Percentages

Video lesson5 worked examples

Theory

Appreciation/Depreciation

You need to calculate compound interest and depreciation over time. It is encouraged to use the most efficient strategy, such as calculating a compound percentage using a decimal multiplier and a power, rather than a year-by-year approach.

Reverse Percentages

You must know how to use reverse percentages to calculate an original quantity before a percentage was added or subtracted (e.g., calculating the price excluding VAT).

The Golden Rule: decide first whether the original amount (100%) is known or unknown. If it is known, apply a multiplier. If it is unknown — the price already includes the change — it is a reverse percentage, so work back from the percentage you are given.

⚠️ Common Examiner Traps

Treating a reverse percentage as a normal one: if £60 is the price after 20% off, that £60 is $80\%$ of the original — you cannot just add 20% of £60 back on.
Adding percentages across years: two years of 10% growth is $\times 1.1^2 = 1.21$ , a 21% rise, not 20%. Compound with a power.
The decrease multiplier: a 15% fall uses $\times 0.85$ , not $\times 1.15$ .
Money to two decimal places: write $£168\,729.60$ , never $£168\,729.6$ .

Worked examples

Example 1

Compound Appreciation

A house is valued at £150,000 and appreciates by 4% each year. What is its value after 3 years?

Step 1: Find the multiplier: 100% + 4% = 104% = 1.04.

Step 2: Apply the power for 3 years: $150{,}000 \times 1.04^3$ .

Answer: £168,729.60.

Example 2

Compound Interest Earned

£2000 is invested at 2.5% interest per year. Calculate the interest earned after 4 years.

Step 1: The multiplier for a 2.5% rise is $1.025$ . Find the total in the account after 4 years: $2000 \times 1.025^4 = £2207.63$ .

Step 2: The question asks for the interest earned, not the total, so subtract the original amount:

2207.63 - 2000 = 207.63

Answer: £207.63.

Example 3

Depreciation

A car worth £18,000 depreciates by 12% each year. Find its value after 3 years.

Step 1: For a decrease, subtract from 100%: $100\% - 12\% = 88\%$ , so the multiplier is $0.88$ — not $1.12$ .

Step 2: Apply the power for 3 years: $18{,}000 \times 0.88^3$ .

Answer: £12,266.50.

Example 4

Reverse Percentages

A jacket is on sale for £60 after a 20% discount. What was the original price?

Step 1: Equate the sale price to the percentage of the original: 80% = £60.

Step 2: Find 1% by dividing by 80, then multiply by 100 to find the original (100%): $\tfrac{60}{80} \times 100$ .

Answer: £75.

Example 5

🎯 Exam-style

£4000 is invested in a fund. It grows by 3% in each of the first two years, but in the third year the fund falls by 5%. Calculate the value at the end of the three years.

Step 1: Apply a separate multiplier for each change. Two years of 3% growth use $1.03$ ; the 5% fall uses $0.95$ .

Step 2: Chain them in one calculation — do not add or average the percentages:

4000 \times 1.03 \times 1.03 \times 0.95 = 4000 \times 1.03^2 \times 0.95

Step 3: Evaluate, keeping full accuracy until the end:

4000 \times 1.0609 \times 0.95 = 4031.42

Answer: £4031.42.

Ready to practise?

18 Percentages questions with answers and video solutions