7 5
7 2515
35
b
b
b
b
=
⋅ = ⋅
⋅=
=
Check: 5 257
5 35 257
25 25 True
b =
⋅
=
The solution is 35.
4. 7 2
10 510 7 10 27 10 7 5
10 27 547
x
x
x
x
− =
− ⋅ = − ⋅
⋅= −
⋅
= −
Check: 7 2
10 57 4 2
10 7 528 270 5
2 2 True5 5
x− =
− ⋅−
=
The solution is 4 .7
−
5. 354
1 1 355 5 4
1 35 4320
x
x
x
x
= −
⋅ = ⋅−
⋅= −
⋅
= −
Check: 354
3 3520 415 320 4
3 3 True4 4
x = −
⋅− −
− −
− = −
The solution is 3 .
20−
6. The LCD of 15 and 5 is 15. 11 315 511 315 1515 511 911 911 11
911
x
x
xx
x
= −
⋅ = ⋅−
= −−
=
= −
The solution is 9 .
11−
7. The LCD of 8 and 4 is 8. 3 2
8 438 8(2)
8 438 8 8(2)
8 46 16
6 6 16 610
y
y
y
yy
y
+ =
⎛ ⎞+ =⎜ ⎟⎝ ⎠
⋅ + ⋅ =
+ =+ − = −
=
Check: 3 2
8 410 3 28 4
10 6 28 8
16 282 2 True
y+ =
+
+
=
The solution is 10.
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Chapter 4: Fractions and Mixed Numbers ISM: Prealgebra
200
8.
15 5
15 55 5
15 5( ) 55 5
5 14 14 14 4
14
x x
x x
x x
x xxx
x
− =
⎛ ⎞ ⎛ ⎞− =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
⎛ ⎞ ⎛ ⎞− =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
− =− =−
=− −
= −
To check, replace x with 14
− in the original
equation to see that a true statement results. The
solution is 1 .4
−
9. The LCD of 2 and 5 is 10. 3
2 5 2310 10
2 5 2310 10 10
2 5 25 2 15
5 2 2 2 153 153 153 3
5
y y
y y
y y
y yy y y y
yy
y
= +
⎛ ⎞ ⎛ ⎞= +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎛ ⎞ ⎛ ⎞ ⎛ ⎞= +⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠
= +− = − +
=
=
=
Check: 3
2 5 25 5 32 5 21 12 1 12 21 12 2 True2 2
y y= +
+
+
=
The solution is 5.
10. 9 9 3 10 27 10 27 10
10 3 10 3 3 10 30 30 30y y y y−
− = ⋅ − ⋅ = − =
Exercise Set 4.8
1. 1 13 3
1 1 1 13 3 3 3
23
x
x
x
+ = −
+ − = − −
= −
Check: 1 13 3
2 1 13 3 32 1 13 3
1 1 True3 3
x + = −
− + −
− +−
− = −
The solution is 2 .3
−
2. 1 79 9
1 1 7 19 9 9 9
89
x
x
x
+ = −
+ − = − −
= −
Check: 1 79 9
8 1 79 9 98 1 79 9
7 7 True9 9
x + = −
− + −
− +−
− = −
The solution is 8 .9
−
3. 3 2
13 133 3 2 3
13 13 13 132 313
113
y
y
y
y
− = −
− + = − +
− +=
=
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201
Check: 3 2
13 131 3 2
13 13 131 3 213 13
2 2 True13 13
y − = −
− −
−−
− = −
The solution is 1 .
13
4. 5 4
14 145 5 4 5
14 14 14 144 5149
14
z
z
z
z
− =
− + = +
+=
=
Check: 5 4
14 149 5 4
14 14 149 5 414 14
4 4 True14 14
z − =
−
−
=
The solution is 9 .
14
5. 1 1 23 25 5 5
1 35 5
1 1 3 15 5 5 5
45
x x
x
x
x
− − = +
− =
− + = +
=
Check: 1 1 23 25 5 5
4 1 4 1 23 25 5 5 5 5
12 1 8 1 25 5 5 5 5
3 3 True5 5
x x− − = +
⋅ − − ⋅ +
− − +
=
The solution is 4 .5
6. 1 2 55 4
11 11 111 311 11
1 1 3 111 11 11 11
411
x x
x
x
x
+ − = −
+ = −
+ − = − −
= −
Check: 1 2 55 4
11 11 114 1 4 2 55 4
11 11 11 11 1120 1 16 2 511 11 11 11 11
3 3 True11 11
x x+ − = −
⎛ ⎞ ⎛ ⎞− + − − −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
− + + −
− = −
The solution is 4 .
11−
7. 1 5
12 61 1 5 1
12 12 6 125 2 16 2 1210 112 121112
x
x
x
x
x
− =
− + = +
⋅= +
⋅
= +
=
Check: 1 5
12 611 1 512 12 6
10 512 6
5 5 True6 6
x − =
−
=
The solution is 11.12
8. 8 19 3
8 8 1 89 9 3 9
1 3 83 3 93 89 9119
y
y
y
y
y
− =
− + = +
⋅= +
⋅
= +
=
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Chapter 4: Fractions and Mixed Numbers ISM: Prealgebra
202
Check: 8 19 3
11 8 19 9 3
3 19 31 1 True3 3
y − =
−
=
The solution is 11.9
9. 2 35 10
2 2 3 25 5 10 5
3 2 210 5 23 4
10 107
10
y
y
y
y
y
+ = −
+ − = − −
⋅= − −
⋅
= − −
= −
Check: 2 35 10
2 7 35 10 10
2 2 7 35 2 10 10
4 7 310 10 10
3 3 True10 10
y+ = −
⎛ ⎞+ − −⎜ ⎟⎝ ⎠⎛ ⎞⋅ + − −⎜ ⎟⎝ ⎠⎛ ⎞+ − −⎜ ⎟⎝ ⎠
− = −
The solution is 7 .
10−
10. 1 32 8
1 1 1 32 2 2 8
1 4 32 4 84 38 878
a
a
a
a
a
+ = −
− + + = − −
⋅= − −
⋅
= − −
= −
Check: 1 32 8
1 7 32 8 8
4 7 38 8 8
3 3 True8 8
a+ = −
⎛ ⎞+ − −⎜ ⎟⎝ ⎠
− −
− = −
The solution is 7 .8
−
11.
1 37 616 4
1 316 4
1 1 3 116 16 4 16
3 4 14 4 1612 116 161116
z z
z
z
z
z
z
+ − =
+ =
+ − = −
⋅= −
⋅
= −
=
Check: 1 37 6
16 411 1 11 37 616 16 16 4
77 1 66 316 16 16 4
12 316 4
3 3 True4 4
z z+ − =
⎛ ⎞ ⎛ ⎞+ −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
+ −
=
The solution is 11.16
12.
