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 ÷ \(\frac{1}{8}\).
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
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. \(\frac{1}{13} \cdot 1 = \frac{1}{13}\)
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. \(\frac{−22}{7} \div \frac{−18}{17}\)
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. \(\frac{2}{9} \div (−8)\)
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. \(\frac{18y^6}{7} \div \frac{14y^4}{9}\)
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. \(\frac{20x^4}{9y^6} \div \frac{14x^2}{17y^4}\)