# Multiplying and Dividing Fractions

Multiplying fractions is easy: just multiply the numerators, and multiply the denominators. (Some people refer to this as multiplying across.)

That is:

$$\frac{A}{B}\cdot\frac{C}{D} = \frac{AC}{BD}$$

Every division problem is a multiplication problem in disguise: to divide by a number means to multiply by its reciprocal.

That is, $\,x\,$ divided by $\,y\,$ is the same as $\,x\,$ times the reciprocal of $\,y\,$.

In symbols:

$$\cssId{s16}{x\div y} \cssId{s17}{= \frac{x}{y}} \cssId{s18}{= x\cdot \frac{1}{y}}$$

Here's what it looks like with fractions:

$$\cssId{s20}{\frac{A}{B}\div\frac{C}{D}} \cssId{s21}{= \frac{A}{B}\cdot\frac{D}{C}} \cssId{s22}{= \frac{AD}{BC}}$$

## Examples

$\displaystyle \frac{1}{3}\cdot\frac{2}{5} =\frac{2}{15}$
$\displaystyle\frac{1}{6}\cdot\frac{3}{7} = \frac{3}{42} = \frac{1}{14}$

$\displaystyle \cssId{s27}{\frac{1}{3}\div\frac{2}{5}} \cssId{s28}{=\frac{1}{3}\cdot\frac{5}{2}} \cssId{s29}{= \frac{5}{6}}$