Ratios and Proportions

What a ratio actually says

A ratio is a comparison written with a colon: 3:2 means three of something for every two of something else. It is written as a pair of numbers rather than a single number, because it describes a relationship between two amounts, not a value in itself.

Order matters. The ratio 3:2 says the first quantity is larger; 2:3 says the second is larger. Both describe the same relationship seen from opposite ends, which is why ratios are never swapped around casually — a bottle diluted 1 part concentrate to 4 parts water is 1:4, not 4:1.

A ratio can also compare different kinds of quantity. "300 g of flour for 4 people" is the ratio 300:4, which is 75 g per person. The fractions, decimals and percentages guide covers the other forms the same relationship can take.

Simplifying a ratio

To simplify a ratio, divide every part by the same number. The best choice is the greatest common divisor of the parts, because it gives the smallest whole-number form. For 12:18, the greatest common divisor is 6, so 12 ÷ 6 = 2 and 18 ÷ 6 = 3, giving 2:3.

The simplest whole-number form is not the only valid form. 2:3, 4:6, 10:15 and 120:180 all describe the same relationship, because multiplying or dividing both sides by the same factor leaves the comparison unchanged. 2:3 is simply the version with the smallest numbers.

One shortcut covers most cases: if both parts are whole numbers and share a factor of 2, 3, 5 or 10, divide by that factor. When the numbers are large or have no obvious factor, the ratio calculator finds the common divisor for you, and the factors and primes guide explains how to do the same by hand.

Ratios that include decimals

A ratio can be given in decimal form. Scale both sides up to whole numbers first, then simplify. For 1.5:2.5, multiply both parts by 10 to get 15:25, and the common divisor is 5, so the simplified ratio is 3:5. Multiplying both parts by the same number never changes the comparison, which is exactly why scaling is safe.

The NumberUtils ratio calculator does this scaling internally, so entering 1.5 and 2.5 returns the same simplified ratio as entering 15 and 25.

Ratios as fractions and percentages

In a two-part ratio a:b, the total is a + b. Each side is a fraction of that total: a ÷ (a + b) and b ÷ (a + b). For 2:3, the total is 5 parts, so the sides are 2/5 and 3/5, which is 40% and 60%.

Worked example: a class of 85 pupils is split in the ratio 2:3. The total is 5 parts, and 85 ÷ 5 = 17 per part. The two groups are 2 × 17 = 34 and 3 × 17 = 51, and 34 + 51 = 85 confirms the split. To turn either share into a percentage instead, use the percentage calculator.

For longer ratios, find one part by dividing the amount by the sum of the parts. 12:2:3 has 17 parts in total, so 85 ÷ 17 = 5 per part, giving 60, 10 and 15. The NumberUtils ratio calculator works with two parts and reports each side's percentage share; three-part sharing is the same method applied by hand.

Solving a proportion

A proportion states that two ratios are equal: a:b = c:x. To find the missing value, cross-multiply — multiply the top-left with the bottom-right, and the bottom-left with the top-right, then divide. In full, a × x = b × c, so x = (b × c) ÷ a.

Worked example: a recipe uses 300 g of flour for 4 people. Set up 4:300 = 6:x and cross-multiply: 4 × x = 300 × 6, so x = 1,800 ÷ 4 = 450 g. Each person uses the same 75 g, and 75 × 6 = 450, which checks the answer.

The proportion calculator applies exactly this cross-multiplication for you. It solves direct proportions, where both quantities rise or fall together, and it cannot divide by zero, so the first value you enter must not be zero.

Map scales and scaling up

A map scale is itself a ratio of the same kind. A scale of 1:25,000 means one unit on the map represents 25,000 of the same units on the ground. For 4 cm on the map: 4 × 25,000 = 100,000 cm, which is 1,000 m, or 1 km.

When a scale factor does not divide exactly, round only at the end and keep the working. 7 cm at 1:50,000 is 350,000 cm, or 3.5 km — a figure that lands exactly. If the division would not, round to a sensible precision such as the nearest metre; the rounding notes in the methodology set out how this site handles precision.

Direct and inverse proportion

Two quantities are in direct proportion when one is a fixed multiple of the other, so y = k × x for some constant k. More people means more flour; a longer route means more fuel. Cross-multiplication solves these, and the constant never changes.

Two quantities are in inverse proportion when one grows as the other shrinks, so x × y = k for a constant k. Work is a fixed amount of effort, so adding people shortens the time. With 3 workers taking 12 days, the total is 3 × 12 = 36 worker-days; with 4 workers it takes 36 ÷ 4 = 9 days.

That last step assumes every worker works at the same rate, which is an assumption rather than a fact about people. Ratios and proportions are exact arithmetic; real-world tasks rarely are, so decide which assumption you are making before you rely on the answer. The ratio and proportion methodology notes set out how these calculations are defined.

Limits worth knowing

The NumberUtils ratio and proportion calculators work with quantities that can sensibly be compared as shares of something, so both reject negative values. A ratio of negative quantities has no meaningful scale, and leaving the sign to fall out of the division would produce an answer the method was never designed to give.

Zero needs more care than a negative. Both calculators reject a pair of zeros, because 0:0 compares nothing with nothing and cannot be simplified. One zero on its own is valid in a ratio: 0:5 simplifies to 0:1. In a proportion a:b = c:x, the first value cannot be zero because it is the divisor, but the others can: 4:0 = 6:x solves to x = 0.