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Exercises on rule of three direct and inverse

Master the simple, direct, and inverse rule of three, and percentages. Clear theory and step-by-step solved exercises, with instantly corrected practice.

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Rule of three: quick theory

The rule of three is used to calculate an unknown quantity from a proportion.

Direct: when one quantity increases, the other increases (more kilos, higher price). You cross-multiply: if a → b, then c → x is calculated as x = (b · c)/a.

Inverse: when one increases, the other decreases (more workers, fewer days). You multiply in line: x = (a · b)/c.

Percentages are a direct rule of three over 100.

Step-by-step solved exercises on rule of three

Solved examples to show you the method. Then, practice with instant correction.

Exercise 1If 3 kg of oranges cost €6, how much do 5 kg cost?

Step-by-step solution

  1. It is direct (more kilos, higher price).
  2. We set it up: 3 kg → 6 € ; 5 kg → x.
  3. We cross-multiply: x = (6 · 5) / 3 = 30/3.
Result: 5 kg cost €10.

Exercise 2Calculate 20% of 150.

Step-by-step solution

  1. A percentage is a rule of three over 100: 100 → 150 ; 20 → x.
  2. We calculate: x = (150 · 20) / 100 = 3000/100.
Result: 20% of 150 is 30.

Exercise 34 workers paint a fence in 6 days. How long would 3 workers take?

Step-by-step solution

  1. It is inverse (fewer workers, more days).
  2. In the inverse proportion, you multiply straight across: x = (4 · 6) / 3 = 24/3.
Result: 3 workers would take 8 days.

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This topic is studied in:

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Frequently asked questions about rule of three

How do I know if a rule of three is direct or inverse?

It is direct if increasing one quantity increases the other (kilos and price). It is inverse if increasing one decreases the other (workers and days, speed and time).

How do you calculate a percentage using the rule of three?

The total is equivalent to 100%. You set up 100 → total ; percentage → x, and cross-multiply: x = (total · percentage) / 100.

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