Compound Percentages
Section: Number and Calculation | Syllabus: Cambridge Lower Secondary Checkpoint Mathematics (0862)
What Are Compound Percentages?
Compound percentage change means each new percentage change is worked out from the latest amount, not the original amount.
- First change: apply the percentage to the original amount.
- Second change: apply the percentage to the new amount.
- Third change: apply the percentage to the amount after the second change.
Multipliers
It is often easier to use a multiplier instead of a percentage.
| Change | Multiplier | Example |
|---|---|---|
| Increase by 20% | 1.20 | New amount = Original × 1.20 |
| Increase by 5% | 1.05 | New amount = Original × 1.05 |
| Decrease by 20% | 0.80 | New amount = Original × 0.80 |
| Decrease by 15% | 0.85 | New amount = Original × 0.85 |
| No change | 1.00 | New amount = Original × 1.00 |
Increase by n%: multiplier = 1 + n/100
Decrease by n%: multiplier = 1 - n/100
Compound Percentage Increase
-
Example: A house worth £200,000 increases by 10% each year for 2 years.
- Step 1: £200,000 × 1.10 = £220,000
- Step 2: £220,000 × 1.10 = £242,000
- Answer: £242,000
-
Example: A population of 50,000 increases by 3% each year for 3 years.
- Method: 50,000 × (1.03)3
- Answer: 54,636 (to the nearest whole number)
Compound Percentage Decrease
-
Example: A car worth £15,000 depreciates by 20% each year for 2 years.
- Step 1: £15,000 × 0.80 = £12,000
- Step 2: £12,000 × 0.80 = £9,600
- Answer: £9,600
General Formula
Final Value = Original Value × (Multiplier)n
- n = number of time periods
- Multiplier = 1 + n/100 for an increase, or 1 - n/100 for a decrease
- This works for years, months, or any repeated time period
Mixed Percentage Changes
If the percentage changes are different, multiply the multipliers in the order the changes happen.
-
Example: Sales of £50,000 increase by 20% in Year 1 and decrease by 10% in Year 2.
- Step 1: £50,000 × 1.20 = £60,000
- Step 2: £60,000 × 0.90 = £54,000
- Answer: £54,000
Does the Order Matter?
The final answer is the same either way because multiplication is commutative.
-
Example: £90 is increased by 25% and decreased by 10%.
- 1.25 × 0.90 = 1.125
- 0.90 × 1.25 = 1.125
- Answer: Both give £101.25
Reverse Percentage Problems
To find the original value, divide by the multiplier instead of multiplying by it.
Original Value = Final Value ÷ (Multiplier)n
-
Example: After a 20% increase, the price of a TV is £360.
- Step 1: Multiplier = 1.20
- Step 2: £360 ÷ 1.20 = £300
- Answer: £300
-
Example: A car is worth £8,000 after a 20% decrease for 2 years.
- Step 1: Multiplier = 0.80
- Step 2: £8,000 ÷ (0.80)2
- Answer: £12,500
Compound Interest
Compound interest is a special case of compound percentages used in finance.
A = P(1 + r/100)n
- A = final amount
- P = principal (starting amount)
- r = interest rate per time period
- n = number of time periods
-
Example: £5,000 is invested at 4% compound interest per year for 3 years.
- Step 1: A = 5000 × (1.04)3
- Step 2: A = £5,624.32
- Interest earned: £624.32
Simple vs Compound
| Feature | Simple Percentage | Compound Percentage |
|---|---|---|
| Calculation base | Always on the original amount | On the previous amount each time |
| Formula | Final = Original × [1 + (r × n)/100] | Final = Original × (1 + r/100)n |
| Growth pattern | Linear | Exponential |
| For increases | Grows by the same amount each time | Grows faster each time |
| Real-world use | Simple interest, basic discounts | Compound interest, growth, depreciation |
Simple growth is a straight line. Compound growth curves upward.
-
Example: Compare £1,000 at 10% for 3 years.
- Simple: 1000 × 1.30 = £1,300
- Compound: 1000 × (1.10)3 = £1,331
- Difference: Compound gives £31 more
Real-World Uses
- Banking: savings accounts with interest
- Loans: interest on borrowed money
- Inflation: prices rising over time
- Depreciation: cars and phones losing value
- Population growth: people increasing each year
Exam Tips
- Look for keywords like each year, annually, or per month.
- Write the multiplier clearly before calculating.
- Use Final = Original × (Multiplier)n.
- For reverse questions, divide by the multiplier.
- Check if your answer makes sense: an increase should give a bigger value.
- Round money to 2 decimal places and populations to the nearest whole number.
Common Mistakes
MistakeUsing the original amount every time
FixEach new percentage uses the latest amount
MistakeAdding percentages together
FixUse multipliers instead
MistakeUsing 0.20 for a 20% decrease
FixUse 0.80 because 100% - 20% = 80%
MistakeForgetting to square the multiplier for 2 years
FixUse the power for the number of time periods
MistakeThinking 10% increase then 10% decrease returns to the original
Fix1.10 × 0.90 = 0.99, so the final value is 1% less
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