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 we 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
   
To this: ad = bc

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! We cannot use it when either of the denominators ("b" and "d" above) are zero. Dividing by zero is "illegal".