Compound Percentages (Appreciation & Depreciation)

Video coming soon3 worked examples

Theory

The Golden Rule: use the most efficient strategy available when answering compound percentage questions — calculate them with a decimal multiplier and a power, rather than a slow, step-by-step, year-by-year approach.

1. Finding the Decimal Multiplier

Before doing any calculations, you must convert the percentage change into a decimal multiplier.

Appreciation (Increase): Add the percentage to 100%, then divide by 100.
Example: A 4% increase $\rightarrow 100\%. + 4\%. = 104\%. \rightarrow 1.04$ multiplier.
Depreciation (Decrease): Subtract the percentage from 100%, then divide by 100.
Example: A 15% decrease $\rightarrow 100\%. - 15\%. = 85\%. \rightarrow 0.85$ multiplier.

2. The Compound Formula

Once you have your multiplier, use the following formula to find the final amount in a single calculation on your calculator:

\text{Original Amount} \times \text{Multiplier}^{\text{Time}}

Time is usually the number of years (or months/days), which you input using the power button (e.g., $x^y$ or $y^x$ ) on your calculator.

3. Mixed Fluctuations

If an amount increases one year and decreases the next, you do not need to do separate calculations. You can simply chain the multipliers together.

\text{Original} \times \text{Multiplier}_1 \times {\text{Multiplier}_2}^{\text{Time}}

⚠️ Common Examiner Traps

The Year-by-Year Trap: Do not calculate the percentage for year 1, add it on, then calculate year 2, add it on, etc. This is best avoided because it consumes far too much exam time and frequently leads to arithmetic and premature rounding errors. Use a multiplier and a power!
The Significant Figures Trap: Compound percentage questions almost always feature an instruction to round your final answer to a specific number of significant figures. You will lose the final mark if you ignore this. Always write down your unrounded answer from the calculator display first, then write your rounded answer underneath.
The Money Formatting Trap: If the question involves calculating a financial value and does not specify rounding to significant figures, you must remember that candidates lose marks for failing to write amounts of money to exactly two decimal places.

Worked examples

Example 1

A city has a population of 125,400. The population is expected to increase by 3.2% each year for the next 4 years. Calculate the expected population of the city after 4 years. Give your answer rounded to 3 significant figures. (4 marks)

Step 1: Find the decimal multiplier for a 3.2% increase.

100\%. + 3.2\%. = 103.2\%. = 1.032

Step 2: Set up the calculation using the original amount, the multiplier, and the power of 4 (for 4 years).

125,400 \times 1.032^4

Step 3: Write down the unrounded answer from your calculator.

= 142,238.163...

Step 4: Round to exactly 3 significant figures (the first three important digits, filling the rest with placeholder zeroes).

Answer: $142,000\text{ people}$ .

Example 2

An electronics company currently sells 350,000 gaming consoles per year. Due to supply issues, it is predicted that sales will:

Decrease by 6.5% in the next year.
Increase by 4.8% in each of the following 2 years.

Calculate the predicted number of console sales after 3 years. Give your answer to 3 significant figures. (4 marks)

Step 1: Find the multiplier for the 6.5% decrease (Year 1).

100\%. - 6.5\%. = 93.5\%. = 0.935

Step 2: Find the multiplier for the 4.8% increase (Years 2 & 3).

100\%. + 4.8\%. = 104.8\%. = 1.048

Step 3: Chain the calculation together. Note that the second multiplier needs a power of 2 because it lasts for two years.

350,000 \times 0.935 \times 1.048^2

Step 4: Evaluate to find the unrounded answer.

= 359,514.808

Step 5: Round to 3 significant figures.

Answer: $360,000\text{ consoles}$ .

Example 3

Jack buys a brand new car for £18,500. The car depreciates in value by 14% each year. Calculate the value of Jack's car after 3 years. (3 marks)

Step 1: Find the multiplier for a 14% decrease.

100\%. - 14\%. = 86\%. = 0.86

Step 2: Set up the calculation with a power of 3.

\pounds18,500 \times 0.86^3

Step 3: Evaluate and ensure the final answer is formatted correctly as money (to two decimal places if required, though here it resolves to exact pence).

= \pounds11,767.09

(The calculator displays 11767.088, which must be rounded to two decimal places for money).

Ready to practise?

13 Compound Percentages questions with answers and video solutions