Number and ratio · GCSE Maths
Reverse percentages
GCSE Maths reverse percentages: recover the original from a sale price or VAT-inclusive total by dividing by the multiplier, with the method written line by line.
The new amount is original × multiplier. To go backwards, divide by that same multiplier — never add the percentage of the new price.
The important bits
What you need to know
- 1
A reverse-percentage question gives the amount after a change and asks for the amount before. You are travelling backwards along the multiplier.
- 2
Write a sentence that names what you have: “£48 is 80% of the original” or “£240 is 120% of the net price.” The sentence chooses the multiplier.
- 3
Decrease by r% uses multiplier (100 − r)/100. Increase by r% uses (100 + r)/100. 20% off is × 0.8; 20% VAT on top is × 1.2.
- 4
Original = new ÷ multiplier. Sale price £68 after 15% off: original = 68 ÷ 0.85. VAT-inclusive £96: net = 96 ÷ 1.2 if VAT is 20%.
- 5
Finding r% of the new amount and adding or subtracting it answers a different question. That is a forward percentage on the wrong starting value.
- 6
Compound or repeated change still reverses by division: after two 10% increases, divide by 1.1², not by 1.2, and not by 1.1 twice in the wrong order of operations.
- 7
Check by going forwards: original × multiplier must recover the given new amount. If it does not, the multiplier was wrong, not the arithmetic.
- 8
On non-calculator papers keep fractions: 15% off is × 17/20, so divide by 17/20 means multiply by 20/17. Exact money answers still need two decimal places when the context is pounds.
Examples worth remembering
Short evidence. Real method.
“Original amount = new amount ÷ multiplier”
This single line is the method mark. Examiners award 68 ÷ 0.85 even if the decimal is mistyped; they cannot award a lone 80 with no story.
“This is 80% of the original.”
Name the percentage you actually have. A 20% sale leaves 80%, not 20%. VAT at 20% leaves 120% of the net, not 20% of the gross.
“Do not add 20% of the sale price back on.”
20% of £48 added to £48 invents a different coat. The discount was 20% of the original, which is the unknown, so you cannot compute it first.
Go deeper
Name the percentage you are holding
Underline the after amount and write what it is as a percentage of the start. Reduced by 20% to £48 means £48 is 80% of the original: original × 0.8 = 48, so original = 48 ÷ 0.8 = £60. Increased by 12% to 280 means 280 is 112% of the start: divide by 1.12. The usual disaster is to take 20% of 48 and add it back, which answers “increase £48 by 20%”, not “undo a 20% decrease”. VAT is the same structure with a different story. A restaurant bill of £54 including 20% VAT is 120% of the food price, so food = 54 ÷ 1.2 = £45. If the question instead gives the food price and asks you to add VAT, you multiply. Direction comes from the sentence; size comes from the multiplier.
Go deeper
VAT, salary and successive changes
Net means before VAT; gross means after. UK VAT at 20% is the multiplier 1.2 in either direction. Gross £180, net = 180 ÷ 1.2 = £150, VAT = £30. Check: 20% of 150 is 30, and 150 + 30 = 180. A salary of £27 000 after a 8% rise was original × 1.08 = 27 000, so original = 27 000 ÷ 1.08 = £25 000. Two successive 10% increases are not a 20% increase: the multiplier is 1.1 × 1.1 = 1.21. To reverse, divide by 1.21. A 10% increase followed by a 10% decrease is not “back to the start”: 1.1 × 0.9 = 0.99, so you finish at 99% of the original. Reverse that by dividing by 0.99, not by pretending the two 10% changes cancel.
Go deeper
Write the equation before you press any keys
Treat reverse percentage as algebra. Let the original be x. After a 15% decrease, 0.85x = 68, so x = 68 ÷ 0.85 = 80. After a 2.5% increase, 1.025x = 410, so x = 410 ÷ 1.025 = 400. The equation earns the method mark if the division is mistyped, and it stops you using 0.15 or 1.15 by accident. On a non-calculator paper, 0.85 is 17/20, and 68 ÷ 17/20 = 68 × 20/17 = 80. Finish with the forward check in the original story: 15% of £80 is £12, and £80 − £12 = £68. If the check fails, the multiplier was the error, not the coat.
See the idea in action
A jacket is reduced by 20% in a sale to £64. Find the original price. Step 1: The sale price is 80% of the original, so the multiplier is 0.8. Step 2: Write original × 0.8 = 64. Step 3: Original = 64 ÷ 0.8 = 80. Step 4: The original price is £80. Check: 20% of £80 = £16, and £80 − £16 = £64. Wrong method to avoid: 20% of 64 is 12.80, then 64 + 12.80 = £76.80, which is not the original.
Exam technique
Turn knowledge into marks
Write the multiplier and the equation original × multiplier = new before you divide. Examiners award 64 ÷ 0.8 even if the money is mistyped; they cannot award a bare 80.
Common mistakes
Do not give these marks away
- 01
Finding 20% of the sale price and adding it back, which increases the new amount instead of reversing the discount.
- 02
Using 0.2 or 1.2 after a 20% decrease, instead of 0.8, because the 20% is the part that has gone.
- 03
Dividing by 1.2 when the amount is VAT-exclusive and the question asked you to add VAT — that is a forward percentage, so multiply.
A coat is reduced by 15% to £68. What was the original price?
A£80
B£78.20
C£58
D£90.67
Show the answer
£80. £68 is 85% of the original, so divide by 0.85: 68 ÷ 0.85 = 80. £78.20 is 68 + 15% of 68, the classic reverse-percentage error. £58 is 68 × 0.85, a forward decrease. £90.67 is 68 ÷ 0.75.
Quick questions
If this is the bit you searched
How do you do reverse percentages GCSE?
Name the percentage the new amount represents, write original × multiplier = new, then original = new ÷ multiplier. Check by going forwards from your answer.
A price includes 20% VAT. How do I find the price before VAT?
The given price is 120% of the net, so divide by 1.2. VAT is then 20% of that net, not 20% of the gross.
Why can I not add 20% of the sale price back on?
The 20% was 20% of the original, which you do not yet know. 20% of the sale price is a smaller number, so adding it back undershoots the original.
How do I reverse two percentage changes in a row?
Multiply the two multipliers first, then divide the final amount by that product. After 10% up then 10% down, divide by 1.1 × 0.9 = 0.99, not by 1.