If £384 is 20% more than last year, what was last year?

£320. This year is last year × 1.20, so £384 = last year × 1.20. To find last year you divide by the multiplier: 384 ÷ 1.20 = 320, and 320 × 1.20 = 384 confirms it. The common wrong answer £307.20 takes 20% off the £384 — but that removes 20% of the wrong figure, and 307.20 × 1.20 = 368.64, not 384.

How do you reverse a percentage decrease?

Build the forward multiplier, then divide. A 12% reduction is × 0.88 (because 100% − 12% = 88%). So if a value of 2 200 000 is a 12% reduction on last year, last year is 2 200 000 ÷ 0.88 = 2 500 000. Check: 2 500 000 × 0.88 = 2 200 000. Taking 12% off 2 200 000 would give 1 936 000, which is wrong.

Is the reduction the same as the new price?

No. If a £24.50 daily rate is reduced by 20%, the reduction is 24.50 × 0.2 = £4.90, but the new price is 24.50 × 0.8 = £19.60. £4.90 is the money taken off; £19.60 is what you pay. They are different numbers — do not hand back the reduction as the price.

GCSE Maths Foundation · AQA · Percentages

Reverse percentages: why you divide by the multiplier, not take the percentage off

Reverse-percentage questions trip students who undo a change by taking the percentage off the figure they were handed. Told £384 is 20% more than last year, they answer £307.20 — $384 - 20\% \text{ of } 384$ — instead of $384 \div 1.20 = 320$ . The £384 is already last year $\times\, 1.20$ , so the way back is to divide.

The thirty-second fix: the value you are given is original × multiplier, so to find the original you divide by that multiplier. A 20% increase is ÷ 1.20; a 12% decrease is ÷ 0.88. Never take the percentage off the figure you were handed.

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How to spot it in your own work

You undid an increase by taking the percentage off the new value — writing $384 - 20\% = 307.20$ instead of $384 \div 1.20 = 320$ .
You reversed a decrease by subtracting the percentage rather than dividing by $0.88$ (for a 12% cut).
You used the same multiplier for an increase and a decrease, instead of $\div 1.20$ for a rise and $\div 0.80$ for a 20% cut.
You read the reduction as the new price — treating $24.50 \times 0.2 = 4.90$ as the rate rather than $24.50 \times 0.8 = 19.60$ .

An exam question that triggers it

Here is the canonical AQA Foundation trigger (Nov24 P2 Q24 shape, calculator):

A town’s population this year is 384 000. That is 20% more than last year. Work out last year’s population.

The misconception answer is $307\,200$ — taking 20% off 384 000. The examiner’s report noted that the majority decided to find 20% and subtract it.

But 384 000 is already last year $\times\, 1.20$ , so divide: $384\,000 \div 1.20 = 320\,000$ .

Why students fall for this

A percentage change is a multiplication. “20% more than last year” means this year = last year × 1.20. So the figure in front of the student is the output of that multiplication, not the starting point. The reverse of multiplying is dividing — but because the student sees the percentage and a number, the instinct is to take the percentage straight off.

The trap also hides a scale error. 20% of the new value is bigger than 20% of the original, because the new value is bigger. Taking 20% off the new figure therefore removes too much, and the answer comes out below the true original.

AQA Foundation calculator papers exploit every face of this: reversing an increase (Nov24 P2 Q24), reversing a reduction (Jun24 P2 Q20: 2 200 000 kg is a 12% reduction), and separating a discount from a discounted price (Jun24 P3 Q22).

The fix: The value you are given is original × multiplier: build the multiplier, then divide

Reverse an increase: divide by 1 + the percentage. For £384 that is 20% more: $384 \div 1.20 = 320$ . The difference taken off, £307.20, answers a different (wrong) question.

Reverse a decrease: divide by 1 − the percentage. A 12% reduction is × 0.88, so $2\,200\,000 \div 0.88 = 2\,500\,000$ . Check it forwards: $2\,500\,000 \times 0.88 = 2\,200\,000$ .

Keep the discount and the discounted price apart. A 20% cut on £24.50 has reduction $24.50 \times 0.2 = 4.90$ and new price $24.50 \times 0.8 = 19.60$ . The £4.90 is what comes off, not what you pay.

Worked example

A factory recycled 2 200 000 kg this year. That is a 12% reduction on last year. How much did it recycle last year?

Build the forward multiplier. A 12% reduction is $100\% - 12\% = 88\%$ , so $\times\, 0.88$ .
$\text{this year} = \text{last year} \times 0.88$
Divide to reverse it.
$2\,200\,000 \div 0.88 = 2\,500\,000$
Check forwards.
$2\,500\,000 \times 0.88 = 2\,200\,000$

The trap answer $1\,936\,000$ takes 12% off 2 200 000 — but that subtracts 12% of the wrong figure. To reverse a percentage change you divide by the multiplier, never take the percentage off the value you were given.

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Common questions

Why can’t I just take the percentage off the value I was given?: Because that value is the original times a multiplier, not the original. £384 is last year × 1.20, so 20% of £384 is 20% of the wrong figure — too much. Dividing reverses the multiplication exactly: $384 \div 1.20 = 320$ , and $320 \times 1.20 = 384$ .
What multiplier do I use for a percentage decrease?: Subtract the percentage from 100% first. A 12% reduction is $100\% - 12\% = 88\% = 0.88$ , a 20% reduction is $0.80$ . Then divide the value you were given by that multiplier to get the original.
A rate of £24.50 is reduced by 20%. What is the new rate?: $24.50 \times 0.8 = 19.60$ . The number $24.50 \times 0.2 = 4.90$ is only the reduction — the money taken off — not the price you pay.
How do I check a reverse-percentage answer?: Put it back through the forward multiplier. If last year was £320 and this year is 20% more, then $320 \times 1.20 = 384$ — it matches the figure you were given, so the answer is right.

Related misconceptions

Percentage change: why the percentage is of the original, not the new valueThe neighbouring percentages misconception — the same 'percentage is of which figure' confusion, met going forwards instead of in reverse.
Compound and repeated percentages: why you multiply the multipliers, not add the percentagesThe next percentages step — once you trust the multiplier going forwards and backwards, repeated changes multiply the multipliers together.

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