Home / IGCSE Mathematics 0580 / Percentages

Cambridge IGCSE Mathematics 0580 · Number

Percentages

Finding a percentage of a quantity, writing one quantity as a percentage of another, percentage increase and decrease (profit and loss, discount, deposits, earnings), and simple and compound interest. Extended candidates also need repeated percentage change and reverse percentages. No formulas are given on the paper, so each method is built from what a percentage means.

C1.13E1.13Core and ExtendedFree lesson

What the syllabus says

C1.13 (Core)

1 Calculate a given percentage of a quantity.
2 Express one quantity as a percentage of another.
3 Calculate percentage increase or decrease.
4 Calculate with simple and compound interest.

Notes and examples

Formulas are not given.
Percentage calculations may include:
deposit
discount
profit and loss (as an amount or a percentage)
earnings
percentages over 100%.

E1.13 (Extended)

1 Calculate a given percentage of a quantity.
2 Express one quantity as a percentage of another.
3 Calculate percentage increase or decrease.
4 Calculate with simple and compound interest.
5 Calculate using reverse percentages.

Notes and examples

Problems may include repeated percentage change.
Formulas are not given.
e.g. find the cost price given the selling price and the percentage profit.
Percentage calculations may include:
deposit
discount
profit and loss (as an amount or a percentage)
earnings
percentages over 100%.

Quoted from the Cambridge IGCSE Mathematics 0580 syllabus for 2025 to 2027.

Percentages, decimals and multipliers

Per cent means 'out of 100'. So 35% is $\frac{35}{100}$ , which is 0.35 as a decimal, and 100% is the whole amount. A percentage over 100%, such as 140%, simply means more than the whole: 140% is 1.4 times the amount.

The decimal form is called the multiplier, and it is the most useful idea in this topic:

to find 35% of an amount, multiply by 0.35;
to increase an amount by 35%, you want $100 % + 35 % = 135 %$ of it, so multiply by 1.35;
to decrease an amount by 35%, you want $100 % - 35 % = 65 %$ of it, so multiply by 0.65.

Be careful with small percentages: 3% is 0.03, not 0.3, and 0.5% is 0.005. A quick check is to move the decimal point two places to the left.

Finding a percentage of a quantity

With a calculator, multiply by the multiplier: 23% of 640 is $0.23 \times 640 = 147.2$ .

Without a calculator (Papers 1 and 2), build the percentage from easy pieces:

10% is the amount divided by 10;
5% is half of 10%;
1% is the amount divided by 100;
50% is half, 25% is a quarter, and 20% is a fifth.

So 35% is $10 % + 10 % + 10 % + 5 %$ , and 17.5% is $10 % + 5 % + 2.5 %$ . Percentages over 100% work the same way: 140% is $100 % + 40 %$ . Write each piece down; an examiner can give method marks for the pieces even if the final addition slips.

Worked example 1

Without a calculator, work out (a) 35% of £240 (b) 140% of 65.

(a) 10% of 240 is 24, so 30% is $3 \times 24 = 72$ .
5% is half of 10%, so 12.
35% is $72 + 12 = 84$ , so £84.
(b) 100% of 65 is 65. 10% is 6.5, so 40% is $4 \times 6.5 = 26$ .
140% is $65 + 26 = 91$ .

Answer: (a) £84 (b) 91

Check yourself

Without a calculator, work out 15% of 360.

Answer it in the portal, where your answer is checked as you type →

Writing one quantity as a percentage of another

To write one quantity as a percentage of another, make a fraction and multiply by 100:

$percentage = \frac{part}{whole} \times 100$

This works because multiplying a fraction by 100 tells you how many hundredths it is. For example, 12 out of 40 is $\frac{12}{40} = 0.3$ , which is $0.3 \times 100 = 30 %$ .

Two rules keep you safe:

Same units first. 45 minutes as a percentage of 2 hours is $\frac{45}{120} \times 100$ , not $\frac{45}{2} \times 100$ .
The whole goes underneath. 'Write A as a percentage of B' means $\frac{A}{B} \times 100$ . If A is bigger than B, the answer is over 100%, and that is fine.

Worked example 2

Write 45 minutes as a percentage of 2 hours.

Same units: 2 hours is 120 minutes.
$\frac{45}{120} \times 100 = 37.5$ .
So 45 minutes is 37.5% of 2 hours.

Answer: 37.5%

Check yourself

Write 18 as a percentage of 75.

