Percentages
Percentage change, reverse percentages, and compound interest.
Watch the lesson
Every point below, worked through in one video, and each point links straight to its own part.
Learn
Percentages, explained point by point
Everything the GCSE specification expects you to be able to do, and how to actually do it - the same lesson a signed-in student studies from.
Convert between fractions, decimals and percentages
A percentage is just a fraction out of 100, so 45% = = 0.45. To turn a fraction into a percentage, divide top by bottom and multiply by 100: = 0.375 = 37.5%. Knowing the common ones by heart - = 50%, = 25%, = 20% - saves time in every paper.
Find a percentage of an amount, with and without a calculator
Without a calculator, build the percentage from easy pieces: 10% of £60 is £6, so 35% is £6 × 3 + £3 = £21. With a calculator, convert to a decimal and multiply: 35% of 60 is 0.35 × 60. Both routes should give exactly the same answer - a good self-check.
Increase or decrease an amount by a percentage using a multiplier
A 12% increase means you end with 112% of the original, so multiply by 1.12; a 12% decrease leaves 88%, so multiply by 0.88. One multiplication does the whole job: £250 increased by 12% is 250 × 1.12 = £280. Multipliers also chain neatly for repeated changes.
Write one quantity as a percentage of another
Divide the part by the whole and multiply by 100: scoring 34 out of 40 is 34 ÷ 40 × 100 = 85%. Make sure both quantities are in the same units first - 30p out of £2 is , which is 15%.
Calculate percentage change - profit, loss and simple interest
Percentage change is the change divided by the ORIGINAL value, times 100. A bike bought for £80 and sold for £92 makes × 100 = 15% profit. Simple interest adds the same amount each year: £400 at 3% simple interest earns £12 every year.
Work backwards from a percentage change to find the original amount (reverse percentages)
If a price AFTER a 20% increase is £54, then £54 is 120% of the original - so divide by the multiplier: 54 ÷ 1.2 = £45. The trap is taking 20% off the new price instead; dividing by the multiplier always undoes the change exactly.
Common mistakes in Percentages
The slips that cost marks year after year, and what to do instead.
Taking the percentage off the new amount in a reverse percentage, so a price of £54 after a 20% rise becomes £43.20.
The new price is 120% of the original, so divide by the multiplier: 54 ÷ 1.2 = £45. Check it: 45 × 1.2 = 54.
Dividing the change by the new amount, so a rise from £80 to £100 comes out as 20% instead of 25%.
Percentage change is the change divided by the original amount, times 100: 20 ÷ 80 × 100 = 25%.
Writing the multiplier for a 5% increase as 1.5, or using 0.15 for a 15% decrease.
Add the change to 100% or take it away, then write that as a decimal: a 5% rise gives 105%, a multiplier of 1.05, and a 15% fall leaves 85%, a multiplier of 0.85.
Adding up successive percentage changes, so a 10% rise followed by a 10% fall looks like no change at all.
Multiply the multipliers instead: 1.1 × 0.9 = 0.99, which is an overall fall of 1%.
Finding 5% of an amount by dividing by 5.
Dividing by 5 finds a fifth, which is 20%. Build 5% from 10% instead: 10% of £60 is £6, so 5% is half of that, £3.
Practice
Try a Percentages question
A GCSE-style original question from this topic. Have a go before you open the working - deciding on an answer first is what makes the working stick.
The price of a phone drops from £250 to £210. Work out the percentage decrease.
- 19%
- 84%
- 40%
- 16%
Show the answer and the working
Answer: 16%
The decrease is £40, and × 100 = 16%.
- Find the actual decrease: 250 − 210 = 40.
- Divide by the original price: 40 ÷ 250 = 0.16.
- Multiply by 100: 0.16 × 100 = 16%.