2 119 87 14
2 117 14
2 2 11 27 7 14 7
11 2 214 7 211 414 141514
x x
x
x
x
x
x
− − =
− =
− + = +
⋅= +
⋅
= +
=
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ISM: Prealgebra Chapter 4: Fractions and Mixed Numbers
203
Check: 2 119 87 14
15 2 15 119 814 7 14 14
135 4 120 1114 14 14 14
11 11 True14 14
x x− − =
⎛ ⎞ ⎛ ⎞− −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
− −
=
The solution is 15 .14
13. 2 59 6
2 5 5 59 6 6 6
2 2 5 39 2 6 3
4 1518 18
1118
x
x
x
x
x
− = −
− + = − +
− ⋅ + ⋅ =
− + =
=
Check: 2 59 62 11 59 18 6
2 2 11 5 39 2 18 6 3
4 11 1518 18 184 4 True
18 18
x− = −
− −
− ⋅ − ⋅
− −
− =−
The solution is 11.18
14. 1 74 10
1 7 7 74 10 10 10
1 5 7 24 5 10 2
5 1420 20
920
y
y
y
y
y
− = −
− + = − +
⋅ ⋅− + =
⋅ ⋅
− + =
=
Check: 1 74 101 9 74 20 10
1 5 9 7 24 5 20 10 2
5 9 1420 20 205 5 True20 20
y− = −
− −
⋅ ⋅− −
⋅ ⋅
− −
− = −
The solution is 9 .20
15. 7 21 17 27 7
27
x
x
x
=
⋅ = ⋅
=
16. 5 41 1( 5 ) 45 5
45
x
x
x
− =
− ⋅ − = − ⋅
= −
17. 1 3414 4 34
12
x
x
x
=
⋅ = ⋅
=
18. 1 6313 3 63
18
x
x
x
=
⋅ = ⋅
=
19. 2 69
9 2 9 62 9 2
9 62
9 2 32
27
y
y
y
y
y
= −
⋅ = ⋅−
⋅= −
⋅ ⋅= −
= −
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Chapter 4: Fractions and Mixed Numbers ISM: Prealgebra
204
20. 4 87
7 4 7 84 7 4
564
14
x
x
x
x
= −
⋅ = ⋅−
= −
= −
21. 4 39 2
9 4 9 34 9 4 2
278
z
z
z
− = −
− ⋅ = − ⋅−
=
22. 11 210 7
10 11 10 211 10 11 7
2077
x
x
x
− = −
− ⋅− = − ⋅−
=
23. 173
1 1 177 7 3
121
a
a
a
=
⋅ = ⋅
=
24. 52
121 1 522 2 12
524
z
z
z
= −
⋅ = ⋅−
= −
25. 63
111 1 633 3 11
63 112
11
x
x
x
x
− = −
− ⋅− = − ⋅−
=⋅
=
26. 12425
1 1 1244 4 25
124 25325
z
z
z
z
− = −
− ⋅− = − ⋅−
=⋅
=
27. 5 39 18
5 318 189 1810 310 310 10
310
x
x
xx
x
= −
⎛ ⎞ ⎛ ⎞= −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
= −−
=
= −
28. 3 75 20
3 720 205 2012 712 712 12
712
y
y
yy
y
= −
⎛ ⎞ ⎛ ⎞= −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
= −−
=
= −
29. 72
3 373 2 3
3 3
3 3 2 73
6 76 6 7 6
1
x
x
x
xx
x
+ =
⎛ ⎞+ = ⋅⎜ ⎟⎝ ⎠
⋅ + ⋅ =
+ =+ − = −
=
30. 71
5 575 1 5
5 5
5 5 1 75
5 75 5 7 5
12
x
x
x
xx
x
− =
⎛ ⎞− = ⋅⎜ ⎟⎝ ⎠
⋅ − ⋅ =
− =− + = +
=
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ISM: Prealgebra Chapter 4: Fractions and Mixed Numbers
205
31.
5 15 10
510 10(1)5 10
510 10 105 10
2 5 102 5 5 10 5
2 152 152 2
152
x
x
x
xx
xx
x
− =
⎛ ⎞− =⎜ ⎟⎝ ⎠
⋅ − ⋅ =
− =− + = +
=
=
=
32. 63
3 3( 6)3
3 3( ) 183
3 182 182 182 2
9
x x
x x
x x
x xxx
x
− = −
⎛ ⎞− = −⎜ ⎟⎝ ⎠
⎛ ⎞ − = −⎜ ⎟⎝ ⎠
− = −− = −− −
=− −
=
33. 1 32 5 10
1 310 102 5 10
1 310 102 5
5 61
x
x
x
xx
− =
⎛ ⎞− = ⋅⎜ ⎟⎝ ⎠
⋅ − ⋅ =
− =− =
34. 2 13 4 12
2 112 123 4 12
2 112 123 4
8 35
x
x
x
xx
− =
⎛ ⎞ ⎛ ⎞− =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
⎛ ⎞ ⎛ ⎞− =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
− ==
35.
23 5
15 15 23 5
5 15 15 25
5 3 305 3 3 3 30
2 302 302 2
15
x x
x x
xx
x xx x x x
xx
x
= −
⎛ ⎞ ⎛ ⎞= −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
= ⋅ − ⋅
= −− = − −
= −−
=
= −
36.
52 7 2
514 142 7 2
57 14 147 2
7 2 357 2 2 2 35
5 355 355 5
7
a a
a a
aa
a aa a a a
aa
a
= +
⎛ ⎞ ⎛ ⎞= +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
= ⋅ + ⋅
= +− = − +
=
=
=
37. 4 3 4 7
7 3 7 3 3 73 2821 213 28
21
x x
x
x
− = ⋅ − ⋅
= −
−=
38. 5 5 8 9 40 9 40 99 8 9 8 8 9 72 72 72
y y y y− +− + = − ⋅ + ⋅ = − + =
39. 5 2 10 105
2 2 1 2 2 2 2y y y y ++ = + ⋅ = + =
40. 7 2 3 7 6 7 6 723 1 3 3 3 3 3x x x x+
+ = ⋅ + = + =
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Chapter 4: Fractions and Mixed Numbers ISM: Prealgebra
206
41. 3 3 3 510 6 10 3 6 5
9 530 3014302 72 15
715
x x x x
x x
x
x
x
+ = ⋅ + ⋅
= +
=
⋅ ⋅=
⋅
=
42. 9 5 9 3 5 4 27 20 78 6 8 3 6 4 24 24 24x x x x x x x− = ⋅ − ⋅ = − =
43. 3 18 2
8 3 8 13 8 3 2
8643
x
x
x
x
=
⋅ = ⋅
=
=
44. 2 35 10
5 2 5 32 5 2 10
152034
y
y
y
y
=
⋅ = ⋅
=
=
45.
2 43 5 15
2 415 153 5 15
215 15 43 5
10 3 410 3 10 4 10
3 63 63 3
2
x
x
x
xx
xx
x
− =
⎛ ⎞− = ⋅⎜ ⎟⎝ ⎠
⋅ − ⋅ =
− =− − = −
− = −− −
=− −
=
46.
