Rule of three calculator
Solve direct and inverse proportions by entering the three known values.
Calculator inputs
Results
Enter your values and press “Calculate” to see the result.
In short
- What it calculates
- Solve direct and inverse proportions by entering the three known values.
- Formula used
Direct proportion:- Example
- Example 1: If 3 kg of apples cost £12, how much do 7 kg cost?
Rule of three calculator
The rule of three solves the most common problem in everyday arithmetic: if I know three values of a proportion, what is the fourth?
The key is telling a direct proportion from an inverse one, because the operation changes completely.
How it works
In a direct proportion both quantities grow together: more kilos, higher price. You cross-multiply and divide by the value paired with the unknown.
In an inverse proportion, when one quantity grows the other shrinks: more workers, fewer days. Here the product of each pair stays constant.
Formula
Direct proportion:
A → B
C → X
X = (B × C) ÷ A
Inverse proportion:
A → B
C → X
X = (A × B) ÷ C
Worked example
Example 1: If 3 kg of apples cost £12, how much do 7 kg cost?
This is a direct proportion:
X = (12 × 7) ÷ 3 = £28
If 4 workers take 12 days, how long would 6 take?
Inverse proportion: X = (4 × 12) ÷ 6 = 8 days
Example 2: A recipe uses 250 g of flour for 12 biscuits. How much flour do you need for 30?
More biscuits need more flour: direct proportion
12 biscuits → 250 g
30 biscuits → X
X = (250 × 30) ÷ 12 = 625 g
Explanation
Direct or inverse: the question that decides everything
In a direct proportion, if one quantity rises so does the other: more kilos of fruit, higher price. In an inverse one, if one rises the other falls: more workers, fewer days on site. Framing an inverse problem as direct is the error that ruins the result, and the check is simple: ask yourself whether doubling one quantity doubles the other or halves it.
Beware of what is not proportional
If one painter takes four hours to paint a room, two painters do not always take two. And nine women do not gestate a baby in a month. Many real situations have diminishing returns, physical limits or tasks that cannot be divided. The rule of three assumes a perfect proportionality that does not always exist.
It is the same operation as a percentage
Calculating fifteen per cent of two hundred is a direct rule of three: if a hundred corresponds to two hundred, fifteen corresponds to thirty. Seeing it that way helps solve percentage problems without memorising separate formulas.
Scaling recipes and mixtures
This is its commonest domestic use: adapting a recipe from four to six people, or working out the water and cement proportion for a different quantity of mortar. In mixtures, keep all the proportions, not just the one being adjusted.
Frequently asked questions
How do I know whether it is direct or inverse?
Think of an extreme case. If doubling A should double the result, it is direct. If doubling A should halve it, it is inverse.
Does it work for percentages?
Yes. “15% of 300” is a direct rule of three: if 100 corresponds to 300, then 15 corresponds to X.
Can I use decimals?
Yes, and negative values too. The only restriction is that the divisor cannot be zero.
Can a rule of three always be applied?
No. Many real situations have diminishing returns or physical limits, and the perfect proportionality the rule assumes does not hold.
What is a compound rule of three?
One involving more than two quantities, such as workers, days and metres. You solve it by chaining simple rules of three, one per quantity, checking each time whether it is direct or inverse.
How can I check the answer?
In a direct proportion the ratio stays the same: 250 ÷ 12 and 625 ÷ 30 both give 20.83 g per biscuit. In an inverse one the product stays the same: 4 × 12 = 6 × 8 = 48.
Does it work for currency or unit conversions?
Yes, any conversion with a fixed factor is a direct proportion. If 1 euro is 1.17 dollars, 40 euros are 40 × 1.17 = 46.80 dollars.
Sources and references
The formula behind this calculator is not ours. These are the sources it comes from, in APA 7 format, in case you want to go to the original or cite them in your own work.
- Weisstein, E. W. (s. f.). Rule of Three. MathWorld—A Wolfram Web Resource. https://mathworld.wolfram.com/RuleofThree.html https://mathworld.wolfram.com/RuleofThree.html The classic method for solving direct and inverse proportions with four terms.
Need to calculate something else?
These tools are often used alongside this calculator.
Percentage
The four most common percentage calculations in a single tool.
Fractions
Add, subtract, multiply or divide two fractions and get the result already simplified.
Average
Paste your list of numbers and get the mean, median, mode, range and standard deviation.
Length
Convert between metres, kilometres, miles, feet, inches and yards.
Calculator
Type the whole expression and get the result instantly, brackets and functions included.
Scientific
Trigonometry, logarithms, exponentials and factorials, with a degrees/radians switch.