GCSE Maths: Percentage problems: choosing the method and comparing (Foundation)
Part of the GCSE Maths percentages chapter (Foundation). Chapter overview · Higher version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Percentage problems: choosing the method and comparing
Exam questions rarely tell you which percentage method to use. Your job is to read the question and decide. The table below is a checklist: look for the key words and pick the matching method.
| The question says… | What to do |
|---|---|
| “Find 35% of £240” | Percentage of an amount: 0.35 × 240, or build up from 10% and 1% |
| “Increase by 8%”, “reduced by 30%”, “add VAT” | Multiplier: 1.08, 0.7, 1.2 — new amount = original × multiplier |
| “What percentage is 18 of 25?” | Part ÷ whole × 100 |
| “Work out the percentage increase/decrease/profit/loss” | Change ÷ original × 100 |
| “After a 20% rise the price is £84. Find the original” | Reverse percentage: divide by the multiplier (1.2) |
| “Simple interest” | Same amount each year: principal × rate × years ÷ 100 |
| “Compound interest”, “grows by 4% each year” | Multiplier to the power n |
One change after another
Suppose a price of £200 goes up by 10% and then the new price goes down by 10%. Does it come back to £200? No. The 10% rise adds £20, giving £220. The 10% fall is 10% of £220, which is £22, giving £198. The two percentages are of different amounts. In multiplier form: 1.1 × 0.9 = 0.99, so the overall change is a 1% decrease. When there is one change after another, multiply the multipliers; never add or subtract the percentages.
Comparing offers
To compare two offers, work out the final price (or the saving) for each, using the same starting amount, and then compare. If no starting price is given, use 100 or an easy number: percentages are the same for any starting amount. A fraction and a percentage can be compared too: one third off (33.3%) is a bigger discount than 30% off.
A percentage of a percentage, and fractions with percentages
Take it one step at a time and keep track of what the 100% is each time. If 45% of a school is in Key Stage 3, and one third of those are in Year 7, then Year 7 is one third of 45% = 15% of the whole school.
Key ideas
- Choose the method from the key words (see the table).
- For several changes in a row, multiply the multipliers. Do not add the percentages.
- A rise of p% followed by a fall of p% is always a (small) loss.
- To compare offers, use the same starting amount (use 100 if none is given).
- Keep track of what the 100% is at each step.
- Always read the last line of the question: it tells you what to give as your final answer.
Method. Find the overall multiplier by multiplying the separate multipliers, then use it on the starting amount.
- Write a multiplier for each change. Each percentage change has its own multiplier (100% plus or minus the change, as a decimal).increase of 10%: 110% = 1.1
decrease of 10%: 90% = 0.9 - Multiply the multipliers together. One change is applied after the other, so the multipliers are multiplied.1.1 × 0.9 = 0.99
- Multiply the starting amount by the overall multiplier. This gives the amount after both changes.£200 × 0.99 = £198
Check. Apply the changes one at a time and you reach the same amount. ✓
Watch out. Do not add the percentages. A 10% rise then a 10% rise is a 21% rise, not 20%.
Method. Find the overall multiplier by multiplying the separate multipliers, then turn it back into a percentage change.
- Write a multiplier for each change. Each percentage change has its own multiplier (100% plus or minus the change, as a decimal).increase of 20%: 120% = 1.2
decrease of 20%: 80% = 0.8 - Multiply the multipliers together. One change is applied after the other, so the multipliers are multiplied.1.2 × 0.8 = 0.96
- Turn the overall multiplier into a percentage change. Subtract 1 (the original, 100%) and multiply by 100.(0.96 − 1) × 100 = −4%
- Say it in words. an overall decrease of 4%
Check. Try it with 100: apply each change in turn to 100 and compare the result with 100 times the overall multiplier. ✓
Watch out. Do not add or subtract the percentages. They are applied to different amounts.
Method. Work out the price in each shop and compare.
- Shop A: find 15% of £400. 10% of 400 = 40 and 5% = 20, so 15% = 40 + 20.£40 + £20 = £60
- Shop A: subtract. The reduction is taken off the price.£400 − £60 = £340
- Shop B: subtract £50. £400 − £50 = £350
- Compare. £340 is less than £350.Shop A is cheaper, by £10.
Check. 0.85 × 400 = 340. ✓
Watch out. A bigger-looking discount is not always the bigger saving: work out the actual prices.
Method. Find the amount in the tank, then find what is used, then subtract.