4 215 4 20
4 2120 205 4 20
420 20 215 4
16 5 2116 5 16 21 16
5 55 55 5
1
x
x
x
xx
xx
x
+ =
⎛ ⎞+ = ⋅⎜ ⎟⎝ ⎠
⋅ + ⋅ =
+ =+ − = −
=
=
=
47. 9 27
14 2014 9 14 279 14 9 20
2 7 9 39 2 102110
z
z
z
z
=
⋅ = ⋅
⋅ ⋅ ⋅=
⋅ ⋅
=
48. 5 5
16 616 5 16 55 16 5 6
2 8 55 2 383
a
a
a
a
=
⋅ = ⋅
⋅ ⋅=
⋅ ⋅
=
49. 43 57487
1 1 488 8 7
1 42 4 71
14
m m
m
m
m
m
− − =
− =
− ⋅− = − ⋅
⋅= −
⋅ ⋅
= −
50. 330 34
2034
201 1 344 4 20
380
n n
n
n
n
− =
− =
− ⋅− = − ⋅
= −
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ISM: Prealgebra Chapter 4: Fractions and Mixed Numbers
207
51. 11
4 414 1 4
4 4
4 4 1 14
4 14 4 1 4
3
x
x
x
xx
x
+ =
⎛ ⎞ ⎛ ⎞+ =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
⋅ + ⋅ =
+ =+ − = −
= −
52. 12
7 717 2 7
7 7
7 7 2 17
14 114 14 1 14
15
y
y
y
yy
y
− =
⎛ ⎞ ⎛ ⎞− =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
⋅ − ⋅ =
− =− + = +
=
53. 5 2 5 2 3 5 6 19 3 9 3 3 9 9 9− = − ⋅ = − = −
54. 8 1 8 2 1 11 16 11 511 2 11 2 2 11 22 22 22
− = ⋅ − ⋅ = − =
55. 1 10515 5 105
50
y
y
y
=
⋅ = ⋅
=
56. 1 2414 4 ( 2)4
8
x
x
x
= −
⋅ = ⋅ −
= −
57. 5 157 49
7 5 7 155 7 5 49
7 155 497 3 55 7 737
y
y
y
y
y
= −
⋅ = ⋅−
⋅= −
⋅⋅ ⋅
= −⋅ ⋅
= −
58. 3 94 2
4 3 4 93 4 3 2
4 93 22 2 3 3
3 26
x
x
x
x
x
− =
− ⋅− = − ⋅
⋅= −
⋅⋅ ⋅ ⋅
= −⋅
= −
59. 22
2 2( 2)2
2 2 42
2 444
1 14
x x
x x
x x
x xxx
x
− = −
⎛ ⎞− = −⎜ ⎟⎝ ⎠
⋅ − ⋅ = −
− = −− = −− −
=− −
=
60. 43
3 3( 4 )3
3( 4) 312 3
3 12 3 32 122 122 2
6
y y
y y
y yy y
y y y yyy
y
= − +
⎛ ⎞ = − +⎜ ⎟⎝ ⎠
= − + ⋅= − +
− = − + −− = −− −
=− −
=
61.
5 3 98 16 165 68 165 38 8
8 5 8 35 8 5 8
8 35 835
y
y
y
y
y
y
− = −
− = −
− = −
− ⋅− = − ⋅−
⋅=
⋅
=
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Chapter 4: Fractions and Mixed Numbers ISM: Prealgebra
208
62. 7 5 49 18 187 99 18
9 7 9 97 9 7 18
9 97 9 29
14
x
x
x
x
x
− = − −
− = −
− ⋅− = − ⋅−
⋅=
⋅ ⋅
=
63. 117 253183
1 1 188 8 3
124
x x
x
x
x
− =
− =
− ⋅− = − ⋅
= −
64. 427 309439
1 1 433 3 9
427
x x
x
x
x
− =
− =
− ⋅− = − ⋅
= −
65.
7 1 26 4 37 1 212 126 4 3
1 214 12 124 3
14 3 814 514 514 14
514
x
x
x
xxx
x
= −
⎛ ⎞⋅ = −⎜ ⎟⎝ ⎠
= ⋅ − ⋅
= −= −−
=
= −
66.
5 1 74 2 10
5 1 720 204 2 10
1 725 20 202 10
25 10 1425 425 425 25
425
y
y
y
yyy
y
= −
⎛ ⎞ ⎛ ⎞= −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
= ⋅ − ⋅
= −= −−
=
= −
67.
24 12 3
212 124 12 3
23 12 1212 3
3 83 8
2 82 82 2
4
b b
b b
bb
b bb b b bbb
b
= +
⎛ ⎞⋅ = +⎜ ⎟⎝ ⎠
= ⋅ + ⋅
= +− = + −
=
=
=
68.
16 3 2
16 66 3 2
16 63 2
2 32 2 3 2
33
1 13
a a
a a
aa
a aa a a a
aa
a
= +
⎛ ⎞⋅ = +⎜ ⎟⎝ ⎠
= ⋅ + ⋅
= +− = + −− =−
=− −
= −
69. 2 83 2
6 2 6 83 2
6 6 2 6 6 83 22 12 3 48
2 12 2 3 48 212 48
12 48 48 4836
x x
x x
x x
x xx x x x
xxx
+ = +
⎛ ⎞ ⎛ ⎞+ = +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
⋅ + ⋅ = ⋅ + ⋅
+ = ++ − = + −
= +− = + −− =
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ISM: Prealgebra Chapter 4: Fractions and Mixed Numbers
209
70.
2 45 3
15 2 15 45 3
15 15 2 15 15 45 33 30 5 60
3 5 30 5 5 602 30 60
2 30 30 60 302 302 302 2
15
y y
y y
y y
y yy y y y
yy
yy
y
− = −
⎛ ⎞ ⎛ ⎞− = −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
⋅ − ⋅ = ⋅ − ⋅
− = −− − = − −− − = −
− − + = − +− = −− −
=− −
=
71. 57,236 rounded to the nearest hundred is 57,200.
72. 576 rounded to the nearest hundred is 600.
73. 327 rounded to the nearest ten is 330.
74. 2333 rounded to the nearest ten is 2330.
75. answers may vary
76.
14 311 8 2
14 388 8811 8 2
14 388 88 4411 8
112 33 44112 33 33 44 33
112 11112 1111 11
11211
x x
x x
x x
x xx x x x
xx
x
+ =
⎛ ⎞+ = ⋅⎜ ⎟⎝ ⎠
⋅ + ⋅ =
+ =+ − = −
=
=
=
77.
19 353 2353 1431 2719 353 231431 143153 1431 27
353 2327 19 1431 14311431 27
513 353 1219513 1219 353 1219 1219
706 353706 353
353 3532
x
x
x
xxxx
x
= +
⎛ ⎞⋅ = +⎜ ⎟⎝ ⎠
⋅ = ⋅ + ⋅
= +− = + −− =−
=
− =
78. 5 3 5 4 5 4 5 4 5
12 4 12 3 12 3 4 3 3 9⋅ ⋅
÷ = ⋅ = = =⋅ ⋅ ⋅
The length is 5 inch.9
Chapter 4 Vocabulary Check
1. Two numbers are reciprocals of each other if their product is 1.
2. A composite number is a natural number greater than 1 that is not prime.
3. Fractions that represent the same portion of a whole are called equivalent fractions.
4. An improper fraction is a fraction whose numerator is greater than or equal to its denominator.
5. A prime number is a natural number greater than 1 whose only factors are 1 and itself.
6. A fraction is in simplest form when the numerator and the denominator have no factors in common other than 1.
7. A proper fraction is one whose numerator is less than its denominator.
8. A mixed number contains a whole number part and a fraction part.
9. In the fraction 7 ,9
the 7 is called the numerator
and the 9 is called the denominator.
