Proportion Calculator (Rule of Three)
Solve a proportion a : b = c : x for the missing value.
The “rule of three” solves proportions: if a is to b as c is to x, this finds x. Enter the three known values to get the fourth — useful for scaling recipes, maps, currencies and mixtures.
Formula / method
Examples
x = 15
x = 160
Solving the rule of three
A proportion states that two ratios are equal, written a to b equals c to x. When one of the four values is unknown, you find it by cross-multiplying: from a : b = c : x you get a times x equals b times c, so x equals (b times c) divided by a. This tool performs that cross-multiplication instantly for whichever value you leave blank.
The method is often called the rule of three because you supply three known numbers to recover the fourth. It only works when the two quantities are genuinely proportional, meaning one grows in direct step with the other.
Everyday uses and cautions
Proportions handle a huge range of practical scaling problems: doubling a recipe, converting a distance on a map using its scale, working out unit prices, or resizing an image while keeping its aspect ratio. If three eggs make twelve pancakes, a proportion tells you how many eggs make thirty.
Take care that both ratios use the same units and the same order on each side; mixing grams with kilograms, or putting a cost where a quantity belongs, produces a technically correct but meaningless answer. For straight percentage questions, the Percentage Calculator is a more direct fit.
Where it's used
- Scaling a recipe up or down for a different number of servings
- Converting a measured distance using a map's scale ratio
- Working out a unit price to compare two package sizes
- Resizing an image while keeping its aspect ratio
- Mixing a solution, paint, or fuel to a fixed ratio
Real-world examples
- If 3 eggs make 12 pancakes, the tool finds that 5 eggs make 20.
- A map scale of 1:50,000 turns a 4 cm measurement into 2 km on the ground.
- A 400 g item costs $3; a proportion gives the price of 650 g.
A bit of history
Solving proportions this way is the centuries-old 'rule of three': known in Chinese and Indian arithmetic well over a thousand years ago, it reached Europe through Arab mathematicians and was so prized by Renaissance merchants that it was nicknamed the 'Golden Rule'.
Did you know?
The rule of three was such a standard skill that some of the oldest surviving examples of Abraham Lincoln's handwriting come from a boyhood arithmetic notebook in which he worked rule-of-three problems.
FAQ
When is the rule of three useful?
Any time two quantities scale together — converting recipe amounts, map distances, exchange rates or mixing ratios. Set up the known ratio, and it finds the unknown.