Edexcel IGCSE

Percentages

A percentage is a number of parts per hundred, so every percentage is also a fraction and a decimal. It is section 1.6 of the Edexcel International GCSE Mathematics A specification, and almost every question on it reduces to the same move: decide which quantity counts as the whole, turn the percentage into a decimal multiplier, and then multiply to go forwards, divide to come back, or raise the multiplier to a power to repeat it.

Edexcel IGCSE 4MA149 approved practice questions

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Before you start

What you need to be able to do already

If any of these is shaky, fix it first — every worked example below leans on all three.

Convert between a fraction, a decimal and a percentage, and simplify the fractionFractions
Write $48%$ as a fraction in its simplest form. If $\frac{12}{25}$ did not come quickly, fix fractions first — several questions here ask for exactly this, and $\frac{48}{100}$ left uncancelled loses the accuracy mark.
Multiply and divide by a decimal, and round money to two decimal placesDecimals
Work out $60 \times 0.85$ , then $54 \div 0.6$ . If either was awkward, the multiplier method below will be too: it is decimal arithmetic from start to finish. In standard money notation, write $51.8$ as £51.80.
Raise a decimal to a power using the power key on a calculatorPowers and roots
Work out $1.0 3^{4}$ . You should get $1.12550881$ . Compound interest, depreciation and repeated change are the largest group of questions on this subtopic, and every one of them is a multiplier raised to a power.

The idea

What is percentages?

Per cent means 'out of one hundred'. So $35%$ means $35$ parts in every $100$ , which is the fraction $\frac{35}{100}$ and the decimal $0.35$ — three ways of writing one number. Because it is a fraction of something, a percentage on its own says nothing: it is always a percentage of some whole, and deciding which quantity is that whole is most of the work. Once you know it, the percentage becomes a decimal you multiply by.

The notation, and how to read it

Every worked solution below is written in these five symbols.

SymbolReads asMeans
$%$ per centOut of one hundred. $35% = \frac{35}{100} = 0.35$ , so a percentage is a fraction and a decimal wearing a different hat.
$\times 1.15$ times one point one fiveApplies a $15%$ increase in a single step. The $1$ keeps the original and the $0.15$ adds the extra, so the answer arrives without a separate addition.
$\times 0.85$ times nought point eight fiveApplies a $15%$ decrease in a single step, because what is left is $100% - 15% = 85%$ . Multiplying by $0.15$ instead gives the discount, not the sale price.
$\div 1.15$ divided by one point one fiveUndoes a $15%$ increase — the reverse percentage. Use it when the amount you are given is the one after the change and the question wants the original.
$1.0 4^{3}$ one point nought four to the power threeThree rounds of $4%$ growth applied one after another. The power counts how many times the change happens, so $n$ years of compound interest is the multiplier to the power $n$ .

The one idea underneath all of it

Fix the whole, turn the percentage into a multiplier, then move. Multiply to go forwards, divide to come back, raise to a power to repeat. Increase and decrease are not two methods but one: the same multiplication, with a multiplier above $1$ or below it. This is why percentages compound rather than add — two changes multiply their multipliers together, and $1.2 \times 0.8 = 0.96$ , not $1$ .

The smallest possible example

A jacket costs £60. In a sale it is reduced by $15%$ .

$60 \times 0.85 = 51$ , so the sale price is £51.00

A $15%$ reduction leaves $85%$ of the price, and $85%$ written as a decimal is $0.85$ . One multiplication finishes the question. You could instead work out the $£9$ discount and subtract it, and you would get the same answer — but the multiplier is the version that keeps working when the change repeats or has to be undone. Every other question on this page is this one move, run forwards, backwards, or several times over.

Key terms

The words a question will use

Exam wording is precise. These five carry the meaning a question depends on.

