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Cross Multiply


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

How Does it Work?

Well, if you multiply the top and bottom of a fraction by the same amount, it doesn't change its value.

Example (from 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.

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

And we would then have:

8 × 3  =  2 × 12
12 × 3 3 × 12

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

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 middle steps and go straight to the "cross-multiplied" form).

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.


Using Variables

Now so far I have used a real example, 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

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
   
And solve x = √16 = 4

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".