Answer it in the portal, where your answer is checked as you type →

Percentage increase and decrease

There are two different jobs here, and it helps to keep them apart.

Changing an amount by a percentage. Use a multiplier. Increasing by 12% means multiplying by 1.12; decreasing by 15% means multiplying by 0.85. On a non-calculator paper you can instead find the percentage and add it on or take it off.

Finding the percentage change. Compare the change with the original amount:

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

The words in a question tell you which job it is, and they come from everyday money:

Profit and loss. Profit = selling price $-$ cost price. Percentage profit is the profit as a percentage of the cost price, because that is the original amount. A question may ask for the profit as an amount or as a percentage.
Discount. A price reduction, often given as a percentage of the normal price.
Deposit. A first payment, often a percentage of the price, followed by instalments. The total cost on credit is the deposit plus all the instalments.
Earnings. A pay rise of 3% multiplies pay by 1.03. Tax or commission is a percentage of earnings.

Worked example 3

A coat costs £64. In a sale its price is reduced by 15%. Work out the sale price.

A 15% reduction leaves $100 % - 15 % = 85 %$ of the price.
Sale price $= 0.85 \times 64 = 54.40$ , so £54.40.
Non-calculator route: 10% of 64 is 6.40 and 5% is 3.20, so 15% is 9.60. Then $64 - 9.60 = 54.40$ .

Answer: £54.40

Worked example 4 (exam style)

A trader buys 40 lamps for £12 each. She sells 32 of them for £18 each and the rest for £9 each. Work out her percentage profit.

Cost price: $40 \times 12 = 480$ , so £480.
Money from sales: $32 \times 18 = 576$ and $8 \times 9 = 72$ , so £648 in total.
Profit: $648 - 480 = 168$ , so £168.
Percentage profit, compared with the cost price: $\frac{168}{480} \times 100 = 35$ .
Dividing by 648 instead would give the profit as a percentage of the selling price, which is not what percentage profit means.

Answer: 35%

Check yourself

Increase 250 by 12%.

Answer it in the portal, where your answer is checked as you type →

Simple interest

When you invest or borrow money, interest is paid as a percentage of it. With simple interest, the interest is worked out on the original amount (the principal) every year, so it is the same each year.

No formula is given in the exam, so build it from the meaning. £1200 at 3% per year earns $0.03 \times 1200 = 36$ each year. Over 5 years that is $5 \times 36 = 180$ , so £180 of interest, and the investment is worth $1200 + 180 = 1380$ .

In general, interest = principal $\times$ rate $\times$ number of years, with the rate as a decimal. Read the question carefully: does it want the interest, or the total amount (principal plus interest)?

Compound interest

With compound interest, the interest each year is added to the account, and next year's interest is worked out on the new, bigger amount. You earn interest on your interest.

Each year the amount is multiplied by the same multiplier. At 2.5% per year the multiplier is 1.025, so after 3 years the amount is

$principal \times 1.025 \times 1.025 \times 1.025 = principal \times {1.025}^{3}$

That is why compound interest is quicker to work out with the power key than year by year, and why it grows faster than simple interest: the interest itself keeps growing.

Do not round the amount at the end of each year. Work out ${1.025}^{3}$ and multiply in one go, then round your final answer to the nearest penny.

Worked example 5

£4000 is invested for 3 years at 2.5% per year. Work out the value of the investment at the end of the 3 years with (a) simple interest (b) compound interest.

(a) Interest each year: $0.025 \times 4000 = 100$ . Over 3 years: $3 \times 100 = 300$ .
Value: $4000 + 300 = 4300$ , so £4300.
(b) Multiplier 1.025, three times: $4000 \times {1.025}^{3} = 4307.5625$ .
To the nearest penny: £4307.56. Compound interest gives £7.56 more, because years 2 and 3 earn interest on interest.

Answer: (a) £4300 (b) £4307.56

Check yourself

Work out the simple interest on £1200 invested for 5 years at 3% per year.

Answer it in the portal, where your answer is checked as you type →

Repeated percentage change Extended

Compound interest is one example of a repeated percentage change. The same idea describes a car that loses 15% of its value every year (depreciation), a population that grows by 2% a year, or a price that is changed twice in a row.

Each change multiplies by its own multiplier, so changes multiply, they do not add:

a 15% fall each year for 3 years multiplies by ${0.85}^{3}$ ;
an increase of 20% followed by a decrease of 20% multiplies by $1.2 \times 0.8 = 0.96$ , which is an overall decrease of 4%, not 'no change'.