10. The prime factorization of a number is the factorization in which all the factors are prime numbers.
11. The fraction 30
is undefined.
12. The fraction 0 0.5=
13. Fractions that have the same denominator are called like fractions.
14. The LCM of the denominators in a list of fractions is called the least common denominator.
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Chapter 4: Fractions and Mixed Numbers ISM: Prealgebra
210
15. A fraction whose numerator or denominator or both numerator and denominator contain fractions is called a complex fraction.
16. In ,a cb d= a ⋅ d and b ⋅ c are called cross products.
Chapter 4 Review
1. 2 out of 6 equal parts are shaded: 26
2. 4 out of 7 equal parts are shaded: 47
3. Each part is 13
of a whole and there are 7 parts shaded, or 2 wholes and 1 more part: 7 1 or 23 3
4. Each part is 14
of a whole and there are 13 parts shaded, or 3 wholes and 1 more part: 13 1 or 34 4
5. successful free throws 11total free throws 12
→→
1112
of the free throws were made.
6. a. 131 − 23 = 108 108 cars are not blue.
b. not blue 108total 131
→→
108131
of the cars on the lot are not blue.
7. 3 13= −
−
8. 20 120−
=−
9. 0 01=
−
10. 4 is undefined0
11.
·
10
12. ¢
1 20
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ISM: Prealgebra Chapter 4: Fractions and Mixed Numbers
211
13. @
1 2 30
14.
Y
1 20
15. 3
4 1512
3
15 334 4=
16. 3
13 39390
39 313
=
17. 1 5 2 1 10 1 1125 5 5 5
⋅ + += = =
18. 8 9 3 8 27 8 3539 9 9 9
⋅ + += = =
19. 12 4 3 428 7 3 7
⋅= =
⋅
20. 15 3 5 527 3 9 9
⋅= =
⋅
21. 225 1 25 1
3 25 375x x
x x xx⋅ ⋅
− = − = −⋅ ⋅ ⋅
22. 3 236 1 36
72 2 36 2y y y y yy y
⋅ ⋅ ⋅ ⋅− = − = −
⋅ ⋅
23. 29 29 2932 32 32ab a babc a b c c
⋅ ⋅= =
⋅ ⋅ ⋅
24. 18 18 1823 23 23xyz x y z zxy x y
⋅ ⋅ ⋅= =
⋅ ⋅
25. 2
3 245 9 5 5
9 327 3x y x x y x
x y y yxy y⋅ ⋅ ⋅ ⋅
= =⋅ ⋅ ⋅ ⋅ ⋅
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Chapter 4: Fractions and Mixed Numbers ISM: Prealgebra
212
26. 2
3 242 6 7 7
6 530 5ab c a b b c b
a b c c cabc c⋅ ⋅ ⋅ ⋅ ⋅
= =⋅ ⋅ ⋅ ⋅ ⋅ ⋅
27. 8 inches 8 4 2 2
12 inches 12 4 3 3⋅
= = =⋅
8 inches represents 23
of a foot.
28. 15 − 6 = 9 cars are not white. 9 non-white cars 9 3 3 3
15 total cars 15 3 5 5⋅
= = =⋅
35
of the cars are not white.
29. Not equivalent, since the cross products are not equal: 34 ⋅ 4 = 136 and 10 ⋅ 14 = 140
30. Equivalent, since the cross products are equal: 30 ⋅ 15 = 450 and 50 ⋅ 9 = 450
31. 3 1 3 1 35 2 5 2 10
⋅⋅ = =
⋅
32. 6 5 6 5 6 5 5 57 12 7 12 7 6 2 7 2 14
⋅ ⋅− ⋅ = − = − = − = −
⋅ ⋅ ⋅ ⋅
33. 3 3
2
24 15 24 155 8 5 8
8 3 5 35 83 3
9
x xx x
xx x x
x x
x
⋅⎛ ⎞− ⋅ − =⎜ ⎟
⋅⎝ ⎠⋅ ⋅ ⋅ ⋅
=⋅ ⋅ ⋅ ⋅⋅
=⋅
=
34. 3 3
2 227 7 27 7 3 9 7
21 3 7 9 2 218 21 18y y y y y y
y yy y⋅ ⋅ ⋅ ⋅ ⋅ ⋅
⋅ = = =⋅ ⋅ ⋅ ⋅ ⋅⋅
35. 31 1 1 1 1 1 1 1
3 3 3 3 3 3 3 27⋅ ⋅⎛ ⎞ ⎛ ⎞⎛ ⎞⎛ ⎞− = − − − = − = −⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⋅ ⋅⎝ ⎠ ⎝ ⎠⎝ ⎠⎝ ⎠
36. 25 5 5 5 5 25
12 12 12 12 12 144⋅⎛ ⎞ ⎛ ⎞⎛ ⎞− = − − = =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⋅⎝ ⎠ ⎝ ⎠⎝ ⎠
37. 3 3 3 8 3 8 3 4 2 2 24 8 4 3 4 3 1 4 3 1
⋅ ⋅ ⋅− ÷ = − ⋅ = − = − = − = −
⋅ ⋅ ⋅
38. 21 7 21 54 5 4 7
21 54 77 3 54 7
3 54
154
a a aa
aaaa
÷ = ⋅
⋅=
⋅⋅ ⋅ ⋅
=⋅ ⋅⋅
=
=
39. 9 1 9 3 9 3 272 3 2 1 2 1 2
⋅⎛ ⎞− ÷ − = − ⋅ − = =⎜ ⎟ ⋅⎝ ⎠
40. 5 5 2 5 1 5 1 523 3 1 3 2 3 2 6
yyy y y
⋅− ÷ = − ÷ = − ⋅ = − = −
⋅
41.
9 37 49 47 39 47 33 3 47 3
3 47
127
x y÷ = ÷
= ⋅
⋅=
⋅⋅ ⋅
=⋅⋅
=
=
42. 9 7 9 7 9 637
10 1 10 1 10 10ab ⋅
= − ⋅ = − ⋅ = − = −⋅
43. 11 7 11 7 77area length width6 8 6 8 48
⋅= ⋅ = ⋅ = =
⋅
The area is 7748
square feet.
44. 2 2 2 2 4area side side3 3 3 3 9
⋅= ⋅ = ⋅ = =
⋅
The area is 49
square meter.
