Proportion Calculator

About this Proportion Calculator

The Proportion Calculator solves a common cross-multiplication problem where three values are known and the fourth is unknown. It can help with scale drawings, recipes, rates, conversions and classroom proportion exercises.

Solving a proportion (a/b = c/x) is the mathematical basis for scaling recipes, converting map distances to real distances, and adjusting ingredient ratios in manufacturing, making it one of the more broadly applicable pieces of basic algebra.

For related tasks, try Ratio Calculator, Fraction Calculator or Percentage Calculator.

How the calculation works

From a/b = c/x, cross multiplication gives a × x = b × c. The calculator therefore solves x = (b × c) ÷ a when a is non-zero.

How to use it

1. Enter values for a, b and c.
2. The tool treats the equation as a/b = c/x.
3. Calculate to solve for x.

Practical example

If 2/5 = 6/x, cross multiplication gives x = 15.

If 3 apples cost 90, then at the same per-unit price 5 apples cost 150; this is a true proportion, unlike a scenario where more workers reduce completion time, which is an inverse relationship rather than a direct proportion.

Things to double-check

Proportions assume a direct (linear) relationship between the quantities involved; they do not apply correctly to relationships that scale differently, such as area (which scales with the square of a linear dimension) or volume (which scales with the cube), so use this tool only for genuinely proportional quantities.

Solving for the missing value

A proportion says that two ratios are equal: a over b equals c over x. Cross multiplying gives x equals b times c divided by a. If 3 kg of rice costs 450, then 5 kg costs 450 times 5 divided by 3, which is 750. The method works whenever the two quantities rise and fall together at a constant rate, such as price and weight at a fixed unit price. Always put the same kinds of quantities in the same positions on each side.

Scaling recipes

Recipes are a natural use. A recipe for 4 people uses 250 g of flour; for 10 people, the amount is 250 times 10 divided by 4, which is 625 g. Apply the same factor, 2.5, to every ingredient. Some ingredients, such as spices, salt and baking powder, do not scale perfectly and are better increased by a smaller factor and adjusted to taste. Cooking times also change with the size of the dish, so treat the calculated quantities as the starting point.

Maps and scale models

Map scales are proportions. On a 1 to 50,000 map, 3.2 cm represents 3.2 times 50,000 cm, which is 160,000 cm or 1.6 km. Model builders and architects use the same principle to convert between a drawing and real life. Make sure the units are the same on both sides before you calculate, and convert centimetres to metres or kilometres at the end.

Direct and inverse proportion

Not all relationships are direct. If 6 workers finish a job in 10 days, 4 workers need 15 days, because the total work, 60 worker-days, stays the same. That is an inverse proportion, and using the direct formula here would give the wrong answer. Ask whether doubling one quantity doubles the other or halves it. Area and volume scale with the square and the cube of length, so a model twice as wide has four times the area, not twice.

Unit rates and best value

A unit rate turns a proportion into a single comparison. If 3 kg costs 450, one kilogram costs 150. A car covering 150 km in 2.5 hours travels at 60 km per hour. Unit rates make shopping comparisons fair: a 500 g pack at 120 costs 24 per 100 g, while a 750 g pack at 168 costs 22.40 per 100 g, so the larger pack is better value if you will use it. Always compare the same unit, check expiry dates and make sure that you will really use the larger amount.

Worked example: paint for a room

A 2.5 litre tin of paint covers 30 square metres. A room has 48 square metres of wall to paint. Setting up the proportion, 2.5 over 30 equals x over 48, so x equals 2.5 times 48 divided by 30, which is 4 litres. For two coats, double that to 8 litres, and add a little for waste. Always check that the units match on both sides, here litres and square metres, and that the relationship is really proportional: paint coverage depends on the surface, so a rough wall may need more. A proportion gives the starting estimate, and the label on the tin gives the details.

Unit rates and comparing deals

Turning a proportion into a unit rate makes comparisons simple. If 150 km takes 2 hours 30 minutes, the speed is 60 km/h. If 500 g of rice costs 120 and 2 kg costs 430, the first is 24 per 100 g and the second 21.50 per 100 g, so the larger pack is cheaper per gram. Currency conversion works the same way: if 1 dollar buys 278 rupees, then 45 dollars buys 12,510. Writing the unit after each number helps you set the proportion in the right order and catches mistakes before they reach the answer.

Inverse proportion

In an inverse proportion, one quantity rises as the other falls so that their product stays constant. If three pumps fill a tank in 8 hours, the total work is 24 pump-hours, so four pumps take 6 hours. The same idea applies to workers on a job, speed and travel time, and the number of people sharing a cost. If a group of 6 splits a bill of 120, each pays 20, and if 8 people share it, each pays 15. Recognising whether a problem is direct or inverse is the key step, and a quick check is to ask whether doubling one quantity should double or halve the other.

Scale models and similar shapes

Proportions also describe scale. On a model at 1 to 20, a real length of 3 metres becomes 15 centimetres. But areas and volumes do not scale the same way: the area of the model is 1/400 of the real area, and its volume is 1/8,000 of the real volume. That is why a model boat made at half the size has only one eighth of the weight, and why large animals need thicker legs than small ones. When you scale a recipe, a drawing or a design, check whether the quantity depends on length, area or volume, and apply the right power.

Key terms explained

A ratio compares two quantities, and a proportion states that two ratios are equal. In direct proportion, two quantities rise and fall together at a constant rate. In inverse proportion, one rises as the other falls so that their product is constant. The constant of proportionality is the fixed factor linking them. Cross-multiplication is the method for solving a proportion. A unit rate is a quantity per one unit, a scale relates a drawing to reality, and similar shapes have equal angles and proportional sides.

Rules of thumb and quick checks

If a over b equals c over d, then a times d equals b times c. For direct proportion, y equals k times x, and for inverse proportion, x times y equals k. Check the units on both sides of the proportion, and ask whether doubling one quantity should double or halve the other. After you solve for the missing value, plug it back into the original ratio to confirm that the two sides match.

Frequently asked questions