- Find the water in the tank. 38 of 800: divide by 8, then multiply by 3.800 ÷ 8 = 100, 100 × 3 = 300 litres
- Find 40% of the water. 10% of 300 = 30, so 40% = 4 lots of 30.4 × 30 = 120 litres used
- Subtract. The water left is what was there take away what was used.300 − 120 = 180 litres
Check. 60% of the water is left: 0.6 × 300 = 180. ✓
Watch out. The 40% is of the 300 litres in the tank, not of the full 800.
Method. Find the number in Year 7, then write it as a percentage of the whole school.
- Find the Key Stage 3 students. 10% of 600 = 60, so 40% = 240 and 5% = 30.240 + 30 = 270
- Find the Year 7 students. One third of 270.270 ÷ 3 = 90
- Write 90 as a percentage of 600. Part over whole, times 100.90600 = 15100 = 15%
Check. One third of 45% is 45% ÷ 3 = 15%. ✓
Watch out. Compare with the whole school (600), not with Key Stage 3 (270).
Watch out
Percentages do not simply add or cancel. “Up 20% then down 20%” is not back to the start. And never apply the second percentage to the original amount: it applies to the new amount after the first change.Exercise 5O Mixed percentage problems (non-calculator)
Non-calculatorMethod. Look at the key words in the question and match them with a method.
- Underline the key words. They tell you what is known and what you must find.A bike costs £250. It is reduced by 12%. Find the sale price.
- Choose the method. Key words “reduced by 12%” mean a percentage decrease of an amount. Use the multiplier 0.88 (or find 12% and subtract).Increase or decrease: multiplier 0.88
Check. Check that the method matches what the question asks you to find. ✓
Method. Look at the key words in the question and match them with a method.
- Underline the key words. They tell you what is known and what you must find.A rent rises from £40 to £46. Find the percentage rise.
- Choose the method. Key words “percentage rise” with an old and a new value mean percentage change: change ÷ original × 100.Percentage change: 6 ÷ 40 × 100
Check. Check that the method matches what the question asks you to find. ✓
Method. Look at the key words in the question and match them with a method.
- Underline the key words. They tell you what is known and what you must find.After a 20% rise the rent is £600. Find the old rent.
- Choose the method. The rent given is the new amount after the rise, so we work backwards: divide by the multiplier 1.2.Reverse percentage: 600 ÷ 1.2
Check. Check that the method matches what the question asks you to find. ✓
Method. Look at the key words in the question and match them with a method.
- Underline the key words. They tell you what is known and what you must find.There are 120 pupils. 35% of them play football. How many play?
- Choose the method. We want a part of the 120 pupils, so find a percentage of an amount: 0.35 × 120.Percentage of an amount: 35% of 120
Check. Check that the method matches what the question asks you to find. ✓
Method. Find the overall multiplier by multiplying the separate multipliers, then use it on the starting amount.
- Write a multiplier for each change. Each percentage change has its own multiplier (100% plus or minus the change, as a decimal).increase of 10%: 110% = 1.1
decrease of 10%: 90% = 0.9 - Multiply the multipliers together. One change is applied after the other, so the multipliers are multiplied.1.1 × 0.9 = 0.99
- Multiply the starting amount by the overall multiplier. This gives the amount after both changes.£200 × 0.99 = £198
Check. Apply the changes one at a time and you reach the same amount. ✓
Watch out. Do not add the percentages. A 10% rise then a 10% rise is a 21% rise, not 20%.
Method. Find the overall multiplier by multiplying the separate multipliers, then use it on the starting amount.
- Write a multiplier for each change. Each percentage change has its own multiplier (100% plus or minus the change, as a decimal).decrease of 20%: 80% = 0.8
increase of 20%: 120% = 1.2 - Multiply the multipliers together. One change is applied after the other, so the multipliers are multiplied.0.8 × 1.2 = 0.96
- Multiply the starting amount by the overall multiplier. This gives the amount after both changes.£500 × 0.96 = £480
Check. Apply the changes one at a time and you reach the same amount. ✓
Watch out. Do not add the percentages. A 10% rise then a 10% rise is a 21% rise, not 20%.
Method. Find the overall multiplier by multiplying the separate multipliers, then use it on the starting amount.
- Write a multiplier for each change. Each percentage change has its own multiplier (100% plus or minus the change, as a decimal).increase of 50%: 150% = 1.5
decrease of 50%: 50% = 0.5 - Multiply the multipliers together. One change is applied after the other, so the multipliers are multiplied.1.5 × 0.5 = 0.75
- Multiply the starting amount by the overall multiplier. This gives the amount after both changes.£80 × 0.75 = £60
Check. Apply the changes one at a time and you reach the same amount. ✓
Watch out. Do not add the percentages. A 10% rise then a 10% rise is a 21% rise, not 20%.