45. 7 3 7 3 1011 11 11 11
++ = =
46. 4 2 4 2 6 2 3 29 9 9 9 3 3 3
+ ⋅+ = = = =
⋅
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ISM: Prealgebra Chapter 4: Fractions and Mixed Numbers
213
47. 1 5 1 5 4 1 4 1
12 12 12 12 3 4 3− − ⋅
− = = = − = −⋅
48. 11 11 12 3 4 415 15 15 15 3 5 5x x x x x x x+ ⋅ ⋅+ = = = =
⋅
49. 4 3 4 321 21 21y y −− =
50. 4 3 2 4 3 2 1 1
15 15 15 15 15 15− − −
− − = = = −
51. 3 = 3 x = x LCD = 3 ⋅ x = 3x
52. 4 = 2 ⋅ 2 8 = 2 ⋅ 2 ⋅ 2 12 = 2 ⋅ 2 ⋅ 3 LCD = 2 ⋅ 2 ⋅ 2 ⋅ 3 = 24
53. 2 2 10 2 10 203 3 10 3 10 30
⋅= ⋅ = =
⋅
54. 5 5 7 5 7 358 8 7 8 7 56
⋅= ⋅ = =
⋅
55. 7 7 7 7 7 496 6 7 6 7 42a a a a⋅= ⋅ = =
⋅
56. 9 9 5 9 5 454 4 5 4 5 20b b b b⋅= ⋅ = =
⋅
57. 4 4 10 4 10 40
5 5 10 5 10 50x x x x⋅
= ⋅ = =⋅
58. 5 5 2 5 2 10
9 9 2 9 2 18y y y y⋅
= ⋅ = =⋅
59. 3 2 1 3 2 1 6 2 3 38 8 8 8 8 2 4 4
+ + ⋅+ + = = = =
⋅
He did 34
of his homework that evening.
60. 9 3 9 3 9 3 9 3
16 16 16 16 1624163 82 832
+ + ++ + + =
=
⋅=
⋅
=
The perimeter is 32
miles.
61. The LCD of 18 and 9 is 18. 7 2 7 2 2 7 4 11
18 9 18 9 2 18 18 18⋅
+ = + = + =⋅
62. The LCD of 13 and 26 is 26. 4 1 4 2 1 8 1 7
13 26 13 2 26 26 26 26⋅
− = − = − =⋅
63. The LCD of 3 and 4 is 12. 1 1 1 4 1 3 4 3 13 4 3 4 4 3 12 12 12
⋅ ⋅− + = − + = − + = −
⋅ ⋅
64. The LCD of 3 and 4 is 12. 2 1 2 4 1 3 8 3 53 4 3 4 4 3 12 12 12
⋅ ⋅− + = − + = − + = −
⋅ ⋅
65. The LCD of 11 and 55 is 55. 5 2 5 5 2 25 2 25 211 55 11 5 55 55 55 55x x x x⋅ ++ = + = + =
⋅
66. The LCD of 15 and 5 is 15. 4 4 3 4 3 4 3
15 5 15 5 3 15 15 15b b b b⋅ +
+ = + = + =⋅
67. The LCD of 12 and 9 is 36. 5 2 5 3 2 4 15 8 712 9 12 3 9 4 36 36 36y y y y y y y⋅ ⋅− = − = − =
⋅ ⋅
68. The LCD of 18 and 9 is 18. 7 2 7 2 2 7 4 1118 9 18 9 2 18 18 18x x x x x x x⋅+ = + = + =
⋅
69. The LCD of 9 and y is 9y. 4 5 4 5 9 4 45 4 459 9 9 9 9 9
y y yy y y y y y
⋅ ⋅ ++ = + = + =
⋅ ⋅
70. The LCD of 14 and 7 is 14. 9 3 9 3 2 9 6 15
14 7 14 7 2 14 14 14⋅
− − = − − = − − = −⋅
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Chapter 4: Fractions and Mixed Numbers ISM: Prealgebra
214
71. The LCD of 25, 75, and 50 is 150. 4 23 7 4 6 23 2 7 325 75 50 25 6 75 2 50 3
24 46 21150 150 15091
150
⋅ ⋅ ⋅+ + = + +
⋅ ⋅ ⋅
= + +
=
72. The LCD of 3, 9, and 6 is 18. 2 2 1 2 6 2 2 1 33 9 6 3 6 9 2 6 3
12 4 318 18 185
18
⋅ ⋅ ⋅− − = − −
⋅ ⋅ ⋅
= − −
=
73. The LCD of 9 and 6 is 18. 2 5 2 5 2 2 5 3 2 2 5 39 6 9 6 9 2 6 3 9 2 6 3
4 15 4 1518 18 18 1838182 192 9
199
⋅ ⋅ ⋅ ⋅+ + + = + + +
⋅ ⋅ ⋅ ⋅
= + + +
=
⋅=
⋅
=
The perimeter is 199
meters.
74. The LCD of 5 and 10 is 10. 1 3 7 1 2 3 2 75 5 10 5 2 5 2 10
2 6 710 10 1015103 52 532
⋅ ⋅+ + = + +
⋅ ⋅
= + +
=
⋅=
⋅
=
The perimeter is 32
feet.
75. The LCD of 25 and 50 is 50. 9 3 9 2 3 18 3 2125 50 25 2 50 50 50 50
⋅+ = + = + =
⋅
2150
of the donors have type A blood.
76. The LCD of 12 and 3 is 12. 2 5 2 4 5 8 5 3 1 3 13 12 3 4 12 12 12 12 4 3 4
⋅ ⋅− = − = − = = =
⋅ ⋅
The difference in length is 14
yard.
77. 257
10
2 75 10
2 105 7
2 105 7
2 5 25 7
47
x x
x
x
x
x
= ÷
= ⋅
⋅=
⋅⋅ ⋅ ⋅
=⋅
=
78. 37
117
3 11 3 7 3 7 37 7 7 11 7 11 11
yy y y y⋅
= ÷ = ⋅ = =⋅
79. ( )( )
2 12 15 25 2
3 7 3 74 10 4 10
2 15 23 74 10
20
20
20 20
20 208 10
15 142
12
−−=
− −
⋅ − ⋅=
⋅ − ⋅−
=−
−=
= −
80. ( )( )5 15 16 46 4
1 112 12
5 16 4
12
12
12 12
110 3
17
17
y y
y
y
y y
y y
y
y
− −
−−=
⋅ − ⋅=
−−
=−
=−
= −
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ISM: Prealgebra Chapter 4: Fractions and Mixed Numbers
215
81.
( )
( )
12
2 43 5
12
2 43 5
2 43 5
30
3015
30 3015
20 24154
xy z
=+ − +
⋅=
− +
=− + ⋅
=− +
=
82.
( )
( )
1 22 3
451 22 3
45
1 22 3
30
30
30 30
6 415 20
24524
x yz
+ −+=
−=
⋅
⋅ − ⋅=
⋅−
=
= −
83. 5 1 4 5 2 4 5 2 4 8
13 2 5 13 1 5 13 1 5 13⋅ ⋅
÷ ⋅ = ⋅ ⋅ = =⋅ ⋅
84. 22 1 2 1
27 3 27 92 1 3
27 9 32 3
27 27127
⎛ ⎞− = −⎜ ⎟⎝ ⎠
⋅= −
⋅
= −
= −
85. 9 1 2 1 9 1 2 1
10 3 5 11 10 3 5 113 2
10 553 11 2 2
10 11 55 233 4
110 11029
110
⋅ ⋅⋅ − ⋅ = −
⋅ ⋅
= −
⋅ ⋅= −
⋅ ⋅
= −
=
86. 2 1 3 2 1 2 37 5 10 7 5 2 10
2 2 37 10 102 57 102 17 217
⋅⎛ ⎞ ⎛ ⎞− ⋅ + = − ⋅ +⎜ ⎟ ⎜ ⎟⋅⎝ ⎠ ⎝ ⎠⎛ ⎞= − ⋅ +⎜ ⎟⎝ ⎠
= − ⋅
= − ⋅
= −
87. 37859613
12+
97242092423
2431 7 719 19 1 2024 24 24
+
= + =
88. 185
rounds to 8.
