SPEC R9 • RATIO, PROPORTION & RATES OF CHANGE
Reverse Percentages: Finding the Original
After a price change, find the value you started with. The method is divide by the multiplier — not subtract the percentage. A reliable 2–3 mark question on every paper.
What Is a Reverse Percentage Problem?
Lesson 3.03 asked: "Start with X. Apply a percentage change. What's the new value?"
Lesson 3.04 asks the reverse: "You see the new value. What was the original?"
Definition — Reverse Percentage
A reverse percentage problem gives you the value after a percentage change and asks for the value before. Typical contexts: sale prices, VAT-inclusive prices, pay rises, and any "before vs after" problem.
Common Exam Wordings
"After a 15% pay rise, a salary is £1,840. What was the original salary?"
"In a sale, prices are reduced by 25%. A coat costs £60 in the sale. Find the original price."
"The price of a TV including 20% VAT is £360. Find the price excluding VAT."
Signal Words
"After…", "including VAT", "in the sale", "the original…" — these phrases all signal a reverse percentage. If you see them, don't subtract — divide.
The Core Relationship
Every percentage change can be written as one equation:
Original
Multiplier
New Value
In a forward percentage problem, you know the original and multiplier — you multiply to find the new value.
In a reverse percentage problem, you know the new value and multiplier — so you rearrange:
New Value
Multiplier
Original
Key Formula
Original = New value ÷ Multiplier
The Two-Step Method
1
Identify the multiplier — for a p% increase, multiplier is (1 + p/100). For a p% decrease, multiplier is (1 − p/100).
2
Divide the new value by the multiplier. The result is the original.
Common Multipliers for Reverse Problems
Change statedMultiplierIncrease by 15% (e.g. pay rise)1.15Increase by 20% (e.g. VAT)1.20Decrease by 25% (e.g. sale)0.75Decrease by 30%0.70Increase by 12.5%1.125
Always Verify
After finding the original, check by multiplying: original × multiplier should give back the new value. If it doesn't, you divided when you should have multiplied, or used the wrong multiplier.
Paper 1 Non-Calculator Trick
On the non-calculator paper, convert the multiplier to a fraction and divide by flipping:
÷ 1.20 = ÷ 6/5 = × 5/6
÷ 0.75 = ÷ 3/4 = × 4/3
÷ 0.80 = ÷ 4/5 = × 5/4
Example: £60 ÷ 0.75 = 60 × 4/3 = £80 ✓ — no long division.
Reverse % After an Increase
EX
After a price rise
After a 20% increase, a product's price is £90. Find the original price.
Multiplier = 1 + 0.20 = 1.20.
Original = 90 ÷ 1.20 = £75.
Check: 75 × 1.20 = £90 ✓
Sanity Check
After an increase, the original must be smaller than the new value.
£75 < £90 ✓ — consistent with a rise. If your answer is bigger than the new value, you made an error.
Reverse % After a Decrease
EX
Sale price
In a sale, prices are reduced by 25%. A coat costs £60 in the sale. Find the original price.
Multiplier = 1 − 0.25 = 0.75.
Original = 60 ÷ 0.75 = £80.
Check: 80 × 0.75 = £60 ✓
Sanity Check
After a decrease, the original must be larger than the new value.
£80 > £60 ✓ — consistent with a reduction. The multiplier 0.75 is < 1, so dividing makes the number larger.
VAT Problems — A Special Case
VAT (Value Added Tax) in the UK is usually 20%. When a price is given "including VAT", the 20% has already been added — so finding the price excluding VAT is a reverse percentage problem.
VAT Method
Pre-VAT price = VAT-inclusive price ÷ 1.20
EX
VAT-inclusive price
The price of a laptop including 20% VAT is £840. Find the price excluding VAT.
VAT multiplier = 1.20 (a 20% increase applied to the pre-VAT price).
Pre-VAT price = 840 ÷ 1.20 = £700.
Check: 700 × 1.20 = £840 ✓. The VAT added was £140.
Common Error
"Just subtract 20% from £840": 840 − 168 = £672. Wrong — that's 20% of £840, not 20% of the pre-VAT price. The correct answer is £700, and the actual VAT was £140.
MCE Worked Example — Pay Rise
EX
3 Marks — Paper 2 or 3 style
After a 15% pay rise, Sara's monthly salary is £1,840. Find her original monthly salary.
M
A 15% rise means the pay-rise multiplier is 1.15. Divide the new salary by 1.15 to recover the original.
C
Multiplier = 1 + 0.15 = 1.15.
C
Original = 1,840 ÷ 1.15 = £1,600.
E
Sara's original monthly salary was £1,600. Check: 1,600 × 1.15 = £1,840 ✓. Sanity: £1,600 < £1,840, consistent with a rise.
Mark Scheme Breakdown
AQA awards M1 for identifying the multiplier (1.15), M1dep for dividing (1840 ÷ 1.15), and A1 for the correct final answer (£1,600). Show your working on its own line for each mark.
Why Subtracting Doesn't Work
The biggest trap in reverse percentages is to subtract the percentage from the new value. Here's why that fails:
The Weak Method
"After a 15% rise, salary is £1,840. Find original."
Weak student computes: 1,840 − 15% of 1,840 = 1,840 − 276 = £1,564. This is wrong — and yet it feels natural. The correct answer is £1,600.