TermWhat it meansNotation
Multiplier
The decimal you multiply by to apply a percentage change in one step. Add the percentage to $100%$ for an increase and subtract it for a decrease, then divide by $100$ : a $15%$ rise is $1.15$ , a $15%$ fall is $0.85$ , and a plain 'find $35%$ of' is $0.35$ . $0.85$
Percentage change
How big a change is, measured against what you started with. Work out the actual change, divide by the original amount, then multiply by $100$ . The original is always the denominator, whether the quantity went up or down. $\frac{change}{original}$
Reverse percentage
The question gives you the amount after a change and asks for the amount before it. Because going forwards was a multiplication, coming back is a division by that same multiplier. Signals to watch for: 'this is $60%$ of the original price', 'after a $20%$ increase', 'the sale price is'. $\div 1.15$
Compound interest
Interest paid on the interest already earned, so each year's growth is worked out on the new, larger balance rather than on the original deposit. That makes it a multiplier raised to a power. Simple interest — the same amount added every year — is a different and smaller thing, and using it here is a standard way to lose every mark. $P (1 + r)^{n}$
Depreciation
Losing a fixed percentage of value each year, as a car or a machine does. It is compound interest with a multiplier below $1$ , so it never reaches zero — and it is not the same as losing the same fixed amount each year. $P (1 - r)^{n}$

The method

How to work out a percentages

Five steps, in this order. The worked examples below point back to these numbers.

Decide which quantity is the whole — the one that counts as $100%$ .Everything else depends on this. In 'a shirt was £60, now £45, find the percentage decrease' the whole is the original £60. In 'the sale price is £54, which is $60%$ of the original' the whole is the thing you are being asked for, which is what makes it a reverse question. Underline it in the question before you touch the calculator.
Turn the percentage into a decimal multiplier.Divide by $100$ , and adjust for direction: an increase of $3%$ gives $1.03$ , a decrease of $15%$ gives $0.85$ , and a bare 'find $35%$ of' gives $0.35$ . The commonest slip on this subtopic is multiplying by the rate instead of the multiplier — $0.15$ gives you the discount, not the sale price, and $0.04$ gives you one year's interest, not the balance.
Apply the multiplier the right way round: multiply forwards, divide backwards, and raise it to a power to repeat it.Forwards is $60 \times 0.85 = 51$ . Backwards is $54 \div 0.6 = 90$ — dividing by $0.6$ , never by $60$ , and never multiplying. Repeated is $1200 \times 1.0 3^{4} = 1350.61$ , where the power is the number of years, not one less and not one more. If instead the question is asking you to find a percentage, this is the step where you divide the change by the whole and multiply by $100$ .
Convert or round into the exact form the question named.Money to two decimal places means £540.80, not £540.8 and not £540.8016. 'In its simplest form' means $\frac{12}{25}$ , not $\frac{48}{100}$ . 'Complete years' means round down, because a year that has not finished does not count. Do not round anything before this step — rounding each year in turn is a recorded misconception and it shifts the final penny.
Check the direction and the size, then re-read what was actually asked.An increase must end up bigger and a decrease smaller; if a $4%$ rise made the number fall, the multiplier went in the wrong direction. Then check you answered the question in front of you: 'the total interest earned' is $£8736.20 - £8000 = £736.20$ , not $£8736.20$ ; and if $68$ of $400$ people prefer tea, the percentage who prefer coffee is $83%$ , not $17%$ .

Use it when

The question gives you a percentage and a quantity for it to act on, and the change happens in one direction: find $35%$ of $200$ people, reduce £60 by $15%$ , grow £1200 by $3%$ a year for four years. Fix the whole, build the multiplier, apply it.

