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