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This page sets out the method for reverse percentages in the order you would use it, then walks through the full lesson explanation. For three complete worked examples with the slips that cost marks, see the Reverse percentages topic page.
When you see 'after a increase/decrease the amount is …, find the original', use the multiplier.
When the numbers are friendly and you have no calculator, work out what one 'chunk' of percent is worth.
After a decrease, divide by the multiplier that is less than 1.
After a 10% increase the price is £66. The original is £66 ÷ 1.1 = £60.
The multiplier for a 20% decrease is 0.8.
The multiplier for a 10% increase is 1.1.
A percentage decrease takes 100 to 80. The decrease is 20%.
A fall uses a multiplier below 1. A rise uses a multiplier above 1.
Read the tense of the sentence. 'The price is now £84' and 'the sale price is £63' and 'the bill including VAT is £96' all hand you the result of a multiplication. The original is the unknown. Finding 20% of £84 and subtracting it answers a different question: it takes 20% off the new price. That is a forwards reduction, and it will not undo a rise.
Write one line that names the percentage of the original. After a 20% rise, the new amount is 120% of the original. After a 30% fall, the new amount is 70% of the original. After VAT at 20% is added, the price you pay is 120% of the price before VAT. That percentage becomes the divisor once you write it as a decimal multiplier.
On a non-calculator paper you can build the same division with a 10% or 1% bar. If £84 is 120%, then 10% is 84 ÷ 12 = 7, and 100% is 7 × 10 = 70. The examples use the single division, because it is the same arithmetic and it stays short when the percentage is awkward.
Method: 1. The given price is the amount after the change, not before it. 2. An increase means the given amount is more than 100%. 3. A reduction means the given amount is less than 100%.
Match the change to its multiplier, then divide. Rise of 20%: divide by 1.2. Fall of 30%: divide by 0.7. Fall of 35%: the remaining share is 65%, so divide by 0.65. Fall of 15%: divide by 0.85. Rise of 5%: divide by 1.05. The original is new amount ÷ multiplier.
84 ÷ 1.2 = 70. The original price was £70.00. 63 ÷ 0.7 = 90. The price before the 30% reduction was £90.00, and the saving was 90 - 63 = 27, so £27.00. 52 ÷ 0.65 = 80. A coat at £52.00 after 35% off started at £80.00. Do the division in full before you subtract anything.
A calculator paper is the same division. Type the new amount, divide by the multiplier, and round money to the nearest penny only if the display is not already a whole number of pence. The examples in this lesson land on whole pounds, so the answer is exact. If you round the multiplier itself, for example writing 1/1.2 as 0.83, the original will drift. Use 1.2, not a chopped reciprocal.
Method: 1. Build the multiplier for the change that has already happened. 2. Divide the given amount by that multiplier. 3. That quotient is the original amount.
Multiply your original by the multiplier and compare with the given amount. 70 × 1.2 = 84. 90 × 0.7 = 63. 80 × 0.65 = 52. 80 × 1.2 = 96. If the forward step misses the given figure, the multiplier is wrong or the division was the wrong way up. Multiplying the given figure by the multiplier makes the number move further from the original.
The saving, the VAT, or the interest is not the original. On the coat, the saving is £28.00 and the original is £80.00. On the laptop at £96.00 including VAT at 20%, the VAT itself is 96 - 80 = 16, so £16.00, and the price before VAT is £80.00. Read whether the question wants the original price or the amount of the tax.
Two different stories can land on the same original. £96.00 with 20% VAT added, and £52.00 after a 35% reduction, both come from £80.00. The multipliers 1.2 and 0.65 are doing different jobs. Write the story next to the multiplier so you do not divide a sale price by 1.2 or a VAT-inclusive price by 0.65.
Method: 1. Multiply your original amount by the multiplier. 2. The result should match the price in the question. 3. If it misses, the multiplier was the wrong way up.
When the remaining percentage splits into friendly pieces, a build-up is a good check. £63.00 is 70% of the original. 10% is 63 ÷ 7 = 9, so £9.00. Then 100% is 10 × £9.00 = £90.00. That matches 63 ÷ 0.7. Use the build-up when you want to see the 10% bar. Use the division when you want one line.
£170.00 after a 15% reduction is 85% of the original. 170 ÷ 0.85 = 200, so the original is £200.00. A build-up: 85% is £170, so 5% is £10.00 and 100% is £200.00. Both methods divide by 85 in disguise. 170 ÷ 85 = 2, and that 2 is 1% in pounds, then × 100.
