Cross Multiply

To cross multiply is to go

from this:

812 = 23

to this:

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 second fraction.

8 × 312 × 3 = 23

Step 2: Multiply the top and bottom of the second fraction by the bottom number that the first fraction had.

8 × 312 × 3 = 2 × 123 × 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 cross multiply is to go from this:ab = cd
To this:ad = bc

To remember think cross (x) multiply:

cross multiply illustration

 

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

Example: Find "x" in x8 = 2x

Start with: x8 = 2x
Cross multiply:x2 = 8 × 2
Calculate:x2 = 16
And solve:x = 4 or −4

Check: Does 48 = 24 and −48 = 2−4 ?

 

Terminology

fraction numerator denominator

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".