3511
rounds to 5.
An estimate is 8 + 5 = 13. 18535
11−
1185515555
−
667551555551255
−
89. 518
rounds to 2.
135
rounds to 3.
An estimate is 2 ⋅ 3 = 6. 5 1 13 16 13 16 13 8 2 26 11 3 58 5 8 5 8 5 8 5 5 5
⋅ ⋅ ⋅⋅ = ⋅ = = = =
⋅ ⋅
90. 3 2 27 9 27 7 9 3 7 21 16 1 54 7 4 7 4 9 4 9 4 4
⋅ ⋅÷ = ÷ = ⋅ = = =
⋅
91. 1 341 31 341 2 11 31 2341 15 222 1 2 1 31 1 31
⋅ ⋅÷ = ÷ = ⋅ = =
⋅
We would expect 22 miles on one gallon.
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Chapter 4: Fractions and Mixed Numbers ISM: Prealgebra
216
92. 1 22 5 22 5 110 27 5 363 3 1 3 1 3 3
⋅⋅ = ⋅ = = =
⋅
There are 110
3 or
2363
grams of fat in a 5-ounce
hamburger patty.
93. 7 3 4 118 10 8 88 8 8 2− = =
1 17 2 17 1 17 18 2 42 2 1 2 2 4 4÷ = ÷ = ⋅ = =
Each measurement is 14 inches.4
94. 3 3 13 3 13 3 2 13 6 71
10 5 10 5 10 5 2 10 10 10⋅
− = − = − = − =⋅
The unknown measurement is 7
10 yard.
95. 1 3 2 3 512 15 12 15 277 14 14 14 14
⎛ ⎞ ⎛ ⎞− + − = − + − = −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
96. 1 7 16 273 25 10 5 10
16 105 27
16 5 25 27
3227
5127
⎛ ⎞ ⎛ ⎞− ÷ − = − ÷ −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
⎛ ⎞= − ⋅ −⎜ ⎟⎝ ⎠
⋅ ⋅=
⋅
=
=
97. 1 3 9 7 63 152 1 34 4 4 4 16 16
− ⋅ = − ⋅ = − = −
98. 7 7 7 723 24 24 238 10 10 8
⎛ ⎞− = − −⎜ ⎟⎝ ⎠
7 28 6824 24 2310 40 407 35 3523 23 238 40 40
3340
− − −
7 7 3323 248 10 40− = −
99. 2 13 6
2 2 1 23 3 6 3
1 2 26 3 21 46 656
a
a
a
a
a
− =
− + = +
⋅= +
⋅
= +
=
100. 1 79 85 10
1 75 10
1 1 7 15 5 10 5
7 210 109
10
x x
x
x
x
x
+ − = −
+ = −
+ − = − −
= − −
= −
101. 3 65
5 3 5 63 5 3
10
x
x
x
− =
− ⋅− = − ⋅
= −
102. 2 49 3
9 2 9 42 9 2 3
9 42 33 3 2 2
2 3616
y
y
y
y
y
y
= −
⋅ = ⋅−
⋅= −
⋅⋅ ⋅ ⋅
= −⋅
= −
= −
103. 63
7 767 3 7
7 7
7 7 3 67
21 621 21 6 21
15
x
x
x
xx
x
− = −
⎛ ⎞ ⎛ ⎞− = −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
⋅ − ⋅ = −
− = −− + = − +
=
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ISM: Prealgebra Chapter 4: Fractions and Mixed Numbers
217
104. 112
5 5115 2 5
5 5
5 5 2 115
10 1110 10 11 10
1
y
y
y
yy
y
+ =
⎛ ⎞ ⎛ ⎞+ =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
⋅ + ⋅ =
+ =+ − = −
=
105.
1 176 4 12
1 1712 126 4 12
112 12 176 4
2 3 172 3 2 17 2
3 153 153 3
5
x
x
x
xxxx
x
+ =
⎛ ⎞+ = ⋅⎜ ⎟⎝ ⎠
⋅ + ⋅ =
+ =+ − = −
=
=
=
106.
5 15 4 2 20
5 120 205 4 2 20
5 120 20 20 205 4 2 20
4 25 10 14 25 10 10 1 10
25 6 125 6 25 1 25
6 246 246 6
4
x x
x x
x x
x xx x x x
xx
xx
x
− = −
⎛ ⎞ ⎛ ⎞− = −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
⋅ − ⋅ = ⋅ − ⋅
− = −− − = − −− − = −
− − + = − +− =−
=− −
= −
107. 6 5 6 5 3 2 5 1
15 8 15 8 5 3 2 4 4⋅ ⋅ ⋅
⋅ = = =⋅ ⋅ ⋅ ⋅
108. 2 3 2 3
3 3
2 3
3
2
5 10 510
510
55 2
2
x x x yy yy x
x yy xx x y y y
y x x xyx
÷ = ⋅
⋅=
⋅⋅ ⋅ ⋅ ⋅ ⋅
=⋅ ⋅ ⋅ ⋅ ⋅
=
109. 3 1 3 1 2 1 2 1
10 10 10 10 5 2 5− ⋅
− = = = =⋅
110. 7 2 7 2 7 2 7
8 3 8 3 2 4 3 12x x x x⋅ ⋅
⋅ − = − = − = −⋅ ⋅ ⋅ ⋅
111. 2 2 4 33 4 3 4 4 3
8 312 128 3
121112
x x x x
x x
x x
x
+ = ⋅ + ⋅
= +
+=
=
112. 5 2 5 5 2 25 2 23
11 55 11 5 55 55 55 55− + = − ⋅ + = − + = −
113. 3 1 8 115 4 5 4
8 45 18 45 132 2or 65 5
− ÷ = − ÷
= − ⋅
⋅= −
⋅
= − −
114. 728
rounds to 3.
192
rounds to 10.
An estimate is 3 + 10 = 13. 728192
+
728498
11 3 311 11 1 128 8 8
+
= + =
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Chapter 4: Fractions and Mixed Numbers ISM: Prealgebra
218
115. 1127
rounds to 12.
395
rounds to 10.
An estimate is 12 − 10 = 2. 1127395
−
512
3521935
−
4011352193519235
−
116. ( )( )
3344
1 18 8
3418
8 22
1 8 1
8 2 8
8 1 816 68 1
22 1 or 37 7
++=
− −
⋅ + ⋅=
⋅ − ⋅+
=−
=
117.
3 2 4 3 2 3 48 3 9 8 3 3 9
3 6 48 9 93 28 93 28 9
3 22 4 3 31
12
⋅⎛ ⎞ ⎛ ⎞− ⋅ − = − −⎜ ⎟ ⎜ ⎟⋅⎝ ⎠ ⎝ ⎠⎛ ⎞= − −⎜ ⎟⎝ ⎠⎛ ⎞= − ⎜ ⎟⎝ ⎠⋅
= −⋅⋅
= −⋅ ⋅ ⋅
= −
118.
