How it works
The rule of three solves any problem with four quantities where three are known and the relationship is proportional. It is the method behind recipe scaling, unit prices, currency conversion at a known rate, map scales and many exam questions.
Direct proportion — “more gives more”
When one quantity doubles, the other doubles too: the ratio B ÷ A stays the same.
A → B, C → x ⇒ x = B × C ÷ AWorked example (direct). 3 kg of rice cost 45. How much do 7 kg cost? x = 45 × 7 ÷ 3 = 105. The price per kg is 45 ÷ 3 = 15, and 7 × 15 = 105.
Worked example (inverse). 4 workers finish a job in 6 days. How long do 3 workers take? x = 4 × 6 ÷ 3 = 8 days — fewer workers, more days.
Inverse proportion — “more gives less”
When one quantity doubles, the other halves: the product A × B stays the same. Typical cases are workers and time, speed and travel time, or the number of people sharing a fixed amount.
A × B = C × x ⇒ x = A × B ÷ CHow to tell which one to use
- Ask: if the first quantity grows, does the second grow (direct) or shrink (inverse)?
- Keep units consistent: if A is in kilograms, C must be in kilograms too.
- Check the answer for sense: buying more should cost more; more workers should take less time.
Comparing two pack sizes? The unit price calculator does that directly. For percentages, which are proportions out of 100, use the percentage calculator.