4.5: Division of Fractions

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Learning Objectives

• be able to determine the reciprocal of a number
• be able to divide one fraction by another

Reciprocals

Definition: Reciprocals

Two numbers whose product is 1 are called reciprocals of each other.

Sample Set A

The following pairs of numbers are reciprocals.

$$\underbrace{\dfrac{3}{4} \text{and} \dfrac{4}{3}}_{\dfrac{3}{4} \cdot \dfrac{4}{3} = 1}$$

Sample Set A

$$\underbrace{\dfrac{7}{16} \text{and} \dfrac{16}{7}}_{\dfrac{7}{16} \cdot \dfrac{16}{7} = 1}$$

Sample Set A

$$\underbrace{\dfrac{1}{6} \text{and} \dfrac{6}{1}}_{\dfrac{1}{6} \cdot \dfrac{6}{1} = 1}$$

Notice that we can find the reciprocal of a nonzero number in fractional form by inverting it (exchanging positions of the numerator and denominator).

Practice Set A

Find the reciprocal of each number.

$$\dfrac{3}{10}$$

$$\dfrac{10}{3}$$

Practice Set A

$$\dfrac{2}{3}$$

$$\dfrac{3}{2}$$

Practice Set A

$$\dfrac{7}{8}$$

$$\dfrac{8}{7}$$

Practice Set A

$$\dfrac{1}{5}$$

5

Practice Set A

$$2 \dfrac{2}{7}$$

Hint

Write this number as an improper fraction first.

$$\dfrac{7}{16}$$

Practice Set A

$$5 \dfrac{1}{4}$$

$$\dfrac{4}{21}$$

Practice Set A

$$10 \dfrac{3}{16}$$

$$\dfrac{16}{163}$$

Dividing Fractions

Our concept of division is that it indicates how many times one quantity is con­tained in another quantity. For example, using the diagram we can see that there are 6 one-thirds in 2.

There are 6 one-thirds in 2.

Since 2 contains six $$\dfrac{1}{3}$$'s we express this as

Using these observations, we can suggest the following method for dividing a number by a fraction.

Dividing One Fraction by Another Fraction
To divide a first fraction by a second, nonzero fraction, multiply the first traction by the reciprocal of the second fraction.

Invert and Multiply
This method is commonly referred to as "invert the divisor and multiply."

Sample Set B

Perform the following divisions.

$$\dfrac{1}{2} \div \dfrac{3}{4}$$. The divisor is $$\dfrac{3}{4}$$. Its reciprocal is $$\dfrac{4}{3}$$. Multiply $$\dfrac{1}{3}$$ by $$\dfrac{4}{3}$$.

$$\dfrac{1}{3} \cdot \dfrac{4}{3} = \dfrac{1 \cdot 4}{3 \cdot 3} = \dfrac{4}{9}$$

$$\dfrac{1}{2} \div \dfrac{3}{4} = \dfrac{4}{9}$$

Sample Set B

$$\dfrac{3}{8} \div \dfrac{5}{4}$$. The divisor is $$\dfrac{5}{4}$$. Its reciprocal is $$\dfrac{4}{5}$$. Multiply $$\dfrac{3}{8}$$ by $$\dfrac{4}{5}$$.

$$\dfrac{3}{\begin{array} {c} {\cancel{8}} \\ {^2} \end{array}} \cdot \dfrac{\begin{array} {c} {^1} \\ {\cancel{4}} \end{array}}{5} = \dfrac{3 \cdot 1}{2 \cdot 5} = \dfrac{3}{10}$$

$$\dfrac{3}{8} \div \dfrac{5}{4} = \dfrac{3}{10}$$

Sample Set B

$$\dfrac{5}{6} \div \dfrac{5}{12}$$. The divisor is $$\dfrac{5}{12}$$. Its reciprocal is $$\dfrac{12}{5}$$. Multiply $$\dfrac{5}{6}$$ by $$\dfrac{12}{5}$$.

$$\dfrac{\begin{array} {c} {^1} \\ {\cancel{5}} \end{array}}{\begin{array} {c} {\cancel{6}} \\ {^1} \end{array}} \cdot \dfrac{\begin{array} {c} {^2} \\ {\cancel{12}} \end{array}}{\begin{array} {c} {\cancel{5}} \\ {^1} \end{array}} = \dfrac{1 \cdot 2}{1 \cdot 1} = \dfrac{2}{1} = 2$$

$$\dfrac{5}{6} \div \dfrac{5}{12} = 2$$

Sample Set B

$$2 \dfrac{2}{9} \div 3 \dfrac{1}{3}$$. Convert each mixed number to an improper fraction.

