Lesson Objectives

- Demonstrate an understanding of how to multiply fractions
- Learn how to find the reciprocal of a number
- Learn how to divide fractions

## How to Divide Fractions

### How to Find the Reciprocal of a Number

Before we can divide fractions, we must first learn how to find the reciprocal of a fraction. To find the reciprocal of a fraction, we simply interchange the numerator and the denominator. In other words, we will flip the fraction. Let's take a look at a few examples:Example 1: Find the reciprocal of 1/8, 14/17, and 9.

To find the reciprocal of 1/8, we interchange the numerator and denominator. This means 8 will go into the numerator and 1 will go into the denominator. The reciprocal of 1/8 is 8/1 or just 8.

To find the reciprocal of 14/17, we interchange the numerator and denominator. This means 17 will go into the numerator and 14 will go into the denominator. The reciprocal of 14/17 is 17/14.

To find the reciprocal of 9, we first write 9 as a fraction. 9 can be written as 9/1, since 9 ÷ 1 = 9. Now, we can just interchange the numerator and denominator. This means 9 will go into the numerator and 1 will go into the denominator. The reciprocal of 9 is 1/9.

Now that we know how to find the reciprocal of a number, we can move on to division of fractions. Let's begin by thinking about a pizza once again. Our pizza above is cut up into 8 equal slices. If we eat one slice of pizza each day for 5 days, the total amount eaten would be 5/8 of the pizza. $$5 \cdot \frac{1}{8}=\frac{5}{8}$$ Since we know that 5 • 1/8 is 5/8, then it should be true that 5/8 ÷ 1/8 is 5. $$\frac{5}{8}\div \frac{1}{8}=5$$ How can we get this result? In other words, we are asking how many equal groups of 1/8 can be made from 5/8? At this point, we have not yet studied complex fractions. This is just a fraction that contains a fraction in its numerator, denominator, or both. Here, we will just write our division using a complex fraction. $$\Large{\frac{\frac{5}{8}}{\frac{1}{8}}}$$ Let's multiply both the numerator and denominator of the complex fraction by the reciprocal of the denominator 8/1. $$\require{cancel}\large{\frac{\frac{5}{\cancel{8}}\cdot \frac{\cancel{8}}{1}}{\frac{1}{\cancel{8}}\cdot \frac{\cancel{8}}{1}}}=\frac{\frac{5}{1}}{\frac{1}{1}}=\frac{5}{1}=5$$ When we divide fractions, we will multiply the leftmost fraction by the reciprocal of the rightmost fraction. $$\frac{5}{8}\div \frac{1}{8}=\frac{5}{8}\cdot \frac{8}{1}=5$$

### How to Divide one Fraction by Another

- Keep the leftmost fraction unchanged
- Find the reciprocal of the rightmost fraction
- Find the product of the leftmost fraction and the reciprocal of the rightmost fraction

Example 2: Find each quotient. $$\frac{3}{14}÷ \frac{9}{28}$$ $$\frac{3}{14}÷ \frac{9}{28}=\frac{3}{14}\cdot \frac{28}{9}$$ $$\require{cancel}\frac{3}{14}\cdot \frac{28}{9}=\frac{\cancel{3}1}{\cancel{14}1}\cdot \frac{\cancel{28}2}{\cancel{9}3}=\frac{2}{3}$$ $$\frac{3}{14}÷ \frac{9}{28}=\frac{2}{3}$$ Example 3: Find each quotient. $$\frac{5}{18}÷ \frac{20}{27}$$ $$\frac{5}{18}÷ \frac{20}{27}=\frac{5}{18}\cdot \frac{27}{20}$$ $$\frac{5}{18}\cdot \frac{27}{20}=\frac{\cancel{5}1}{\cancel{18}2}\cdot \frac{\cancel{27}3}{\cancel{20}4}=\frac{3}{8}$$ $$\frac{5}{18}÷ \frac{27}{20}=\frac{3}{8}$$

### Dividing a Fraction by a Whole Number

In some cases, we will either divide a whole number by a fraction or a fraction by a whole number. When this situation arises, we can write our whole number as a fraction with a denominator of 1.Example 4: Find each quotient. $$\frac{12}{19}÷ 4$$ $$\frac{12}{19}÷ \frac{4}{1}=\frac{12}{19}\cdot \frac{1}{4}$$ $$\frac{12}{19}\cdot \frac{1}{4}=\frac{\cancel{12}3}{19}\cdot \frac{1}{\cancel{4}1}=\frac{3}{19}$$ $$\frac{12}{19}÷ 4=\frac{3}{19}$$ Example 5: Find each quotient. $$6 ÷ \frac{3}{5}$$ $$6 ÷ \frac{3}{5}=\frac{6}{1}\cdot \frac{5}{3}$$ $$\frac{\cancel{6}2}{1}\cdot \frac{5}{\cancel{3}1}=10$$ $$6 ÷ \frac{3}{5}=10$$

#### Skills Check:

Example #1

Find each quotient. $$\frac{44}{32}\div \frac{38}{25}$$

Please choose the best answer.

A

$$\frac{275}{304}$$

B

$$\frac{51}{59}$$

C

$$\frac{17}{103}$$

D

$$\frac{19}{39}$$

E

$$\frac{401}{551}$$

Example #2

Find each quotient. $$\frac{16}{18}\div \frac{24}{45}$$

Please choose the best answer.

A

$$\frac{51}{43}$$

B

$$\frac{3}{7}$$

C

$$\frac{4}{5}$$

D

$$\frac{5}{3}$$

E

$$\frac{12}{5}$$

Example #3

Find each quotient. $$\frac{6}{9}\div 10$$

Please choose the best answer.

A

$$\frac{5}{14}$$

B

$$\frac{9}{5}$$

C

$$\frac{3}{2}$$

D

$$\frac{1}{15}$$

E

$$\frac{12}{13}$$

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