Suppose you are making orange squash. The bottle says: for every 1 cup of orange concentrate, add 4 cups of water. That little recipe — 1 of these for every 4 of those — is a ratio. A ratio is a way of comparing two amounts by saying how much of one you have for a fixed amount of the other.

We write a ratio with a small two dots, like 1 : 4, and we read it out loud as "one to four". So 3 : 2 means "3 of these for every 2 of those", and 5 : 1 means "5 of these for every 1 of those".

The big rule to remember: the order matters. 3 : 2 is not the same as 2 : 3 — the first one means more of the first thing, the second one means more of the second thing. A ratio is like a sentence: swap the words around and you change the meaning.

Part-to-part, and part-to-whole

Most ratios you meet are part-to-part: they compare one group with another group. "3 red sweets to 2 green sweets" is the part-to-part ratio 3 : 2 — it tells you nothing on its own about how many sweets there are altogether, only how the reds compare to the greens.

But a ratio hides a clue about the whole too. Just add the parts. If there are 3 reds for every 2 greens, then in each batch there are 3 + 2 = 5 sweets. So out of every 5 sweets, 3 are red — and as a fraction the reds are \tfrac{3}{5} of the whole, and the greens are \tfrac{2}{5}.

See it: counters in two groups

The easiest way to feel a ratio is to line up the two groups as coloured counters. The top row is Group A, the bottom row is Group B: count each row and read the ratio straight off. Press Refresh for a brand-new ratio to read.

Notice you can see the order: a row with more counters is the bigger part of the ratio. Swap which row is which and the ratio flips — that is the "order matters" rule, right in front of you.

The same ratio as one bar

You can also picture the ratio 3 : 2 as a single bar of equal blocks — three of one colour, two of another. This view makes the whole jump out: the blocks join into one bar, and you can see the first colour fills three of its five parts. Step through to count.

Five equal parts in all: the first colour is \tfrac{3}{5} of the bar, the second is \tfrac{2}{5}.

Three worked examples

1. A fruit bowl. There are 4 apples and 6 oranges. The ratio of apples to oranges is 4 : 6. Altogether that is 4 + 6 = 10 pieces of fruit, so the apples are \tfrac{4}{10} of the bowl. (Later you will learn to tidy 4 : 6 into its simplest form, 2 : 3 — the same recipe with smaller numbers.)

2. A class. In a class there are 3 girls for every 2 boys, a ratio of 3 : 2. Each "batch" is 3 + 2 = 5 children. So the girls are \tfrac{3}{5} of the class and the boys \tfrac{2}{5}. If the class has 20 children — four batches of 5 — that is 4 \times 3 = 12 girls and 4 \times 2 = 8 boys.

3. The squash again. Concentrate to water is 1 : 4. That is 1 + 4 = 5 parts of liquid, so the concentrate is only \tfrac{1}{5} of the whole drink — and the water is a thirsty \tfrac{4}{5}.