The Rule of Three: How to Use It for Direct and Inverse Proportions, with Worked Examples

September 29, 2026

The rule of three finds an unknown value from three known values that are in proportion. It is one of the most useful calculations in real life: prices, recipes, working time, fuel use, percentages. The trick is knowing whether the proportion is direct or inverse.

How to set it up

Put the two quantities in two columns, with matching values on the same row:

Quantity 1	Quantity 2
A	B
C	X

Then ask: if quantity 1 doubles, what happens to quantity 2?

If it doubles too, the proportion is direct: X = B × C ÷ A (cross-multiply).
If it halves, it is inverse: X = A × B ÷ C (multiply along the row).

Direct proportion examples

Price per kilo

If 3 kg cost 12, how much do 7 kg cost? More kilos, higher price – direct. X = 12 × 7 ÷ 3 = 28.

Recipe

A recipe uses 2 cups of flour for 10 servings. For 25 servings: X = 2 × 25 ÷ 10 = 5 cups.

Inverse proportion examples

Workers and time

4 builders put up a wall in 6 days. How long do 8 builders take? More people, fewer days – inverse. X = 4 × 6 ÷ 8 = 3 days.

Taps and time

5 taps fill a tank in 12 hours. With 6 taps: X = 5 × 12 ÷ 6 = 10 hours.

Speed and time

At 60 km/h a trip takes 2 hours. At 80 km/h: X = 60 × 2 ÷ 80 = 1.5 hours (1 h 30 min).

The rule of three for percentages

Put 100% next to the total. If 100% is 180, what is 25%? X = 180 × 25 ÷ 100 = 45. And the other way round: 45 is what % of 180? X = 100 × 45 ÷ 180 = 25%. More in the percentages guide.

How to check the answer

Direct: the ratios must match – 3 ÷ 12 = 7 ÷ 28.
Inverse: the products must match – 4 × 6 = 8 × 3.

Solve yours

The rule of three calculator solves direct and inverse proportions, shows the working and the check, with ready-made examples.

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