Method. Find the overall multiplier by multiplying the separate multipliers, then use it on the starting amount.
- Write a multiplier for each change. Each percentage change has its own multiplier (100% plus or minus the change, as a decimal).increase of 10%: 110% = 1.1
increase of 10%: 110% = 1.1 - Multiply the multipliers together. One change is applied after the other, so the multipliers are multiplied.1.1 × 1.1 = 1.21
- Multiply the starting amount by the overall multiplier. This gives the amount after both changes.£1,000 × 1.21 = £1,210
Check. Apply the changes one at a time and you reach the same amount. ✓
Watch out. Do not add the percentages. A 10% rise then a 10% rise is a 21% rise, not 20%.
Method. Work out the price in each shop.
- Shop A: find 12%. 10% of 600 = 60 and 2% = 2 lots of 1% = 2 × 6 = 12, so 12% = 72.£60 + £12 = £72
- Shop A: price. Subtract the reduction.£600 − £72 = £528
- Shop B: price. Subtract £70.£600 − £70 = £530
- Compare. 528 is less than 530.Shop A is cheaper by £530 − £528 = £2.
Check. 0.88 × 600 = 528. ✓
Watch out. Give both the shop and the difference, since the question asks for both.
Method. Work out the saving in each shop and compare.
- Shop A: 25% is a quarter. Divide by 4.£120 ÷ 4 = £30
- Shop B: one third. Divide by 3.£120 ÷ 3 = £40
- Compare. 40 is more than 30.Shop B saves £40 − £30 = £10 more.
Check. One third is 33.3%, which is bigger than 25%. ✓
Watch out. Convert a fraction to a percentage (or the other way round) to compare.
Method. Find the water in the tank, then the water used, then subtract.
- Find the water in the tank. 25 of 600: divide by 5, then multiply by 2.600 ÷ 5 = 120, 120 × 2 = 240 litres
- Find 30% of the water. 10% of 240 = 24, so 30% = 3 lots of 24.3 × 24 = 72 litres used
- Subtract. The water left is what was there take away what was used.240 − 72 = 168 litres
Check. 70% of the water is left: 0.7 × 240 = 168. ✓
Watch out. The 30% is of the water in the tank (240), not of 600.
Method. Find the number of boys, then the number who wear glasses.
- Find the number of boys. 10% of 30 = 3, so 40% = 4 lots of 3.4 × 3 = 12 boys
- Find 25% of the boys. 25% is a quarter.12 ÷ 4 = 3
Check. 0.25 × 12 = 3. ✓
Watch out. The 25% is of the boys, not of the whole class.
Method. Write 3 as a percentage of 30.
- Write 3 out of 30 as a fraction. Part over whole.330
- Simplify the fraction. Divide the top and the bottom by 3.330 = 110
- Make the denominator 100. Multiply the top and the bottom by 10.110 = 10100 = 10%
Check. 10% of 30 = 3. ✓
Method. Take off the two 10% reductions one at a time.
- First reduction. 10% of £20 is £2.£20 − £2 = £18
- Second reduction. This time 10% of the new price £18 is £1.80.£18 − £1.80 = £16.20
- Compare with Dan’s answer. £16.20 is not £16.Dan is wrong. The second 10% was of a smaller amount.
Check. 0.9 × 0.9 = 0.81, and 0.81 × 20 = 16.20. ✓
Watch out. Two 10% reductions are not a 20% reduction.
Method. Find the overall multiplier by multiplying the separate multipliers, then use it on the starting amount.
- Write a multiplier for each change. Each percentage change has its own multiplier (100% plus or minus the change, as a decimal).decrease of 20%: 80% = 0.8
decrease of 10%: 90% = 0.9 - Multiply the multipliers together. One change is applied after the other, so the multipliers are multiplied.0.8 × 0.9 = 0.72
- Multiply the starting amount by the overall multiplier. This gives the amount after both changes.£90 × 0.72 = £64.80
Check. Apply the changes one at a time and you reach the same amount. ✓
Watch out. Do not add the percentages. A 10% rise then a 10% rise is a 21% rise, not 20%.
Method. Work out each new salary, then find the difference.
