4.4: Dividing Fractions

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.

Figure \(\PageIndex{1}\): 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 ÷ \(\frac{1}{8}\).

On the other hand, to find the number of one-eights in four, Figure \(\PageIndex{1}\) 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,

\[\begin{align*} 4 ÷ 1/8 &= 4 \cdot 8 \\[4pt] &= 32. \end{align*}\]

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,

\[ \frac{a}{b} \cdot 1 = \frac{a}{b} \text{ and } 1 \cdot \frac{a}{b} = \frac{a}{b}.\nonumber \]

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.

\[ \frac{a}{b} \cdot \frac{b}{a} = 1\nonumber \]

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

\[ 8 \cdot \frac{1}{8} = 1.\nonumber \]

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 \(\PageIndex{1}\)

Find the multiplicative inverses (reciprocals) of: (a) 2/3, (b) −3/5, and (c) −12.

Solution

a) Because

\[ \frac{2}{3} \cdot \frac{3}{2} = 1,\nonumber \]

the multiplicative inverse (reciprocal) of 2/3 is 3/2.

b) Because

\[ - \frac{3}{5} \cdot \left( - \frac{5}{3} \right) = 1,\nonumber \]

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

\[ -12 \cdot \left( - \frac{1}{12} \right) = 1, \nonumber \]

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 \(\PageIndex{1}\)

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:

\[ \begin{align*} 4 ÷ \frac{1}{8} &= 4 · 8 \\[4pt] &= 32.\end{align*}\]

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

\[ \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}.\nonumber \]

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 \(\PageIndex{2}\)

Divide 1/2 by 3/5.

Solution

To divide 1/2 by 3/5, invert the divisor (second number), then multiply.

\[ \begin{align*} \frac{1}{2} \div \frac{3}{5} &= \frac{1}{2} \cdot \frac{5}{3} ~ && \textcolor{red}{ \text{ Invert the divisor (second number).}} \\[4pt] &= \frac{5}{6} ~&& \textcolor{red}{ \text{ Multiply.}} \end{align*}\]

Exercise \(\PageIndex{2}\)

Divide:

\[ \frac{2}{3} \div \frac{10}{3}\nonumber \]

Answer

1/5

Example \(\PageIndex{3}\)

Simplify the following expressions: (a) 3 ÷ \(\frac{2}{3}\) and (b) \(\frac{4}{5}\) ÷ 5.

Solution

In each case, invert the divisor (second number), then multiply.

a) Note that 3 is understood to be 3/1.

\[ \begin{align*} 3 \div \frac{2}{3} &= \frac{3}{1} \cdot \frac{3}{2} ~ && \textcolor{red}{ \text{ Invert the divisor (second number).}} \\[4pt] &= \frac{9}{2} ~ && \textcolor{red}{ \text{ Multiply numerators; multiply denominators.}} \end{align*} \]

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

\[ \begin{align*} \frac{4}{5} \div 5 &= \frac{4}{5} \cdot \frac{1}{5} ~ && \textcolor{red}{ \text{ Invert the divisor (second number).}} \\[4pt] &= \frac{4}{25} ~ && \textcolor{red}{ \text{ Multiply numerators; multiply denominators.}} \end{align*}\]

Exercise \(\PageIndex{3}\)

Divide:

\[ \frac{15}{7} \div 5\nonumber \]

Answer

\(\frac{3}{7}\)

After inverting, you may need to factor and cancel, as we learned to do in Section 4.2.

Example \(\PageIndex{4}\)

Divide −6/35 by 33/55.

Solution

Invert, multiply, factor, and cancel common factors.

\[ \begin{aligned} - \frac{6}{35} \div \frac{33}{55} = - \frac{6}{35} \cdot \frac{55}{33} ~ & \textcolor{red}{ \text{ Invert the divisor (second number).}} \\ = - \frac{6 \cdot 55}{35 \cdot 33} ~ & \textcolor{red}{ \text{ Multiply numerators; multiply denominators.}} \\ = - \frac{(2 \cdot 3) \cdot (5 \cdot 11)}{(5 \cdot 7) \cdot (3 \cdot 11)} ~ & \textcolor{red}{ \text{ Factor numerators and denominators.}} \\ = - \frac{2 \cdot \cancel{3} \cdot \cancel{5} \cdot \cancel{11}}{ \cancel{5} \cdot 7 \cdot \cancel{3} \cdot \cancel{11}} ~ & \textcolor{red}{ \text{ Cancel common factors.}} \\ = - \frac{2}{7} ~ & \textcolor{red}{ \text{ Remaining factors.}} \end{aligned}\nonumber \]

Note that unlike signs produce a negative answer.

Exercise \(\PageIndex{4}\)

Divide:

\[ \frac{6}{15} \div \left( - \frac{42}{35} \right)\nonumber \]

Answer

-1/3

Of course, you can also choose to factor numerators and denominators in place, then cancel common factors.

