Percentages
A percentage is a number out of 100. The quickest route through nearly every percentage question is a multiplier: a single decimal that does the increase or decrease in one step, and that you can divide by to work backwards.
Part 1 of 3: Learn it
In short
- Find a multiplier: a 12% increase is × 1.12, a 12% decrease is × 0.88.
- Percentage change = change ÷ original × 100.
- Reverse percentage: divide the new amount by the multiplier to get the original.
Where this is in your specification
Spec points: DfE N12 and R9 (repeated percentage change and compound interest: R16)
| Board | Topic: Fractions, decimals and percentages |
|---|---|
| DfE content | N8, N10, N11, N12, R9 |
| AQA 8300 | Number; Ratio, proportion and rates of change |
| Edexcel 1MA1 | Number; Ratio, proportion and rates of change |
| OCR J560 | 2 Fractions, decimals and percentages |
| Eduqas C300 | HN8, HN10 to HN12, HR10 |
| Cambridge IGCSE 0580 | E1.4, E1.13 |
| National 5 C847 75 | Numerical skills: fractions, percentages, appreciation and depreciation |
Percentages, decimals and fractions
| Percentage | Decimal | Fraction |
|---|---|---|
| 50% | 0.5 | 1/2 |
| 25% | 0.25 | 1/4 |
| 20% | 0.2 | 1/5 |
| 12.5% | 0.125 | 1/8 |
| 5% | 0.05 | 1/20 |
To change a percentage to a decimal, divide by 100. Without a calculator, build answers from 10% (divide by 10) and 1% (divide by 100): 35% of 260 is 3 × 26 + 13 = 91.
Multipliers
Start from 100%. An increase of r% gives (100 + r)%, a decrease gives (100 − r)%. Write that as a decimal and multiply once.
- Increase by 8%: × 1.08
- Decrease by 30%: × 0.7
- Increase by 2.5%: × 1.025
Percentage change
Always divide by the original value, not the new one. A negative answer is a decrease. The same idea gives percentage profit and percentage error.
Reverse percentages
If you know the amount after a change and want the amount before, divide by the multiplier. After a 15% rise the price is 115% of the original, so divide by 1.15.
Simple interest adds the same amount every year. Compound interest multiplies by the same multiplier every year: £P after n years at r% is P × (1 + r/100)ⁿ.
Write down the multiplier for a 7% decrease.
Show the answer
100 − 7 = 93%, so × 0.93.
Part 2 of 3: See it worked
Worked examples
Example 1
A train ticket cost £250 last year and £215 this year. Find the percentage change.
- Change = 215 − 250 = −35
- −35 ÷ 250 = −0.14
- −0.14 × 100 = −14%
Answer: A 14% decrease.
Example 2
In a sale everything is reduced by 20%. A jacket costs £68 in the sale. What was its original price?
- Sale price = 80% of the original, multiplier 0.8
- Original = 68 ÷ 0.8
- = 85
Answer: £85.
Example 3
£2400 is invested at 3% compound interest per year. How much is it worth after 4 years, to the nearest penny?
- Multiplier for each year: 1.03
- 2400 × 1.03⁴ = 2400 × 1.12550881
- = 2701.2211...
Answer: £2701.22.
Common mistakes
- Working out a reverse percentage by taking the percentage off the new price. 20% of £68 is not the amount that was taken off.
- Dividing by the new value in a percentage change.
- Using 1.3 for a 3% increase. 3% is 0.03, so the multiplier is 1.03.
- Using simple interest when the question says compound.
A coat costs £126 after a 5% price rise. What did it cost before?
Show the answer
126 ÷ 1.05 = £120.
Part 3 of 3: Test yourself
Check yourself
Answer each one in your head or on paper first, then open it to check.
Write down the multiplier for a 7% decrease.
100 − 7 = 93%, so × 0.93.
A coat costs £126 after a 5% price rise. What did it cost before?
126 ÷ 1.05 = £120.
A plant grows from 40 cm to 46 cm. Find the percentage increase.
6 ÷ 40 × 100 = 15%.
Jobs that use this
Each link opens the job profile on the National Careers Service (England). In the rest of the UK: My World of Work (Scotland), Careers Wales, nidirect careers (Northern Ireland).
Full lessons and marked practice for this course are coming soon to Brainlag Learn. See courses