How to use the rule of three
This rule of three calculator solves direct and inverse proportions involving two quantities. Enter three related values, select which of the four is unknown, and see the setup, rearranged equation and calculation. A direct relationship assumes a constant ratio; an inverse relationship assumes a constant product. You can enter comma or dot decimals and simple fractions. The type of relationship depends on the real situation and cannot be inferred from the numbers alone.
Direct: if 3 kg cost €12, then 5 kg cost (12 × 5) ÷ 3 = €20. Inverse: if 4 people finish in 6 days, 8 people would take (4 × 6) ÷ 8 = 3 days, assuming constant productivity.
Results are estimates and depend on the supplied values.
A practical guide to the rule of three
What is the rule of three?
The rule of three finds a fourth number when two quantities are proportional. Identify what each column means and keep each matching pair in one row. This method only applies when the relationship really is proportional.
A ↔ B and C ↔ D · Three known values → one unknown
Direct proportion: how to calculate it
When you buy more weight at the same unit price, the price increases. If 3 kg cost €12, then 5 kg cost €20: multiply 12 by 5 and divide by 3. If another position is missing, this calculator rearranges the equation for you.
D = (B × C) ÷ A · Example: (12 × 5) ÷ 3 = €20
Inverse proportion: how to calculate it
In an inverse relationship, one quantity goes up as the other goes down, keeping their product constant. If 4 people take 6 days for a task, 8 people would take 3 days, assuming equal productivity and no interruptions.
D = (A × B) ÷ C · Example: (4 × 6) ÷ 8 = 3 days
How to find A, B, C or D
The missing value does not have to be the last one. If 3 kg cost €12 and another purchase costs €20, choose C to work out the missing quantity. The calculator rearranges the expression and keeps the other known values when you change the unknown.
Direct: A × D = B × C · Inverse: A × B = C × D
Common rule of three mistakes
Choosing direct when the relationship is inverse changes the answer. Mixing units between pairs or using the method for a non-proportional relationship can also give misleading results. If a zero enters the denominator, there may be no solution or no unique solution.
More weight → higher price: direct · More workers → less time: inverse
Rule of three versus percentages
Percentages are a special case of direct proportion: 100% represents the whole. For discounts, percentage increases or finding what percent one amount is of another, a dedicated percentage calculator is often simpler.
If 100% is €80, then 25% is (80 × 25) ÷ 100 = €20
Need to calculate discounts or percentages? Try our percentage calculator.
Frequently asked questions
How do I know whether to use direct or inverse proportion?
If doubling one quantity also doubles the other, use direct proportion. If doubling one halves the other, use inverse proportion. These rules assume a proportional relationship over the interval.
Can I use decimal values?
Yes. Enter a dot or comma as the decimal separator, such as 2.5 or 2,5. The calculator handles the operation without needing to convert to integers.
Does the rule of three calculator accept fractions?
Yes. Enter 3/4, 1/2 or 1.5/2 in any known-value field. They are evaluated numerically and displayed results may be rounded.
Can I solve for a value other than D?
Yes. Select A, B, C or D. For example, choose C to find the quantity that corresponds to a known price.
What happens if one of my numbers is zero?
Zero can be valid in some proportions, but division by zero does not produce a unique result. Depending on the other values, the equation may be impossible or indeterminate. The calculator explains the issue.
Can it solve compound proportions?
Not yet. This tool handles simple proportional relationships between two quantities. Compound proportions require more than two quantities and their separate relationships.