$$2 \dfrac{2}{9} = \dfrac{9 \cdot 2 + 2}{9} = \dfrac{20}{9}$$.

$$3 \dfrac{1}{3} = \dfrac{3 \cdot 3 + 1}{3} = \dfrac{10}{3}$$.

$$\dfrac{20}{9} \div \dfrac{10}{3}$$ The divisor is $$\dfrac{10}{3}$$. Its reciprocal is $$\dfrac{3}{10}$$. Multiply $$\dfrac{20}{9}$$ by $$\dfrac{3}{10}$$.

$$\dfrac{\begin{array} {c} {^2} \\ {\cancel{20}} \end{array}}{\begin{array} {c} {\cancel{9}} \\ {^3} \end{array}} \cdot \dfrac{\begin{array} {c} {^1} \\ {\cancel{3}} \end{array}}{\begin{array} {c} {\cancel{10}} \\ {^1} \end{array}} = \dfrac{2 \cdot 1}{3 \cdot 1} = \dfrac{2}{3}$$

$$2 \dfrac{2}{9} \div 3 \dfrac{1}{3} = \dfrac{2}{3}$$

Sample Set B

$$\dfrac{12}{11} \div 8$$. First conveniently write 8 as $$\dfrac{8}{1}$$.

$$\dfrac{12}{11} \div \dfrac{8}{1}$$. The divisor is $$\dfrac{8}{1}$$. Its reciprocal is $$\dfrac{1}{8}$$. Multiply $$\dfrac{12}{11}$$ by $$\dfrac{1}{8}$$.

$$\dfrac{\begin{array} {c} {^3} \\ {\cancel{12}} \end{array}}{11} \cdot \dfrac{1}{\begin{array} {c} {\cancel{8}} \\ {^2} \end{array}} = \dfrac{3 \cdot 1}{11 \cdot 2} = \dfrac{3}{22}$$

$$\dfrac{12}{11} \div 8 = \dfrac{3}{22}$$

Sample Set B

$$\dfrac{7}{8} \div \dfrac{21}{20} \cdot \dfrac{3}{35}$$. The divisor is $$\dfrac{21}{20}$$. Its reciprocal is $$\dfrac{20}{21}$$.

$$\dfrac{\begin{array} {c} {^1} \\ {\cancel{7}} \end{array}}{\begin{array} {c} {\cancel{8}} \\ {^2} \end{array}} \cdot \dfrac{\begin{array} {c} {^{^1}} \\ {^{\cancel{5}}} \\ {\cancel{20}} \end{array}}{\begin{array} {c} {\cancel{21}} \\ {^{\cancel{3}}} \\ {^{^1}} \end{array}} \dfrac{\begin{array} {c} {^1} \\ {\cancel{3}} \end{array}}{\begin{array} {c} {\cancel{35}} \\ {^7} \end{array}} = \dfrac{1 \cdot 1 \cdot 1}{2 \cdot 1 \cdot 7} = \dfrac{1}{14}$$

$$\dfrac{7}{8} \div \dfrac{21}{20} \cdot \dfrac{3}{25} = \dfrac{1}{14}$$

Sample Set B

How many $$2 \dfrac{3}{8}$$ inch-wide packages can be placed in a box 19 inches wide?

The problem is to determine how many two and three eighths are contained in 19, that is, what is $$19 \div 2 \dfrac{3}{8}$$?

$$2\dfrac{3}{8} = \dfrac{19}{8}$$ Convert the divisor $$2 \dfrac{3}{8}$$ to an improper fraction.

$$19 = \dfrac{19}{1}$$ Write the dividend 19 as $$\dfrac{19}{1}$$.

$$\dfrac{19}{1} \div \dfrac{19}{8}$$ The divisor is $$\dfrac{19}{8}$$. Its reciprocal is $$\dfrac{8}{19}$$.

$$\dfrac{\begin{array} {c} {^1} \\ {\cancel{19}} \end{array}}{1} \cdot \dfrac{8}{\begin{array} {c} {\cancel{19}} \\ {^1} \end{array}} = \dfrac{1 \cdot 8}{1 \cdot 1} = \dfrac{8}{1} = 8$$

Thus, 8 packages will fit into the box.

