Cross Multiply
To cross multiply is to go from this: 


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): 

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): 

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 "crossmultiplied" 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: 


To this:  ad = bc 
How to remember: "cross" multiply: 
Example
Cross multiplication can help speed up a solution. Like in this example:
Find "x": 


Let's cross multiply:  x^{2} = 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".