R9: Percentages
Define percentage; interpret percentages and percentage changes; percentage increase/decrease; original value problems; compound interest
📋 Key Concepts
Definition: A percentage is a fraction out of 100. 1% = 1/100 = 0.01.
Key Conversions
| Percentage | Decimal | Fraction |
|---|---|---|
| 10% | 0.1 | 1/10 |
| 25% | 0.25 | 1/4 |
| 50% | 0.5 | 1/2 |
| 75% | 0.75 | 3/4 |
| 20% | 0.2 | 1/5 |
📝 Finding Percentages
Method: Convert the percentage to a decimal, then multiply.
Example 1
Find 35% of £240.
Solution:
35% = 0.35
35% of £240 = 0.35 × 240 = £84
Example 2
Find 17.5% of £600.
Solution:
17.5% = 0.175
0.175 × 600 = £105
📝 Percentage Increase and Decrease
Multipliers:
- Percentage increase: multiply by (1 + percentage as decimal)
- Percentage decrease: multiply by (1 - percentage as decimal)
Example 3
Increase £80 by 15%.
Solution:
Method 1: Find 15% first
15% of £80 = 0.15 × 80 = £12
New value = £80 + £12 = £92
Method 2: Use multiplier
£80 × 1.15 = £92
Example 4
Decrease £250 by 20%.
Solution:
Method 1: Find 20% first
20% of £250 = £50
New value = £250 - £50 = £200
Method 2: Use multiplier
£250 × 0.8 = £200
📝 Percentage Change
Percentage Change = (Change ÷ Original) × 100
Example 5
A coat increases in price from £45 to £54. Find the percentage increase.
Solution:
Change = £54 - £45 = £9
Percentage increase = (9 ÷ 45) × 100 = 20%
Example 6
A car's value decreases from £12000 to £9000. Find the percentage decrease.
Solution:
Change = £12000 - £9000 = £3000
Percentage decrease = (3000 ÷ 12000) × 100 = 25%
📝 Original Value Problems
Method: Work backwards using the inverse multiplier (divide by the multiplier).
Example 7
After a 20% increase, a price is £96. Find the original price.
Solution:
After 20% increase: new = original × 1.20
Original = £96 ÷ 1.20 = £80
Example 8
In a sale, an item costs £68 after a 15% discount. What was the original price?
Solution:
After 15% decrease: new = original × 0.85
Original = £68 ÷ 0.85 = £80
📝 Compound Interest
Compound Interest: Final = Initial × (1 + r)^n
Where r is the interest rate (as decimal) and n is the number of periods
Where r is the interest rate (as decimal) and n is the number of periods
Example 9
£5000 is invested at 4% compound interest per year. Find the value after 3 years.
Solution:
Multiplier = 1.04
Value after 3 years = £5000 × 1.04³
= £5000 × 1.124864
= £5624.32
Example 10
A car depreciates at 12% per year. If it cost £15000 new, what is its value after 2 years?
Solution:
Multiplier = 1 - 0.12 = 0.88
Value = £15000 × 0.88²
= £15000 × 0.7744
= £11616
❓ Practice Questions
Q1: Find 28% of £350.
Q2: Increase £720 by 35%.
Q3: A price changes from £40 to £52. Calculate the percentage increase.
Q4: After a 25% discount, a TV costs £420. Find the original price.
Q5: £3000 is invested at 5% compound interest. Find the value after 4 years.
✅ Answers
- £98
- £972
- 30%
- £560
- £3646.52 (3000 × 1.05⁴)
🎯 Exam Tips
- Percentage increase multiplier: 1 + (percentage/100)
- Percentage decrease multiplier: 1 - (percentage/100)
- For original value: divide by the multiplier, not multiply
- Compound interest uses powers, simple interest does not
- Check your answer is sensible - increasing should give a larger value
🧠 Problem-Solving Strategies
Problem-Solving
Use multipliers for percentage calculations: increase by 15% → ×1.15, decrease by 8% → ×0.92. For reverse percentages, divide by the multiplier to find the original. For compound changes, multiply the multipliers together. For percentage change: change ÷ original × 100.
Multi-Step Problem
A TV is reduced by 20% then reduced by a further 15%. The final price is £544. What was the original price?
Solution: After 20% off: ×0.80. After 15% off: ×0.85. Combined multiplier = 0.80 × 0.85 = 0.68. Original = 544 ÷ 0.68 = £800.
⚠️ Common Errors
Watch Out!
1. Wrong: Adding two percentage decreases together: 20% + 15% = 35% off, so £544 is 65% of original = £836.92 Correct: The reductions compound: 0.80 × 0.85 = 0.68, so £544 is 68% of original = £800
2. Wrong: Finding 23% of 200 by calculating 200 ÷ 0.23 Correct: 23% of 200 = 0.23 × 200 = 46. Division by 0.23 would give a reverse percentage.
3. Wrong: Saying an item that increases by 50% then decreases by 50% returns to its original price Correct: £100 → £150 (×1.5) → £75 (×0.5). The result is £75, not £100. Percentage changes of equal size don't cancel out.
✍️ 6-Mark Exam Question
Extended Answer
6 marks: A house was bought for £180,000 in 2018. It increased in value by 4.5% each year for 3 years, then decreased by 2.8% in the fourth year. (a) Calculate the value after 4 years. (b) A second house was bought for £195,000 and is now worth £210,060 after 4 years. Which house had the greater overall percentage increase?
(a) After 3 years: 180,000 × 1.045³ = 180,000 × 1.14117 = £205,410 (nearest £). After 4th year: 205,410 × 0.972 = £199,658 (nearest £).
(b) House 1: percentage increase = (199,658 − 180,000) ÷ 180,000 × 100 = 10.9%. House 2: (210,060 − 195,000) ÷ 195,000 × 100 = 7.7%. House 1 had the greater percentage increase.
Mark scheme: M1 for compound multiplier, A1 for 1.045³, A1 for 3-year value, M1 for 4th year decrease, A1 for final value, M1 for both % calculations, A1 for comparison
📊 AO3: Reason & Interpret
Reasoning and Interpretation
Shop A has a jacket priced at £120 with "30% off". Shop B has the same jacket priced at £95 with "15% off".
(a) Which shop offers the lower sale price?
(b) A student says "Shop A is better because 30% off is more than 15% off." Is this reasoning always valid?
(c) Shop A then offers an additional 10% off the already-reduced price. Is this the same as 40% off the original? Explain.
Answers: (a) Shop A: 120 × 0.70 = £84. Shop B: 95 × 0.85 = £80.75. Shop B is cheaper. (b) No — the percentage discount is applied to different original prices. A larger % off a higher price may still cost more. (c) No — 120 × 0.70 × 0.90 = £75.60, but 120 × 0.60 = £72. Compound reductions give a different (higher) price than adding the percentages.
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