Current for 2026•As of: July 2026

Rule of Three Calculator solve proportions.

Direct and inverse – with calculation method and step-by-step explanation

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Rule of Three Calculator

Calculate direct and inverse rule of three – with the full calculation method.

If a gets bigger, b also gets bigger (e.g. price & quantity).

Desired value (x)

2.5

Direct rule of three

Calculation method

3 → 1.5, i.e. 5 → 5 × 1.5 ÷ 3 = 2.5

Rule of three: method and application

Understanding direct and inverse proportions

The rule of three is one of the most fundamental calculation methods in mathematics and indispensable every day – when shopping, cooking, in technical fields, and at work. The principle: a fourth unknown value is calculated from three known values. The method works because two quantities are related to each other in a fixed ratio.

The direct rule of three applies wherever more of one quantity also means more of the other. If 1 kg of flour costs €1.20, 3 kg costs exactly €3.60. The price grows in the same ratio as the quantity. Formula: x = c × b ÷ a. You first calculate the unit value (b ÷ a) and then multiply by the desired quantity c.

The inverse rule of three is used when an increase in one quantity causes a decrease in the other. If 4 workers need 6 days for a task, 12 workers complete the same task in just 2 days. Formula: x = a × b ÷ c. Memory aid: the product of both value pairs is always equal (a × b = c × x).

In everyday life and at work you encounter the rule of three constantly: scale calculations for maps and building plans, recipe quantities for different numbers of people, currency conversions, speed-distance-time calculations, and material calculations. Our calculator shows you not only the result but also the complete calculation method.

Calculation examples

Direct: 3 kg → €1.50, how much does 5 kg cost?

Direct: 3 kg → €1.50, how much does 5 kg cost?
ItemAmount
Known pair3 kg → €1.50
Unit value1 kg → €0.50
Desired quantity5 kg
Result x€2.50

Inverse: 4 workers × 6 days, how long do 12 workers need?

Inverse: 4 workers × 6 days, how long do 12 workers need?
ItemAmount
a × b4 × 6 = 24
New quantity c12 workers
Formula24 ÷ 12
Result x2 days

Frequently asked questions about the rule of three calculator

Everything about direct and inverse rule of three

The rule of three (also called cross-multiplication) is a calculation method that lets you determine an unknown fourth value from three known values. It is used whenever two quantities are related in a fixed ratio. Typical applications: price calculations for quantities, travel times, recipe quantities, or scale conversions.

With the direct rule of three, one quantity grows in the same ratio as the other: if you buy twice as many apples, you pay twice as much. With the inverse rule of three, the opposite applies: if twice as many workers complete a task, the time needed is halved. The formula for the direct rule of three is x = c × b ÷ a, for the inverse rule of three x = a × b ÷ c.

Step 1: Set up the known values (e.g. 3 kg costs €1.50). Step 2: Reduce to 1 (1 kg costs 1.50 ÷ 3 = €0.50). Step 3: Multiply by the desired quantity (5 kg costs 0.50 × 5 = €2.50). Our calculator performs all steps automatically and shows the calculation method.

You use the inverse rule of three when two quantities behave inversely to each other. Classic situations: machines and production time (more machines = shorter time), workers and construction time, speed and travel time. If a × b = c × x holds, the rule of three is inverse.

Direct: x = (b × c) ÷ a. Inverse: x = (a × b) ÷ c. Here a and b are the known pair of values, c is the new quantity, and x is the desired value. Memory aid: for the inverse rule of three, you invert the second ratio – instead of multiplying, you divide.

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