Enter three known values to find the fourth using direct or inverse proportion.
A rule of three finds an unknown fourth value from three known ones that form a proportion. In a direct rule of three, when A corresponds to B, then C corresponds to X in the same direction — as one increases, so does the other (X = B × C ÷ A). In an inverse rule of three, the relationship reverses — as one increases, the other decreases (X = A × B ÷ C), as with workers and days needed to finish a job.
To solve a rule of three, identify whether the relationship is direct (both quantities increase together) or inverse (one increases as the other decreases), then apply the matching formula to the three known values to find the fourth. Example (direct): if 5 kg costs 20, then 8 kg costs 20 × 8 ÷ 5 = 32.
| Type | A : B = C : X | Result |
|---|---|---|
| Direct | 5 : 20 = 8 : X | X = 32 |
| Direct | 3 : 45 = 7 : X | X = 105 |
| Inverse | 4 workers : 12 days = 6 workers : X days | X = 8 |
| Inverse | 6 workers : 10 days = 15 workers : X days | X = 4 |
| Direct | 10 : 2.5 = 4 : X | X = 1 |
Ask what happens to the second quantity when the first one increases. If it also increases (more hours worked, more pay earned), the relationship is direct. If it decreases (more workers, fewer days needed), the relationship is inverse.
Division by zero is undefined, so A cannot be zero in a direct rule of three — it represents a known nonzero reference quantity. The same applies to C in inverse mode, since it's the divisor.
Yes. The rule of three works with any positive real numbers, not just whole numbers — enter decimals directly (e.g. 2.5).
The classic case is work and time: if 4 workers finish a job in 12 days, 6 workers (more people) will finish it in fewer days — 8 days — because the quantities move in opposite directions.
This calculator assumes a simple (two-variable) proportion. Compound rule-of-three problems with three or more varying quantities require chaining multiple direct or inverse steps, which this tool does not automate.