Proportion

What is a proportion?

Four numbers $a, b, c, d$ are said to be in proportion, written $a : b :: c : d$, when the ratio of the first pair equals the ratio of the second pair:

That cross-multiplied form (the product of the outer (extreme) terms equals the product of the inner (middle) terms) is the fastest way to test or solve a proportion.

Worked example: testing a proportion

Are 20, 18, 5, 6 in proportion?

Solution: Check whether $20 \times 6 = 18 \times 5$:

120 ≠ 90, so 20, 18, 5, 6 are not in proportion.

Compare that with 30, 40, 45, 60:

These match, so 30, 40, 45, 60 are in proportion.

Worked example: solving for a missing term

If $36 : 81 :: x : 63$, find $x$.

Solution: Cross-multiplying, $36 \times 63 = 81 \times x$: