Numbers & Percentages
Rule of Three Calculator
Find the fourth value from three known ones by direct or inverse proportion, with the working shown.
Updated:
Result
Updates as you typeX (what C corresponds to)
20
- Proportion
- 3 → 12 · 5 → 20
- Unit value (B / A)
- 4
- Calculation
- 12 × 5 / 3
How it's calculated
The rule of three finds an unknown value from three known ones, using the way two quantities change together. First decide the type: if both quantities rise together it is direct proportion; if one rises as the other falls it is inverse proportion. Then enter A, B and C.
Worked example
3 kg of apples cost 12. What do 5 kg cost? More apples cost more, so it is direct proportion:
- A = 3, B = 12, C = 5
- X = 12 × 5 / 3 = 60 / 3
- X = 20 (4 per kg)
Everyday rule of three problems
| Problem | Type | Answer |
|---|---|---|
| Problem: 3 kg of apples cost 12. What do 5 kg cost? | TypeDirect | Answer20 |
| Problem: A car uses 7 litres per 100 km. How much for 350 km? | TypeDirect | Answer24.5 litres |
| Problem: A recipe for 4 needs 300 g of flour. How much for 6? | TypeDirect | Answer450 g |
| Problem: 4 workers finish a job in 12 days. How long for 6 workers? | TypeInverse | Answer8 days |
| Problem: At 60 km/h a trip takes 5 hours. How long at 75 km/h? | TypeInverse | Answer4 hours |
Frequently asked questions
How does the rule of three work?
If A corresponds to B, the rule of three finds what C corresponds to. With direct proportion X = B × C / A: if 3 kg cost 12, then 5 kg cost 12 × 5 / 3 = 20.
What is cross-multiplication?
In the proportion A / B = C / X, the product of the outer terms equals the product of the inner ones: A × X = B × C. Divide B × C by A and you have X. Every direct proportion problem is solved in this one step.
How do I know if it is direct or inverse proportion?
If one value goes up when the other goes up by the same factor, it is direct: more apples, more money. If one goes up while the other goes down, it is inverse: more workers, fewer days. In inverse proportion the product stays the same, so 4 workers taking 12 days means 6 workers take 4 × 12 / 6 = 8 days.
Is a percentage a rule of three?
Yes. "What is 15% of 200?" is the direct proportion "100 is to 15 as 200 is to X": X = 15 × 200 / 100 = 30. For percentage changes and discounts the percentage calculator is quicker.
How do I solve a compound rule of three?
When more than two quantities change (workers, days and amount of work, say), split the problem into two simple proportions: hold one quantity fixed and solve the first, then use that answer in the second. Check direct or inverse separately at each step.
Simple rule of three arithmetic; the answer assumes the two quantities really are proportional.