What is a proportion?

A proportion is a statement that two ratios are equal — written a/b = c/d. It means the relationship between a and b is the same as the relationship between c and d. Proportions appear everywhere from recipe scaling ("if 2 eggs make 8 biscuits, how many eggs for 20?") to map reading, currency conversion and geometry.

How does cross-multiplication work?

Cross-multiplication is the standard algebraic method for solving proportions. Starting from a/b = c/d, multiply both sides by b and by d to get a·d = b·c. Rearrange for the unknown: to find a use a = (b·c)/d; to find b use b = (a·d)/c; to find c use c = (a·d)/b; to find d use d = (b·c)/a. This calculator applies whichever rearrangement matches your chosen unknown.

Can I use decimals or negative numbers?

Yes. The formula works for any real numbers. Negative values are valid — for example a proportion involving temperatures below zero or debt ratios. The only restriction is that a denominator (b or d) must not be zero, because division by zero is undefined.

What is the difference between a ratio and a proportion?

A ratio compares two quantities (e.g. 3:4). A proportion says two ratios are equal (e.g. 3/4 = 9/12). A ratio is a single comparison; a proportion is an equation between two ratios. This tool solves proportions — it finds the fourth value when three are known.

How do I use this for recipe scaling?

Set a = original ingredient amount, b = original recipe yield, c = desired yield, solve for d = scaled ingredient amount. For example: a recipe uses 250 g flour (a=250) to make 12 cookies (b=12) and you want 30 cookies (c=30). Solving for d gives d = (b·c)/a... wait, the proportion is a/b = d/c, so rearrange: d = (a·c)/b = (250×30)/12 = 625 g. You can also set up the proportion with a and c swapped depending on which quantity you know.

Is this the same as solving similar triangles?

Yes. In similar triangles corresponding sides are proportional — side a of triangle 1 over side b of triangle 2 equals side c of triangle 1 over side d of triangle 2. Enter the three known side lengths, choose the unknown, and the calculator gives the missing side using the same cross-multiplication formula.

Proportion Calculator

A proportion states that two fractions are equal: a/b = c/d. Given any three of those four numbers, the fourth is determined by cross-multiplication — a fast algebraic rearrangement that needs only one arithmetic step to produce an exact answer. This calculator lets you pick which of the four values is unknown, type the other three, and see the result update in real time, together with a step-by-step working pane you can expand and a one-click copy button.

What proportions are used for

Proportions are one of the most widely-used tools in everyday arithmetic and professional maths:

Recipe scaling — if 300 g of flour makes 12 cookies, how much makes 20?
Map and model reading — a 1:50,000 map means 2 cm on the map equals 1 km on the ground.
Currency and unit conversion — if £1 = 1.27 USD, how many dollars is £350?
Similar-triangle geometry — corresponding sides of similar shapes share the same ratio.
Percentage problems — “what percentage of 80 is 12?” is a proportion (12/80 = x/100).
Mixing and dilution — if a 1:4 bleach solution requires 200 ml bleach for 800 ml water, how much bleach for 3 litres of solution?
Speed, distance, time — if a car travels 90 km in 1.5 hours, how far in 2.5 hours?

How the maths works

The proportion equation is:

a / b = c / d

Cross-multiply both sides by b and by d:

a · d = b · c

This single identity — the cross-product rule — is the engine behind all four solve-for-variable rearrangements:

Unknown	Formula
a	a = (b × c) / d
b	b = (a × d) / c
c	c = (a × d) / b
d	d = (b × c) / a

The calculator applies whichever row matches your selected unknown, then checks the result by verifying that a·d = b·c to confirm the proportion balances.

Worked example

Problem: A car travels 150 km in 2 hours. At the same speed, how far does it travel in 3.5 hours?

Set up the proportion:

150 km / 2 h = d km / 3.5 h

Here a = 150, b = 2, c = 3.5, solve for d:

d = (b × c) / a … no — the proportion is a/b = d/c, so cross-multiply: a · c = b · d d = (a × c) / b = (150 × 3.5) / 2 = 525 / 2 = 262.5 km

Alternatively, set a = 150, b = 2, c = 262.5, d = 3.5 and press Proportion check to confirm 150 × 3.5 = 2 × 262.5 (both = 525). The tool shows this balance automatically once all four fields are populated.

Formula note

Cross-multiplication is valid whenever b ≠ 0 and d ≠ 0 — division by zero is undefined. The formula handles any real numbers including decimals, fractions expressed as decimals, and negative values. It does not require the numbers to be integers. All computation happens in 64-bit floating-point arithmetic inside your browser; results are displayed to up to eight significant figures with trailing zeros stripped.