How Ratios and Proportions Work — A Complete Guide

A recipe calls for two parts flour to three parts sugar. A map says one centimetre stands for fifty thousand centimetres on the ground. A bill of 600 is split between three people in the ratio 2:3:5. All three situations are the same idea wearing different clothes: a ratio compares amounts, and a proportion says that two ratios are equal. Once you can read a ratio, simplify it and scale it up or down, a surprising amount of everyday maths falls into place.

This guide explains what a ratio means, how to simplify one, how equivalent ratios let you scale a recipe or a drawing, and how ratios connect to fractions and percentages. It works through sharing an amount in a ratio, reading a map scale, and comparing prices fairly, and it shows where people most often go wrong, so you can check your own answers with confidence.

What a ratio means

A ratio compares two or more amounts by showing how many times bigger one is than another. It is usually written with a colon, such as 3:2, and read as “three to two”. If a fruit drink is mixed in the ratio 3:2, juice to water, then for every 3 parts of juice there are 2 parts of water. The parts can be any size — spoonfuls, cups, litres — as long as both sides are measured in the same unit and the same size of part is used throughout.

The order matters, and it is the first thing to get right. A ratio of 3:2 for juice to water is not the same mixture as 2:3 for juice to water: the first has more juice than water, and the second has more water than juice. Always read the words before the ratio, because they tell you which amount comes first. A ratio is a part-to-part comparison: it compares one part with another part. That is different from comparing one part with the whole, which is where fractions and percentages come in later in this guide.

Simplifying ratios

Ratios are usually written in their simplest form, just as fractions are. To simplify a ratio, divide every part by the same number, and keep dividing until the parts share no common factor other than 1. Dividing all parts by the same number does not change the comparison, because every part is shrunk by the same amount. The mixture stays the same; only the numbers get smaller and easier to work with.

Take the ratio 12:18. Both numbers divide by 6, so 12 ÷ 6 is 2 and 18 ÷ 6 is 3, giving 2:3. The ratio 24:36:48 has three parts, and all three divide by 12, giving 2:3:4. A useful check is to ask whether any number divides into every part: if one does, the ratio is not yet in its simplest form. Simplifying first also makes the later steps easier, because sharing an amount or scaling a recipe with small numbers leaves less room for arithmetic slips.

Equivalent ratios: scaling up and down

Multiplying or dividing every part of a ratio by the same number gives an equivalent ratio, one that describes exactly the same comparison. This is how recipes are scaled. If a recipe for 5 servings uses flour and sugar in the ratio 2:3, then doubling the recipe for 10 servings uses 4:6, which is the same ratio with both parts multiplied by 2. Halving it for a smaller batch gives 1:1.5, which is the same ratio with both parts divided by 2.

Scaling is also how you find a missing value. If 2 parts of a mixture weigh 300 grams, then 1 part weighs 150 grams, and 3 parts weigh 450 grams, so the ratio 2:3 corresponds to 300 grams to 450 grams. The key idea is that the relationship between the parts never changes: whatever you do to one side of the ratio, you do to every side. Adding a number to only one side, or multiplying only one side, breaks the comparison and gives a different mixture, a different price, or a different map.

Ratios, fractions and percentages

A ratio can be turned into fractions and percentages by looking at the whole. In the ratio 3:2, there are 3 + 2, which is 5, equal parts in total. The first part is 3 out of 5, which is the fraction 3/5, or 60%, and the second part is 2 out of 5, which is 2/5, or 40%. This part-to-whole step is the bridge between ratios and the other two ideas: the ratio compares parts with each other, while the fraction and the percentage compare one part with the total.

A class with girls and boys in the ratio 4:5 makes the same point. There are 9 equal parts in total, so girls are 4/9 of the class and boys are 5/9. If the class has 27 pupils, each part is worth 27 ÷ 9, which is 3 pupils, so there are 12 girls and 15 boys. The guide to fractions, decimals and percentages explains the conversions in more detail, and the Percentage Calculator can check a ratio turned into a percentage.

Worked example 1: sharing an amount in a ratio

An amount of 600 is shared between three people in the ratio 2:3:5. First, add the parts: 2 + 3 + 5 is 10 equal parts. Next, find what one part is worth: 600 ÷ 10 is 60. Then multiply: the first person gets 2 × 60, which is 120, the second gets 3 × 60, which is 180, and the third gets 5 × 60, which is 300.

Checking the answer is quick and worth doing every time. Add the shares: 120 + 180 + 300 is 600, which is the amount that was shared, so nothing has been lost or invented. The shares also keep the original ratio: dividing 120:180:300 by 60 gives back 2:3:5. If the shares do not add up to the total, or do not simplify back to the original ratio, something has gone wrong in the middle step, and the most likely culprit is the number of parts or the value of one part.

