How do you share £240 in the ratio 1:3?

Add the parts first: 1 + 3 = 4 equal shares. Divide the total by that: 240 ÷ 4 = £60 a share. Then multiply each ratio number: 1 × £60 = £60 and 3 × £60 = £180, and they add back to £240. The common wrong answer £80 comes from dividing by one of the ratio numbers, 240 ÷ 3 — but that ignores one of the four shares and never reconstructs the total.

Why do you divide by the sum of the parts, not by one of the numbers?

Because a ratio like 1:3 describes how many equal shares there are altogether — 1 + 3 = 4 of them — not what to divide by. Dividing the total by the number of shares gives the value of one share, which you then multiply up. Dividing by a single ratio number leaves out the other parts, so the shares no longer add back to the total.

How do you split 360° round a point in the ratio 2:7?

There are 2 + 7 = 9 equal parts, so one part is 360 ÷ 9 = 40°. The two angles are 2 × 40 = 80° and 7 × 40 = 280°, which add to 360°. Dividing 360 ÷ 7 = 51.4° uses one ratio number and is the trap.

GCSE Maths Foundation · AQA · Ratio

Sharing in a ratio: why £240 in 1:3 is £60 and £180, not £80

Sharing trips students who read $1:3$ as “one then three” and divide the total by one of the ratio numbers — answering $240 \div 3 = \pounds 80$ — without counting how many equal shares the ratio really makes. The ratio $1:3$ describes four equal shares ( $1 + 3 = 4$ ), so one share is $240 \div 4 = \pounds 60$ and the larger is $3 \times \pounds 60 = \pounds 180$ .

The thirty-second fix: to share in a ratio, count the parts and divide by their SUM to find one share, then multiply. $\pounds 240$ in $1:3$ → $240 \div 4 = \pounds 60$ a share → $\pounds 60$ and $\pounds 180$ , never $240 \div 3 = \pounds 80$ .

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How to spot it in your own work

You divided the total by one of the ratio numbers — writing $240 \div 3 = \pounds 80$ instead of $240 \div 4 = \pounds 60$ a share.
You never added the parts — $1:3$ is $1 + 3 = 4$ equal shares, not 3.
Your shares do not add back to the total — a sure sign a part was left out of the division.
You found one share correctly but multiplied the larger part by the wrong ratio number — the larger share of $\pounds 240$ in $1:3$ is $3 \times 60 = \pounds 180$ , not $2 \times 60$ .
You stopped at the value of one share without multiplying it back up to each person’s amount.

An exam question that triggers it

Here is the canonical AQA Foundation trigger (Jun24 P1 Q13a shape, non-calculator):

Share £240 in the ratio 1:3. Write down the larger share.

The misconception answer is $240 \div 3 = \pounds 80$ — dividing by one ratio number. But £80 a share for a 1:3 split does not add back to £240.

Add the parts: $1 + 3 = 4$ equal shares, so $240 \div 4 = \pounds 60$ a share, and the larger is $3 \times \pounds 60 = \pounds 180$ (and $\pounds 60 + \pounds 180 = \pounds 240$ ).

Why students fall for this

The numbers in a ratio look like instructions to divide, so students reach for the nearest one. But $1:3$ is not “divide by 3”; it says how many equal shares there are in total — $1 + 3 = 4$ of them. You divide the total by the number of shares to find the size of one share, then multiply it back up. The tell is the add-back: real shares reconstruct the total, so $\pounds 60 + \pounds 180 = \pounds 240$ , whereas the trap $240 \div 3 = \pounds 80$ leaves a part out.

A bar model makes the count visible: $1:3$ draws as $1 + 3 = 4$ equal boxes, all worth the same. One box is the total divided by the number of boxes, so $240 \div 4 = \pounds 60$ — never the total divided by one of the ratio numbers.

AQA Foundation papers exploit this on the non-calculator paper — sharing money (Jun24 P1 Q13a, £240 in 1:3), worded “does each get more than ...?” decisions (Jun23 P3 Q21, £2450 shared 2:5 between brothers), and angles round a point (Jun23 P3 Q11, splitting 360° in 2:7). Each one rewards dividing by the sum of the parts and punishes dividing by one part.

The fix: To share in a ratio, divide by the SUM of the parts to find one share, then multiply

Add the parts first. $1:3$ is $1 + 3 = 4$ equal shares. One share is $240 \div 4 = \pounds 60$ . If you divided by one of the ratio numbers, the shares will not add back to the total.

Multiply each ratio number by one share. The shares are $1 \times 60 = \pounds 60$ and $3 \times 60 = \pounds 180$ .

Check the add-back. $\pounds 60 + \pounds 180 = \pounds 240$ — the shares reconstruct the total, so the split is right.

Worked example

Angles round a point are in the ratio 2:7. Work out the larger angle.

Add the parts. Angles round a point total $360^\circ$ , and there are $2 + 7 = 9$ equal parts.
Find one part.
$360 \div 9 = 40^\circ \text{ per part}$
Multiply up and check. The angles are $2 \times 40 = 80^\circ$ and $7 \times 40 = 280^\circ$ , and $80 + 280 = 360^\circ$ . The larger angle is $280^\circ$ .

The trap $360 \div 7 = 51.4^\circ$ divides by one ratio number and never adds back to 360°. Add the parts, divide, then multiply.

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Common questions

Why can’t I just divide by one of the ratio numbers?: Because the ratio counts how many equal shares there are altogether, not what to divide by. $1:3$ is $1 + 3 = 4$ shares, so one share is $240 \div 4 = \pounds 60$ . Dividing $240 \div 3 = \pounds 80$ leaves out a share, so the amounts no longer add back to £240.
How do I know which share is the larger one?: The larger ratio number gives the larger share. In $1:3$ with one share worth £60, the shares are $1 \times 60 = \pounds 60$ and $3 \times 60 = \pounds 180$ , so the larger is £180.
How do I check a sharing answer quickly?: Add the shares back up — they must equal the total. For £240 in 1:3, $\pounds 60 + \pounds 180 = \pounds 240$ works, so the split is right. If the shares do not reconstruct the total, you divided by the wrong number.
Does £2450 shared 2:5 give a brother more than £430?: Yes. There are $2 + 5 = 7$ parts, so one share is $2450 \div 7 = \pounds 350$ . The shares are $2 \times 350 = \pounds 700$ and $5 \times 350 = \pounds 1750$ , both above £430. The trap $2450 \div 5 = \pounds 490$ divides by one ratio number.

Related misconceptions

Ratio is not the denominatorThe partner trap — in 1:3 the first share is 1 out of 1 + 3 = 4, the fraction 1/4, not 1/3; the ratio number is not the denominator.
Scaling a ratio: multiply, don't addKeeps a ratio in proportion when you scale it — multiply both parts by the same number rather than adding the same amount to each.
Direct and inverse proportion: not everything scales the same wayA neighbouring Foundation skill — another place where finding the value of one unit first is the move that keeps the relationship right.

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