# Cross Multiply

To cross multiply is to go from this:
 8 = 2 12 3

To this: 8 × 3 = 12 × 2

## How Does it Work?

Step 1: When you multiply the top and bottom of a fraction by the same amount, it doesn't change its value.

Example (first fraction above):
 8 = 8 × 3 12 12 × 3

In that example I multiplied the top and bottom of the first fraction
by the bottom number of the second fraction.

Step 2: We could also multiply the top and bottom of the second fraction by the bottom number of the first fraction.

Example (second fraction above):
 2 = 2 × 12 3 3 × 12

Step 3: And we would then have:

 8 × 3 = 2 × 12 12 × 3 3 × 12

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

Step 4: We can get rid of the 12 × 3 (because 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

So far I have used numbers, but we can state it more generally using variables:

To cross multiply is to go from this:
 a = c b d

 How to remember: "cross" multiply:

## Example

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

Find "x":

 x = 2 8 x

Let's cross multiply: x2 = 8 × 2 = 16

And solve x = 4 or -4

## Terminology

I have been saying "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! You cannot use it if either of the denominators ("b" and "d" above) are zero. Dividing by zero is "illegal".