Lesson 25: Dividing Fractions

New Concept

To divide by a fraction, multiply by its reciprocal. This method simplifies complex division into a more familiar multiplication problem.

To find the quotient of two fractions, multiply the dividend by the reciprocal of the divisor.

What’s next

This rule is the key to the entire lesson. Next, you'll see worked examples that break down the process and solve challenge problems using this method.

Property

To find the quotient of two fractions, multiply the dividend by the reciprocal of the divisor. This rule turns a tricky division problem into a simple multiplication one. For example: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$.

Examples

• $\frac{3}{5} \div \frac{2}{3} = \frac{3}{5} \times \frac{3}{2} = \frac{9}{10}$
• $\frac{7}{8} \div \frac{1}{4} = \frac{7}{8} \times \frac{4}{1} = \frac{28}{8} = \frac{7}{2}$
• $6 \div \frac{2}{3} = 6 \times \frac{3}{2} = \frac{18}{2} = 9$

Explanation

Think of it like finding how many quarters are in five dollars. First, you find how many quarters are in one dollar (the reciprocal, 4), then multiply by five. This two-step thinking makes dividing fractions a piece of cake!

Property

When you divide the number 1 by any fraction, the answer is always the reciprocal of that fraction. So, for any fraction $\frac{a}{b}$, the result of $1 \div \frac{a}{b}$ is simply $\frac{b}{a}$.

Examples

• How many $\frac{2}{3}$s are in 1? $1 \div \frac{2}{3} = \frac{3}{2}$
• How many $\frac{9}{10}$s are in 1? $1 \div \frac{9}{10} = \frac{10}{9}$

Explanation

Imagine asking, “How many $\frac{1}{4}$s are in 1?” The answer is 4, which is the reciprocal of $\frac{1}{4}$. This trick works every time! Dividing 1 by a fraction is as simple as flipping that fraction upside down.

Property

A compound fraction is a fraction where the numerator, denominator, or both are also fractions. It looks complicated, but it is just another way to write a division problem, like this: $\frac{\frac{3}{4}}{\frac{9}{10}}$.

Examples

• $\frac{\frac{2}{5}}{\frac{3}{7}} = \frac{2}{5} \div \frac{3}{7} = \frac{2}{5} \times \frac{7}{3} = \frac{14}{15}$
• $\frac{6}{\frac{2}{3}} = 6 \div \frac{2}{3} = 6 \times \frac{3}{2} = 9$

Explanation

Don't be intimidated by these giant fractions! The main fraction bar is just a secret division sign. To solve it, rewrite the expression as a standard division problem, then use the 'flip and multiply' rule you already know.