Proportion Calculator

What is a proportion?

A proportion is a statement that two ratios or fractions are equal. If a recipe uses 2 cups of rice for every 3 cups of water, then a larger batch that uses 4 cups of rice needs 6 cups of water, because 2/3 and 4/6 describe the same relationship.

a ÷ b = c ÷ da and d are the extremes, b and c are the means.

Proportions are the standard tool for scaling. They turn up in recipes, map scales, currency exchange, unit prices, similar triangles, dosage tables and anywhere one quantity changes in step with another. A proportion is built from two ratios, and each side can also be read as a fraction.

How to use the calculator

1. Write the first ratio in the Proportion 1 boxes, with the numerator on top and the denominator below.
2. Write the second ratio in the Proportion 2 boxes in the same order.
3. Leave the one term you want to find empty and press Calculate.

Exactly one box must be empty. If all four are filled the calculator asks you to clear one, and if two or more are empty there is not enough information to solve. A box containing 0 is treated as empty, so a zero cannot be used as a term. Negative numbers and decimals are accepted, and answers are shown with up to 14 significant digits.

How to solve a proportion

The method is cross multiplication. Multiply the numerator of each fraction by the denominator of the other. In a true proportion the two products are equal, which gives a simple equation for the unknown.

a × d = b × c

Dividing by the coefficient of the unknown term gives one formula for each position:

• missing a = b × c ÷ d
• missing b = a × d ÷ c
• missing c = a × d ÷ b
• missing d = b × c ÷ a

You can also reason with a scale factor. In 3/12 = x/30 the denominator grows from 12 to 30, a factor of 2.5, so the numerator must grow by the same factor: 3 × 2.5 = 7.5.

The constant of proportionality

Two quantities are directly proportional when their ratio never changes. That fixed ratio is called the constant of proportionality, usually written k, and it links the two quantities through a straight line that passes through the origin.

y = k × x, so k = y ÷ x

Example: 5 notebooks cost 12.50, so k = 12.50 ÷ 5 = 2.50 per notebook. Eight notebooks then cost 2.50 × 8 = 20.00. Every pair of values from the same table gives the same k, which is exactly what makes the two ratios equal.

In an inverse proportion the product stays fixed instead: x × y = k. If 6 workers finish a job in 10 days, the fixed product is 60 worker-days, so 15 workers need 60 ÷ 15 = 4 days. For inverse problems, flip one of the two ratios before entering them in the calculator: 6/15 = x/10 gives x = 4.

Common mistakes

• Mixing the order. Keep the same quantity on top in both fractions. 3 cups / 12 muffins must be matched with x cups / 30 muffins, not 30 muffins / x cups.
• Mismatched units. Convert first, so that minutes sit against minutes and metres against metres.
• Assuming every relationship is proportional. A taxi fare with a fixed starting charge does not double when the distance doubles, so a proportion gives the wrong answer there.
• Rounding too early. Keep the full value through the calculation and round only the final answer.

When the question is about a share of a total, a percentage is often easier. To reduce a fraction after solving, try the simplify fractions calculator.