Lesson 7: Divide Fractions

Property

The reciprocal of a non-zero fraction $\frac{a}{b}$ is the fraction $\frac{b}{a}$. A number multiplied by its reciprocal always equals 1.

Examples

• The reciprocal of $\frac{2}{3}$ is $\frac{3}{2}$.
• The reciprocal of the whole number $5$ (written as $\frac{5}{1}$) is $\frac{1}{5}$.
• To find the reciprocal of a mixed number like $1\frac{1}{4}$, first convert it to an improper fraction, $\frac{5}{4}$. The reciprocal is $\frac{4}{5}$.

Explanation

The reciprocal of a fraction is what you get when you "flip" the numerator and the denominator. This is also known as the multiplicative inverse. The key property of a reciprocal is that when you multiply a number by its reciprocal, the result is always 1. Understanding reciprocals is the first step to learning how to divide fractions, as division is the same as multiplying by the reciprocal.

Property

To divide a fraction by another fraction, multiply the first fraction by the reciprocal of the second fraction.

Examples

• $\frac{2}{3} \div \frac{1}{5} = \frac{2}{3} \times \frac{5}{1} = \frac{10}{3}$
• $\frac{3}{4} \div \frac{9}{8} = \frac{3}{4} \times \frac{8}{9} = \frac{24}{36} = \frac{2}{3}$

Explanation

Dividing by a fraction is the same as multiplying by its reciprocal. To solve a fraction division problem, you keep the first fraction the same, change the division sign to a multiplication sign, and flip the second fraction to find its reciprocal. After rewriting the problem as multiplication, multiply the numerators and the denominators, and simplify the result if necessary.