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Rule of Three Calculator

Solve the rule of three online — free and step by step. Direct and inverse proportion, with the working shown via the value for one. Great for grade 6–7.

Quick answer
How does the rule of three work?
For a direct proportion, first scale down to 1, then up to the quantity you want: if 3 kg cost 12 €, then 1 kg costs 12 ÷ 3 = 4 €, so 5 kg cost 5 · 4 = 20 €. For an inverse proportion (more means less), the product stays constant: if 4 workers take 6 hours, that’s 24 worker-hours, so 3 workers finish in 24 ÷ 3 = 8 hours.
The tool

Enter values — get full working

How does b relate to a? — more means more
Comma or dot as decimal separator, negative values allowed.
Step-by-step
Press Calculate to see every step.
HowTo

Rule of three — 4 steps

Direct proportion: “3 kg cost 12 €, what do 5 kg cost?”
  1. 1
    Step 1 of 4

    Write the known pairing

    3 kg → 12 €. Note which quantity is on the left (amount) and which on the right (value).

  2. 2
    Step 2 of 4

    Check the proportionality

    More kilograms means more euros — that’s directly proportional. If more means less, you work inversely.

  3. 3
    Step 3 of 4

    Scale down to 1

    1 kg → 12 ÷ 3 = 4 €. This value-for-one is the key step of the direct rule of three.

  4. 4
    Step 4 of 4

    Scale up to the wanted amount

    5 kg → 5 · 4 = 20 €. Answer: 5 kg cost 20 €.

Examples

Rule of three — worked examples

Direct and inverse proportion
3 kg → 12 €, 5 kg → ?
1 kg → 12 ÷ 3 = 4 €
5 kg → 5 · 4
20 €
4 books → 6 €, 10 books → ?
1 book → 6 ÷ 4 = 1.5 €
10 · 1.5
15 €
250 km on 5 L, 400 km → ?
1 km → 5 ÷ 250 = 0.02 L
400 · 0.02
8 L
4 workers → 6 h, 3 workers → ? (inverse)
4 · 6 = 24 worker-hours
24 ÷ 3
8 h
6 pumps → 8 h, 4 pumps → ? (inverse)
6 · 8 = 48
48 ÷ 4
12 h
2 m fabric → 9 €, 7 m → ?
1 m → 9 ÷ 2 = 4.5 €
7 · 4.5
31.5 €
Theory

Direct and inverse rule of three

The rule of three solves problems where two quantities are proportional and three of the four values are known — hence the name. In a direct (proportional) rule of three, both quantities grow in the same ratio: twice as many kilograms cost twice as much. You work through the value for one unit (scale down to 1, then up to the wanted amount). In an inverse (antiproportional) rule of three the relationship runs the other way: more workers means less time. Here the product of the two quantities stays constant (4 workers · 6 hours = 24 worker-hours), and you divide that product by the new amount. The crucial first move is recognising the kind of proportionality: if one quantity rises and the other rises with it, it’s direct; if one rises while the other falls, it’s inverse. From grade 6–7 the rule of three is the standard tool for percentage, scale, mixture and speed problems.

Pitfalls

Common mistakes with the rule of three

Misjudging the proportionality

“More workers need more time” is wrong — more workers need less time. Settle the direction first.

Multiplying instead of dividing in the direct case

Scaling down to 1 means dividing (12 ÷ 3), not multiplying.

Swapping the units

Amount and value must stay consistently on the same side. Lay out the table first, then compute.

Treating an inverse problem like a direct one

Inverse means: form the product and divide, not scale up through a value-for-one.
FAQ

Frequently asked questions about the rule of three

Glossary

Glossary — key terms explained simply

Rule of three
A method to find the fourth value of a proportion from three known values.
Directly proportional
Both quantities grow in the same ratio.
Inversely proportional
As one quantity rises the other falls; their product stays constant.
Value for one
The value of a single unit, the intermediate step in the direct rule of three.
Proportion
Equality of two ratios, e.g. 3:12 = 5:20.
Ratio
The relationship between two quantities, written a:b.