Compound interest is a core topic in GCSE Maths, appearing on all major exam boards (AQA, Edexcel, OCR). It's a real-world concept used in savings, loans, and investments. Many students find it tricky at first, but once you understand the multiplier method, it becomes straightforward. This guide will walk you through it from first principles to the efficient formula.
What is Compound Interest?
Simple interest is calculated only on the original amount. Compound interest is different: interest is calculated on the original amount and on any interest already added. This means your money grows faster (or a debt increases faster) because you earn "interest on interest".
The Long Way: Understanding the Process
Let's start with an example to see what's actually happening. This is crucial for understanding, but you wouldn't want to do it this way in an exam!
Example 1: £800 is invested in a savings account paying 3.5% compound interest per annum (per year). Find the value after 5 years.
Step-by-step (the long method):
Start: £800
- Year 1: Interest = 3.5% of £800 = 0.035 × £800 = £28.
New total = £800 + £28 = £828.
- Year 2: We start with £828.
Interest = 3.5% of £828 = 0.035 × £828 = £28.98.
New total = £828 + £28.98 = £856.98.
- Year 3: Start with £856.98.
Interest = 3.5% of £856.98 ≈ £29.9943 ≈ £30.00 (to nearest penny).
New total = £856.98 + £30.00 = £886.98.
- Year 4: Start with £886.98.
Interest = 3.5% of £886.98 ≈ £31.0443 ≈ £31.04.
New total = £886.98 + £31.04 = £918.02.
- Year 5: Start with £918.02.
Interest = 3.5% of £918.02 ≈ £32.1307 ≈ £32.13.
New total = £918.02 + £32.13 = £950.15.
After 5 years, the investment is worth £950.15.
See how tedious that was? Just five years involved lots of steps and rounding. For 10 or 20 years, it would be a nightmare! This shows why we need a better method.
The Efficient Method: The Multiplier
The key is to spot a pattern. Each year, we are finding 103.5% of the previous year's amount.
Finding 103.5% is the same as multiplying by 1.035 (since 100% + 3.5% = 103.5%, and 103.5100 = 1.035).
- This number (1.035) is called the multiplier.
Let's retry the same problem using this powerful idea.
Original Amount: £800
Interest Rate: 3.5%
Multiplier: 1 + (3.5100) = 1.035
Time: 5 years
Instead of doing 5 separate steps, we can do one calculation:
Final Amount = Original Amount × (Multiplier)Number of Years
Or, using the common formula:
A = P(1 + r100)n
Where:
A = final amount
P = principal (original amount)
r = percentage rate
n= number of time periods (years)
Calculation:
Final Amount = £800 × (1.035)⁵
We now use our calculator carefully:
Calculate 1.035⁵.
Type: 1.035, then the xy or ^ button, then 5, then =.
You should get: 1.187686 (keep this on your calculator screen).
Now multiply by 800.
With the previous answer still on screen, press × 800 =.
Answer: 950.1488...
Round to the nearest penny (two decimal places for currency).
- £950.15
We got the same answer with one calculation! This method is essential for the exam.
flowchart TD
A["Start: Principal P, Rate r%, Time n"] --> B["Calculate multiplier: 1 + r/100"]
B --> C["Calculate multiplier^n (use x^y button)"]
C --> D["Multiply by P"]
D --> E["Round to 2 d.p. for currency"]
E --> F["Final Amount A"]
What About Depreciation?
Depreciation (like a car losing value) works exactly the same way, but the multiplier is less than 1.
For interest (increase): Multiplier = 1 + (percentage100)
- For depreciation (decrease): Multiplier = 1 – (percentage100)
Example 2 (Depreciation): A new car costs £15,000. It depreciates in value by 12% per year. Find its value after 6 years.
Method:
P = £15,000
Rate of decrease = 12%
Multiplier = 1 – (12100) = 1 – 0.12 = 0.88
- n = 6 years
Calculation:
Final Value = £15,000 × (0.88)⁶
Calculate 0.88⁶ = 0.464404
- Multiply by 15,000 = 6966.06
After 6 years, the car is worth £6,966.06.
Key Takeaways and Exam Tips
Identify Increase or Decrease: Is it interest (growth) or depreciation (loss)? This decides if you add or subtract the percentage.
Find the Multiplier: This is the most important step.
Growth: 1 + (r100)
Decay: 1 – (r100)
Use the Formula: Final Amount = Original Amount × (Multiplier)n
Calculator Skills: Practise using the power button (xy or ^). Always do the power calculation first, then multiply by the principal.
Rounding: For money, round your final answer to 2 decimal places.
- Units: Make sure the time period (n) matches the interest period. If interest is 4% per month for 2 years, then n = 24 months.
Practice Question
Try this yourself before looking at the answer.
Question: £2,500 is invested at 4.2% compound interest for 8 years. Calculate the total amount in the account after 8 years.
Click to reveal answer
Answer: £3,458.49
Working:
Multiplier = 1 + (4.2100) = 1.042
Final Amount = £2,500 × (1.042)⁸
= £2,500 × 1.383396...
= £3,458.49 (to nearest penny)
- By mastering the multiplier method, you can solve any compound interest or depreciation problem quickly and accurately. Remember: understand the long way, but always use the formula in your exam. Good luck!