2 1311 107 14
2 137 14
2 2 13 27 7 14 7
13 2 214 7 213 414 149
14
x x
x
x
x
x
x
− − = −
− = −
− + = − +
⋅= − +
⋅
= − +
= −
119. 3 45 15
5 3 5 43 5 3 15
5 43 155 4
3 5 349
x
x
x
x
x
− =
− ⋅− = − ⋅
⋅= −
⋅⋅
= −⋅ ⋅
= −
120. 5 3
12 6 45 312 12
12 6 4512 12 9
12 610 9
10 10 9 1019
x
x
x
xx
x
+ = −
⎛ ⎞ ⎛ ⎞+ = −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
⋅ + ⋅ = −
+ = −+ − = − −
= −
121. 1 50 1150 52 1 2
50 2 111 2 2
100 112 2
892
1442
− = −
⋅= −
⋅
= −
=
=
The length of the remaining piece is 1442
yards.
122. 1 4 11 81 11 81 81 15 7 or 402 11 2 11 2 11 2 2
⋅⋅ = ⋅ = =
⋅
The area is 812
or 1402
square feet.
Chapter 4 Test
1. 7 of the 16 equal parts are shaded: 7 .
16
2. 2 3 7 2 21 2 2373 3 3 3
⋅ + += = =
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ISM: Prealgebra Chapter 4: Fractions and Mixed Numbers
219
3. 18
4 7543532
3
75 3184 4=
4. 24 6 4 4210 6 7 5 35
⋅= =
⋅ ⋅
5. 42 3 14 370 5 14 5x x x⋅ ⋅
− = − = −⋅
6. 5 ⋅ 11 = 55 7 ⋅ 8 = 56 Since 55 ≠ 56, the fractions are not equivalent.
7. 6 ⋅ 63 = 378 27 ⋅ 14 = 378 Since 378 = 378, the fractions are equivalent.
8. 284 2 42 2 2 21 2 2 3 7 2 3 7= ⋅ = ⋅ ⋅ = ⋅ ⋅ ⋅ = ⋅ ⋅
9. 2495 3 165 3 3 55 3 3 5 11 3 5 11= ⋅ = ⋅ ⋅ = ⋅ ⋅ ⋅ = ⋅ ⋅
10. 4 3 4 4 4 4 44 4 4 3 4 3 3
⋅÷ = ⋅ = =
⋅
11. 4 4 4 4 43 4 3 4 3
⋅− ⋅ = − = −
⋅
12. 7 7 89 9 9 9x x x x x++ = =
13. The LCD of 7 and x is 7x. 1 3 1 3 7 21 217 7 7 7 7 7
x x xx x x x x x
−− = ⋅ − ⋅ = − =
14. 3 3
21
xy z xy z x y y y z y y yz xy z xy x y z
⋅ ⋅ ⋅ ⋅ ⋅ ⋅⋅ = = = =
⋅ ⋅ ⋅
15. 2 8 2 8 163 15 3 15 45
⋅− ⋅ − = =
⋅
16. 9 2 9 2 2 9 4 9 410 5 10 5 2 10 10 10a a a a⋅ ++ = + = + =
⋅
17. 8 2 8 2 10 2 5 2
15 15 15 15 3 5 3y y y y y y− − − ⋅
− − = = = − = −⋅ ⋅
18. 3 3 23 16 3 16 3 8 2 1 18 8 2 36 8 6a a a
a a a a aa a a⋅ ⋅ ⋅ ⋅
⋅ = = = =⋅ ⋅ ⋅ ⋅ ⋅ ⋅⋅
19.
11 3 5 11 2 3 3 512 8 24 12 2 8 3 24
22 9 524 24 2422 9 5
2418243 64 634
⋅ ⋅− + = − +
⋅ ⋅
= − +
− +=
=
⋅=
⋅
=
20. 738275324
+
35340167403024081 1 112 12 2 1440 40 40
+
= + =
21. 1932
11−
11181132
11816
11
−
22. 16 3 16 123 12 3 3
16 123 3
16 3 43 3
64 1or 213 3
− ÷− = − ⋅−
⋅=
⋅⋅ ⋅
=⋅
=
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Chapter 4: Fractions and Mixed Numbers ISM: Prealgebra
220
23. 1 3 10 273 63 4 3 4
10 273 4
2 5 3 93 2 2
5 92
45 1 or 222 2
⋅ = ⋅
⋅=
⋅⋅ ⋅ ⋅
=⋅ ⋅⋅
=
=
24.
2 1 2 6 167 6 7 1 6
2 6 6 17 1 6 62 36 17 6 62 357 62 357 62 7 57 2 35 2 or 13 3
⎛ ⎞ ⎛ ⎞− ⋅ − = − ⋅ −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
⋅⎛ ⎞= − ⋅ −⎜ ⎟⋅⎝ ⎠⎛ ⎞= − ⋅ −⎜ ⎟⎝ ⎠
= − ⋅
⋅= −
⋅⋅ ⋅
= −⋅ ⋅
= − −
25. 1 2 3 1 3 3 1 3 3 92 3 4 2 2 4 2 2 4 16
⋅ ⋅÷ ⋅ = ⋅ ⋅ = =
⋅ ⋅
26.
2 2
2
2
3 2 5 3 2 2 54 3 6 4 3 2 6
3 4 54 6 63 94 6
9 916 69 6
16 99 2 32 8 938
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞− ÷ + = − ÷ ⋅ +⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
⎛ ⎞ ⎛ ⎞= − ÷ +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
⎛ ⎞= − ÷⎜ ⎟⎝ ⎠
= ÷
= ⋅
⋅ ⋅=
⋅ ⋅
=
27.
5 4 7 5 2 4 4 73 36 3 12 6 2 3 4 12
10 16 7 312 12 12
33 312 133 112 333 112 311 3 112 3
1112
⋅ ⋅⎛ ⎞ ⎛ ⎞+ + ÷ = + + ÷⎜ ⎟ ⎜ ⎟⋅ ⋅⎝ ⎠ ⎝ ⎠⎛ ⎞= + + ÷⎜ ⎟⎝ ⎠
= ÷
= ⋅
⋅=
⋅⋅ ⋅
=⋅
=
28. 2
5 27
2021
2
5 207 21
5 217 205 3 7
7 4 53
4
x
x
x x
xx
xx x
x
= ÷
= ⋅
⋅ ⋅ ⋅=
⋅ ⋅ ⋅ ⋅
=
29. ( )( )
33 377 7
1 112 22
14 55 14 5 14 70 6 7628 7 212 14 2 1414 2
++ ⋅ + ⋅ += = = =
−− ⋅ − ⋅−
30. 3 38 4
8 3 8 33 8 3 4
8 33 44 2 33 4
2
x
x
x
x
x
− =
− ⋅− = − ⋅
⋅= −
⋅⋅ ⋅
= −⋅
= −
31. 24
5 5245 5
5 5
5 5 245
5 246 246 246 6
4
x x
x x
x x
x xxx
x
+ = −
⎛ ⎞ ⎛ ⎞+ = −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
⋅ + ⋅ = −
+ = −= −−
=
= −
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ISM: Prealgebra Chapter 4: Fractions and Mixed Numbers
221
32.
