How do I recognise a rule of three word problem?

Look for a known relationship between two quantities and a question asking for a fourth value when one quantity changes. Recipes, prices, speeds and workloads are typical.

How do I know if a word problem is direct or inverse?

Ask whether more of the first quantity should give more or less of the second. More servings means more flour, so that is direct. More workers means fewer days, so that is inverse.

Guide · Practice

Rule of Three Word Problems

Q: Can I check my answer to a word problem?

Yes. Enter your three known values in the calculator, leave the unknown blank, and compare its answer and working with yours.

Updated: June 2026

The hardest part of a word problem is not the arithmetic — it is turning sentences into a proportion and deciding whether it is direct or inverse. These ten worked examples cover the situations that come up most: recipes, prices, currency, speed, workers and mixtures. Each one shows the setup, the method and the answer, so you can copy the pattern onto your own questions.

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Direct proportion problems

1. Recipe. A cake for 4 needs 200 g of sugar. For 10 people? 200 × 10 ÷ 4 = 500 g.

2. Shopping. 3 kg of rice costs £4.50. What do 7 kg cost? 4.50 × 7 ÷ 3 = £10.50.

3. Currency. €50 buys $54. How many dollars for €130? 54 × 130 ÷ 50 = $140.40.

4. Fuel. A car uses 6 litres per 100 km. How much for 250 km? 6 × 250 ÷ 100 = 15 litres.

5. Printing. 8 pages take 24 seconds. How long for 30 pages? 24 × 30 ÷ 8 = 90 seconds.

In every case more of the first quantity means more of the second, so you cross-multiply diagonally.

Inverse proportion problems

6. Workers. 4 builders finish a wall in 9 days. How long for 6 builders? 4 × 9 ÷ 6 = 6 days.

7. Speed. A trip takes 3 hours at 80 km/h. How long at 120 km/h? 80 × 3 ÷ 120 = 2 hours.

8. Taps. 2 taps fill a tank in 90 minutes. With 5 taps? 2 × 90 ÷ 5 = 36 minutes.

9. Food supply. Food lasts 30 days for 20 people. For 25 people? 20 × 30 ÷ 25 = 24 days.

Here more of the first quantity means less of the second, so you multiply across each row instead.

A compound problem

10. If 5 machines produce 600 parts in 4 hours, how long do 3 machines need for 900 parts?

Machines — fewer machines, more time, so inverse: 5 ÷ 3.
Parts — more parts, more time, so direct: 900 ÷ 600.

hours = 4 × (5 ÷ 3) × (900 ÷ 600) = 4 × 1.6667 × 1.5 = 10 hours

Treating each factor on its own turns a tangled sentence into two simple ratios.

How to translate any word problem

Name the two quantities and the units involved.
Write the known pair as the first ratio, with units matched top and bottom.
Decide direct or inverse by asking what doubling the first does to the second.
Solve and sanity-check — does the size of the answer make sense in the story?

That last check is your safety net. An answer where more workers somehow take longer, or more rice costs less, tells you the relationship was set up the wrong way round.

Frequently asked questions

How do I recognise a rule of three problem?

Look for a known relationship between two quantities and a question for a fourth value when one quantity changes — recipes, prices, speeds, workloads.

How do I know if it is direct or inverse?

Ask whether more of the first quantity gives more or less of the second. More servings, more flour is direct; more workers, fewer days is inverse.

Can I check my answer?

Yes — enter your three known values in the calculator, leave the unknown blank, and compare its answer and working with yours.