Do not use it when

Do not apply a multiplier when the question asks you to find a percentage rather than use one. 'A shirt was £60 and is now £45 — work out the percentage decrease' is a division: the change is $£15$ , and $15 \div 60 = 0.25 = 25%$ . The original is the denominator; dividing by the new price gives $33.3%$ and scores nothing. Do not use a single multiplier where the change repeats: a car bought for £16000 losing $15%$ a year is worth $16000 \times 0.8 5^{3} = £9826.00$ after three years, not $16000 - 3 \times 2400 = £8800$ , which is what subtracting the same amount each year would give. And do not assume opposite changes cancel — a price raised $20%$ and then cut $20%$ is $\times 1.2 \times 0.8 = \times 0.96$ , so it ends $4%$ below where it started, not back at the start.

Worked examples

6 IGCSE Maths practice questions on percentages, fully worked

One per objective, so the range is covered rather than the same question repeated. Every step is written out, and the answer is shown — this is study material, not a test.

Example 1

In a survey of 200 people, 35% said they prefer tea over coffee. What is the number of people in the survey who prefer tea? [2 marks]

Understand that 'percentage' means 'number of parts per 100'

Worked solution

Step 1: Recognise that '35%' means 35 out of every 100 parts.
Step 2: Write 35% as the fraction $\frac{35}{100}$ .
Step 3: Multiply the total number of people (200) by the fraction $\frac{35}{100}$ : $200 \times \frac{35}{100}$ .
Step 4: Simplify: $200 \div 100 = 2$ , then $2 \times 35 = 70$ .

Answer70

Example 2

A student scores 15 out of 60 on a test. Express 15 as a percentage of 60. [1 mark]

Express a given number as a percentage of another number

Worked solution

Step 1: Percentages mean 'out of 100', so we need to find what fraction 15 out of 60 is as a number out of 100.
Step 2: First write the fraction: $\frac{15}{60}$ .
Step 3: Simplify the fraction. Both numbers can be divided by 15: $15 \div 15 = 1$ and $60 \div 15 = 4$ , so the simplified fraction is $\frac{1}{4}$ .
Step 4: Convert $\frac{1}{4}$ to a percentage: $\frac{1}{4} = 25$ % because 1 out of 4 equals 25 out of 100.

Answer25%

Example 3

Write down 35% as a fraction in its simplest form. [1 mark]

Express a percentage as a fraction and as a decimal

Worked solution

Step 1: Write 35% as a fraction with denominator 100: $\frac{35}{100}$ .
Step 2: Simplify the fraction by dividing the numerator and denominator by their highest common factor, which is 5.
Step 3: $35 \div 5 = 7$ and $100 \div 5 = 20$ , so the simplified fraction is $\frac{7}{20}$ .

Answer

\frac{7}{20}

Show 3 more worked examples

Example 4

A shop has a sale with 15% off all items. A jacket costs £60. Work out the sale price of the jacket. [2 marks]

Understand the multiplicative nature of percentages as operators

Worked solution

Step 1: First, find 15% of £60. 10% is $60 \div 10 = 6.5$ % is half of 10%, so $6 \div 2 = 3$ .
Step 2: So 15% = 10% + 5% = $6 + 3 = 9$ .
Step 3: The discount is £9. Subtract this from the original price: 60 − $9 = 51$ .
Step 4: The sale price is £51.

Answer£51

Example 5

A shop has a sale. All items are reduced by 15%. A coat originally costs £60. Work out the sale price of the coat. Give your answer in pounds to 2 decimal places with the £ sign, e.g. £51.00. [2 marks]

Solve simple percentage problems, including percentage increase and decrease

Worked solution

Step 1: The original price is £60.
Step 2: The reduction is 15%.
Step 3: Find 15% of £60: 15% as a decimal is 0.15, so $0.15 \times 60 = 9$ .
Step 4: Subtract the reduction from the original price: £60 − £9 = £51.
Step 5: Write the answer with the £ sign and two decimal places: £51.00.

