Reverse percentages and compound interest, with multipliers
Learn how to find original prices and calculate compound interest using the price-tag machine.
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Download the PDF (620 KB)A jumper costs £36 in a 20% off sale. Why was it NOT £43.20 before the sale?
Reverse percentages and compound interest can feel tricky because you're working backwards. Think of each percentage change as a price-tag machine. To find the original price, run the machine backwards.
The picture to keep in your head: The price-tag machine
Every percentage change is a machine you feed a price into. A 20% increase is the 'times 1.2' machine; a 20% decrease is the 'times 0.8' machine, because you keep 80%. Reverse percentages run the machine backwards: divide by the multiplier. Compound interest is the same machine run once every year, so 3 years at 5% is times 1.05 three times, or times 1.053.
The method, every time
- Use the price-tag machine to find the new price after a percentage change.
- To reverse a percentage change, divide by the multiplier used in the machine.
- For compound interest, apply the multiplier each year and raise it to the power of years.
- Check your answer by reversing the steps or using the original value.
- Always round your final answer to two decimal places for money.
Words you'll see
- multiplier
- A number that shows the effect of a percentage change, like 1.2 for 20% increase.
- reverse
- Working backwards to find the original value before a percentage change.
- compound
- Interest that grows on both the original amount and any previous interest.
- original
- The starting value before any percentage change is applied.
- round
- Make sure to round money to two decimal places at the end of calculations.
Watch it explained
- The percentage flip trick (one-minute video)
One number does the whole change
The price-tag machine does all the work. Whether you increase or decrease a price, the machine multiplies it by a special number. For a 15% decrease, the machine is times 0.85. This method is quick and works for any percentage change. Just remember: increase by r% is multiply by 1 + r/100, and decrease by r% is multiply by 1 - r/100.
Watch out
- Using 0.15 instead of 0.85 for a decrease. Remember: decrease by r% is multiply by 1 - r/100
- Writing 1.5 for a 5% increase instead of 1.05. Increase by r% is multiply by 1 + r/100
Change over original
The price-tag machine shows how much the price has changed. To find the percentage change, you compare the change to the original price. Always divide by the original amount. Percentage change = change ÷ original × 100.
Watch out
- Dividing by the new amount. Always divide by the original amount, not the new one
- Forgetting to multiply by 100. Multiply by 100 to convert the fraction to a percentage
Run the machine backwards
The price-tag machine can run backwards. If you know the final price after a percentage change, you can reverse the machine by dividing by the multiplier. This gives you the original price before the change. Never take the percentage of the new amount and add or subtract it.
Watch out
- Adding 20% of £36. Don't add the percentage to the new amount
- Dividing by 0.2. Use the multiplier for the change, not its percentage
- Taking the percentage of the new amount. Always use the original amount for percentage changes
The same machine, once a year
The price-tag machine runs once each year for compound interest. You multiply the original amount by the yearly multiplier, repeated for each year. The formula is amount = original × multiplieryears. This grows faster than simple interest because you earn interest on your interest.
Watch out
- Adding 3% of £2000 four times. Use the multiplier, not simple interest
- Rounding each year instead of at the end. Round only at the end to avoid errors
22 exam-style reverse percentages and compound interest questions
Show your working: most marks are for the method. Marks are a guide to how much working an exam would expect. Questions written by Bloom Education in the style of GCSE (AQA, Edexcel, OCR) and Cambridge IGCSE papers.
A · Multipliers
- 1Write the multiplier for a 12% increase.[2 marks]
Show the answer
1.12
Keep 100% and add 12%: 1 + 0.12 = 1.12. - 2Write the multiplier for a 35% decrease.[2 marks]
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0.65
Keep 100% - 35% = 65%: 1 - 0.35 = 0.65. - 3Write the multiplier for a 2.5% increase.[2 marks]
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1.025
1 + 0.025 = 1.025. - 4Increase £40 by 150%.[2 marks]
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£100
Multiplier 1 + 1.5 = 2.5; 40 × 2.5 = 100. - 5Decrease £80 by 2.5%.[2 marks]
Show the answer
£78
Multiplier 1 - 0.025 = 0.975; 80 × 0.975 = 78.
