Rule of Three Calculator

How this is calculated

X = B × C ÷ A (cross-multiplication)

Worked example: 3 kg costs R45 — what do 5 kg cost? X = 45 × 5 ÷ 3 = R75. The per-unit line (R15/kg here) is the same answer wearing shop clothes, and comparing per-unit prices is the fastest honest way to read supermarket shelf tags. Works identically for servings, map scales, paint coverage, and exchange amounts — anything directly proportional.

The oldest calculator still in daily use

The rule of three solves the most common shape of everyday arithmetic: three quantities are known, a fourth is proportional to them, find it. If a is to b as c is to x, then x = b × c ÷ a. Medieval merchants called it the Golden Rule and taught it as the practical summit of commercial mathematics, because with it you could price any quantity from any other.

The method is cross-multiplication, and it works because a proportion is an equation between two fractions. Multiply both sides by the denominators and the unknown is isolated. Nothing more sophisticated is required, which is exactly why it has outlasted every notation it was written in.

Direct and inverse proportion are different problems

In direct proportion, both quantities move together: more fabric costs more money, more distance takes more fuel. Cross-multiply as above.

In inverse proportion they move against each other: more workers means less time, more speed means less travel time. Here the product stays constant rather than the ratio, so a × b = c × x, and x = a × b ÷ c. Six painters take four days, so twelve painters take (6 × 4) ÷ 12 = two days. Applying the direct formula to an inverse problem gives eight days, and it is a mistake people make under time pressure because both problems look identical on the page.

The test is a sentence: "if this goes up, does that go up or down?" Answer it before reaching for a formula, and the rest is arithmetic.

Worked examples across different domains

Recipe scaling. A recipe for 4 needs 300 g of rice; for 7 people you need 300 × 7 ÷ 4 = 525 g. Map distance. On a 1:25,000 map, 6 cm represents 6 × 25,000 = 150,000 cm = 1.5 km. Paint coverage. If 2.5 litres covers 30 m², then 46 m² needs 2.5 × 46 ÷ 30 = 3.83 litres, so buy four. Unit price. If 750 ml costs £2.40, a litre costs 2.40 × 1000 ÷ 750 = £3.20, which is how you compare two package sizes honestly.

Where proportion quietly fails

Not everything scales linearly, and the rule of three cannot tell you when it does not. Nine women do not make a baby in one month. Doubling a cake tin's diameter quadruples its area, so baking times do not scale with the ingredient quantities. Cooking a joint twice the weight does not take twice the time, because heat penetrates by surface area and distance rather than mass. Drug doses often scale with body surface area rather than weight for the same reason.

Before cross-multiplying, ask whether the relationship really is proportional — whether doubling one side genuinely doubles the other. When it does, the rule of three is the fastest tool available. When it does not, no amount of correct arithmetic saves the answer.