The Rule of Three Explained: Direct and Inverse Proportion

The rule of three in three steps: direct and inverse proportion with examples, a shortcut formula and the compound rule of three.

Math & Technology2 linked calculatorsReading time: 5 min.

The rule of three lets you get from a known quantity to an unknown one – by way of a single unit. It lies behind wage and price calculations, recipe quantities and percentage problems.

Key takeaways

  • The rule of three always goes via one unit: first convert to 1, then to the quantity you want.
  • Direct proportion: if one quantity doubles, so does the other – on the way to the unit you divide.
  • Inverse proportion: if one quantity doubles, the other halves – on the way to the unit you multiply.
  • Before calculating, check: “the more, the more” or “the more, the less”?
  • The compound rule of three chains several simple ones together.

The principle in three steps

The German name, Dreisatz, refers to the three statements in which the calculation is written:

  1. Write down what you know: quantity a goes with value b.
  2. Convert to one unit: what goes with 1?
  3. Scale up to the quantity c you want.

Keep left and right apart

The quantity you are looking for always goes on the right. On the left are the quantities you know or choose – hours, items, people.

Direct proportion: the more, the more

In a direct proportion, both quantities grow in the same ratio. Example using the German statutory minimum wage for 2026:

StepCalculation
Known8 hours → €111.20
One unit (divided by the hours)1 hour → €13.90
Wanted (times the new hours)6.5 hours → €90.35

Shortcut: x = b ÷ a × c.

Rule of three with working

Direct or inverse – enter three values and the result appears with all three steps.

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Inverse proportion: the more, the less

In an inverse proportion, the product of both quantities stays the same: more people need less time.

4 painters need 6 days for a house, 24 working days in total. One person alone would need 24 days, so 3 painters need 8 days (assuming everyone works at the same pace).

Shortcut: x = a × b ÷ c.

Compound rule of three

If the result depends on more than one quantity, you chain several rules of three. Example: 3 machines produce 240 parts in 4 hours. One machine makes 80 parts in 4 hours, i.e. 20 parts in 1 hour. So 5 machines make 5 × 6 × 20 = 600 parts in 6 hours.

Rule of three and percentages

Percentage problems are rules of three with the base value as 100%: if €12 is exactly 8%, then 1% is €1.50 and 100% is €150.

Solve percentage problems

Percentage value, rate and base value with the working.

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Frequently asked questions

When is a rule of three inverse?
When more of one quantity means less of the other, such as more workers and fewer days, or higher speed and shorter travel time.
What is the rule of three formula?
Direct: x = b ÷ a × c. Inverse: x = a × b ÷ c. Here b goes with a, and x is the value you want for c.
Can every relationship be solved with the rule of three?
No, only if the quantities really are directly or inversely proportional. Volume discounts, base fees or tiered prices break that assumption.

Sources

Every figure in this article comes from these primary sources or from the linked calculators, which are based on the same sources.

  1. Section 1 MiLoV5 – statutory minimum wage (accessed September 30, 2026)
Onur Cirakoglu — Full-Stack Developer & Founder of HEADON.pro
Onur CirakogluSources verified

Full-Stack Developer & Founder of HEADON.pro

Full-stack developer and founder of HEADON.pro. Developer of Rechnerzentrale.