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.

### 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 $$6 ÷ \frac{3}{5}$$ $$6 ÷ \frac{3}{5} = \frac{6}{1} \cdot \frac{5}{3}$$ $$\frac{\cancel{6}2}{1} \cdot \frac{5}{\cancel{3}1} = 10$$ $$6 ÷ \frac{3}{5} = 10$$

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