Percentage Change, Reverse Percentage and Compound Interest
Percentages appear in every IGCSE Maths paper and in real questions about money, populations and depreciation. The topic has one recurring trap — knowing which number to divide by — and one formula that must be used exactly as written.
1. Multipliers — learn these first
Almost every percentage question becomes one multiplication once you can write the multiplier.
Increase of r% → multiply by (100 + r)/100 Decrease of r% → multiply by (100 − r)/100
| Change | Multiplier |
|---|---|
| Increase 15% | 1.15 |
| Increase 7% | 1.07 |
| Increase 100% | 2.00 |
| Decrease 15% | 0.85 |
| Decrease 8% | 0.92 |
| Decrease 40% | 0.60 |
Example: increase 240 by 15% → 240 × 1.15 = 276
A decrease multiplier is 1 minus the rate, not the rate itself. To decrease by 15% you multiply by 0.85, not 0.15. Multiplying by 0.15 gives you the discount, not the new price — a distinction that caused repeated errors.
Finding the amount vs finding the new total. “How much is the discount?” wants 240 × 0.15 = 36. “What is the sale price?” wants 240 × 0.85 = 204. Read which one the question asks for.
2. Percentage change
Percentage change = (change / original) × 100
where change = new value − original value.
Example: a price rises from 80 to 92.
- change = 92 − 80 = 12
- percentage change = (12/80) × 100 = 15% increase
Always divide by the ORIGINAL value — the one you started from. Dividing by the new value is the most common error in the entire topic. If a price goes from 80 to 92, you divide by 80.
The “original” is whatever came first in time. For a decrease from 92 to 80, you still divide by 92, because that is where you started.
3. Profit and loss
Percentage profit = (profit / cost price) × 100 profit = selling price − cost price
Example: bought for 250, sold for 300.
- profit = 50
- percentage profit = (50/250) × 100 = 20%
Divide by the COST price, not the selling price. This is the same “divide by the original” rule wearing different clothes, and it was got wrong repeatedly.
Find the profit first. Dividing the selling price by the cost price gives 120%, not 20% — you must subtract before you divide.
4. Reverse percentages — finding the original
This is the type students find hardest, and it is very easy to spot: you are given the value AFTER a change and asked for the value BEFORE.
Original = final ÷ multiplier
Example: a coat costs 84 after a 30% discount. What was the original price?
- 84 is 70% of the original, so the multiplier is 0.7
- original = 84 ÷ 0.7 = 120
Example: a price including 20% profit is 300. Find the cost price.
- multiplier = 1.2
- cost price = 300 ÷ 1.2 = 250
You divide, not multiply. Adding 30% back onto 84 gives 109.20, which is wrong — because the 30% was taken off the larger original, not off 84.
How to recognise it: the question gives you a price after a discount, a population after growth, or an amount including tax, and asks for the value before. The word “was” or “original” is usually there.
Check your answer by applying the change forwards: 120 × 0.7 = 84. ✓
5. Simple interest
Simple interest = (P × R × T) / 100
where P = principal, R = rate per year (%), T = time in years.
Interest is calculated on the original amount every year, so it is the same each year.
Example: 2000 at 5% for 3 years.
- interest = (2000 × 5 × 3)/100 = 300
- total = 2000 + 300 = 2300
This formula gives the INTEREST, not the total. If the question asks how much is in the account, you must add the principal back. Confusing these two was a documented error.
6. Compound interest
Interest is added to the balance, so the following year’s interest is calculated on a larger amount.
Final amount = P × (1 + r/100)ⁿ
where P = principal, r = rate per year (%), n = number of years.
Example: 2000 at 5% compound for 3 years.
- amount = 2000 × (1.05)³ = 2000 × 1.157625 = 2315.25
- interest earned = 2315.25 − 2000 = 315.25
The formula gives the TOTAL amount. To find the interest alone, subtract the principal. This is the mirror image of the simple interest trap, and the two get swapped constantly.
It is a power, not a multiplication. (1.05)³ means 1.05 × 1.05 × 1.05 = 1.157625. Multiplying by 3, or by 1.05 × 3, is wrong — and “forgot to apply the power” appeared many times in lessons.
