IGCSE Maths percentages: compound interest, depreciation and reverse percentages

What each tier expects

The key ideas

A multiplier turns a percentage change into one multiplication. Increase by r%: multiply by 1 + r/100. Decrease by r%: multiply by 1 - r/100. So a 5% VAT charge is × 1.05 and a 30% discount is × 0.7.

Simple interest is the same amount every year: principal × rate × years. Compound interest earns interest on interest, so the amount after n years is principal × (1 + r/100)^n. Depreciation is the decreasing version: value × (1 - r/100)^n. Cambridge Extended calls both exponential growth and decay.

Repeated changes multiply, they do not add. A 30% increase followed by a 20% decrease is × 1.3 × 0.8 = × 1.04, an overall increase of 4%, not 10%.

Reverse percentages work backwards from a value after a change. If a price after a 30% discount is 17.50, then original × 0.7 = 17.50, so the original is 17.50 ÷ 0.7 = 25. Taking 30% of 17.50 and adding it back is the classic wrong method.

Worked example 1: compound against simple interest

1. $2000 is invested for 4 years at 3.5% per year. Compare simple and compound interest.
2. Simple interest: 2000 × 0.035 × 4 = $280, so the total is $2280.
3. Compound: 2000 × 1.035^4 = 2000 × 1.14752... = $2295.05 (to the nearest cent).
4. Compound interest earned = 2295.05 - 2000 = $295.05, which is $15.05 more than simple interest.

Worked example 2: reverse percentage

1. In a sale, prices are reduced by 30%. A jacket costs 17.50 in the sale. Find the original price.
2. Sale price = original × 0.7, so original = 17.50 ÷ 0.7 = 25.00.
3. Check: 30% of 25 is 7.50, and 25 - 7.50 = 17.50. Correct.
4. A Gulf example: a bill of AED 210 includes 5% VAT. The price before VAT is 210 ÷ 1.05 = AED 200.

Worked example 3: depreciation and repeated change

1. A car bought for AED 80 000 loses 15% of its value each year. Find its value after 3 years.
2. Multiplier 0.85, three times: 80 000 × 0.85^3 = 80 000 × 0.614125 = AED 49 130.
3. Percentage change in a single step: from 250 to 290 is a change of 40, and 40 ÷ 250 × 100 = 16% increase.
4. Repeated change: an increase of 30% followed by a decrease of 20% is 1.3 × 0.8 = 1.04, so the overall change is a 4% increase.

Common mistakes that cost marks

• Finding the percentage of the new value in a reverse percentage question instead of dividing by the multiplier.
• Adding successive percentage changes instead of multiplying the multipliers.
• Dividing by the new value when calculating percentage change; always divide by the original.
• Giving the total amount when the question asks for the interest, or the reverse.
• Using simple interest when the question says 'compound', or 'per annum compound' when it says 'simple'.
• Rounding money to 1 decimal place: give money to the nearest cent unless told otherwise.

Exam technique and how a tutor helps

Percentages appear on every paper and are often the first part of a longer money question. Write the multiplier explicitly, such as '× 1.035^4', because that shows the method even if you mistype into the calculator. On the Cambridge non-calculator papers, work with easy percentages built from 10%, 5% and 1%, and expect numbers that divide cleanly.

Read for the trigger words: 'original' or 'before the increase' signals a reverse percentage; 'each year' or 'per annum compound' signals a power; 'overall' signals multiplying several multipliers.

Most students can do a single percentage; the marks are lost when two ideas are combined. In one-to-one lessons a tutor focuses on choosing the right method from the wording, using mixed real-world questions such as VAT, salary rises and car depreciation, and shows how the multiplier method covers almost every case so there is one approach to rely on.

Self-check: can you do these?

• Find 15% of 120. (Answer: 18)
• Express 45 as a percentage of 60. (Answer: 75%)
• A price of 64 is increased by 25%. Find the new price. (Answer: 80)
• A population of 50 000 grows by 2% a year. Find it after 10 years, to the nearest whole number. (Answer: 60 950)
• After a 20% increase, a salary is 4800. What was it before? (Answer: 4000)