# Cross Multiply

To cross multiply is to go

\frac{8}{12} = \frac{2}{3}

8 × 3 = 12 × 2

## How Does it Work?

Multiplying the top *and* bottom of a fraction by the same amount doesn't change its value.

Step 1: Multiply the top and bottom of the * first* fraction by the bottom number of the

*fraction.*

**second**\frac{8 × 3}{12 × 3} = \frac{2}{3}

Step 2: Multiply the top and bottom of the * second* fraction by the bottom number that the

*fraction had.*

**first**\frac{8 × 3}{12 × 3} = \frac{2 × 12}{3 × 12}

And Magic! The bottom of **both** fractions is now 12 × 3

Step 3: We can get rid of the 12 × 3 (as we are dividing both sides by the same amount) and the equation is still true:

8 × 3 = 12 × 2

**Job Done!**

In practice, though, it is easier to skip the steps and go straight to the "cross-multiplied" form.

## Using Variables

The general case, using variables instead of numbers, is:

To remember think **cross** (x) multiply:

Cross multiplication can help speed up a solution. Like in this example:

### Example: Find "x" in \frac{x}{8} = \frac{2}{x}

^{2}= 8 × 2

^{2}= 16

Check: Does \frac{4}{8} = \frac{2}{4} and \frac{−4}{8} = \frac{2}{−4} ?

## Terminology

I said "top" and "bottom" of the fractions ... but the correct words are **numerator** and **denominator**, OK? (I just wanted to keep it simple.)

## Caution: Zero

Be careful, though!

We cannot use it when a denominator ("b" and "d" above) is zero, as dividing by zero is "illegal".