Compare: over 3 years at 5% on 2000, simple interest gives 300 and compound gives 315.25. Compound is always more (for a positive rate), and the gap widens with time.
Depreciation and decay
Final amount = P × (1 − r/100)ⁿ
Example: a car worth 15 000 depreciating 12% per year for 4 years.
- value = 15000 × (0.88)⁴ = 9004.53 (2 d.p.)
Same formula, minus instead of plus. Use it for depreciation, population decline and radioactive-style decay questions.
Repeated changes do not add up
Two increases of 10% are not a single increase of 20%.
100 → ×1.1 → 110 → ×1.1 → 121, which is a 21% increase overall. Combine percentage changes by multiplying the multipliers: 1.1 × 1.1 = 1.21.
Similarly, a 20% rise followed by a 20% fall gives 1.2 × 0.8 = 0.96 — a 4% loss, not a return to the start.
This “percentages don’t add” idea is one of the most-tested points in the topic, and it was a specific confusion recorded in lessons.
Finding the number of years
If a question asks how many years until an amount exceeds some value, the reliable approach at IGCSE is trial and improvement — try n = 3, n = 4, and so on until the amount passes the target.
Round the number of years UP. If the balance passes the target during year 5, the answer is 5 years — a fractional year isn’t a meaningful answer.
Some tutors demonstrated logarithms for this. Logarithms are not required on 0580 — if you have met them, they are fine to use, but trial and improvement earns exactly the same marks.
7. Method and accuracy
Do not round intermediate values in compound interest. Carry the full figure through the powers and round only at the end.
Money is normally given to 2 decimal places. Check whether the question wants the nearest whole unit instead.
Show your working, including the multiplier. Writing ”× 1.05³” earns method marks even if the final arithmetic slips.
8. Mistakes that cost marks
Dividing by the new value instead of the original.
Dividing by the selling price instead of the cost price.
Using the selling price in place of the profit.
Multiplying by 0.15 to decrease by 15% instead of 0.85.
Multiplying instead of dividing in a reverse percentage.
Adding the percentage back to find an original value.
Giving interest when the total was asked for, or the reverse.
Using simple interest where compound was meant, or vice versa.
Forgetting the power in the compound formula.
Adding successive percentage changes instead of multiplying the multipliers.
Rounding the number of years down.
Rounding intermediate values.
Frequently asked questions
How do I increase a number by a percentage? Multiply by (100 + r)/100 — e.g. ×1.15 for a 15% increase.
How do I calculate percentage change? (change ÷ original) × 100, always dividing by the original value.
How do I calculate percentage profit? Find the profit (selling − cost), divide by the cost price, multiply by 100.
What is a reverse percentage? Finding the value before a change. Divide the final value by the multiplier.
What is the compound interest formula? Amount = P(1 + r/100)ⁿ — it gives the total, so subtract P for the interest.
What is the difference between simple and compound interest? Simple interest is always calculated on the original amount. Compound interest is calculated on the growing balance, so it earns more.
How do I calculate depreciation? P(1 − r/100)ⁿ — the same formula with a minus.
Is a 10% rise followed by another 10% rise the same as 20%? No — it is a 21% rise. Multiply the multipliers: 1.1 × 1.1 = 1.21.
Do I need logarithms for compound interest? No. Trial and improvement is enough on 0580.
Quick revision checklist
- I can write the multiplier for any increase or decrease
- I know a 15% decrease means ×0.85
- I can tell “find the discount” from “find the new price”
- I always divide by the original in percentage change
- I divide by the cost price for percentage profit, after finding the profit
- I can recognise a reverse percentage question
- I divide by the multiplier to find an original value, and check forwards
- I know the simple interest formula gives interest, not the total
- I know the compound formula gives the total, not the interest
- I apply the power correctly
- I can use the formula for depreciation and decay
- I know percentage changes multiply, not add
- I round years up
- I keep full accuracy until the final answer
- I show the multiplier in my working
These notes cover percentage change, reverse percentages and compound interest in the Cambridge IGCSE Mathematics (0580) syllabus, and are written for Grade 9–11 / Year 10–11 students. They are based on teaching patterns observed across a large set of one-to-one IGCSE Maths lessons, with particular attention to the errors students make most often and the wording examiners reward. Always check the current syllabus and formula list for your own exam series.
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