Dividing in a Ratio

A ratio can be used to share out an amount fairly. Imagine two friends agree to split some sweets 3 : 2 — for every 3 sweets the first friend takes, the second takes 2. The clever trick is to think of the ratio as a recipe of equal parts: work out what one part is worth, then build each share from it.

Add the parts of the ratio to get the number of equal shares.
Divide the amount by that number to find the value of one part.
Multiply each number in the ratio by that value to get each share.

For example, share \pounds 20 in the ratio 3 : 2. There are 3 + 2 = 5 equal shares, so one part is \pounds 20 \div 5 = \pounds 4. The shares are then 3 \times \pounds 4 = \pounds 12 and 2 \times \pounds 4 = \pounds 8 — and notice they add back to \pounds 12 + \pounds 8 = \pounds 20. That last check is your safety net: if the shares don't add to the total, something has gone wrong.

To share an amount in a given ratio:

the number of equal shares is the sum of the ratio numbers — \text{shares} = a + b;
one part is the amount divided by that number — \text{one part} = \dfrac{\text{amount}}{a + b};
each share is its ratio number times one part — a \times \text{one part} and b \times \text{one part};
the shares always add back to the original amount.

A few worked examples

Once you know the three steps, every sharing problem is the same dance. Watch the pattern:

Share 24 sweets between two friends in the ratio 5 : 1. There are 5 + 1 = 6 equal shares, so one part is 24 \div 6 = 4 sweets. The shares are 5 \times 4 = 20 and 1 \times 4 = 4. Check: 20 + 4 = 24.
Share \pounds 45 in the ratio 2 : 3 : 4. A ratio can have more than two numbers! There are 2 + 3 + 4 = 9 equal shares, so one part is \pounds 45 \div 9 = \pounds 5. The three shares are \pounds 10, \pounds 15 and \pounds 20. Check: 10 + 15 + 20 = 45.
Share 30 marbles in the ratio 1 : 1. That just means "share equally": 1 + 1 = 2 shares, one part is 30 \div 2 = 15, so each child gets 15. A ratio of 1 : 1 is the same as splitting in half.

The two traps that catch everyone:

Add the ratio parts FIRST to find how many shares there are. To split in 3 : 2, divide by 3 + 2 = 5 — not by 2 (the number of people) and not by 3. Dividing by the wrong number is the most common mistake of all.
Always check the shares add back to the total. If you share \pounds 20 as \pounds 12 and \pounds 8, then 12 + 8 = 20 — good. If they don't add up, you've divided by the wrong number.

You and a friend find a bag of 10 sweets and agree to share them 3 : 2, because you spotted the bag first. Five equal shares means one part is 10 \div 5 = 2 sweets. You get 3 \times 2 = 6 and your friend gets 2 \times 2 = 4 — and together that is all 10.

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Six for you, four for your friend — a fair 3 : 2 split.

Two pirates win 12 gold coins and divide them in the ratio 2 : 1, because one did twice as much rowing. Three equal shares, so one part is 12 \div 3 = 4 coins. The shares are 2 \times 4 = 8 and 1 \times 4 = 4. The scales balance: 8 + 4 = 12.

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See it as a bar

Step through how a bar of £20 splits into a 3 : 2 share.

Try it yourself

Here the bar is cut into equal blocks — one for each share — and every block is worth the same amount. The left colour is the first part of the ratio, the right colour the second, and each side is totalled up. Press Refresh for a brand-new amount and ratio, and check that the two side totals always add back to the whole.