Division rules

To divide by a fraction, multiply by its reciprocal, meaning you flip the second fraction and multiply. In symbols, $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}$.

For instance, $\frac{2}{3} \div \frac{4}{5} = \frac{2}{3} \times \frac{5}{4} = \frac{10}{12} = \frac{5}{6}$. Flip the divisor, multiply straight across, then simplify.

A stacked fraction (a fraction on top of a fraction) means exactly the same thing: the top divided by the bottom. So $\frac{\frac{2}{4}}{\frac{6}{5}} = \frac{2}{4} \times \frac{5}{6} = \frac{10}{24} = \frac{5}{12}$.

When only one part is a fraction, write the whole number as itself over $1$ and use the same rule. A number over a fraction: $\frac{a}{\frac{b}{c}} = \frac{\frac{a}{1}}{\frac{b}{c}} = \frac{a}{1} \times \frac{c}{b} = \frac{ac}{b}$. A fraction over a number: $\frac{\frac{a}{b}}{c} = \frac{\frac{a}{b}}{\frac{c}{1}} = \frac{a}{b} \times \frac{1}{c} = \frac{a}{bc}$.

The rule works with variables in the same way, which is how many algebra problems get simplified. For example $\frac{\frac{a}{b}}{\frac{c}{d}} = \frac{ad}{bc}$, so flipping the bottom fraction turns a messy division into a single product.

What is $\frac{3}{4} \div \frac{2}{5}$?
Multiply by the reciprocal: $\frac{3}{4} \times \frac{5}{2} = \frac{15}{8}$.

Simplify the stacked fraction $\frac{\frac{2}{4}}{\frac{6}{5}}$.
Top divided by bottom: $\frac{2}{4} \times \frac{5}{6} = \frac{10}{24} = \frac{5}{12}$.

Simplify $\frac{5}{\frac{2}{3}}$, a whole number over a fraction.
Write $5$ as $\frac{5}{1}$: $\frac{\frac{5}{1}}{\frac{2}{3}} = \frac{5}{1} \times \frac{3}{2} = \frac{15}{2}$.

Simplify $\frac{\frac{3}{4}}{2}$, a fraction over a whole number.
Write $2$ as $\frac{2}{1}$: $\frac{\frac{3}{4}}{\frac{2}{1}} = \frac{3}{4} \times \frac{1}{2} = \frac{3}{8}$.

Simplify $\frac{\frac{m}{no}}{\frac{n}{mo}}$.
Multiply the top by the reciprocal of the bottom: $\frac{m}{no} \times \frac{mo}{n} = \frac{m^2 o}{n^2 o} = \frac{m^2}{n^2}$.