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 of the first fraction had.

8 × 3 12 × 3 = 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

OK, we have seen it with numbers, but now we can do it more generally using variables:

To cross multiply is to go from this:  

ab = cd

To this:   ad = bc
     

To remember think "cross" multiply:

 

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

Example: Find "x" here:

  x8 = 2x
     
Let's cross multiply:   x2 = 8 × 2
     
Calculate:   x2 = 16
And solve:   x = 4 or −4

Check: Does 48 = 24 and −48 = 2−4 ? Yes!

 

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".