Ratio and Proportion

A ratio is a way of comparing two or more amounts. If a fruit bowl holds 3 apples for every 2 oranges, we write the ratio of apples to oranges as $3 : 2$, said "three to two".

The colon is read as "to". The order matters a great deal — $3 : 2$ tells you apples come first, so swapping it to $2 : 3$ would describe a completely different bowl.

Ratios turn up everywhere — mixing paint colours, following a recipe, sharing a prize, or setting the scale on a map. They are useful because they describe a relationship rather than a fixed amount, so the same ratio still works whether you have a small bowl or a giant crate.

A ratio compares part to part. This is slightly different from a fraction, which compares a part to the whole — you have already met fractions, and ratios sit neatly alongside them.

A ratio like $12 : 8$ can be written more simply. Just as with fractions, you divide every part by a common factor.

Both 12 and 8 divide by 4, so $12 : 8$ becomes $3 : 2$. The comparison is unchanged — for every 3 of the first thing there are still 2 of the second — but the numbers are now as small as they can be.

A ratio is in its simplest form when the only number that divides every part is 1. Using the highest common factor gets you there in one step, but dividing in stages works too — $12 : 8$ to $6 : 4$ to $3 : 2$.

One thing to watch: every part must be divided by the same number, and the parts should be in the same units first. A ratio of 1 hour to 30 minutes should be written as $60 : 30$, which simplifies to $2 : 1$.

Just as $\frac{1}{2}$ and $\frac{2}{4}$ are equivalent fractions, ratios have equivalent versions too.

$1 : 3$, $2 : 6$ and $5 : 15$ are all equivalent ratios. Each describes the very same comparison — for every 1 of the first thing there are 3 of the second.

To build an equivalent ratio, multiply every part by the same number — that is exactly the reverse of simplifying. From $1 : 3$, multiplying both parts by 4 gives $4 : 12$.

Equivalent ratios are how you scale recipes and mixtures. If a smoothie needs juice and yoghurt in the ratio $2 : 1$ and you want to use 6 cups of juice, scale $2 : 1$ up by 3 to get $6 : 3$ — so 3 cups of yoghurt.

A very common task is splitting an amount into a given ratio. The method has three clear steps.

Suppose $40 is shared between Amy and Ben in the ratio $3 : 2$.

1. Add the parts to find the total number of shares: $3 + 2 = 5$.
2. Find one share by dividing the amount by that total: $40 \div 5 = 8$.
3. Multiply each person's parts by the value of one share.

So Amy gets 3 \times 8 = \24$and Ben gets$2 \times 8 = $16$.

A brilliant check: the two answers should add back to the original amount. $24 + 16 = 40$, so it works. If your shares do not add up, a step has gone wrong.

Two quantities are in direct proportion when they grow or shrink together at the same rate. Double one and the other doubles too; halve one and the other halves.

If 4 pens cost $6, then 8 pens cost $12 and 2 pens cost $3 — the cost stays in step with the number of pens.

The most reliable way to solve proportion questions is the unitary method. The word "unitary" comes from "unit" — meaning one. The idea is to find the value of one item first, then scale up.

For the pens: 4 pens cost $6, so one pen costs 6 \div 4 = \1.50$. Now any amount is easy — 10 pens cost$10 \times 1.50 = $15$.

This "find one, then scale" approach handles recipes, speeds, currency exchange and best-buy comparisons. Whenever a question gives you a quantity and asks about a different quantity, reach for the unitary method.

Ratio and proportion are some of the most practical skills in maths:

• Scale drawings and maps use ratio to shrink real distances down to a page, such as a scale of $1 : 50\,000$.
• Similar shapes in geometry have sides in equal ratios — proportion is what makes one a clean enlargement of the other.
• Science is full of proportion, from mixing solutions to comparing speeds and densities.
• Everyday decisions lean on the unitary method — working out the best-value pack at the shop is a proportion question in disguise.

If a later topic about scaling or comparing feels tricky, it is often a ratio or proportion step underneath. The "find one, then scale" habit is worth practising until it feels natural.