Answer£51.00

Example 6

A shirt costs £54 in a sale. This is 60% of the original price. Work out the original price of the shirt. [2 marks]

Use reverse percentages

Worked solution

Step 1: The shirt costs £54 in the sale, and £54 is 60% of the original price (so the reduction was 40%).
Step 2: To find the original price, divide the sale price by the percentage as a decimal: $54 \div 0.6$ .
Step 3: $54 \div 0.6 = 90$ .
Step 4: So the original price is £90.

Answer£90

Common mistakes

Where marks actually get lost

Learnly tags 20 distinct misconceptions on this subtopic. These are errors that show up in real student work — not generic revision advice.

Examiner traps

Forgetting to simplify fraction
What goes wrong: Students write the percentage as a fraction over 100 but do not simplify to simplest form, e.g., $\frac{35}{100}$ instead of $\frac{7}{20}$ .
What to do instead: Always simplify the fraction by dividing numerator and denominator by their greatest common factor.
Incorrect decimal shift
What goes wrong: Students move the decimal point the wrong way, e.g., writing 35% as 3.5 or 0.035 instead of 0.35.
What to do instead: To convert a percentage to a decimal, divide by 100: move the decimal point two places to the left.
Fraction numerator and denominator swapped
What goes wrong: Students write the percentage as denominator over 100, e.g., $\frac{100}{35}$ instead of $\frac{35}{100}$ .
What to do instead: The percentage is the numerator over 100. For 35%, write $\frac{35}{100}$ , then simplify.
Treating percent as decimal over 1
What goes wrong: Students write the percentage as a fraction with denominator 1, e.g., $\frac{35}{1}$ , misunderstanding the meaning of percent.
What to do instead: Percent means 'out of 100', so always write the number over 100 first, then simplify if needed.
Adding percentage instead of multiplying
What goes wrong: Students add the percentage increase directly to the original amount instead of calculating the multiplier. For example, for a 15% increase, they do $60 + 15 = 75$ .
What to do instead: To increase by 15%, multiply by 1.15, not add 15. For a 15% decrease, multiply by 0.85.
Subtracting percentage from 100% incorrectly
What goes wrong: Students subtract the percentage from 100 but then multiply by the original incorrectly, or they subtract the percentage value from the original without converting to a decimal.
What to do instead: A 15% decrease means you pay 85% of the original. Multiply the original by 0.85, not subtract 15.

