How to Solve Proportions

Step 1 — set up the proportion

Write two ratios as fractions and make them equal. The golden rule is consistency: the same kind of quantity must occupy the same position in both fractions. If kilometres go on top in the first ratio, they go on top in the second; hours on the bottom in both.

Setting up the fraction correctly is where almost all mistakes happen. A clear label for the top and bottom — “km per hour” here — keeps the layout honest.

Step 2 — cross-multiply

Multiply each numerator by the opposite denominator to clear the fractions:

This converts the proportion into a plain linear equation, which is far easier to handle than two fractions.

Step 3 — solve for the unknown

Divide both sides by the number multiplying x:

At a steady speed, five hours covers 300 km. Whatever position the unknown sits in, the move is the same: get x on its own by dividing.

Step 4 — check your answer

Substitute back and compare the two ratios as decimals. 120 ÷ 2 = 60 and 300 ÷ 5 = 60 — they match, so the proportion holds. Alternatively, confirm the two cross products are equal: 120 × 5 = 600 and 2 × 300 = 600. A check costs seconds and catches setup errors before they matter.

Watch the direction of the relationship

Everything above assumes a direct proportion, where both quantities move together. If the situation is inverse — more of one means less of the other, like workers and time — you do not cross-multiply diagonally. Instead the products of each pair are equal: a × b = c × x. Always pause to ask whether doubling the first quantity should double or halve the second; that question decides which method to apply, and it is the most common place students go wrong.

A quick reference

The calculator runs steps 2 and 3 instantly and shows the working, so you can focus on the setup and the check — the parts that actually need your judgement.