Reverse percentages

A reverse percentage question gives you a value after a percentage increase or decrease and asks for the original value. The quick method: write the new value as a percentage of the original, turn that into a multiplier, and divide. For example, if a price is £54 after a 20% increase, the new price is 120% of the original, so the original is £54 ÷ 1.2 = £45.

How to do reverse percentages

1. Decide what percentage of the original you have. After a 20% increase you have 100% + 20% = 120%. After a 15% decrease you have 100% − 15% = 85%.
2. Write it as a multiplier. 120% = 1.2 and 85% = 0.85.
3. Divide the new value by the multiplier. Original = new value ÷ multiplier.
4. Check: multiply your answer by the multiplier; you should get the new value back.

AQA GCSE Maths lists "original value problems" alongside percentage increase and decrease, in the content for both tiers.

Why you divide, not subtract

The 20% is 20% of the original, not of the £54. The bar model shows it: the £54 is six equal parts, each worth 20% of the original, so one part is £9 and the original five parts are £45.

The common wrong answer takes 20% off £54: £54 × 0.8 = £43.20. That is wrong because 20% of £43.20 is £8.64, and £43.20 + £8.64 = £51.84, not £54.

Reverse percentages without a calculator

On a non-calculator paper, use the unitary method: find 1% of the original, then 100%.

1. After 15% off, a price is £34, so £34 is 85% of the original.
2. Divide by 85 to find 1%: £34 ÷ 85 = £0.40.
3. Multiply by 100 to find 100%: £0.40 × 100 = £40.

When the percentage divides 100 neatly, a bar model is quicker: 85% splits into 17 parts of 5%, so each 5% is £34 ÷ 17 = £2 and the original 20 parts are £40.

Multipliers for reverse percentages

Worked examples

Example 1: price before VAT

The standard rate of VAT in the UK is 20%. A coat costs £54 including VAT. What was the price before VAT?

• Price with VAT = 120% of the price without it, so divide by 1.2: £54 ÷ 1.2 = £45.
• Check: £45 × 1.2 = £54.

Example 2: original price before a sale

In a sale, a pair of trainers costs £34 after 15% off. What was the original price?

• After 15% off, the price is 85% of the original: £34 ÷ 0.85 = £40.
• Check: £40 × 0.85 = £34.

Example 3: a decrease in mass

A dog loses 10% of its mass on a diet and now has a mass of 28.8 kg. What was its mass before?

• 28.8 is 90% of the original: 28.8 ÷ 0.9 = 32 kg.

Example 4: an increase over 100%

After a 150% increase, a ticket costs £35. What did it cost before?

• A 150% increase means the new price is 100% + 150% = 250% of the old one: £35 ÷ 2.5 = £14.

Example 5: repeated percentage change (Higher)

Money in a savings account earns 4% compound interest a year. After 3 years the balance is £1,687.30, to the nearest penny. How much was put in?

1. Each year multiplies the balance by 1.04, so 3 years multiply it by 1.04³ = 1.124864.
2. Original = £1,687.30 ÷ 1.124864 = £1,500.00 to the nearest penny, so £1,500 was put in.
3. Do not divide by 1.12: 4% for 3 years is not the same as 12% once.

Reverse percentage questions to try

1. A laptop costs £540 including VAT at 20%. Find the price before VAT.
2. After a 12% discount, a jacket costs £220. Find the original price.
3. A town's population rose by 15% to 12,650. What was it before?
4. A car is worth £8,400 after losing 30% of its value. What was it worth new?
5. Higher: a population of bacteria grows by 5% each hour. After 2 hours there are 66,150 bacteria. How many were there at the start?

Answers

1. £540 ÷ 1.2 = £450.
2. £220 ÷ 0.88 = £250.
3. 12,650 ÷ 1.15 = 11,000.
4. £8,400 ÷ 0.7 = £12,000.
5. 66,150 ÷ 1.05² = 66,150 ÷ 1.1025 = 60,000.

Common mistakes

• Taking the percentage of the new value. The percentage is of the original, so divide by the multiplier instead.
• Using the wrong multiplier for a decrease. After 15% off, divide by 0.85, not by 0.15 or 1.15.
• Rounding too early. Keep full values on the calculator and round the final answer, to the nearest penny for money.
• Adding repeated percentages. Three years of 4% is a multiplier of 1.04³, not 1.12.
• Not checking. Multiply your answer by the multiplier: it should give the value in the question.

Practise with a short timed paper

Original value problems come up at both tiers of GCSE Maths. The daily GCSE Maths Foundation and GCSE Maths Higher papers are short timed papers of 8 questions, with a new one every day.

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