Compound Interest and Depreciation

Multipliers are commutative, meaning you can multiply them together to find the overall effect of multiple successive percentage changes. A $10\%$ increase followed by a $20\%$ decrease is simply $1.10 \times 0.80 = 0.88$, representing a $12\%$ overall decrease.

Compound Interest and Exponential Growth

Every time you leave money in a savings account, you do not just earn interest on your original deposit — you earn interest on your previously accumulated interest. This is called compound interest, which represents exponential growth. To calculate the final amount efficiently without year-on-year rounding errors, you must use the iterative multiplier formula:

$$A = P \times m^n$$

Where:

• $A$ = Final amount (total balance)
• $P$ = Principal (the original investment)
• $m$ = Decimal multiplier
• $n$ = Number of time periods (usually years)

In OCR exam questions, you will frequently see rates given as per annum (p.a.), which is a Latin term simply meaning "per year".

Worked Example 1: Compound Interest

Mateo invests £4,200 in a savings account paying 3.5% compound interest p.a. Calculate the total value of the investment after 6 years.

Step 1: Calculate the decimal multiplier.

• $100\% + 3.5\% = 103.5\% \rightarrow 1.035$

Step 2: Substitute the values into the compound interest formula.

• $A = 4200 \times 1.035^6$

Step 3: Calculate the final amount and round to 2 decimal places (nearest penny).

• $A = 4200 \times 1.229255...$
• $A = £5162.8731...$
• Final Answer: £5,162.87

Depreciation and Exponential Decay

Why does a brand-new server system lose a massive chunk of its resale value in its first year, but lose much less in its fifth year? This happens because of depreciation, which is the decrease in value of an asset over time. Depreciation operates as exponential decay, meaning the item loses a fixed percentage of its current value each year, not its original value.

To find the value after several years, you use the exact same iterative formula ($V = P \times m^n$), but your multiplier will always be less than 1.

Worked Example 2: Depreciation

A server system is bought for £24,000. It depreciates by 18% each year. Find its value after 4 years, rounding your answer to the nearest pound.

Step 1: Calculate the decay multiplier.

• $100\% - 18\% = 82\% \rightarrow 0.82$

Step 2: Substitute into the formula.

• $V = 24000 \times 0.82^4$

Step 3: Calculate the final value and round to the nearest pound as requested.

• $V = 24000 \times 0.452121...$
• $V = £10850.92...$
• Final Answer: £10,851

Fractional Multipliers (Non-Calculator)

Understanding how to use fractions instead of decimals explains why some seemingly impossible non-calculator questions are actually straightforward. OCR requires you to be able to use a fractional multiplier for exact calculations. Instead of a decimal, you express the percentage change as an improper fraction or simplified fraction.

For an increase, place $100 + \text{percentage}$ over 100 (e.g., a $20\%$ increase is $\frac{120}{100}$ or $$). For a decrease, place over 100. You can then raise this fraction to a power just like a decimal multiplier, keeping all numbers mathematically exact.

Worked Example 3: Fractional Multipliers

An antique vase worth £500 increases in value by 10% each year. Find its value after 2 years without using a calculator.

Step 1: Find the simplified fractional multiplier.

• $100\% + 10\% = 110\% \rightarrow \frac{110}{100} = \frac{11}{10}$

Step 2: Substitute into the iterative formula.

• $A = 500 \times (\frac{11}{10})^2$

Step 3: Square the fraction and calculate the final value.

• $A = 500 \times \frac{121}{100}$
• $A = 5 \times 121$

Advanced: Reverse Compound Interest

You can mathematically travel backwards in time to find out what an investment started at. This is called reverse compound interest. If you know the final amount ($A$) and want to find the original principal ($P$), you rearrange the iterative formula by dividing the final amount by the multiplier to the power of $n$.

$$P = \frac{A}{m^n}$$