Rule of Three Calculator
Solve direct and inverse proportions. If A is to B, then C is to X.
The rule of three is a mathematical procedure for finding an unknown value when three proportionally related values are known. It is one of the most widely used tools in practical mathematics: from calculating wholesale prices to scaling recipe ingredients, converting currencies or calculating distances on a map.
There are two types: the direct rule of three (when both quantities grow or decrease in the same proportion) and the inverse rule of three (when one quantity increases while the other decreases). This calculator solves both cases immediately: enter the three known values, select the type and you will get the result instantly.
What the rule of three is and how to solve it
The rule of three is a method for finding an unknown value from three known values that share a proportion. You start from a relationship 'A is to B' and look for the term that completes 'C is to X'. It is direct when both quantities grow or shrink together, and inverse when one rises while the other falls.
For a direct proportion you calculate X = (B × C) / A. For an inverse one you calculate X = (A × B) / C. Choose the type according to the relationship between the quantities and the calculator applies the matching formula.
Example
Direct: if 5 kg cost €20, then 8 kg cost X = (20 × 8) / 5 = €32. Inverse: if 4 workers take 10 days, then 8 workers take X = (4 × 10) / 8 = 5 days.
Direct and inverse rule of three: how to tell them apart
In a direct rule of three, both quantities grow or shrink together: if you buy twice as many kilos of fruit, you pay twice as much. The unknown is solved by cross-multiplying, with the formula x = (b × c) / a. In an inverse rule of three, when one quantity increases the other decreases in the same proportion: if you put twice as many workers on a job, it takes half the time. Here the operation changes and you multiply the two known figures of the same quantity: x = (a × b) / c. Correctly identifying the type of relationship before calculating is the step that determines whether the result is right.
Worked example
Direct: if 3 kilos of apples cost €4.50, how much do 5 kilos cost? Since more kilos means more money, it is direct: x = (4.50 × 5) / 3 = 22.50 / 3 = €7.50. Inverse: if 4 painters take 9 days to paint a building, how long will 6 painters take? Since more painters means fewer days, it is inverse: x = (4 × 9) / 6 = 36 / 6 = 6 days. Note that in the direct case you cross-multiply and in the inverse case you multiply the two figures on the same row.
Most common direct and inverse relationships
| Situation | Type of relationship |
|---|---|
| Kilos of fruit and price paid | Direct |
| Litres of fuel and distance covered | Direct |
| Hours worked and wages earned | Direct |
| Number of workers and days of work | Inverse |
| Speed and journey time | Inverse |
| Taps open and time to fill | Inverse |
How to check the result is correct
A quick and very reliable check is to compare the size of the result with the reference figure. In a direct relationship, if the value you enter is larger than the reference, the result must also be larger; if it is smaller, the result must be smaller. In an inverse relationship exactly the opposite happens. If you get a result that goes in the opposite direction to what you expected, you have almost certainly confused the type of proportionality. Another useful check is to calculate the constant of proportionality: in a direct rule, the ratio between the two quantities must stay constant, while in an inverse rule what stays constant is the product.
Direct or inverse: telling them apart without error
The question that settles the doubt is always the same: if one quantity increases, does the other increase too, or decrease? If they rise together, the proportion is direct and you cross-multiply. If one rises as the other falls, it is inverse and what stays constant is the product of both. Three kilos of oranges cost three times as much as one: direct. Three painters take a third of the time one takes: inverse. The classic mistake is applying the direct rule to everything out of habit, which gives absurd results that a quick check reveals: if adding workers yields more time, the setup is inverted. There is a third case worth recognising: quantities that are not proportional at all. A baby taking nine months to gestate does not mean nine babies take eighty-one months, or that nine mothers take one month.
Four everyday situations solved
Discount: an €80 jacket is reduced by 35%. The discount is 80 × 35 / 100 = €28, so you pay €52. Recipe: a recipe for 4 people uses 320 g of rice and you want to cook for 6. Being a direct proportion, 320 × 6 / 4 = 480 g. Map scale: on a 1:25,000 map, each centimetre on paper is 25,000 real cm, that is 250 metres; a distance of 3.2 cm equals 800 metres. Work rate, the inverse case: if 4 workers unload a lorry in 3 hours, the total work is 12 worker-hours, so 6 workers would take 12 / 6 = 2 hours. Note that the first three cross-multiply and the last conserves the product: that is exactly the difference between direct and inverse proportion.
Results are mathematically exact. If A is 0, the operation is undefined. Verify the type of proportion before applying the result.