- Anna: find 4% of £30,000. 1% = 300, so 4% = 4 lots.4 × £300 = £1,200
- Anna: new salary. Add the rise.£30,000 + £1,200 = £31,200
- Ben: find 8% of £28,000. 1% = 280, so 8% = 8 lots.8 × £280 = £2,240
- Ben: new salary. Add the rise.£28,000 + £2,240 = £30,240
- Compare. 31,200 is more than 30,240.£31,200 − £30,240 = £960: Anna earns £960 more.
Check. 1.04 × 30,000 = 31,200 and 1.08 × 28,000 = 30,240. ✓
Watch out. A bigger percentage rise does not always give the bigger salary.
Method. Find the percentage that is on number, then find that percentage of 40.
- Add the percentages for algebra and geometry. The topics do not overlap.30% + 25% = 55%
- Find the percentage on number. The whole test is 100%.100% − 55% = 45%
- Find 45% of 40. 10% of 40 = 4, so 40% = 16 and 5% = 2.16 + 2 = 18
Check. 30% of 40 = 12 and 25% of 40 = 10; 40 − 12 − 10 = 18. ✓
Watch out. Make sure your answer is for number, not for algebra and geometry.
Method. Find the overall multiplier by multiplying the separate multipliers, then turn it back into a percentage change.
- Write a multiplier for each change. Each percentage change has its own multiplier (100% plus or minus the change, as a decimal).increase of 20%: 120% = 1.2
decrease of 20%: 80% = 0.8 - Multiply the multipliers together. One change is applied after the other, so the multipliers are multiplied.1.2 × 0.8 = 0.96
- Turn the overall multiplier into a percentage change. Subtract 1 (the original, 100%) and multiply by 100.(0.96 − 1) × 100 = −4%
- Say it in words. an overall decrease of 4%
Check. Try it with 100: apply each change in turn to 100 and compare the result with 100 times the overall multiplier. ✓
Watch out. Do not add or subtract the percentages. They are applied to different amounts.
Method. Give C a simple value, such as 100, and work out B and then A.
- Let C = 100. Percentages are the same for any starting value, and 100 is easy.C = 100
- Find B. B is 20% less than C: 100 − 20.B = 100 − 20 = 80
- Find A. A is 20% more than B: 20% of 80 = 16, then add.A = 80 + 16 = 96
- Compare A with C. 96 is 4 less than 100.A is 4% less than C.
Check. Multipliers: 0.8 × 1.2 = 0.96. ✓
Watch out. A is 20% more than B, but B is smaller than C, so the 20% is of a smaller amount.
Exercise 5P Mixed percentage problems (calculator)
Calculator allowedMethod. Write the percentage as a decimal, then multiply it by the amount.
- Change the percentage into a decimal. Per cent means “out of 100”, so divide by 100.15% = 15 ÷ 100 = 0.15
- Write the multiplication. To find a percentage of an amount, multiply the amount by the percentage written as a decimal (“of” means multiply).15% of £1,250 = 0.15 × £1,250
- Work it out on the calculator. Type the multiplication in carefully.0.15 × £1,250 = £187.50
Check. Estimate: 10% of 1,250 is 125, so 15% is 1.5 lots of that: about 187.50. ✓
Watch out. Divide the percentage by 100 first. 8% is 0.08, not 0.8.
Method. Subtract the deposit from the price, then divide by 5.
- Find the amount left to pay. Total take away deposit.£1,250 − £187.50 = £1,062.50
- Divide by 5. The payments are equal.£1,062.50 ÷ 5 = £212.50
Check. 5 × £212.50 = £1,062.50, and £1,062.50 + £187.50 = £1,250. ✓
Watch out. Do not forget the deposit: it is paid in addition to the 5 payments.
Method. Use two multipliers: 1.2 for the VAT, then 0.95 for the discount.
- Add the VAT. 20% increase: multiplier 1.2.£480 × 1.2 = £576
- Take off the discount. 5% decrease: multiplier 0.95, applied to the price including VAT.£576 × 0.95 = £547.20
Check. Overall multiplier: 1.2 × 0.95 = 1.14, and £480 × 1.14 = £547.20. ✓
Watch out. The 5% is of £576, not of £480.
Method. Work out the total in each account and compare.
- Account A: interest for one year. 3% of £4,000.0.03 × £4,000 = £120
- Account A: total after 5 years. Simple interest is the same each year: 5 × £120, added to the principal.£4,000 + 5 × £120 = £4,600
- Account B: total after 5 years. Multiplier 1.025 to the power 5.£4,000 × 1.0255 = £4,000 × 1.131408… = £4,525.63
- Compare. Account A has more.£4,600 − £4,525.63 = £74.37
Check. Account B interest is only £525.63, less than the £600 from Account A. ✓
Watch out. Compound interest is not always better: the rates are different here.
