Rule of Three Calculator
Solve direct or inverse proportion problems: if A is to B, what is C to X?
Inputs
Allowed range: -1000000000 to 1000000000
Allowed range: -1000000000 to 1000000000
Allowed range: -1000000000 to 1000000000
Results
How it works
The rule of three solves a proportion with three known values and one unknown. Direct proportion: A/B = C/X ⇒ X = (B·C)/A. Inverse proportion: A·B = C·X ⇒ X = (A·B)/C.
Complete guide
The rule of three (regra de três) is one of the most useful arithmetic tools ever invented — and one of the oldest. Versions of it appear in the Rhind Mathematical Papyrus from ancient Egypt and in classical Indian and Arabic mathematics, where it was called the 'golden rule' precisely because it solved so many everyday problems: pricing goods by weight, scaling recipes, converting currencies, dividing work among laborers.
It applies whenever three quantities in a problem are known and the fourth is missing, and the relationship between them is either directly or inversely proportional. Direct proportion means both quantities move together: more flour means more cost. Inverse proportion means they move in opposite directions: more workers means fewer days to finish the same job.
Direct-proportion worked example: 2 kg of flour costs $6. How much does 5 kg cost? Set up A/B = C/X, with A = 2 kg, B = $6, C = 5 kg. Solving for X: X = (B × C) / A = (6 × 5) / 2 = $15. The unit price ($3/kg) is implicit in the proportion — you do not have to compute it separately.
Inverse-proportion worked example: 4 workers finish a wall in 6 days. How many days for 8 workers (working at the same pace)? Total work is constant, so A × B = C × X with A = 4, B = 6, C = 8. Solving: X = (A × B) / C = (4 × 6) / 8 = 3 days. Notice doubling the workers exactly halves the time — the signature of inverse proportion.
How to use this calculator: enter A and B (the two known values that go together — same row of the proportion), enter C (the value whose pair you want to find), and pick the proportion type. We return X and show the explicit formula so you can audit the answer by hand.
How to tell which type to use: imagine doubling A. If the answer (X) should also double, the relationship is direct. If the answer should halve, it is inverse. Direct: cost vs quantity, distance vs time at fixed speed, ingredients in a recipe. Inverse: workers vs days, speed vs travel time at fixed distance, pressure vs volume of a gas at fixed temperature.
Common mistakes: mixing units (kilograms in A but grams in C — convert first), confusing direct with inverse (always sanity-check by doubling one quantity in your head), and forcing the rule of three onto problems that are not actually proportional (compound interest, taxes with brackets, anything with a fixed cost component). When in doubt, write the units out explicitly — they tell you whether the cross-multiplication makes sense.
Frequently asked questions
- How do I know whether to pick direct or inverse?
- Ask: if the first quantity doubles, does the answer also double, or does it halve? Doubles → direct. Halves → inverse. If neither, the relationship is not a simple proportion and the rule of three does not apply.
- Can it handle negative numbers?
- Mathematically yes, but most real proportion problems involve positive physical quantities (quantities, prices, times, distances). If you get a negative answer, double-check that the problem actually involves directed quantities — otherwise it usually means a sign error in the inputs.
- What is a compound rule of three?
- A version with more than three knowns, used when an outcome depends on multiple proportional inputs at once (e.g., '5 workers in 8 hours build 200 m of wall; how long for 8 workers to build 500 m?'). This calculator handles only the simple case; compound problems are usually broken down into two simple steps.
- Why is it called 'the rule of three'?
- Because three of the four quantities in the proportion are given and only one is unknown. In medieval European arithmetic textbooks it was often the most-practiced rule, since most commerce reduced to it.
- Is the rule of three the same as cross-multiplication?
- Yes — cross-multiplication is the mechanical step you use to solve a direct-proportion rule of three. The rule names the situation; cross-multiplication is the algebra that finishes it.