What is the rule of three?

A method for finding a fourth number when three are known and the quantities are proportional. Its name is medieval and its use is daily: scaling a recipe, converting a price per unit, reading a scale drawing. The reliable way to do it is to find the unit rate first — what one of something gives — then multiply.

How do I know whether it is direct or inverse?

Ask what happens to the answer if you double C. If it should double, the relationship is direct. If it should halve, it is inverse. Ingredients and servings are direct; workers and duration are inverse; speed and travel time are inverse.

Why show the unit rate?

Because it is the check. "4 square metres per tin" is a statement you can judge against the tin in your hand, whereas "28" on its own is just a number. If the unit rate is implausible, two of your three inputs are in the wrong places.

Math

Rule of Three Calculator

If A corresponds to B, what corresponds to C? Both proportions are solved at once.

The formula

direct: x = B × C ÷ A

inverse: x = A × B ÷ C

unit rate = B ÷ A

Worked example

If 3 tins cover 12 m², what do 7 tins cover?

12 ÷ 3 = 4 m² per tin
4 × 7 = 28 m²
Inverse instead: 3 × 12 ÷ 7 = 5.1429 — the answer to a different question

Where this goes wrong

Using direct proportion where the relationship is inverse

If 3 painters take 12 days, 7 painters do not take 28 days — they take 12 × 3 ÷ 7, about 5.14. Direct proportion applies when more of one means more of the other (tins and coverage); inverse applies when more of one means less (workers and time, speed and journey length). The arithmetic cannot tell which you meant, so this page computes both and makes you choose. Check the unit rate: if "4 m² per tin" reads sensibly, you want the direct answer.

Questions

What is the rule of three?: A method for finding a fourth number when three are known and the quantities are proportional. Its name is medieval and its use is daily: scaling a recipe, converting a price per unit, reading a scale drawing. The reliable way to do it is to find the unit rate first — what one of something gives — then multiply.
How do I know whether it is direct or inverse?: Ask what happens to the answer if you double C. If it should double, the relationship is direct. If it should halve, it is inverse. Ingredients and servings are direct; workers and duration are inverse; speed and travel time are inverse.
Why show the unit rate?: Because it is the check. "4 square metres per tin" is a statement you can judge against the tin in your hand, whereas "28" on its own is just a number. If the unit rate is implausible, two of your three inputs are in the wrong places.