Suppose you're making a salad dressing — one spoon of vinegar for every three spoons of oil. You could write that as 1 : 3, a vinegar-to-oil ratio. The little colon is doing real work: it says these two amounts go together in a fixed pairing. One vinegar, three oil. Always. Whether you make a tiny jar or a vat for a party.
Here is the part worth slowing down for. A ratio is not the same as a fraction, even though they look like cousins. A fraction tells you a part of the whole — “two thirds of the class”. A ratio tells you how two parts compare to each other — “two girls for every three boys”. In that class, the ratio of girls to boys is 2 : 3, but the fraction who are girls is two fifths, because there are five children in every group of five. Same situation, two different jobs.
If the question pits one group against another — paint to water, cats to dogs, sand to cement — it's a ratio. If it pits one group against the whole — “what fraction of the mix is sand?” — it's a fraction. The trick is to find the total first: add the ratio parts (2 + 3 = 5) and that total becomes the bottom of the fraction.
The colon we use for ratios — that pair of dots — was borrowed from old printing in the 1600s, where it already meant “a pause that balances two halves”. Mathematicians liked that it sat evenly between two numbers without favouring either, which is exactly what a ratio does: it holds two amounts in balance, neither one the boss.
A ratio of 8 : 12 and a ratio of 2 : 3 describe exactly the same balance — for every two of the first thing there are three of the second, whether you count in fours or in ones. Simplifying just finds the smallest whole numbers that keep the balance, by dividing both sides by the same thing.
A ratio can hide a unit-price secret. “3 : 2” on a recipe and “3 for £2” on a market stall are the same shape of thought — a fixed pairing you can scale up or down. Market traders who can simplify a ratio in their heads can spot, in a glance, which of two stalls is really the cheaper. The maths on the page is the maths in the marketplace.
Two friends do a car-wash job together. Mia does most of the scrubbing, Sam does the rest, and they agree to split the £40 they earn in the ratio 3 : 2 — three parts to Mia, two to Sam. Not equally; fairly, by effort. Here is the calm, reliable way to do it, and it always runs in the same three steps.
Almost every ratio question secretly asks you to find the value of one part first. Total amount, divided by total number of parts. Once you know what one part is worth, every share is just “how many parts × one part”. It's the same idea behind the unitary method on the next page — get down to one, then build back up to whatever you need.
The oldest written ratios we have are about beer. Clay tablets from ancient Mesopotamia, four thousand years old, record grain-to-beer ratios for paying workers — so many measures of barley made so many jugs of ale. Ratio may be the first piece of maths humans ever wrote down, and it was invented to keep a brewery honest.
Two amounts are in direct proportion when one is always a fixed multiple of the other — double one and the other doubles too; halve one and the other halves. Buy twice as many apples and you pay twice as much. The price and the number of apples rise and fall together, locked in step. Most “if this costs that, then how much for so many” questions are direct proportion, and the cleanest tool for them is the unitary method: get down to one, then scale up.
| Apples | 1 | 6 | 10 |
|---|---|---|---|
| Cost | 25p | £1.50 | £2.50 |
Notice how the two rows move in lock-step: as the top doubles, so does the bottom; the link between them — 25p per apple — never changes. That fixed link is the heart of proportion. Spot it, find its value for one, and the rest is multiplying.
The word proportion comes from the Latin pro portione — “for its share”. The Romans used it for splitting land and water fairly between farmers: each got an amount in proportion to the field they worked. The maths you do to scale a recipe is, word for word, the maths they did to share a river.
The reason ratio is worth keeping in your head is that it's everywhere the moment you stop looking for it on a worksheet. Three places it lives, that you will meet for the rest of your life:
| Where you meet it | What the ratio fixes | An example |
|---|---|---|
| Maps & models | real size to drawn size | 1 : 50 000 |
| Mixing paint, squash, concrete | one ingredient to another | paint 1 : 4 water |
| Recipes scaled up or down | how amounts grow together | flour : milk = 2 : 1 |
Take the map. A scale of 1 : 50 000 means one unit on the paper stands for fifty thousand of the same units on the ground. So 2 cm on the map is 100 000 cm in the real world — which is 1 000 m, or 1 km. The ratio lets a whole county fit on a fold-out sheet without ever lying about the distances. And mixing is the same idea wearing overalls: a paint label that says “dilute 1 : 4” means one part paint to four parts water — get the ratio wrong and the colour comes out wrong, every time.
If a recipe is 2 : 1 flour to milk, it works whether you make a little or a lot — as long as you keep the balance. Double both and it's fine. But change just one — extra milk, same flour — and you've changed the ratio, and the batter is ruined. That's the whole power of ratio thinking: it's the balance that matters, not the size.
Mixing concrete is ratio you can stand on. A common mix is 1 : 2 : 3 — one part cement, two parts sand, three parts gravel. Builders carry it as a chant, not a calculation, because the strength of the finished concrete depends entirely on the balance. Too much sand and a wall can crumble; the ratio is quite literally load-bearing.
Here is the £40 car-wash bar from earlier, shared 3 : 2 between Mia and Sam, with four pins. Pick a label below, then drop it on the matching pin. Two of the chips belong to a different lesson entirely — read each one before you place it.
Fresh one. A box has 10 red pens and 15 blue pens. Write red : blue in simplest form.
Fresh one. £24 is shared in the ratio 1 : 3. How much is the larger share?
Fresh one. 4 pens cost £3.00. Using the unitary method, what would 6 pens cost?
There's a famous ratio that artists and architects have chased for centuries — the golden ratio, roughly 1 : 1.618. It turns up in the spiral of a snail's shell, the proportions of the Parthenon, and the way sunflower seeds pack their heads. Nobody fully agrees why it pleases the eye, only that, again and again, it does.
A jug of squash is mixed in the ratio 1 : 5 — one part cordial to five parts water — and that makes exactly enough for one person. You need enough for a party of nine people, and your bottle holds 300ml of cordial. Work out how much cordial and how much water you need for nine, say whether your 300ml bottle is enough, and explain what would go wrong if you kept the water the same but doubled only the cordial. Show your steps.
First find one person's share, then multiply both parts by nine. Then check the cordial total against 300ml. Last, say in words why changing one part breaks the mix.
strong You scaled both parts by nine, not just one — cordial and water rising together. That's the whole idea of keeping a ratio, and you held it. Multiplying the 1 : 5 up to 9 : 45 is exactly the right move.
try this State the “is it enough?” answer in one plain line — nine parts of cordial against your 300ml bottle — so a reader sees the verdict, not just the sum. The number alone isn't the answer; the yes or no is.
to add On the last part, name why doubling only the cordial ruins it: you'd change the balance from 1 : 5 to 2 : 5, so the squash comes out far too strong. Saying it changes the ratio is the word the marker is listening for.
Two short films to watch alongside today's lesson — each shows you something the words and pictures can't.
You learned that a ratio compares two parts to each other, while a fraction compares one part to the whole. You can simplify a ratio, share an amount in a given ratio by finding one part first, and use the unitary method to scale anything in proportion. Next time you mix a squash, read a map, or double a recipe, Florence, you'll know exactly what's being held in balance — and how to keep it there.
Your whole body is built to a set of rough ratios. Stretch your arms out wide and the span, fingertip to fingertip, is very close to your height — a 1 : 1 ratio Leonardo da Vinci drew famously in his Vitruvian Man. Your foot is about the length of your forearm, too. Proportion isn't just on the page; it's stitched into you.