Percentage Change and Reverse Percentages
The Skill
Percentage of an Amount
To find a percentage of an amount: $$\text{Percentage} \times \text{Amount} = \frac{\text{Percentage}}{100} \times \text{Amount}$$
Example: Find 15% of £80 $$15\% \times 80 = 0.15 \times 80 = £12$$
Percentage Increase and Decrease
Increase by a percentage: $$\text{New amount} = \text{Original} \times (1 + \frac{\text{percentage}}{100})$$
Example: Increase £50 by 20%
- Multiplier: $1 + 0.20 = 1.20$
- New amount: $50 \times 1.20 = £60$
Decrease by a percentage: $$\text{New amount} = \text{Original} \times (1 - \frac{\text{percentage}}{100})$$
Example: Decrease £50 by 20%
- Multiplier: $1 - 0.20 = 0.80$
- New amount: $50 \times 0.80 = £40$
Percentage Change Formula
$$\text{Percentage change} = \frac{\text{Change}}{\text{Original}} \times 100$$
Reverse Percentages
When you know the final amount after a percentage change, work backwards to find the original.
Example: After a 20% increase, a price is £60. Find the original price.
- £60 represents 120% (100% + 20%)
- Original = $60 \div 1.20 = £50$
The Traps
Common misconceptions and how to avoid them.
Adding the percentage instead of multiplying "The Add-On Error"
The Mistake in Action
Increase £80 by 15%.
Wrong: $80 + 15 = £95$
Why It Happens
Students see "increase by 15" and instinctively add 15 to the number. They're treating the percentage as an absolute amount rather than a proportion of the original.
The Fix
15% means "15 per hundred" — it's a fraction of the amount, not £15.
Correct method: Step 1: Find 15% of £80 $$15\% \times 80 = 0.15 \times 80 = £12$$
Step 2: Add to original $$80 + 12 = £92$$
Or use the multiplier: $$80 \times 1.15 = £92$$
Spot the Mistake
Increase £80 by 15%
$80 + 15 = £95$
Click on the line that contains the error.
Dividing by the new value instead of the original "The Denominator Disaster"
The Mistake in Action
A phone's price increased from £400 to £500. Calculate the percentage increase.
Wrong: $$\frac{100}{500} \times 100 = 20\%$$
Why It Happens
Students divide by the new value (500) instead of the original value (400).
The Fix
In percentage change, you always divide by the ORIGINAL value.
$$\text{Percentage change} = \frac{\text{Change}}{\text{ORIGINAL}} \times 100$$
Correct calculation:
- Change = $500 - 400 = £100$
- Original = £400
- Percentage increase = $\frac{100}{400} \times 100 = 25\%$
Memory aid: "Original" starts with "O" — the "O"riginal goes "O"n the bottom.
Spot the Mistake
A phone's price increased from £400 to £500. Find the percentage increase.
Change = $500 - 400 = £100$
Percentage = $\frac{100}{500} \times 100$
$= 20\%$
Click on the line that contains the error.
Finding percentage of final amount for reverse problems "The Backwards Blunder"
The Mistake in Action
After a 20% discount, a jacket costs £48. Find the original price.
Wrong: $20\%$ of $48 = £9.60$ Original = $48 + 9.60 = £57.60$
Why It Happens
Students find 20% of the sale price and add it back, but £48 is not 100% — it's already reduced!
The Fix
After a 20% discount, the price represents 80% of the original (100% - 20% = 80%).
Correct method:
- £48 = 80% of original
- 1% of original = $48 \div 80 = £0.60$
- 100% (original) = $0.60 \times 100 = £60$
Or use the multiplier:
- Multiplier for 20% decrease = 0.80
- Original = $48 \div 0.80 = £60$
Check: $60 \times 0.80 = £48$ ✓
Spot the Mistake
After a 20% discount, a jacket costs £48. Find the original price.
20% of 48 = £9.60
Original = $48 + 9.60 = £57.60$
Click on the line that contains the error.
The Deep Dive
Apply your knowledge with these exam-style problems.
Level 1: Fully Worked
Complete solutions with commentary on each step.
Question
A train ticket costs £45. The price increases by 8%. What is the new price?
Solution
Method 1: Two-step Find 8% of £45: $$8\% \times 45 = 0.08 \times 45 = £3.60$$
Add to original: $$45 + 3.60 = £48.60$$
Method 2: Multiplier The multiplier for an 8% increase is: $$1 + 0.08 = 1.08$$
New price: $$45 \times 1.08 = £48.60$$
Answer: £48.60
Question
A shop has a sale with 35% off everything. A coat originally costs £120. What is the sale price?
Solution
Method 1: Two-step Find 35% of £120: $$35\% \times 120 = 0.35 \times 120 = £42$$
Subtract from original: $$120 - 42 = £78$$
Method 2: Multiplier The multiplier for a 35% decrease is: $$1 - 0.35 = 0.65$$
Sale price: $$120 \times 0.65 = £78$$
Answer: £78
Question
The number of students in a club increases from 25 to 30. Calculate the percentage increase.
Solution
Step 1: Find the change. $$30 - 25 = 5$$
Step 2: Divide by the ORIGINAL value. $$\frac{5}{25} = 0.2$$
Step 3: Convert to a percentage. $$0.2 \times 100 = 20\%$$
Answer: 20% increase
Level 2: Scaffolded
Fill in the key steps.
Question
After a 15% pay rise, Emma earns £34,500. What was her salary before the increase?
Level 3: Solo
Try it yourself!
Question
A price including 20% VAT is £96. What is the price before VAT?
Show Solution
The price with VAT represents 120% of the original (100% + 20%).
Price before VAT = $96 \div 1.20 = £80$
Answer: £80
Examiner's View
Mark allocation: Percentage change questions are typically 2-3 marks. Reverse percentage questions are often 3-4 marks.
Common errors examiners see:
- Adding/subtracting the percentage instead of multiplying
- Using the wrong value in the percentage change formula (new instead of original)
- For reverse percentages: finding the percentage of the final amount instead of working backwards
What gains marks:
- Clearly showing the multiplier
- Stating the formula for percentage change
- For reverse percentages: identifying what percentage the final value represents
AQA Notes
AQA commonly asks percentage change in real-world contexts (sales, population, prices).