Do not take the percentage of the new amount off and call it the original. 20% of £84.00 is £16.80, and £84.00 minus £16.80 is £67.20. That is not £70.00. The £16.80 was 20% of the wrong base. The percentage in the question was 20% of the original, and the original is what you are trying to find. Divide by 1.2.
Method: 1. Find what percentage of the original the given amount represents. 2. Find 1% or 10%, then scale up to 100%. 3. Check by taking the stated percentage of your original answer.
Real exam questions rarely say reverse percentage. They hide it in a short sentence such as "The price of a bag increases by 15% to £13.80" or "In a sale the normal price is reduced by 20%. The sale price is £64". The word to, the phrase sale price, the words including VAT and the word now all tell you that the number in the question is the amount after the change. What you are asked for is the original, normal or before price.
A quick test is to ask whether your answer should be bigger or smaller than the number you were given. After an increase the given number is bigger than the original, so your answer must be smaller. After a reduction the given number is smaller, so your answer must be larger. If your answer fails that test, you have probably taken the percentage off the wrong amount. Write the sentence "given amount is some percent of the original" before you calculate.
Method: 1. Underline the amount given and decide whether it is the original or the new amount. 2. Write whether the change was an increase or a decrease and name the multiplier, for example 1.15 or 0.8. 3. Write new amount equals original times multiplier as a line of working. 4. Divide the given amount by the multiplier, then check the answer is bigger after a reduction and smaller after an increase.
Tip: The words to, sale price and including nearly always mark the new amount.
On a non-calculator paper, dividing by a decimal such as 0.85 is awkward. The percentage method avoids it. After a 15% reduction the sale price is 85% of the original. If a sofa costs £510 in the sale, then 85% is £510. Divide by 85 to find 1%, which is £6, and multiply by 100 to get 100%, which is £600. For a 20% reduction the sale price is 80%, so one tenth of that is 10% and ten tenths make the whole.
This method also shows the structure clearly, which is why a mark scheme gives a method mark for a correct statement such as "80% = £64" even when the later arithmetic slips. It works well for percentages that pair with friendly factors, such as 10%, 20%, 25%, 50% and 5%. Whichever method you choose, finish with a check: apply the percentage change to your answer and see the given number come back.
Method: 1. Write the percentage the given amount represents, for example 100% - 15% = 85%. 2. Divide the given amount by that percentage to find 1%, or find 10% or 5% if that is friendlier. 3. Multiply up to 100%. 4. Check by applying the original change to your answer.
Higher papers often chain two changes, such as a car that loses 25% of its value in the first year and a further 10% in the second. The final value is original times 0.75 times 0.9. To reverse it, divide by the overall multiplier 0.675, or divide by 0.9 first and then by 0.75. Either way you must undo both changes. Never add 25 and 10 to make one change of 35%, because the second percentage is taken from a smaller amount.
A mixed chain, such as an increase followed by a reduction, works the same way. Multiply the multipliers, write the product, and only then divide. Many questions finish by asking for the overall percentage change: compare the product with 1 and turn it into a percentage. A product of 0.765 means an overall reduction of 23.5%. Keep full accuracy in the middle of the working and round only the final answer.
Method: 1. Write a multiplier for each change, in the order they happened. 2. Multiply the multipliers to get one overall multiplier. 3. Divide the final amount by the overall multiplier to get the starting amount. 4. For an overall percentage, subtract the overall multiplier from 1 and write the result as a percentage.
Tip: Percentages from different starting amounts can never be added. Multiply the multipliers instead.
Some questions ask you to show that a price is right or to say whether a claim is true. Suppose a pay rise of 4% gives a new wage of £31 200 and Sam says the old wage was 4% less, so £29 952. To test this, divide to find the true old wage, which is £30 000. Then compare it with Sam's number. They differ because 4% of the new wage is bigger than 4% of the old wage. Show both numbers so the comparison is clear.
Money answers need two decimal places, such as £16.50 and not £16.5. If the division is not exact, round to the nearest penny at the end, or to the accuracy the question asks for. In a show-that question, work forwards from the answer you are given and arrive at the number in the question. That counts as full proof, so finish with a short sentence saying what you have shown.
Method: 1. Find the true original by dividing by the multiplier. 2. Work out what the claim says and write it as a number. 3. Compare the two numbers and say in words whether the claim is right. 4. Round money to the nearest penny only at the very end.
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