2 53 4 12 2
2 512 123 4 12 2
2 512 12 12 123 4 12 2
8 3 5 68 3 6 5 6 6
8 3 58 8 3 5 8
3 33 33 3
1
x x
x x
x x
x xx x x x
xxxx
x
+ = +
⎛ ⎞ ⎛ ⎞+ = +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
⋅ + ⋅ = ⋅ + ⋅
+ = ++ − = + −
− =− − = −
− = −− −
=− −
=
33. 1 5 1 5 1 55 52 1 2 1 2 2
x ⋅⎛ ⎞− = − − = ⋅ = =⎜ ⎟ ⋅⎝ ⎠
34.
1 732 81 312 81 82 311 82 311 2 42 31431
x y÷ = ÷
= ÷
= ⋅
⋅=
⋅⋅ ⋅
=⋅
=
35. 1 2 66 6 52 4 43 3 32 2 24 4 4
334
− − −
The remaining piece is 334
feet.
36. Housing: 825
Food: 750
8 7 8 2 7 16 7 16 7 2325 50 25 2 50 50 50 50 50
⋅ ++ = + = + = =
⋅
2350
of spending goes for housing and food
combined.
37. Education: 150
Transportation: 15
Clothing: 125
1 1 1 1 1 10 1 250 5 25 50 5 10 25 2
1 10 250 50 501350
+ + = + ⋅ + ⋅
= + +
=
1350
of spending goes for education,
transportation, and clothing.
38. 3 3 47,000 3 50 94047,000 2820
50 50 1 50 1⋅ ⋅
⋅ = ⋅ = =⋅
Expect to spend $2820 on health care.
39. 2 2perimeter 1 13 3
3 2 3 23 3 3 3103133
= + + +
= + + +
=
=
2 2area length width 13 3
= ⋅ = ⋅ =
The perimeter is 133
feet and the area is
23
square feet.
40. 3 258 43 258 4 43 6 4258 10 244 1 4 1 43 1 43
⋅ ⋅÷ = ÷ = ⋅ = =
⋅
Expect to travel 24 miles on 1 gallon of gas.
Cumulative Review Chapters 1−4
1. 546 in words is five hundred forty-six.
2. 115 in words is one hundred fifteen.
3. 27,034 in words is twenty-seven thousand, thirty-four.
4. 6573 in words is six thousand five hundred seventy-three.
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Chapter 4: Fractions and Mixed Numbers ISM: Prealgebra
222
5. 46713759
+
6. 58744
631+
7. 54329
514−
Check: 51429
543+
8. 99562
933−
Check: 93362
995+
9. 278,362 rounded to the nearest thousand is 278,000.
10. 1436 rounded to the nearest ten is 1440.
11. 480012
9 60048,00057,600
×
Therefore, 12 DVDs can hold 57,600 megabytes of information.
12. 4353
1305×
He travels 1305 miles in 3 days.
13. 7089
8 56,717560 7
07164
7772
5
56,717 ÷ 8 = 7089 R 5 Check: 7089 × 8 + 5 = 56,712 + 5 = 56,717
14. 379
12 455836958411810810
4558 ÷ 12 = 379 R 10 Check: 379 × 12 + 10 = 4548 + 10 = 4558
15. 37 7 7 7⋅ ⋅ =
16. 27 7 7⋅ =
17. 4 33 3 3 3 9 9 9 3 9⋅ ⋅ ⋅ ⋅ ⋅ ⋅ = ⋅
18. 4 29 9 9 9 5 5 9 5⋅ ⋅ ⋅ ⋅ ⋅ = ⋅
19. 2(x − y) = 2(6 − 3) = 2(3) = 6
20. 8 3( 5) 8 5 3(9 5)8 5 3 440 1252
a b+ − = ⋅ + −= ⋅ + ⋅= +=
21. Let 0 represent the surface of the earth. Then 6824 feet below the surface is represented as −6824.
22. Let 0 represent a temperature of 0°F. Then 21°F below zero is represented as −21.
23. 7 3 4− + =
24. −3 + 8 = 5
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ISM: Prealgebra Chapter 4: Fractions and Mixed Numbers
223
25. 7 8 ( 5) 1 7 8 5 17 ( 8) 5 ( 1)
1 5 ( 1)4 ( 1)3
− − − − = − + −= + − + + −= − + + −= + −=
26. 6 ( 8) ( 9) 3 6 ( 8) 9 32 9 3
7 310
+ − − − + = + − + += − + += +=
27. 2( 5) ( 5)( 5) 25− = − − =
28. 42 2 2 2 2 16− = − ⋅ ⋅ ⋅ = −
29. 3(4 7) ( 2) 5 3( 3) ( 2) 59 ( 2) ( 5)11 ( 5)16
− + − − = − + − −= − + − + −= − + −= −
30. 2 2 2 2(20 5 ) (20 25) ( 5) 25− = − = − =
31. 2 6 4 8 (2 4 ) ( 6 8) 6 2y y y y y− + + = + + − + = +
32. 5x − 1 + x + 10 = (5x + x) + (−1 + 10) = 6x + 9
33. 5 2 4 7 192 12
2 2 12 214
x xx
xx
+ − = −+ = −
+ − = − −= −
34. 9 1 8 3 201 17
1 1 17 118
y yy
yy
+ − = −+ = −
+ − = − −= −
35. 17 7 3 3 21 320 7 21 6
20 7 7 21 6 720 21
20 21 21 211 or 1
x x xx x
x x x xx
xx x
− + = − + −− = −
− + = − += +
− = − +− = = −
36. 9 2 7 249 7 2 7 7 24
2 2 242 2 2 24 2
2 222 222 2
11
x xx x x x
xx
xx
x
− = −− − = − −
− = −− + = − +
= −−
=
= −
37. Two of five equal parts are shaded: 25
38. 156 2 78
2 2 39
2 2 3 13
= ⋅↓ ↓⋅ ⋅
↓ ↓ ↓⋅ ⋅ ⋅
2156 2 3 13= ⋅ ⋅
39. a. 2 9 4 2 36 2 3849 9 9 9
⋅ + += = =
b. 8 11 1 8 11 8 191
11 11 11 11⋅ + +
= = =
40. 4 5 7 4 35 4 3975 5 5 5
⋅ + += = =
41. 42 6 7 766 6 11 11x x x⋅ ⋅= =
⋅
42. 70 35 2 2
105 35 3 3y y y⋅
= =⋅ ⋅
43. 1 7 10 733 8 3 8
10 73 82 5 73 2 45 73 435 11or 212 12
⋅ = ⋅
⋅=
⋅⋅ ⋅
=⋅ ⋅⋅
=⋅
=
44. 2 2 4 2 4 8 24 or 23 3 1 3 1 3 3
⋅⋅ = ⋅ = =
⋅
45. 5 3 5 4 5 4 5 4 5 5
16 4 16 3 16 3 4 4 3 4 3 12⋅ ⋅
÷ = ⋅ = = = =⋅ ⋅ ⋅ ⋅
46. 1 3 11 28 11 5 11 5 111 5
10 5 10 5 10 21 5 2 28 56⋅
÷ = ÷ = ⋅ = =⋅ ⋅
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