2.4: Dividing Fractions
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- Jan 31, 2020
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Suppose that you have four pizzas and each of the pizzas has been sliced into eight equal slices. Therefore, each slice of pizza represents 1/8 of a whole pizza.

Now for the question: How many one-eighths are there in four? This is a division statement. To find how many one-eighths there are in 4, divide 4 by 1/8. That is,
Number of one-eighths in four = 4 ÷ 18.
On the other hand, to find the number of one-eights in four, Figure 2.4.1 clearly demonstrates that this is equivalent to asking how many slices of pizza are there in four pizzas. Since there are 8 slices per pizza and four pizzas,
Number of pizza slices = 4 · 8.
The conclusion is the fact that 4 ÷ (1/8) is equivalent to 4 · 8. That is,
4÷1/8=4⋅8=32.
Therefore, we conclude that there are 32 one-eighths in 4.
Reciprocals
The number 1 is still the multiplicative identity for fractions.
Multiplicative Identity Property
Let a/b be any fraction. Then,
ab⋅1=ab and 1⋅ab=ab.
The number 1 is called the multiplicative identity because the identical number is returned when you multiply by 1.
Next, if we invert 3/4, that is, if we turn 3/4 upside down, we get 4/3. Note what happens when we multiply 3/4 by 4/3.
The number 4/3 is called the multiplicative inverse or reciprocal of 3/4. The product of reciprocals is always 1.
Multiplicative Inverse Property
Let a/b be any fraction. The number b/a is called the multiplicative inverse or reciprocal of a/b. The product of reciprocals is 1.
ab⋅ba=1
Note: To find the multiplicative inverse (reciprocal) of a number, simply invert the number (turn it upside down).
For example, the number 1/8 is the multiplicative inverse (reciprocal) of 8 because
8⋅18=1.
Note that 8 can be thought of as 8/1. Invert this number (turn it upside down) to find its multiplicative inverse (reciprocal) 1/8.
Example 2.4.1
Find the multiplicative inverses (reciprocals) of: (a) 2/3, (b) −3/5, and (c) −12.
Solution
a) Because
23⋅32=1,
the multiplicative inverse (reciprocal) of 2/3 is 3/2.
b) Because
−35⋅(−53)=1,
the multiplicative inverse (reciprocal) of −3/5 is −5/3. Again, note that we simply inverted the number −3/5 to get its reciprocal −5/3.
c) Because
−12⋅(−112)=1,
the multiplicative inverse (reciprocal) of −12 is −1/12. Again, note that we simply inverted the number −12 (understood to equal −12/1) to get its reciprocal −1/12.
Exercise 2.4.1
Find the reciprocals of: (a) −3/7 and (b) 15
- Answer
-
(a) −7/3, (b) 1/15
Division
Recall that we computed the number of one-eighths in four by doing this calculation:
4÷18=4·8=32.
Note how we inverted the divisor (second number), then changed the division to multiplication. This motivates the following definition of division.
Division Definition
If a/b and c/d are any fractions, then
ab÷cd=ab⋅dc.
That is, we invert the divisor (second number) and change the division to multiplication. Note: We like to use the phrase “invert and multiply” as a memory aid for this definition.
Example 2.4.2
Divide 1/2 by 3/5.
Solution
To divide 1/2 by 3/5, invert the divisor (second number), then multiply.
12÷35=12⋅53 Invert the divisor (second number).=56 Multiply.
Exercise 2.4.2
Divide:
23÷103
- Answer
-
1/5
Example 2.4.3
Simplify the following expressions: (a) 3 ÷ 23 and (b) 45 ÷ 5.
Solution
In each case, invert the divisor (second number), then multiply.
a) Note that 3 is understood to be 3/1.
3÷23=31⋅32 Invert the divisor (second number).=92 Multiply numerators; multiply denominators.
b) Note that 5 is understood to be 5/1.
45÷5=45⋅15 Invert the divisor (second number).=425 Multiply numerators; multiply denominators.
Exercise 2.4.3
Divide:
157÷5
Answer
-
37
After inverting, you may need to factor and cancel, as we learned to do in Section 4.2.
Example 2.4.4
Divide −6/35 by 33/55.
Solution
Invert, multiply, factor, and cancel common factors.
−635÷3355=−635⋅5533 Invert the divisor (second number).=−6⋅5535⋅33 Multiply numerators; multiply denominators.=−(2⋅3)⋅(5⋅11)(5⋅7)⋅(3⋅11) Factor numerators and denominators.=−2⋅3⋅5⋅115⋅7⋅3⋅11 Cancel common factors.=−27 Remaining factors.
Note that unlike signs produce a negative answer.