Practice Set B

Perform the following divisions.

$$\dfrac{1}{2} \div \dfrac{9}{8}$$

$$\dfrac{4}{9}$$

Practice Set B

$$\dfrac{3}{8} \div \dfrac{9}{24}$$

1

Practice Set B

$$\dfrac{7}{15} \div \dfrac{14}{15}$$

$$\dfrac{1}{2}$$

Practice Set B

$$8 \div \dfrac{8}{15}$$

15

Practice Set B

$$6 \dfrac{1}{4} \div \dfrac{5}{12}$$

15

Practice Set B

$$3 \dfrac{1}{3} \div 1 \dfrac{2}{3}$$

2

Practice Set B

$$\dfrac{5}{6} \div \dfrac{2}{3} \cdot \dfrac{8}{25}$$

$$\dfrac{2}{5}$$

Practice Set B

A container will hold 106 ounces of grape juice. How many $$6 \dfrac{5}{8}$$ -ounce glasses of grape juice can be served from this container?

16 glasses

Determine each of the following quotients and then write a rule for this type of division.

Practice Set B

$$1 \div \dfrac{2}{3}$$

$$\dfrac{3}{2}$$

Practice Set B

$$1 \div \dfrac{3}{8}$$

$$\dfrac{8}{3}$$

Practice Set B

$$1 \div \dfrac{3}{4}$$

$$\dfrac{4}{3}$$

Practice Set B

$$1 \div \dfrac{5}{2}$$

$$\dfrac{2}{5}$$

Practice Set B

When dividing 1 by a fraction, the quotient is the .

is the reciprocal of the fraction.

Exercises

For the following problems, find the reciprocal of each number.

Exercise $$\PageIndex{1}$$

$$\dfrac{4}{5}$$

$$\dfrac{5}{4}$$ or $$1 \dfrac{1}{4}$$

Exercise $$\PageIndex{2}$$

$$\dfrac{8}{11}$$

Exercise $$\PageIndex{3}$$

$$\dfrac{2}{9}$$

$$\dfrac{9}{2}$$ or $$4 \dfrac{1}{2}$$

Exercise $$\PageIndex{4}$$

$$\dfrac{1}{5}$$

Exercise $$\PageIndex{5}$$

$$3\dfrac{1}{4}$$

$$\dfrac{4}{13}$$

Exercise $$\PageIndex{6}$$

$$8 \dfrac{1}{4}$$

Exercise $$\PageIndex{7}$$

$$3\dfrac{2}{7}$$

$$\dfrac{7}{23}$$

Exercise $$\PageIndex{8}$$

$$5 \dfrac{3}{4}$$

Exercise $$\PageIndex{9}$$

1

1

Exercise $$\PageIndex{10}$$

4

For the following problems, find each value.

