How to Solve a Proportion

A proportion $a : b = c : x$ has one unknown term. The product of the outer terms equals the product of the inner terms, which gives $ax = bc$. Dividing both sides by $a$ isolates $x$.

$a$ and $b$ form the ratio that is known, $c$ is the matching value and $x$ is what you are after. The value of the ratio, $b \div a$, is shown as well.

Example

Solving $3 : 4 = 9 : x$, the outer product is $3x$ and the inner product is $4 \times 9 = 36$. Setting them equal gives $3x = 36$, so $x = 12$. The value of the ratio is $4 \div 3$, about 1.333, the factor that takes the left term to the right one.

Where it is used

Scaling a recipe is a proportion. If 300 g of flour serves two, then five are served by solving $2 : 300 = 5 : x$, which gives 750 g.

Map scales, enlarging a photograph and converting currency all take the same form. Once one matching pair is known, the rest follows.

Notes

Line the terms up so that $a$ and $c$ are the same kind of quantity, and $b$ and $x$ likewise. Swapping a count for a weight changes the answer.

If $a$ is 0 there is nothing to divide by, so $x$ is not determined.