See 14 more examiner traps

Misconverting percentage to decimal
What goes wrong: Students incorrectly convert percentages to decimals, e.g., writing 15% as 0.15 but then using it in the wrong operation or forgetting to multiply.
What to do instead: 15% as a decimal is 0.15. For a 15% decrease, multiply by $0.85 (1 - 0.15)$ . For increase, multiply by 1.15.
Using original amount as final answer
What goes wrong: After calculating the percentage change, students mistakenly give the original amount as the answer instead of the new amount.
What to do instead: Read the question carefully: if it asks for the sale price, you must subtract the discount from the original.
Multiplying by percentage rate
What goes wrong: Students multiply the original value by the percentage rate (e.g., 15%) instead of the multiplier (e.g., 0.85).
What to do instead: For a 15% loss, the multiplier is 0.85, not 0.15. Multiply by 0.85 each year, not 0.15.
Assuming linear depreciation
What goes wrong: Students subtract the same amount each year (e.g., 15% of original) instead of applying compound percentage.
What to do instead: Depreciation is compound: each year you lose 15% of the current value, not the original. Use repeated multiplication.
Incorrect order of operations
What goes wrong: Students multiply the original value by the percentage and then subtract, but do not apply the multiplier correctly.
What to do instead: Use the formula: final = original × (1 - rate)^years. For 15% loss, multiply by 0.85 three times.
Rounding intermediate steps
What goes wrong: Students round the value after each year, leading to inaccurate final answer.
What to do instead: Keep full calculator precision until the final step. Only round to 2 decimal places at the end.
Dividing by the decimal multiplier
What goes wrong: Students incorrectly divide the final amount by the percentage (e.g., $600 \div 20$ ) instead of dividing by the decimal multiplier (0.8).
What to do instead: To find the original after a percentage decrease, divide the final amount by (1 - percentage as a decimal). For 20% off, divide by 0.8.
Adding percentage to final amount
What goes wrong: Students add the percentage of the final amount back to the final amount (e.g., $600 + 20$ % of 600) instead of using the correct multiplier.
What to do instead: Don't add the percentage of the new amount. Use the multiplier method: original = final ÷ (1 ± percentage as decimal).
Multiplying instead of dividing
What goes wrong: Students multiply the final amount by the decimal multiplier (e.g., $600 \times 0.8$ ) instead of dividing, getting a smaller number.
What to do instead: For a decrease, the original is larger than the final. So you must divide by the multiplier, not multiply.
Using wrong multiplier for increase
What goes wrong: When the change is a decrease, students use the multiplier for an increase (e.g., 1.2 instead of 0.8).
What to do instead: For a percentage decrease, subtract the percentage from 100% then convert to decimal. E.g., 20% off means multiplier = 0.8.
Forgetting to round to 2 dp
What goes wrong: Students give the answer as a whole number or with too many decimal places, ignoring the instruction to round to 2 decimal places.
What to do instead: Always check the question for rounding instructions. Here, give your answer in pounds to 2 decimal places, e.g., £750.00.
Adding percentage each year
What goes wrong: Students incorrectly add the percentage each year (e.g., 4% of £5000 = £200, then multiply by 3 years) instead of using compound interest.
What to do instead: Don't just add the same amount each year. Each year the interest is calculated on the new total, so the amount grows faster.
Using wrong multiplier
What goes wrong: Students use 1.04 for increase but then multiply by 0.04 or forget to add 1, e.g., using 0.04 instead of 1.04.
What to do instead: For a 4% increase, multiply by 1.04 (100% + 4%). Never multiply by 0.04 alone.
Off-by-one year error
What goes wrong: Students raise the multiplier to the power of n-1 or n+1 instead of n, e.g., using 2 years for 3 years of growth.
What to do instead: For 3 years, raise the multiplier to the power of 3. The exponent equals the number of years.

Exam technique

How the marks are actually awarded

Method marks are given for working a marker can follow. This is what to put on the page.

Where the marks are: Questions in this section run from $1$ to $4$ marks. One-mark questions are conversions — 'write $48%$ as a fraction in its simplest form' — and carry no method mark, so the answer has to be right. Two- and three-mark questions give a method mark for the multiplier or the division and an accuracy mark for the final figure, so write $1200 \times 1.0 3^{4}$ on the page before you reach for the calculator: that expression earns the method mark even if the arithmetic afterwards slips. Four-mark 'after how many complete years' questions expect the year-by-year working to be visible, not just the final count. 'Show that' questions award nothing for restating the printed answer — the marks live entirely in the working that reaches it, so every intermediate line has to be written down.
Command words to expect: Work outCalculateFindWrite downExpress as a percentage ofShow thatGive your answer in pounds to 2 decimal placesGive your answer as a fraction in its simplest formYou must show all your working
Accuracy and rounding: Use standard money notation with two decimal places when presenting currency: £540.80 rather than £540.8. Where a question names a different accuracy — 'to the nearest penny', 'to the nearest milligram', 'as an integer' — that instruction overrides the default, and 'after how many complete years' means count only completed years. Keep full accuracy inside the calculation and round only at the very end; rounding the balance at each year in turn is a recorded misconception here and it moves the final answer. A fraction asked for 'in its simplest form' must be fully cancelled, and a percentage answer should carry the $%$ sign.
Calculator: A calculator is allowed in every paper of this qualification, on both tiers, which makes the power key the single most useful button on this subtopic: $16000 \times 0.8 5^{3}$ is one keystroke sequence rather than three multiplications, and it removes the temptation to round between years. Use the answer key to carry the unrounded value forward. What the calculator will not do is decide which quantity is the whole or which direction the change goes, so settle both before you type.

