Edexcel GCSE Maths Foundation (1MA1), Foundation tier · Ratio, proportion and rates of change
Practise Percentages and reverse percentages. 2 exam-style questions plus unlimited generated ones on this subtopic, at up to four difficulty levels, with full mark schemes and a progress tracker. Free, no account needed.
A percentage is a number of parts per hundred. You need to find percentages of amounts, increase or decrease by a percentage, work out percentage change, profit and simple interest, and work backwards to an original amount (reverse percentages). Expect these on both calculator and non-calculator papers, on both tiers.
Grade by grade
What you need to be able to do, from the first marks up to the top grade.
2
Find percentages of amounts without a calculatorBuild up from 10%, 5% and 1%, e.g. 35% of £60 = £18 + £3 = £21.
3
Convert between fractions, decimals and percentagesFor example, \(\frac{3}{8}\) = 0.375 = 37.5%, and 120% = 1.2.
3
Write one amount as a percentage of anotherDivide the part by the whole and multiply by 100, e.g. 18 out of 24 is 75%.
4
Increase or decrease by a percentageUse a multiplier, e.g. × 1.15 for a 15% increase or × 0.85 for a 15% decrease.
4
Calculate simple interestFind one year's interest on the original amount and multiply by the number of years.
5
Work out percentage change, profit and lossDivide the change by the original amount and multiply by 100.
5
Find the original amount (reverse percentages)Divide the new amount by the multiplier, e.g. £68 after a 15% reduction was 68 ÷ 0.85 = £80.
Notes
Finding percentages
Per cent means 'out of 100': 37% = \(\frac{37}{100}\) = 0.37.
Without a calculator, build up from 10% (÷ 10), 5% (half of 10%) and 1% (÷ 100). 35% of £60 = £18 + £3 = £21.
With a calculator, multiply by the decimal: 17.5% of 240 = 0.175 × 240 = 42.
One quantity as a percentage of another: \(\frac{\text{part}}{\text{whole}} \times 100\), with both in the same units. 18 out of 24 is 75%.
Percentages over 100% are fine: 150% of 40 = 1.5 × 40 = 60.
Multipliers
Increase by r%: multiply by \(1 + \frac{r}{100}\). An 8% rise is × 1.08; adding VAT at 20% is × 1.2.
Decrease by r%: multiply by \(1 - \frac{r}{100}\). An 8% fall is × 0.92; a 35% discount is × 0.65.
Changes one after another: multiply the multipliers. A 10% rise then a 10% fall is × 1.1 × 0.9 = × 0.99, which is a 1% fall overall.
Percentage change, profit and simple interest
Percentage change = \(\frac{\text{change}}{\text{original}} \times 100\). Always divide by the original amount.
Profit or loss uses the cost price as the original: bought for £40 and sold for £52 is a profit of \(\frac{12}{40} \times 100\) = 30%.
Simple interest is paid on the original amount only, so it is the same every year. £600 at 4% for 3 years earns £24 a year, £72 in total.
Reverse percentages
If you know the amount after a change, divide by the multiplier to find the original amount.
After a 20% increase a price is £90, so the original price = 90 ÷ 1.2 = £75.
Do not take 20% of £90 and subtract it: that gives £72, which is wrong.
Without a calculator: £90 is 120% of the original, so 10% = £7.50 and 100% = £75.
Cheatsheet
Per cent = out of 100: 37% = \(\frac{37}{100}\) = 0.37
Write the multiplier for the change, e.g. a 30% decrease gives 0.7.
Divide the new amount by the multiplier (never multiply).
Check by applying the change forwards to your answer.
Example. In a sale, all prices are reduced by 30%. The sale price of a coat is £59.50. Work out the price of the coat before the sale.
Show the model answer
100% − 30% = 70%, so the multiplier is 0.7 (M1) 59.50 ÷ 0.7 (M1) = £85 (A1) Check: 0.7 × 85 = 59.50
Work out simple interest
2 to 3 marks4
Find one year's interest: the rate as a decimal × the amount invested.
Multiply by the number of years.
Add the amount invested only if the question asks for the total in the account.
Example. Ravi invests £2400 for 5 years in an account that pays 3.5% per year simple interest. Work out the total amount of money in the account at the end of the 5 years.
Show the model answer
One year's interest = 0.035 × 2400 = £84 (M1) Interest for 5 years = 5 × 84 = £420 (M1) Total = 2400 + 420 = £2820 (A1)
Shortcuts and memory tricks
Multiplier fast: work out 100 plus or minus the percentage, then divide by 100, e.g. a 12.5% decrease gives 87.5 ÷ 100 = 0.875.
Percentages swap round: 8% of 25 = 25% of 8 = 2. Use whichever is easier.
Check a reverse percentage by applying the change forwards: 0.7 × £85 = £59.50.
Percentage change is 'difference over original'.
Sense check: after a decrease the original amount must be bigger than the new amount.
Where marks are lost
Dividing by the new amount in a percentage change question: always divide by the original.
Reverse percentages: taking the percentage of the new amount and adding it on (or taking it off).
Using the wrong decimal, e.g. 0.5 for 5% (it is 0.05) or 1.5 for a 5% increase (it is 1.05).
Giving the interest or discount when the question asks for the final amount, or the other way round.
Adding percentages for changes one after another: a 10% rise then a 10% fall is not 0% overall.
Writing money with one decimal place: £4.5 should be £4.50.
Exam technique
Underline what the question wants: the change (discount, profit, interest) or the new amount.
On the non-calculator paper, write down each step (10%, 5%, 1%), as each one can earn method marks.
For 'Is she correct?' questions, work it out fully, then give a clear yes or no with the figures that prove it.
Say 'increase' or 'decrease' (or 'profit' or 'loss') with a percentage change when the question asks you to.
Sample questions
Written for this site in the style of Edexcel exam questions. They are not taken from real past papers.
Question 1Easy4 marks
(a) Increase 320 by 40%.[2]
(b) Work out 65% of £80.[2]
Show the answer and mark scheme
(a)Answer: 448
M1 for 40% of 320 = 128 or \(320 \times 1.4\)
A1 for 448 cao
Worked solution: 40% of 320 = 128, so the answer is \(320 + 128 = 448\).
(b)Answer: £52
M1 for a correct method, e.g. 10% = £8 and 60% = 6 × £8 and 5% = £4, or \(0.65 \times 80\)
A1 for £52 cao
Worked solution: 10% = £8, 60% = 6 × £8 = £48, 5% = £4. So 65% of £80 = £48 + £4 = £52.
Question 2Medium3 marks
In a sale, all prices are reduced by 20%. A jacket costs £36 in the sale. Tom says, 'The price before the sale was \(36 \times 1.2\) = £43.20.'
(a) Explain Tom's mistake.[1]
(b) Work out the price of the jacket before the sale.[2]
Show the answer and mark scheme
(a)Answer: The 20% was taken off the original price, not the sale price. £36 is 80% of the original price, so he should divide by 0.8.
C1 for explaining that £36 is 80% of the original price (the 20% is of the original price, not of £36)
Worked solution: Adding 20% of £36 is not the same as undoing a 20% reduction of the original price: 20% of the original price is more than 20% of £36.
(b)Answer: £45
M1 for \(36 \div 0.8\) or \(36 \div 80 \times 100\)
A1 for £45
Worked solution: 80% of the price is £36, so 1% is \(36 \div 80 = 0.45\) and 100% is £45.