Worked example 2: reading a map scale

A map scale is a ratio comparing a distance on the map with the matching distance on the ground. A scale of 1:50,000 means that 1 unit on the map stands for 50,000 of the same units on the ground. If two places are 4 centimetres apart on this map, the real distance is 4 × 50,000, which is 200,000 centimetres. Converting to kilometres, 200,000 centimetres is 2,000 metres, which is 2 kilometres.

The unit conversion is where map problems usually fail, not the ratio itself. The ratio only works when both sides use the same unit, so the answer comes out in centimetres and must then be converted: 100 centimetres make 1 metre, and 1,000 metres make 1 kilometre. On a 1:25,000 map, the same check gives a distance of 6 centimetres as 6 × 25,000, which is 150,000 centimetres, or 1.5 kilometres. A larger second number in the scale means a smaller, more zoomed-out map: 1:50,000 shows a wider area than 1:25,000, with less detail.

Worked example 3: proportions and comparing prices

A proportion is an equation that says two ratios are equal, and it is the tool for “if this much costs this, how much does that much cost?” problems. If 3 pens cost 12, then 7 pens cost an unknown amount, x, and the proportion is 3/12 = 7/x. Cross-multiplying gives 3 × x = 12 × 7, which is 3x = 84, so x is 28. Seven pens cost 28, which is also what you get by finding the unit rate first: one pen costs 12 ÷ 3, which is 4, and 7 × 4 is 28.

Unit rates make fair comparisons between packs of different sizes. A 2-kilogram bag costs 8, which is 4 per kilogram, while a 3-kilogram bag costs 11, which is about 3.67 per kilogram. The larger bag is cheaper per kilogram even though its total price is higher, because the proportion compares price per equal amount, not total price. Cross-multiplication also gives a built-in check: in the pens example, 3 × 28 is 84 and 12 × 7 is 84, so the two ratios are genuinely equal. If the cross-products disagree, the proportion is wrong.

Common mistakes to avoid

Writing the ratio in the wrong order is the most common error: 2:3 for boys to girls becomes a different answer if it is read as girls to boys. The second is scaling only one side, such as doubling the flour in a recipe but not the sugar, which changes the mixture instead of scaling it. The third is confusing part-to-part with part-to-whole: in the ratio 3:2, the first part is 3/5 of the total, not 3/2 of it, and 3/2 is the ratio written as a fraction comparing one part directly with the other.

Two more traps are worth watching for. One is forgetting to simplify before sharing or scaling, which leaves large numbers that invite slips; simplifying 12:18 to 2:3 first makes every later step easier. The other is mixing units, especially in scale and rate problems: a map distance in centimetres and a real distance in kilometres must be converted to the same unit before the ratio is used, and a price per kilogram cannot be compared with a price per 500 grams until one of them is converted. Writing the unit beside every number makes both traps visible before they do damage.

Frequently asked questions

What is the difference between a ratio and a fraction?

A ratio compares parts with each other, such as 3 parts juice to 2 parts water, while a fraction usually compares one part with the whole, such as 3/5 of the drink being juice. The two are connected: from the ratio 3:2 you can work out that the whole has 5 parts, so the fractions are 3/5 and 2/5. A ratio can also be written as a fraction, 3/2, when it compares one part directly with the other, but that fraction is not a share of the whole.

Can a ratio contain decimals?

Yes, but it is usually tidied up first. A ratio of 1.5:2 describes a real comparison, and multiplying both sides by 2 clears the decimal to give 3:4, which is easier to use. The same applies to a ratio with fractions: multiplying both sides by a common denominator turns it into whole numbers. The comparison is unchanged in both cases, because every part was multiplied by the same number.

What if the total is not given, only one share?

Work out the value of one part from the share you know, then scale up. If one person receives 120 in the ratio 2:3:5, then 2 parts are worth 120, so 1 part is worth 60, and the other shares are 180 and 300, with a total of 600. The same method works in reverse: if you know the total and the ratio, add the parts first and divide the total by that number, as in the first worked example.

Do ratios work with more than two parts, or with three-way comparisons?

Yes. A three-part ratio such as 2:3:5 is handled exactly like a two-part one: add all the parts to find the total number of equal shares, find the value of one part, and multiply. The only extra care needed is to keep all three parts in the same order throughout, and to check at the end that the shares add up to the total and simplify back to the original ratio.

If you want to check a result, try the matching tools on this site: the Percentage Calculator handles ratio-to-percentage conversions, and the Scientific Calculator handles the multiplication and division for larger amounts and scale problems.