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Rule of Three Calculator

Solve any rule of three, direct or inverse: enter the three known values and get the fourth, with the proportion and the formula used explained step by step.

If 3 kg cost €12, then 5 kg cost X.

X: the result
20

How the result is reached

3 is to 12 as 5 is to X · X = B × C ÷ A = 20

Educational arithmetic tool. The result is an exact proportion between the values entered.

1

What the rule of three is

The rule of three solves proportions: given three known values, it finds the fourth. If 3 kg of oranges cost €12, what do 5 kg cost? You set up "3 is to 12 as 5 is to X" and solve for X. It is the same sum behind unit prices, recipe scaling, map scales and currency conversion.

2

Direct or inverse: the only decision that matters

In the direct rule of three the two quantities grow together: more kilos, more euros. X is found by cross-multiplying and dividing: X = B × C ÷ A. In the inverse rule they move in opposite directions: more workers, fewer days of work. There, X = A × B ÷ C. Picking the wrong type is the classic mistake, which is why the calculator shows an example of each next to the selector.

3

How to tell whether your problem is direct or inverse

Ask a single question: if the first quantity increases, does the second increase or decrease? If buying more kilos means paying more euros, it is direct. If adding workers means the job takes fewer days, it is inverse. Nothing else needs memorising: that question decides the formula.

4

The rule of three and percentages are the same sum

Working out 20% of 250 is a direct rule of three: 100 is to 20 as 250 is to X. That is why this calculator and the percentage calculator complement each other: when the question is phrased in percentage terms, the percentage calculator gets there in fewer steps; when it comes with units (kilos, euros, days), the rule of three is more natural.

Worked example

Direct example: 3 kg cost €12; 5 kg cost X = 12 × 5 ÷ 3 = €20. Inverse example: 4 workers finish a job in 6 days; 8 workers finish it in X = 4 × 6 ÷ 8 = 3 days. Note that in the first case X rises relative to B and in the second it falls: that is the difference between direct and inverse.

Frequently asked questions

How do you do a simple rule of three?
Place the three known values in the proportion "A is to B as C is to X". If the relationship is direct (more A, more B), cross-multiply and divide: X = B × C ÷ A. In the calculator, enter A, B and C and choose the type.
When is the inverse rule of three used?
When the two quantities move in opposite directions: more speed, less travel time; more workers, fewer days; more taps, fewer hours to fill a tank. In that case X = A × B ÷ C.
What is the compound rule of three?
A proportion with more than two quantities: for instance, how many days 6 workers doing 7-hour shifts need for what 4 workers did in 9 days at 8 hours. It is solved by chaining simple rules of three, one per quantity, deciding for each whether it is direct or inverse. You can solve it with this calculator by applying it twice.
Does the rule of three work for percentages?
Yes: a percentage is a direct rule of three with 100 as the first value. 20% of 250 is set up as "100 is to 20 as 250 is to X" and gives 50. For that specific case, the percentage calculator solves it in fewer steps.
Why does my rule of three give an absurd result?
Almost always for one of two reasons: direct was chosen when the relationship was inverse (or the other way round), or the values A and B do not belong to the same pair of quantities. Check with the key question: if A increases, does B increase (direct) or decrease (inverse)?

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Author: Thorben Rasmus Idel · Reviewed by: Nahar Geva · Last reviewed: