Ratio
A ratio compares two amounts by counting them against each other: 3 to 9 means three of one thing for every nine of the other. It is a division, a fraction and a comparison all at once.
The ratio tool
Split the whole
Drag the divider to change the two shares
4 8 1 : 2
Twelve equal parts, split into two shares.
How ratio works
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Write the two amounts in order
Three parts to nine parts is written 3 : 9, read "three to nine". Order carries information: 9 : 3 describes the same two amounts the other way round, and it is a different comparison. The divider on this page keeps both shares in whole parts so the order stays honest.
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Simplify by dividing both sides
3 : 9 and 1 : 3 compare the same two shares, because both sides were divided by three. Simplifying a ratio is simplifying a fraction: divide both numbers by the same number, and keep doing it until nothing divides both of them.
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Count the parts to share a total
To share 12 in the ratio 1 : 2, first add the parts: 1 + 2 = 3 parts altogether. Then divide the total by the number of parts: 12 ÷ 3 = 4 in each part. One share gets 4 and the other gets two of them, which is 8.
Worked examples
3 : 9
The same as 1 : 3. Both numbers divide by three, so three parts to nine is one part to three. Nothing about the actual amounts has changed; the comparison has simply been written with smaller numbers, just as 3/9 simplifies to 1/3.
Share 12 in the ratio 1 : 2
Four and eight. The ratio has 1 + 2 = 3 parts, and 12 ÷ 3 = 4 in each part, so the first share takes 1 × 4 = 4 and the second takes 2 × 4 = 8. Checking: 4 + 8 = 12, and 4 : 8 simplifies to 1 : 2.
Squash: 1 part to 4 parts water
The ratio is 1 : 4, which is 5 parts in total. To make 500 ml you divide 500 by 5 to get 100 ml per part, so 100 ml of squash and 400 ml of water. That is why 100 and 400 are the amounts, not 1 and 4.
Where it goes wrong
- Reversing the order 3 : 9 and 9 : 3 describe different situations: one says there is three times as much of the second thing, the other says the opposite. Read the sentence that goes with the ratio before writing the numbers down.
- Simplifying by subtracting Subtracting gives 3 : 9 → 6, which is not a comparison at all. Ratios simplify by division, because the aim is to keep the same relationship in smaller numbers: 3 and 9 both divide by 3.
- Treating the first number as the fraction of the whole In the ratio 3 : 9 the whole is 12 parts, so the first share is 3/12 — a quarter of the total — not 3/9. Adding both sides to find the number of parts is the step that stops this going wrong.
Questions
What does a ratio tell you?
It tells you how two amounts compare, and in which order. A recipe with a ratio of 1 : 4 uses one unit of the first thing for every four of the second, so the amounts scale together: double the drink and you double both parts. A ratio describes a relationship rather than a fixed quantity.
How do I share an amount in a ratio?
Add the parts of the ratio to find how many there are altogether, divide the total by that number to find the size of one part, then give each share the number of parts it is owed. Sharing 12 in the ratio 1 : 2 has three parts, so each part is 4, giving 4 and 8.
How is a ratio different from a fraction?
A ratio compares one part with another part; a fraction compares a part with the whole. In the ratio 3 : 9 there are 12 parts in total, so the first share is 3/12 of the whole. The ratio 3 : 9 and the fraction 3/9 are not interchangeable, and mixing them up is the commonest error in this topic.
Does the order in a ratio matter?
It matters completely. 3 : 9 means three of the first thing for every nine of the second; 9 : 3 means nine for every three. Both simplify — to 1 : 3 and 3 : 1 — and those describe opposite situations, so the order is part of the meaning.
Why do recipes and mixtures use ratios?
Because they scale cleanly. Concentrate to water is roughly 1 : 4 whatever size of jug you use, so the ratio stays the same while the amounts change: 100 ml to 400 ml, or 250 ml to a litre. A ratio is the instruction for how to make more or less without tasting anything.