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
In other words, to divide fractions we multiply the first fraction (leftmost) by the reciprocal of the second fraction (rightmost). Let's take a look at a few examples.
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. $$\frac{44}{32}\div \frac{38}{25}$$

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. $$\frac{16}{18}\div \frac{24}{45}$$

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. $$\frac{6}{9}\div 10$$

A
$$\frac{5}{14}$$
B
$$\frac{9}{5}$$
C
$$\frac{3}{2}$$
D
$$\frac{1}{15}$$
E
$$\frac{12}{13}$$