Cambridge IGCSE Mathematics 0580

Percentages, interest and reverse percentages

Every percentage question on Cambridge IGCSE Maths 0580 is the same calculation: multiply by a multiplier. An increase of 8% is $\times 1.08$ , a decrease of 8% is $\times 0.92$ , $n$ years of compound change is $\times$ that multiplier to the power $n$ , and a reverse percentage — “the price after a 20% rise is $60, what was it before?” — is dividing by the multiplier instead of multiplying.

Compound interest alone carries 66 marks across the 42 papers Quanta has mapped, and with percentage change, reverse percentages and exponential growth and decay the group is over 150 marks in section 1. The mistake that costs most of them is answering a reverse percentage by taking a percentage off.

Updated 15 September 2026

The multiplier

Convert the percentage change into a single number to multiply by:

Change	Multiplier	Because
Increase 8%	1.08	100% + 8% = 108%
Decrease 8%	0.92	100% − 8% = 92%
Increase 150%	2.5	100% + 150% = 250%
Decrease 4.5%	0.955	100% − 4.5% = 95.5%
Find 8% of	0.08	8% on its own, not a change

The last row is the one to keep separate. “Find 8% of $250” is $0.08 \times 250 = \$20$ ; “increase $250 by 8%” is $1.08 \times 250 = \$270$ . Reading which is being asked is worth more marks than any technique on this page.

Percentage increase and decrease

\text{percentage change} = \frac{\text{change}}{\text{original}} \times 100

The denominator is always the original value — the one before the change — and getting that wrong turns a 25% rise into a 20% one. Profit and loss questions are the same formula with cost price as the original:

\text{percentage profit} = \frac{\text{selling price} - \text{cost price}}{\text{cost price}} \times 100

Compound interest, growth and decay

\text{final} = \text{initial} \times \left(1 + \frac{r}{100}\right)^{n}

“Compound” means each year’s change applies to the new amount, not the original. $2 000 at 3% for 4 years is $2000 \times 1.03^4 = \$2251.02$ . The interest earned is that minus the original — $\$251.02$ — and questions ask for one or the other, so check which.

Depreciation and decay use the same formula with a multiplier below 1: a car worth $18 000 losing 15% a year is worth $18000 \times 0.85^5 = \$7986.70$ after five years. Population growth, bacterial growth and radioactive decay are all this formula with different words.

Simple interest, for contrast

Simple interest is charged on the original only: $I = \dfrac{P r t}{100}$ . Over 4 years at 3%, $2 000 earns $\$240$ — $11 less than compound. Questions often ask for the difference between the two, which is exactly that subtraction.

Finding n or r

Extended papers ask how many years until a value passes a threshold. Trial and improvement with the multiplier is accepted: keep multiplying until you cross it, and state the first year that does. To find the rate instead, divide final by initial, take the $n$ th root, subtract 1 and multiply by 100.

Reverse percentages

When the value you are given is the one after the change, you divide by the multiplier:

\text{original} = \frac{\text{new value}}{\text{multiplier}}

A coat costs $60 in a 20% off sale. The multiplier was $0.8$ , so the original price was $60 \div 0.8 = \$75$ . Taking 20% off $60 gives $48, which is wrong — and the check is immediate: 20% off $75 is $60. ✓

The signal words are “after”, “in a sale”, “including tax”, “had increased by” — anything saying the figure you have already contains the change. Extended papers also compound it: divide by the multiplier raised to $n$ to get back to the starting value $n$ years ago.

Worked example

A laptop is sold for $828 after a 15% discount. (a) Find the price before the discount. (b) The shop bought it for $600. Find the percentage profit on the sale price of $828. (c) A second shop invests $828 at 4% compound interest. Find the value after 3 years, and the interest earned.

(a) The $828 is after the change, so divide. Multiplier $= 0.85$ :

828 \div 0.85 = \$974.12

Check: $974.12 \times 0.85 = 828.00$ . ✓

(b) Profit = $828 - 600 = \$228$ , on a cost price of $600:

\frac{228}{600} \times 100 = 38\%

(c) $828 \times 1.04^3 = \$931.39$ , so the interest earned is $931.39 - 828 = \$103.39$ .

Money answers go to 2 decimal places; other answers to 3 significant figures unless told otherwise. Keep the full value in the calculator between parts — rounding $974.12 to $974 and reusing it drifts the later answers.

Common mistakes

1.Reverse percentage done by taking the percentage off
If the figure is already after the change, divide by the multiplier. Then check by applying the change forwards — it takes five seconds and catches the error every time.
2.Percentage change divided by the new value
The denominator is the original. Change ÷ original × 100.
3.Compound interest worked out year by year and rounded each time
Use the power: $P \times 1.04^3$ in one step. Rounding at each stage accumulates error and can lose the accuracy mark.
4.Giving the total when the question wanted the interest
“How much interest is earned” means final minus initial. “What is it worth” means the final value.
5.Percentage changes added together
A 10% rise then a 10% fall is not back to the start: $1.1 \times 0.9 = 0.99$ , a 1% loss overall. Multiply multipliers, never add percentages.

Common questions

How do you do a reverse percentage?

Divide by the multiplier. If $60 is the price after 20% off, the multiplier was 0.8 and the original was $60 \div 0.8 = \$75$ .

What is the compound interest formula?

$\text{final} = P\left(1 + \frac{r}{100}\right)^n$ , where $P$ is the starting amount, $r$ the percentage rate and $n$ the number of periods. For depreciation, subtract instead of adding — the multiplier drops below 1.

What is the difference between simple and compound interest?

Simple interest is calculated on the original amount every period; compound interest is calculated on the running total, so it grows faster. Over four years at 3%, $2 000 earns $240 simple and $251.02 compound.

How do you find the percentage increase between two numbers?

Subtract to get the change, divide by the original value, and multiply by 100. From 40 to 50 is $\tfrac{10}{40} \times 100 = 25\%$ .

Practise percentages, interest and reverse percentages against real mark schemes

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