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Reverse Percentages - Edexcel GCSE Higher Maths

Typically 3-4 marks per paper, usually as one problem-solving question · Spec 1MA1

A reverse percentage question gives you the value after a change and asks for the value before it. On Edexcel's 1MA1 specification it sits inside spec reference R9, the same reference as ordinary percentage change, and it is common content, so both tiers can be asked it. What marks the Higher versions out is where the reversal is hidden: a price quoted as including VAT, a car two years into its depreciation, a population figure recorded after a rise, or a sale price with a second discount already applied on top of the first.

Edexcel sets it on all three papers. On Papers 2 and 3 the calculator makes the division itself trivial, so what the question tests is whether you know which number to divide by, and the figures are picked so that multiplying instead produces a tidy wrong answer that looks right. Paper 1 is non-calculator, where the percentage is chosen so unitary working stays clean: if 80 percent of a price is £24 then 1 percent is 30p and 100 percent is £30. A bare reversal is 2-3 marks; folded into a comparison or a two-stage change it runs to 4 or 5.

The whole topic is one decision, divide by the multiplier or multiply by it, and nearly every lost mark comes from getting that backwards. Undoing a 10 percent fall by adding 10 percent back is the version examiners see most, and it fails because the 10 percent was taken off the larger original, not off the reduced value. Before you touch the calculator, write the sentence down as an equation: original times multiplier equals the amount the question gave you. Then divide, and put your answer back through the change as a check. If it does not land on the number printed in the question, you divided by the wrong thing.

Worked example

A dealer buys a van. In the first year the van loses 25 percent of its value. In the second year it loses a further 8 percent of its value. At the end of the two years the van is worth £13800. Work out the value of the van when the dealer bought it.
[4 marks]
  1. Write each year as a multiplier. Losing 25 percent leaves 75 percent, so the first multiplier is 0.750.75; losing 8 percent leaves 92 percent, so the second is 0.920.92.
    P1: both multipliers correct, as decimals under 1 because both years are decreases
  2. Combine them for the two years: 0.75×0.92=0.690.75 \times 0.92 = 0.69.
    P1: multiplying the multipliers; adding 25 and 8 to make 33 percent would give 0.67 and scores nothing here
  3. The £13800 is 69 percent of the starting value, so value×0.69=13800\text{value} \times 0.69 = 13800.
    P1: setting up the reverse step, which is the mark for treating 13800 as the after value rather than the before value
  4. Divide: 13800÷0.69=2000013800 \div 0.69 = 20000, so the dealer paid £20000. Check: 20000×0.75=1500020000 \times 0.75 = 15000 and 15000×0.92=1380015000 \times 0.92 = 13800.
    A1: 20000 from a correct division; the check costs ten seconds and catches a multiply-instead-of-divide slip

Practice questions

These are original questions written in Edexcel 1MA1 Higher style, covering sale prices back to the original, VAT-inclusive totals back to the net price, populations and values after a rise or a fall, and two-stage reversals. A few of them are ordinary percentage questions dressed up to look like reversals, so read the stem for which value came first. Money answers are marked to the nearest penny, so type £48.60 as 48.6 or 48.60, either is accepted.

No account needed: the questions below are always free to check, plus 8 extra questions today. Your score so far updates as you go.

  1. Question 1 Warm-up [1 mark]
    A price is reduced by 35 percent. What percentage of the original price is the new price?
  2. Question 2 Exam pace [3 marks]
    In a sale, all prices in a shop are reduced by 24 percent. Aisha pays £57 for a coat in the sale. Work out the price of the coat before the sale, in pounds.
  3. Question 3 Exam pace [3 marks]
    A promotional bag of pasta contains 780 g, which is 4 percent more than a standard bag. Work out the mass of pasta in a standard bag, in grams.
  4. Question 4 Exam pace [2 marks]
    In a school, 60 percent of the students are girls. There are 468 girls. Work out the total number of students in the school.
  5. Question 5 Stretch [4 marks]
    The price of a bicycle is increased by 25 percent. The new price is then reduced by 12 percent in a sale. The sale price is £275. Work out the price of the bicycle before the increase, in pounds.
  6. Question 6 Stretch [4 marks]
    In a sale, a jeweller takes 20 percent off all prices. Members of the shop's loyalty scheme then get a further 10 percent off the sale price. A member pays £151.20 for a watch. Work out the price of the watch before the sale, in pounds.

Common mistakes examiners see

  • Undoing a decrease by adding the same percentage back on. A car worth £10200 after a 15 percent fall becomes 10200×1.15=1173010200 \times 1.15 = 11730 instead of £12000.

    The 15 percent came off the original value, which was bigger, so adding 15 percent of the smaller number never gets you back. Divide by the multiplier for the change that happened: 10200÷0.8510200 \div 0.85.

  • Removing 20 percent VAT by multiplying by 0.8. A bill of £294 including VAT gives £235.20 rather than £245.

    Adding VAT multiplies by 1.2, so stripping it divides by 1.2. Multiplying by 0.8 lands at 96 percent of the right figure every time, because 1.2×0.8=0.961.2 \times 0.8 = 0.96.

  • Dividing by the wrong multiplier when the change is a decrease. For a 35 percent reduction to £390, dividing by 1.35 gives £288.89, which is smaller than the sale price.

    A reduction means the multiplier is under 1, so use 390÷0.65=600390 \div 0.65 = 600. Sense-check the direction first: after a discount the original must be larger than the price given.

  • Reading a stated change as the new total. When a season ticket rises by 8 percent and the rise is £96, dividing 96 by 1.08 gives £88.89 instead of the £1200 the ticket cost.

    Decide what the number in front of you actually is. If it is the size of the increase, divide by 0.08; if it is the price after the increase, divide by 1.08.

  • Adding the two percentages in a two-stage reversal. Prices cut by 20 percent and then by a further 10 percent are undone by dividing by 0.7, when the combined multiplier is 0.72.

    Multiply the multipliers before you reverse anything: 0.8×0.9=0.720.8 \times 0.9 = 0.72, then divide the final price by 0.72. The second cut is taken from the already reduced price, so the two percentages never simply add.

Frequently asked questions

How do you know when a question is a reverse percentage?
The number you are given comes after the change and the number you are asked for comes before it. Words that give it away are original, normal price, before, was, or a value described as including VAT or as a sale price. If the amount printed in the question is the starting value, it is an ordinary percentage question and you multiply instead.
Do you divide or multiply for reverse percentages?
You divide, by the multiplier for the change that has already happened. Twenty percent added means divide by 1.2, and 20 percent taken off means divide by 0.8. Multiplying by 0.8 to strip VAT is the most common wrong move and always lands at 96 percent of the correct answer.
Are reverse percentages on the Foundation paper too?
Yes. Spec reference R9 is common content, so it is examined at both tiers. On Higher the reversal is usually one step inside a longer problem, or there are two changes to undo at once, which is why the marks run to 4 or 5 rather than 2.
Is there a reverse percentage formula on the Edexcel formulae sheet?
No. The Exam Aid sheet issued with 1MA1 papers for the 2025 to 2027 windows carries the quadratic formula, the trigonometry rules, the circle and trapezium formulae and the compound interest formula, but nothing for percentage work. You are expected to know that the amount given equals the original multiplied by the multiplier, and to rearrange it yourself.

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