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Rule of Three Calculator

Enter the three known values and the calculator finds the fourth: the direct proportion ("if 3 kg cost €12, 5 kg cost X") and the inverse one ("if 4 workers take 6 days, 8 workers take X"), with the formula and the working written out.

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 a proportion is

A proportion is an equality between two ratios: A is to B as C is to X. You know three values and want the fourth. It is the same reasoning as the rule of three: cross-multiply and divide. In a direct proportion X = B × C ÷ A; in an inverse one X = A × B ÷ C.

2

Direct or inverse: how to tell

The proportion is direct when the two quantities grow together: more kilos, more euros; more hours worked, more pay. It is inverse when one grows and the other shrinks: more workers, fewer days; more speed, less time. Get the direction wrong and the calculator still gives you a number, just the wrong one: before entering values, ask whether "more" on one side means "more" or "less" on the other.

3

Percentages are a proportion with 100

A percentage is a direct proportion whose first quantity is 100: 100 is to 20 as 250 is to X, so X = 50. When the problem comes in per cent, use the percentage calculator; when it comes in kilos, euros, hours or days, the proportion is the natural route.

4

When the proportion does not exist

If one of the known values is zero the proportion is undefined: dividing by zero has no meaning, and the calculator says so instead of inventing a number. It also fails silently when the quantities are not truly proportional, such as costs with a fixed component: doubling the kilometres does not double a hire with a base fee.

Worked example

Direct example: 3 kg of apples cost €12. How much do 5 kg cost? 12 × 5 ÷ 3 = €20. Inverse example: 4 workers finish a wall in 6 days. How long do 8 workers take? 4 × 6 ÷ 8 = 3 days. In the first case the quantity rose and so did the price; in the second the workers rose and the days fell.

Frequently asked questions

How do you solve a proportion?
Write A : B = C : X and solve for X. If the proportion is direct, X = B × C ÷ A (cross-multiply and divide by the known term). If it is inverse, X = A × B ÷ C.
Are a proportion and the rule of three the same thing?
Yes. "Rule of three" is the name of the practical method for solving a proportion with three known values: cross-multiply and divide. The calculator does exactly that and shows the step.
How do I know whether it is direct or inverse?
Ask whether the two quantities grow together (direct: more quantity, more price) or in opposite directions (inverse: more workers, fewer days). The direction changes the formula, not just the number.
Why does the calculator say the proportion is undefined?
Because one of the values you divide by is zero. A proportion with a zero in the denominator has no solution; check the inputs.

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