Example \(\PageIndex{5}\)

Divide \(−6/x\) by \(−12/x^2\).

Solution

Invert, factor numerators and denominators, cancel common factors, then multiply.

\[ \begin{align*} - \frac{6}{x} \div \left( - \frac{12}{x^2} \right) &= - \frac{6}{x} \cdot \left( - \frac{x^2}{12} \right) ~ && \textcolor{red}{ \text{ Invert second number.}} \\[4pt] &= - \frac{2 \cdot 3}{x} \cdot - \frac{x \cdot x}{2 \cdot 2 \cdot 3} ~ && \textcolor{red}{ \text{ Factor numerators and denominators.}} \\[4pt] &= - \frac{ \cancel{2} \cdot \cancel{3}}{ \cancel{x}} \cdot - \frac{ \cancel{x} \cdot x}{2 \cdot \cancel{2} \cdot \cancel{3}} ~ && \textcolor{red}{ \text{ Cancel common factors.}} \\[4pt] &= \frac{x}{2} ~ && \textcolor{red}{ \text{ Multiply.}} \end{align*}\]

Note that like signs produce a positive answer.

Exercise \(\PageIndex{5}\)

Divide:

\[ - \frac{12}{a} \div \left( - \frac{15}{a^3} \right)\nonumber \]

Answer

\[ - \frac{4 a^2}{5}\nonumber \]

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

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In Exercises 17-32, determine which property of multiplication is depicted by the given identity.

17. \(\frac{2}{9} \cdot \frac{9}{2} = 1\)

18. \(\frac{12}{19} \cdot \frac{19}{12} = 1\)

19. \( \frac{−19}{12} \cdot 1 = \frac{−19}{12}\)

20. \(\frac{−19}{8} \cdot 1 = \frac{−19}{8}\)

21. \(−6 \cdot \left( − \frac{1}{6} \right) = 1\)

22. \(−19 \cdot \left( − \frac{1}{19} \right) = 1\)

23. \( \frac{−16}{11} \cdot 1 = \frac{−16}{11}\)

24. \(\frac{−7}{6} \cdot 1 = \frac{−7}{6}\)

25. \(− \frac{4}{1} \cdot \left( − \frac{1}{4} \right) = 1\)

26. \(− \frac{9}{10} \cdot \left( − \frac{10}{9} \right) = 1\)

27. \( \frac{8}{1} \cdot 1 = \frac{8}{1}\)

28. \(\frac{13}{15} \cdot 1 = \frac{13}{15}\)

29. \(14 \cdot \frac{1}{14} = 1\)

30. \(4 \cdot \frac{1}{4} = 1\)

31. \( \frac{13}{8} \cdot 1 = \frac{13}{8}\)

32. \(\frac{1}{13} \cdot 1 = \frac{1}{13}\)

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In Exercises 33-56, divide the fractions, and simplify your result.

33. \(\frac{8}{23} \div \frac{−6}{11}\)

34. \(\frac{−10}{21} \div \frac{−6}{5}\)

35. \(\frac{18}{19} \div \frac{−16}{23}\)

36. \(\frac{13}{10} \div \frac{17}{18}\)

37. \(\frac{4}{21} \div \frac{−6}{5}\)

38. \(\frac{2}{9} \div \frac{−12}{19}\)

39. \(\frac{−1}{9} \div \frac{8}{3}\)

40. \(\frac{1}{2} \div \frac{−15}{8}\)

41. \(\frac{−21}{11} \div \frac{3}{10}\)

42. \(\frac{7}{24} \div \frac{−23}{2}\)

43. \(\frac{−12}{7} \div \frac{2}{3}\)

44. \(\frac{−9}{16} \div \frac{6}{7}\)

45. \(\frac{2}{19} \div \frac{24}{23}\)

46. \(\frac{7}{3} \div \frac{−10}{21}\)

47. \(\frac{−9}{5} \div \frac{−24}{19}\)

48. \(\frac{14}{17} \div \frac{−22}{21}\)

49. \(\frac{18}{11} \div \frac{14}{9}\)

50. \(\frac{5}{6} \div \frac{20}{19}\)

51. \(\frac{13}{18} \div \frac{4}{9}\)

52. \(\frac{−3}{2} \div \frac{−7}{12}\)

53. \(\frac{11}{2} \div \frac{−21}{10}\)

54. \(\frac{−9}{2} \div \frac{−13}{22}\)

55. \(\frac{3}{10} \div \frac{12}{5}\)

56. \(\frac{−22}{7} \div \frac{−18}{17}\)

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In Exercises 57-68, divide the fractions, and simplify your result.

57. \(\frac{20}{17} \div 5\)

58. \(\frac{21}{8} \div 7\)

59. \(−7 \div \frac{21}{20}\)

60. \(−3 \div \frac{12}{17}\)

61. \(\frac{8}{21} \div 2\)

62. \(\frac{−3}{4} \div (−6)\)

63. \(8 \div \frac{−10}{17}\)

64. \(−6 \div \frac{20}{21}\)

65. \(−8 \div \frac{18}{5}\)

66. \(6 \div \frac{−21}{8}\)

67. \(\frac{3}{4} \div (−9)\)

68. \(\frac{2}{9} \div (−8)\)

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In Exercises 69-80, divide the fractions, and simplify your result.

69. \(\frac{11x^2}{12} \div \frac{8x^4}{3}\)

70. \(\frac{−4x^2}{3} \div \frac{11x^6}{6}\)

71. \(\frac{17y}{9} \div \frac{10y^6}{3}\)

72. \(\frac{−5y}{12} \div \frac{−3y^5}{2}\)

73. \(\frac{−22x^4}{13} \div \frac{12x}{11}\)

74. \(\frac{−9y^6}{4} \div \frac{24y^5}{13}\)

75. \(\frac{−3x^4}{10} \div \frac{−4x}{5}\)

76. \(\frac{18y^4}{11} \div \frac{4y^2}{7}\)

77. \(\frac{−15y^2}{14} \div \frac{−10y^5}{13}\)

78. \(\frac{3x}{20} \div \frac{2x^3}{5}\)

79. \(\frac{−15x^5}{13} \div \frac{20x^2}{19}\)

80. \(\frac{18y^6}{7} \div \frac{14y^4}{9}\)

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In Exercises 81-96, divide the fractions, and simplify your result.

81. \(\frac{11y^4}{14x^2} \div \frac{−9y^2}{7x^3}\)

82. \(\frac{−5x^2}{12y^3} \div \frac{−22x}{21y^5}\)

83. \(\frac{10x^4}{3y^4} \div \frac{7x^5}{24y^2}\)

84. \(\frac{20x^3}{11y^5} \div \frac{5x^5}{6y^3}\)

85. \(\frac{22y^4}{21x^5} \div \frac{−5y^2}{6x^4}\)

86. \(\frac{−7y^5}{8x^6} \div \frac{21y}{5x^5}\)

87. \(\frac{−22x^4}{21y^3} \div \frac{−17x^3}{3y^4}\)

88. \(\frac{−7y^4}{4x} \div \frac{−15y}{22x^4}\)

89. \(\frac{−16y^2}{3x^3} \div \frac{2y^6}{11x^5}\)

90. \(\frac{−20x}{21y^2} \div \frac{−22x^5}{y^6}\)

91. \(\frac{−x}{12y^4} \div \frac{−23x^3}{16y^3}\)

92. \(\frac{20x^2}{17y^3} \div \frac{8x^3}{15y}\)

93. \(\frac{y^2}{4x} \div \frac{−9y^5}{8x^3}\)

94. \(\frac{−10y^4}{13x^2} \div \frac{−5y^6}{6x^3}\)

95. \(\frac{−18x^6}{13y^4} \div \frac{3x}{y^2}\)

96. \(\frac{20x^4}{9y^6} \div \frac{14x^2}{17y^4}\)

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Answers

1. \(− \frac{5}{16}\)

3. \(− \frac{1}{17}\)

5. \(\frac{16}{15}\)

7. \(\frac{1}{30}\)

9. \(− \frac{1}{46}\)

11. \(− \frac{19}{9}\)

13. \(\frac{17}{3}\)

15. \(\frac{1}{11}\)

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. \(− \frac{44}{69}\)

35. \(− \frac{207}{152}\)

37. \(− \frac{10}{63}\)

39. \(− \frac{1}{24}\)

41. \(− \frac{70}{11}\)

43. \(− \frac{18}{7}\)

45. \( \frac{23}{228}\)

47. \( \frac{57}{40}\)

49. \(\frac{81}{77}\)

51. \(\frac{13}{8}\)

53. \(− \frac{55}{21}\)

55. \(\frac{1}{8}\)

57. \(\frac{4}{17}\)

59. \(− \frac{20}{3}\)

61. \(\frac{4}{21}\)

63. \(− \frac{68}{5}\)

65. \(− \frac{20}{9}\)

67. \(− \frac{1}{12}\)

69. \(\frac{11}{32x^2}\)

71. \(\frac{17}{30y^5}\)

73. \(− \frac{121x^3}{78}\)

75. \(\frac{3x^3}{8}\)

77. \( \frac{39}{28y^3}\)

79. \(− \frac{57x^3}{52}\)

81. \(− \frac{11y^2x}{18}\)

83. \(\frac{80}{7xy^2}\)

85. \(− \frac{44y^2}{35x}\)

87. \(\frac{22xy}{119}\)

89. \(− \frac{88x^2}{3y^4}\)

91. \(\frac{4}{69x^2y}\)

93. \(− \frac{2x^2}{9y^3}\)

95. \(− \frac{6x^5}{13y^2}\)