B · Percentage change
- 6A bike's price goes up from £250 to £290. Find the percentage increase.[3 marks]
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16%
Change 40; 40 ÷ 250 × 100 = 16. - 7A car worth £12 000 is now worth £9600. Find the percentage decrease.[3 marks]
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20%
Change 2400; 2400 ÷ 12 000 × 100 = 20. - 8A shop buys a lamp for £40 and sells it for £58. Find the percentage profit.[3 marks]
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45%
Profit 18; 18 ÷ 40 × 100 = 45. - 9Sam's time for a run falls from 25 minutes to 22 minutes. Find the percentage decrease.[3 marks]
Show the answer
12%
Change 3; 3 ÷ 25 × 100 = 12.
C · Reverse
- 10A jacket costs £240 after a 20% discount. What was the original price?[3 marks]
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£300
Multiplier 0.8; 240 ÷ 0.8 = 300. - 11A laptop is sold for £630, making a 10% profit. What did it cost the shop?[3 marks]
Show the answer
£572.73
Multiplier 1.1; 630 ÷ 1.1 = 572.727… - 12A salary is increased by 4% to £3120. What was the original salary?[3 marks]
Show the answer
£3000
Multiplier 1.04; 3120 ÷ 1.04 = 3000. - 13A TV costs £450 including 20% VAT. What was the price before VAT?[3 marks]
Show the answer
£375
Multiplier 1.2; 450 ÷ 1.2 = 375. - 14A car is sold for £14 520 after a 12% increase. What was the original price?[3 marks]
Show the answer
£12 964.29
Multiplier 1.12; 14 520 ÷ 1.12 = 12 964.285…
D · Compound
- 15£5000 is invested at 2% compound interest per year. Find its value after 3 years.[3 marks]
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£5306.04
5000 × 1.02³ = 5306.04. - 16A car worth £18 000 loses 15% of its value each year. Find its value after 2 years.[3 marks]
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£13 005
18 000 × 0.85² = 13 005. - 17A town of 24 000 people grows by 3% a year. Estimate its population after 5 years, to the nearest whole number.[3 marks]
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27 823
24 000 × 1.03⁵ = 27 822.56… - 18A population of 10 000 grows by 5% each year. After how many years will it first be more than 13 000?[3 marks]
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6 years
1.05⁵ gives 12 763, 1.05⁶ gives 13 401. - 19A population of 8000 grows by 4% each year. After how many years will it first be more than 10 000?[3 marks]
Show the answer
6 years
8000 × 1.04⁵ = 9733, 8000 × 1.04⁶ = 10 123.
E · Mixed (Higher)
- 20A price goes up by 10%, then down by 10%. Is it back where it started? Explain.[4 marks]
Show the answer
No, it is 1% lower
× 1.1 then × 0.9 is × 0.99: 99% of the start. - 21£1000 is invested for 3 years at 5% per year. How much more does compound interest earn than simple interest?[4 marks]
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£7.63
Compound: 1000 × 1.05³ = 1157.63. Simple: 1000 + 3 × 50 = 1150. - 22After 2 years of 4% compound interest, an account holds £5408. How much was invested?[4 marks]
Show the answer
£5000
Run the machine backwards twice: 5408 ÷ 1.04² = 5000.
Reverse percentages and compound interest exam checklist
- Increase by r%: multiply by 1 + r/100. Decrease: multiply by 1 - r/100.
- Percentage change = change ÷ original × 100.
- Reverse percentage: divide by the multiplier, never take the percentage of the new amount.
- Compound interest: original × multiplieryears.
- Keep full values in your calculator; round money to 2 decimal places at the end.
- Check by running the machine forwards again.
Got stuck on a question?
Find the part of this guide it belongs to and go through the worked example again, line by line. Then put your answer back into the question: if it doesn't work, look for a sign or a step you skipped.
All of this page as a 10-page guide, with every answer at the back. No sign-up.
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