The second change is a percentage of a different amount from the first, which is why the two 20% changes do not cancel.

To find when a value first passes a target, keep multiplying by the multiplier and check each year, or use trial with the power key: try $n = 8$ , $n = 9$ and so on until the value first crosses the target.

Worked example 6 Extended

A car is bought for £20 000. Its value falls by 15% each year. Work out its value after 3 years.

Each year the value is multiplied by $1 - 0.15 = 0.85$ .
After 3 years: $20 000 \times {0.85}^{3} = 12 282.5$ .
The value is £12 282.50.
Taking off $3 \times 15 % = 45 %$ of £20 000 would give £11 000, which is wrong: each year's 15% is taken from a smaller value.

Answer: £12 282.50

Reverse percentages Extended

In a reverse percentage question you are given the amount after a percentage change and asked for the amount before it, such as the original price before a sale, or the cost price given the selling price and the percentage profit.

Write the change as a multiplier and work backwards:

$original \times multiplier = new so original = new \div multiplier$

If a price after a 30% reduction is £56, then $0.7 \times original = 56$ , so the original is $56 \div 0.7 = 80$ , which is £80.

The classic wrong answer is to take 30% of the new price and add it back on, giving £72.80. That fails because the 30% was 30% of the original price, not of the sale price. A useful non-calculator method is to say: the sale price is 70% of the original, so 1% is $56 \div 70 = 0.8$ and 100% is 80.

Worked example 7 Extended

A shopkeeper sells a radio for £66, making a profit of 20% on the cost price. Work out the cost price.

The selling price is $100 % + 20 % = 120 %$ of the cost price, so the multiplier is 1.2.
Cost price $\times 1.2 = 66$ , so cost price $= 66 \div 1.2 = 55$ .
Check: 20% of 55 is 11, and $55 + 11 = 66$ .
Finding 20% of £66 and subtracting gives £52.80, which is wrong: the profit is 20% of the cost price, not of the selling price.

Answer: £55

Check yourself Extended

After an increase of 25%, a price is £95. Work out the price before the increase.

Answer it in the portal, where your answer is checked as you type →

Key points

The multiplier for an increase of $r %$ is $1 + \frac{r}{100}$ ; for a decrease it is $1 - \frac{r}{100}$ .
Percentage of another: $\frac{part}{whole} \times 100$ , with both in the same units.
Percentage change and percentage profit compare the change with the original (cost) amount.
Simple interest is the same every year; compound interest multiplies by the same multiplier every year.
Read whether the question wants the interest or the total amount.
Extended: repeated changes multiply, so $+ 20 %$ then $- 20 %$ is an overall decrease of 4%.
Extended: reverse percentages divide the new amount by the multiplier.

Where marks are lost

Using 0.3 for 3%, or 1.3 for a 3% increase. 3% is 0.03.
Giving the increase or the discount when the question asks for the new amount, or the other way round.
Working out percentage profit as a percentage of the selling price instead of the cost price.
Forgetting the deposit when finding the total cost on credit.
Using simple interest when the question says compound, or rounding at the end of each year.
Giving the interest when the question asks for the total value of the investment.
Extended: adding repeated percentage changes ( $3 \times 15 % = 45 %$ ) instead of multiplying the multipliers.
Extended: in reverse percentages, taking the percentage of the new amount and adding it back.

Related IGCSE Mathematics 0580 lessons

C1.14 · E1.14

All IGCSE Mathematics 0580 lessons →

An original lesson written to the Cambridge IGCSE Mathematics 0580 syllabus (2025 to 2027). The worked examples and questions are our own, not Cambridge questions. Back to every IGCSE Mathematics 0580 lesson.

Percentages

Percentages, decimals and multipliers

Finding a percentage of a quantity

Worked example 1

Check yourself

Writing one quantity as a percentage of another

Worked example 2

Check yourself

Percentage increase and decrease

Worked example 3

Worked example 4 (exam style)

Check yourself

Simple interest

Compound interest

Worked example 5

Check yourself

Repeated percentage change Extended

Worked example 6 Extended

Reverse percentages Extended

Worked example 7 Extended

Check yourself Extended

Key points

Where marks are lost

Related IGCSE Mathematics 0580 lessons

Using a calculator

Rates

Time

Ratio and proportion