Method. Find the number who walk and the number who drive, then subtract from the total.
- Number who walk. 35% as a decimal is 0.35.0.35 × 360 = 126
- Number who drive. 38 of 360: divide by 8, multiply by 3.360 ÷ 8 = 45, 45 × 3 = 135
- Number who cycle. Everyone else cycles.360 − 126 − 135 = 99
Check. 38 = 37.5%, so 35% + 37.5% = 72.5% and 27.5% cycle: 0.275 × 360 = 99. ✓
Watch out. Change everything into a number of people before subtracting.
Method. To increase an amount by a percentage, find the multiplier and multiply. The multiplier for a 36% increase is 1.36.
- Find the percentage the new amount will be. An increase of 36% means we add 36% of the original. The original is 100%, so the new amount is 100% + 36%.100% + 36% = 136% of the original
- Write that percentage as a decimal: this is the multiplier. Divide by 100.136% = 136 ÷ 100 = 1.36
- Multiply the original amount by the multiplier. One multiplication gives the new amount straight away.£25 × 1.36 = £34
Check. Other way: 36% of £25 = 9; £25 + 9 = 34. ✓
Watch out. Do not multiply by 0.36 only: that finds the 36% but not the new amount. The multiplier includes the original 100%.
Method. Find the original price first, then apply the other reduction.
- Find the original price. After a 15% decrease the price is 85% of the original: divide by 0.85.£51 ÷ 0.85 = £60
- Apply a 25% reduction to the original. 25% off: multiplier 0.75.£60 × 0.75 = £45
Check. 0.85 × 60 = 51. ✓
Watch out. The second reduction applies to the original price, not to £51.
Method. Split the income into the two tax bands and find the tax on each.
- Tax on the first £37,700. 20% = 0.2.0.2 × £37,700 = £7,540
- Find the income in the higher band. Everything above £37,700.£50,000 − £37,700 = £12,300
- Tax on the higher band. 40% = 0.4.0.4 × £12,300 = £4,920
- Add the two amounts of tax. £7,540 + £4,920 = £12,460
Check. If all of £50,000 were taxed at 20% the tax would be £10,000; the extra 20% on £12,300 is £2,460; £10,000 + £2,460 = £12,460. ✓
Watch out. Each rate applies only to its own band of income.
Method. Multiply the two multipliers and turn the result back into a percentage.
- Multiplier for the rise. 112% = 1.12.1.12
- Multiplier for the fall. 88% = 0.88.0.88
- Overall multiplier. Multiply.1.12 × 0.88 = 0.9856
- Change to a percentage. 1 − 0.9856 = 0.0144.0.0144 × 100 = 1.44%
- Conclude. Hannah is wrong: it is a 1.44% decrease overall.
Check. Start at 100: up 12% to 112, then down 12% of 112 = 13.44, giving 98.56. ✓
Watch out. The fall is 12% of a bigger number than the rise.
Method. If you know what a percentage of the whole is, find 1% first, then multiply by 100 to find the whole (100%).
- Write down what we know. Part of the whole has been given as a percentage.40% of the whole = 12 litres
- Find 1%. Divide by 40. That takes us from 40% to 1%.12 litres ÷ 40 = 0.3 litres
- Find 100%. The whole is 100%, so multiply 1% by 100.0.3 litres × 100 = 30 litres
Check. 40% of 30 litres = 0.4 × 30 litres = 12 litres. ✓
Watch out. The whole is 100%. Do not stop at 1%.
Method. Subtract the blue paint from the total.
- Write the total and the blue paint. From part (a) the total is 30 litres.30 litres, 12 litres blue
- Subtract. The rest is yellow.30 − 12 = 18 litres
Check. Yellow is 60% of 30 = 18. ✓
Method. Use a starting price of 100 and find the percentage fall needed.
- Increase 100 by 25%. 25% of 100 = 25.100 + 25 = 125
- Find the fall needed to get back to 100. The price must drop by 25 from 125.125 − 100 = 25
- Write the fall as a percentage of the price it falls from (125). The fall starts at the new price, 125.25 ÷ 125 × 100 = 20%
Check. 1.25 × 0.8 = 1, so the overall change is nothing. ✓
Watch out. A fall that undoes a rise is a smaller percentage, because it is a percentage of the bigger number.