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2.4: Dividing Fractions

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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.

Screen Shot 2019-08-30 at 2.22.06 PM.png
Figure 2.4.1: One slice of pizza is 1/8 of one 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=48=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,

ab1=ab and 1ab=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.

abba=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

818=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

2332=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=abdc.

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=1253  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=3132  Invert the divisor (second number).=92  Multiply numerators; multiply denominators.

b) Note that 5 is understood to be 5/1.

45÷5=4515  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=6355533  Invert the divisor (second number).=6553533  Multiply numerators; multiply denominators.=(23)(511)(57)(311)  Factor numerators and denominators.=2351157311  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.=23xxx223  Factor numerators and denominators.=23xxx223  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. 2992=1

18. 12191912=1

19. 19121=1912

20. 1981=198

21. 6(16)=1

22. 19(119)=1

23. 16111=1611

24. 761=76

25. 41(14)=1

26. 910(109)=1

27. 811=81

28. 13151=1315

29. 14114=1

30. 414=1

31. 1381=138

32. 1131=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


This page titled 2.4: Dividing Fractions is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by David Arnold.

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