Exercise $$\PageIndex{11}$$

$$\dfrac{3}{8} \div \dfrac{3}{5}$$

$$\dfrac{5}{8}$$

Exercise $$\PageIndex{12}$$

$$\dfrac{5}{9} \div \dfrac{5}{6}$$

Exercise $$\PageIndex{13}$$

$$\dfrac{9}{16} \div \dfrac{15}{8}$$

$$\dfrac{3}{10}$$

Exercise $$\PageIndex{14}$$

$$\dfrac{4}{9} \div \dfrac{6}{15}$$

Exercise $$\PageIndex{15}$$

$$\dfrac{25}{49} \div \dfrac{4}{9}$$

$$\dfrac{225}{196}$$ or $$1 \dfrac{29}{196}$$

Exercise $$\PageIndex{16}$$

$$\dfrac{15}{4} \div \dfrac{27}{8}$$

Exercise $$\PageIndex{17}$$

$$\dfrac{24}{75} \div \dfrac{8}{15}$$

$$\dfrac{3}{5}$$

Exercise $$\PageIndex{18}$$

$$\dfrac{5}{7} \div 0$$

Exercise $$\PageIndex{19}$$

$$\dfrac{7}{8} \div \dfrac{7}{8}$$

1

Exercise $$\PageIndex{20}$$

$$0 \div \dfrac{3}{5}$$

Exercise $$\PageIndex{21}$$

$$\dfrac{4}{11} \div \dfrac{4}{11}$$

1

Exercise $$\PageIndex{22}$$

$$\dfrac{2}{3} \div \dfrac{2}{3}$$

Exercise $$\PageIndex{23}$$

$$\dfrac{7}{10} \div \dfrac{10}{7}$$

$$\dfrac{49}{100}$$

Exercise $$\PageIndex{24}$$

$$\dfrac{3}{4} \div 6$$

Exercise $$\PageIndex{25}$$

$$\dfrac{9}{5} \div 3$$

$$\dfrac{3}{5}$$

Exercise $$\PageIndex{26}$$

$$4 \dfrac{1}{6} \div 3 \dfrac{1}{3}$$

Exercise $$\PageIndex{27}$$

$$7 \dfrac{1}{7} \div 8 \dfrac{1}{3}$$

$$\dfrac{6}{7}$$

Exercise $$\PageIndex{28}$$

$$1 \dfrac{1}{2} \div 1 \dfrac{1}{5}$$

Exercise $$\PageIndex{29}$$

$$3 \dfrac{2}{5} \div \dfrac{6}{25}$$

$$\dfrac{85}{6}$$ or $$14 \dfrac{1}{6}$$

Exercise $$\PageIndex{30}$$

$$5 \dfrac{1}{6} \div \dfrac{31}{6}$$

Exercise $$\PageIndex{31}$$

$$\dfrac{35}{6} \div 3 \dfrac{3}{4}$$

$$\dfrac{28}{18} = \dfrac{14}{9}$$ or $$1 \dfrac{5}{9}$$

Exercise $$\PageIndex{32}$$

$$5 \dfrac{1}{9} \div \dfrac{1}{18}$$

Exercise $$\PageIndex{33}$$

$$8 \dfrac{3}{4} \div \dfrac{7}{8}$$

10

Exercise $$\PageIndex{34}$$

$$\dfrac{12}{8} \div 1 \dfrac{1}{2}$$

Exercise $$\PageIndex{35}$$

$$3 \dfrac{1}{8} \div \dfrac{15}{16}$$

$$\dfrac{10}{3}$$ or $$3 \dfrac{1}{3}$$

Exercise $$\PageIndex{36}$$

$$11 \dfrac{11}{12} \div 9 \dfrac{5}{8}$$

Exercise $$\PageIndex{37}$$

$$2 \dfrac{2}{9} \div 11 \dfrac{2}{3}$$

$$\dfrac{4}{21}$$

Exercise $$\PageIndex{38}$$

$$\dfrac{16}{3} \div 6 \dfrac{2}{5}$$

Exercise $$\PageIndex{39}$$

$$4 \dfrac{3}{25} \div 2 \dfrac{56}{75}$$

$$\dfrac{3}{2}$$ or $$1 \dfrac{1}{2}$$

Exercise $$\PageIndex{40}$$

$$\dfrac{1}{1000} \div \dfrac{1}{100}$$

Exercise $$\PageIndex{41}$$

$$\dfrac{3}{8} \div \dfrac{9}{16} \cdot \dfrac{6}{5}$$

$$\dfrac{4}{5}$$

Exercise $$\PageIndex{42}$$

$$\dfrac{3}{16} \cdot \dfrac{9}{8} \cdot \dfrac{6}{5}$$

Exercise $$\PageIndex{43}$$

$$\dfrac{4}{15} \div \dfrac{2}{25} \cdot \dfrac{9}{10}$$

3

Exercise $$\PageIndex{44}$$

$$\dfrac{21}{30} \cdot 1 \dfrac{1}{4} \div \dfrac{9}{10}$$

Exercise $$\PageIndex{45}$$

$$8 \dfrac{1}{3} \cdot \dfrac{36}{75} \div 4$$

1

Exercises for Review

Exercise $$\PageIndex{46}$$

What is the value of 5 in the number 504,216?

Exercise $$\PageIndex{47}$$

Find the product of 2,010 and 160.

321,600

Exercise $$\PageIndex{48}$$

Use the numbers 8 and 5 to illustrate the commutative property of multiplication.

Exercise $$\PageIndex{49}$$

Find the least common multiple of 6, 16, and 72.

Exercise $$\PageIndex{50}$$
Find $$\dfrac{8}{9}$$ of $$6 \dfrac{3}{4}$$