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.

The general case, using variables instead of numbers, is:

To cross multiply is to go from this:ab = cdTo this:ad = bc

To remember think cross (x) multiply:

Cross multiplication arrows connecting top-left to bottom-right and bottom-left to top-right

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

Start with:x10 = 35Cross multiply:5x = 10 × 3Calculate:5x = 30Solve:x = 6

Check: Does 610 = 35 ?

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

Check: Does 48 = 24 and −48 = 2−4 ?

Fraction 3 over 4 with labels identifying 3 as the numerator and 4 as the 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.)

Be careful, though!

We can't use it when a denominator ("b" or "d" above) is zero, as dividing by zero is illegal.

334,335, 2045, 2046, 3174, 3175, 3176, 3876, 3877, 3878