Check yourself

You should now be able to:

Explain that a percentage is a number of parts per hundred, and find a percentage of a quantity — $35%$ of $200$ people is $70$ .
Convert a percentage to a fraction in its simplest form and to a decimal, in both directions — $37.5% = \frac{3}{8} = 0.375$ .
Express one number as a percentage of another, including when the two numbers are not out of $100$ — $28$ out of $80$ is $35%$ .
Build the multiplier for an increase or a decrease, and say why a $15%$ cut uses $0.85$ rather than $0.15$ .
Apply a percentage increase or decrease in one multiplication, and give a money answer to two decimal places.
Work out a percentage change by dividing the change by the original amount, and recognise that the original is always the denominator.
Solve a reverse percentage by dividing by the multiplier, and identify from the wording that the amount given is the one after the change.
Work out compound interest and depreciation as a multiplier raised to the power of the number of years, and distinguish both from simple interest.
Handle repeated percentage change, including finding how many complete years pass before a total first exceeds a given value.
Explain why a percentage increase followed by an equal percentage decrease does not return to the starting value.

Specification coverage

The 9 things 4MA1 asks you to do

The assessable objectives for numbers and the number system — percentages, in the order the specification lists them.

Understand that 'percentage' means 'number of parts per 100'
Express a given number as a percentage of another number
Express a percentage as a fraction and as a decimal
Understand the multiplicative nature of percentages as operators
Solve simple percentage problems, including percentage increase and decrease
Use reverse percentages

Show 3 more objectives

Use compound interest and depreciation
Use repeated percentage change
Solve compound interest problems

See who is stuck on percentages before you teach it.

Create a class, share the 8-character code, and set percentages as practice. Your students work through all 49 approved practice questions — not just the 6 on this page — and you see which objectives the class has not got yet. Free for individual students and teachers.

How Learnly works for teachers

Set up your class — free Practise on your own

What you need to be able to do already

Convert between a fraction, a decimal and a percentage, and simplify the fractionFractions

Multiply and divide by a decimal, and round money to two decimal placesDecimals

Raise a decimal to a power using the power key on a calculatorPowers and roots

What is percentages?

The notation, and how to read it

The words a question will use

Multiplier

Percentage change

Reverse percentage

Compound interest

Depreciation

How to work out a percentages

6 IGCSE Maths practice questions on percentages, fully worked

In a survey of 200 people, 35% said they prefer tea over coffee. What is the number of people in the survey who prefer tea? [2 marks]

A student scores 15 out of 60 on a test. Express 15 as a percentage of 60. [1 mark]

Write down 35% as a fraction in its simplest form. [1 mark]

A shop has a sale with 15% off all items. A jacket costs £60. Work out the sale price of the jacket. [2 marks]

A shop has a sale. All items are reduced by 15%. A coat originally costs £60. Work out the sale price of the coat. Give your answer in pounds to 2 decimal places with the £ sign, e.g. £51.00. [2 marks]

A shirt costs £54 in a sale. This is 60% of the original price. Work out the original price of the shirt. [2 marks]

Where marks actually get lost

Forgetting to simplify fraction

Incorrect decimal shift

Fraction numerator and denominator swapped

Treating percent as decimal over 1

Adding percentage instead of multiplying

Subtracting percentage from 100% incorrectly

Misconverting percentage to decimal

Using original amount as final answer

Multiplying by percentage rate

Assuming linear depreciation

Incorrect order of operations

Rounding intermediate steps

Dividing by the decimal multiplier

Adding percentage to final amount

Multiplying instead of dividing

Using wrong multiplier for increase

Forgetting to round to 2 dp

Adding percentage each year

Using wrong multiplier

Off-by-one year error

How the marks are actually awarded

You should now be able to:

The 9 things 4MA1 asks you to do

See who is stuck on percentages before you teach it.