The short answer
- “Per cent” means “out of 100”, so 20% is 20 ÷ 100, or 0.2.
- X% of Y = X ÷ 100 × Y.
- Percentage change = (new − old) ÷ old × 100.
- To undo a percentage increase, divide, do not subtract.
Finding a percentage of a number
- 20 ÷ 1000.2
- 0.2 × 15030
The same method finds a 17.5% share of 2,400, which is 420.
What percentage one number is of another
- 45 ÷ 600.75
- × 10075%
This is how exam marks, survey results and budget shares are worked out.
Percentage increase and decrease
Always divide by the starting number.
| From | To | Change |
|---|---|---|
| 80 | 100 | +25% |
| 100 | 80 | −20% |
| 1,250 | 1,400 | +12% |
Notice that going from 80 to 100 is a 25% rise, but going back from 100 to 80 is only a 20% fall, because the starting point is different.
Adding and taking off a percentage
- Multiply by
- 1.15
- Answer
- 287.5
- Multiply by
- 0.75
- Answer
- 60
Multiplying by one number is quicker and avoids mistakes. A 20% discount means paying 80%, so multiply by 0.8.
Reverse percentages
To find the original amount before a percentage was added, divide by (1 + the percentage ÷ 100). Taking the percentage off the final amount gives the wrong answer.
- Divide by 1.245
- VAT included9
The common error
Taking 20% off 54 gives 43.20, which is wrong. The VAT was 20% of 45, not of 54.
The same applies to pay rises. If your salary is 36,000 after a 4.5% rise, your old salary was 34,449.76, not 34,380.
Percentage points
When a rate changes, the difference is measured in percentage points. If an interest rate rises from 4% to 5%, it has gone up by 1 percentage point, but by 25% in relative terms. News reports often mix the two, which can make changes sound bigger or smaller than they are.
Several changes in a row
Percentage changes multiply rather than add.
| Changes | Overall change |
|---|---|
| +50% then −50% | −25% |
| +10% then −10% | −1% |
| −20% then +25% | 0% |
| +3% for three years | +9.27% |
This is why inflation of 3% a year adds up to more than 9% over three years, and why a share that falls 20% needs to rise 25% to get back to where it started.
Percentages in everyday money
- Sales tax and VAT: to add a tax of 20%, multiply by 1.2; to remove it, divide by 1.2. For a 7.25% sales tax, multiply or divide by 1.0725.
- Pay rises: a 5% rise on 30,000 is 1,500 before tax.
- Discounts: 30% off then a further 10% off is 37% off, not 40%.
- Interest: rates are usually yearly, so 4% on 1,000 is about 40 a year.
- Splitting bills: dividing 100 three ways gives 33.34, 33.33 and 33.33, so the shares add up.
Mental maths shortcuts
- 10%: move the decimal point one place left. 10% of 85 is 8.50.
- 5%: half of 10%. 1%: move the decimal point two places.
- Percentages are reversible: 8% of 50 is the same as 50% of 8, which is 4.
- 25% is a quarter, 50% a half, 75% three quarters, 20% a fifth.
Common mistakes
- Dividing by the new number instead of the old one when working out a change.
- Subtracting a percentage to undo an increase instead of dividing.
- Adding successive percentages instead of multiplying.
- Confusing percentage points with percentages.
Percentages, fractions and decimals
| Percentage | Decimal | Fraction |
|---|---|---|
| 1% | 0.01 | 1/100 |
| 5% | 0.05 | 1/20 |
| 10% | 0.1 | 1/10 |
| 12.5% | 0.125 | 1/8 |
| 20% | 0.2 | 1/5 |
| 25% | 0.25 | 1/4 |
| 33.3% | 0.333 | 1/3 |
| 50% | 0.5 | 1/2 |
| 75% | 0.75 | 3/4 |
To turn a fraction into a percentage, divide the top by the bottom and multiply by 100. To turn a percentage into a decimal, divide by 100.
Exam marks and grades
A mark of 54 out of 72 is 75%. Grade boundaries for GCSEs and A levels are set each year as raw marks, not fixed percentages, so the same percentage can lead to different grades in different years or subjects. University degree classes, by contrast, often use percentage bands, such as 70% and above for a first.
Tips and service charges
A tip or service charge is a percentage of the bill. A 12.5% service charge on a bill of 64 is 8, taking the total to 72; an 18% tip on the same bill is 11.52. Customs differ: in the US a tip of 15% to 20% is expected, while in the UK a service charge of 10% to 12.5% is often added to the bill and is optional. Our tip calculator splits a bill between people.
APR, AER and interest rates
- Means
- Annual equivalent rate
- Shows
- Interest over a year, including compounding
- Means
- Annual percentage rate
- Shows
- The yearly cost including interest and most fees
Compare savings accounts by AER and loans or credit cards by APR, because both convert different payment patterns into a single yearly percentage. The compound interest calculator shows how rates grow over time.
Reading percentages in the news
- Ask “a percentage of what?”. A 50% rise in a rare risk may still be a tiny risk.
- Check whether a change is in percentages or percentage points.
- Inflation figures show how much prices have risen over 12 months, not how high prices are.
- Small samples can give large percentage swings, so look at the actual numbers too.
Percentages in a spreadsheet
In Excel or Google Sheets, type =A1*20% for 20% of a cell, =A1/B1 and format as a percentage for a share, and =(B1-A1)/A1 for a percentage change. To remove 20% VAT, use =A1/1.2. Format cells as percentages rather than multiplying by 100, to avoid errors when the numbers are used again.