Why It Fails — the percentage is OF the original
The 15% is 15% of the original salary, not 15% of the new salary.
15% of the correct answer (£1,600) = £240. Check: 1,600 + 240 = 1,840 ✓.
15% of the new value (£1,840) = £276. Wrong base — it's bigger than the real rise.
The Fix
Always divide by the multiplier. Never subtract. The percentage change is defined relative to the original, so you need to reverse the multiplication, not reverse the addition.
Examiner Alert — AQA Jun23 Paper 2 Q4
Authentic Exam Evidence
A 2-mark reverse percentage question in AQA June 2023 Higher Paper 2 gave the value after a 12.5% increase and asked for the original. The examiner report stated: "A majority of students recognised the 'reverse' nature of the question and most of these were awarded both marks. Some identified 112.5% but then decreased the new value by 12.5% — this was the most common wrong method."
What Went Wrong
Students correctly spotted the multiplier (1.125 = 112.5%), but then made a second-step error: they multiplied by 0.875 (decreasing by 12.5%) instead of dividing by 1.125.
These two operations are not inverses. "Decrease by 12.5%" undoes a 12.5% decrease, not a 12.5% increase.
Remember
To reverse a multiplication by 1.125, you must divide by 1.125 — not multiply by anything else. The inverse of × 1.125 is ÷ 1.125, not × 0.875.
Examiner Alert — Wrong Multiplier for Decrease
Sale-Price Trap
"Prices are reduced by 25%. A coat costs £60 in the sale. Find the original price." Weak student writes: 60 ÷ 0.25 = £240. That's wrong — the multiplier is 0.75, not 0.25.
The Fix
For a decrease of p%Use multiplier25%0.75 (not 0.25)30%0.70 (not 0.30)10%0.90 (not 0.10)The multiplier is always 1 − p/100, representing what's left, not what was taken away.
Sanity Check
For the 25%-off coat: if you divide by 0.25 and get £240, ask "does 25% off £240 give £60?" No — 25% off £240 = £180. So £240 is wrong. Only £80 passes the check.
Weak vs Strong Answers
Weak ApproachStrong Approach
Subtracts the % from the new value e.g. 1,840 − 15% of 1,840 = 1,564Identifies multiplier, divides 1,840 ÷ 1.15 = £1,600 ✓
Uses p/100 as the multiplier for decrease 60 ÷ 0.25 = £240Uses (1 − p/100) = 0.75 60 ÷ 0.75 = £80 ✓
"Decreases the new value by p%" 1,840 × 0.85 = £1,564Divides by the multiplier 1,840 ÷ 1.15 = £1,600 ✓
Doesn't check with the inverse calculationVerifies: original × multiplier = new value
The Three-Word Fix
"Divide by multiplier." Say it every time you see "after a % change, find the original". It replaces every weak method with the correct one.
Practice Question
In a sale, all prices are reduced by 20%. A television costs £240 in the sale.
(a) Find the original price of the television. [3 marks]
(b) A customer thinks that to find the original price they should add 20% of £240 to £240. Explain why that method gives the wrong answer. [1 mark]
M
(a) A 20% reduction means multiplier 0.80. Divide the sale price by 0.80 to recover the original. (b) Compute what "add 20% of £240" gives, then check whether a 20% reduction of that returns to £240.
C
(a) Multiplier = 1 − 0.20 = 0.80. Original = 240 ÷ 0.80 = £300.
C
(b) Customer's method: 240 + (20% of 240) = 240 + 48 = £288. Check: 20% off £288 = 288 × 0.80 = £230.40, NOT £240. So £288 cannot be the original.
E
(a) £300. (b) The 20% discount is 20% of the original price, not 20% of the sale price. 20% of £300 = £60, and £300 − £60 = £240 ✓. 20% of £240 is only £48, which is the wrong amount. Always divide, don't add the percentage back.
Reverse Percentage Pitfalls: Quick Recap
TrapFix
Subtracting p% from the new valueDivide the new value by the multiplier
Using 0.25 instead of 0.75 as the multiplier for a 25% decreaseMultiplier = 1 − p/100 , not p/100
Multiplying by 0.875 instead of dividing by 1.125 (Jun23 P2 Q4)The inverse of × M is ÷ M, not × (1 − something)
"Adding back the %" to recover the original after a saleAdding 20% to sale price ≠ original — divide by 0.80 instead
Not sanity-checking: original × multiplier = new valueAlways verify the answer with the forward calculation
Three-Step Self-Check
Did I identify the correct multiplier?
Did I divide (not multiply, not subtract)?
Does original × multiplier give back the new value?
Lesson Summary
✓ Recognise Reverse Percentage Problems
Signal words: "after a…", "including VAT", "in the sale", "the original…". You are given the new value and asked for the original.
✓ The Formula
Original = New value ÷ Multiplier. One division. No subtraction, no second multiplication.
✓ Identify the Multiplier
Increase by p% → multiplier (1 + p/100). Decrease by p% → multiplier (1 − p/100). VAT 20% → multiplier 1.20.
✓ Never Subtract; Never Multiply by the Inverse Change
The percentage is always of the original, not of the new value. Subtracting or multiplying by (1 − p/100) after the change are both wrong.
✓ Always Verify
Multiply your answer by the multiplier. It should give back the new value. If not, retry.
Next: What Is a Reverse Percentage Problem?