3-4 Divide Fractions by Fractions

Property

Fraction division can be represented visually using area models, number lines, and partitioning diagrams.
The division $\frac{a}{b} \div \frac{c}{d}$ asks "how many groups of $\frac{c}{d}$ fit into $\frac{a}{b}$?"
Visual models help demonstrate why we multiply by the reciprocal: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$.

Examples

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.