Exercise 2.4.4
Divide:
615÷(−4235)
- Answer
-
-1/3
Of course, you can also choose to factor numerators and denominators in place, then cancel common factors.
Example 2.4.5
Divide −6/x by −12/x2.
Solution
Invert, factor numerators and denominators, cancel common factors, then multiply.
−6x÷(−12x2)=−6x⋅(−x212) Invert second number.=−2⋅3x⋅−x⋅x2⋅2⋅3 Factor numerators and denominators.=−2⋅3x⋅−x⋅x2⋅2⋅3 Cancel common factors.=x2 Multiply.
Note that like signs produce a positive answer.
Exercise 2.4.5
Divide:
−12a÷(−15a3)
- Answer
-
−4a25
Exercises
In Exercises 1-16, find the reciprocal of the given number.
1. −16/5
2. −3/20
3. −17
4. −16
5. 15/16
6. 7/9
7. 30
8. 28
9. −46
10. −50
11. −9/19
12. −4/7
13. 3/17
14. 3/5
15. 11
16. 48
In Exercises 17-32, determine which property of multiplication is depicted by the given identity.
17. 29⋅92=1
18. 1219⋅1912=1
19. −1912⋅1=−1912
20. −198⋅1=−198
21. −6⋅(−16)=1
22. −19⋅(−119)=1
23. −1611⋅1=−1611
24. −76⋅1=−76
25. −41⋅(−14)=1
26. −910⋅(−109)=1
27. 81⋅1=81
28. 1315⋅1=1315
29. 14⋅114=1
30. 4⋅14=1
31. 138⋅1=138
32. 113⋅1=113
In Exercises 33-56, divide the fractions, and simplify your result.
33. 823÷−611
34. −1021÷−65
35. 1819÷−1623
36. 1310÷1718
37. 421÷−65
38. 29÷−1219
39. −19÷83
40. 12÷−158
41. −2111÷310
42. 724÷−232
43. −127÷23
44. −916÷67
45. 219÷2423
46. 73÷−1021
47. −95÷−2419
48. 1417÷−2221
49. 1811÷149
50. 56÷2019
51. 1318÷49
52. −32÷−712
53. 112÷−2110
54. −92÷−1322
55. 310÷125
56. −227÷−1817
In Exercises 57-68, divide the fractions, and simplify your result.
57. 2017÷5
58. 218÷7
59. −7÷2120
60. −3÷1217
61. 821÷2
62. −34÷(−6)
63. 8÷−1017
64. −6÷2021
65. −8÷185
66. 6÷−218
67. 34÷(−9)
68. 29÷(−8)
In Exercises 69-80, divide the fractions, and simplify your result.
69. 11x212÷8x43
70. −4x23÷11x66
71. 17y9÷10y63
72. −5y12÷−3y52
73. −22x413÷12x11
74. −9y64÷24y513
75. −3x410÷−4x5
76. 18y411÷4y27
77. −15y214÷−10y513
78. 3x20÷2x35
79. −15x513÷20x219
80. 18y67÷14y49
In Exercises 81-96, divide the fractions, and simplify your result.
81. 11y414x2÷−9y27x3
82. −5x212y3÷−22x21y5
83. 10x43y4÷7x524y2
84. 20x311y5÷5x56y3
85. 22y421x5÷−5y26x4
86. −7y58x6÷21y5x5
87. −22x421y3÷−17x33y4
88. −7y44x÷−15y22x4
89. −16y23x3÷2y611x5
90. −20x21y2÷−22x5y6
91. −x12y4÷−23x316y3
92. 20x217y3÷8x315y
93. y24x÷−9y58x3
94. −10y413x2÷−5y66x3
95. −18x613y4÷3xy2
96. 20x49y6÷14x217y4
Answers
1. −516
3. −117
5. 1615
7. 130
9. −146
11. −199
13. 173
15. 111
17. multiplicative inverse property
19. multiplicative identity property
21. multiplicative inverse property
23. multiplicative identity property
25. multiplicative inverse property
27. multiplicative identity property
29. multiplicative inverse property
31. multiplicative identity property
33. −4469
35. −207152
37. −1063
39. −124
41. −7011
43. −187
45. 23228
47. 5740
49. 8177
51. 138
53. −5521
55. 18
57. 417
59. −203
61. 421
63. −685
65. −209
67. −112
69. 1132x2
71. 1730y5
73. −121x378
75. 3x38
77. 3928y3
79. −57x352
81. −11y2x18
83. 807xy2
85. −44y235x
87. 22xy119
89. −88x23y4
91. 469x2y
93. −2x29y3
95. −6x513y2