Percentage formulas · examples · rules of thumb
Percentage formulas – all the formulas on one page
Four basic formulas cover almost everything, plus the derived formulas you need most often, worked exercises with solutions and the most common mistakes. Try each formula in practice with the calculator below.
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What a percentage means
A per cent means a hundredth. When you say "25%" you mean 25 hundredths, i.e. the fraction
25/100 = 0.25. This one insight solves most percentage problems: a percentage is a
multiplier. Once you convert a percentage to a decimal, the calculation becomes ordinary multiplication or
division.
| Percentage | Decimal | Fraction | Rule of thumb |
|---|---|---|---|
| 1% | 0.01 | 1/100 | Divide by a hundred |
| 10% | 0.10 | 1/10 | Move the point one step left |
| 25% | 0.25 | 1/4 | A quarter |
| 33.33% | 0.3333 | 1/3 | A third |
| 50% | 0.50 | 1/2 | Half |
| 75% | 0.75 | 3/4 | Three quarters |
| 150% | 1.50 | 3/2 | One and a half times |
In UK English the percent sign is written closed up to the number: 25%, not
25 %.
The four basic formulas
1. How much is X% of Y?
result = Y × X / 100
Example. A 15% tip on a restaurant bill of £68:
68 × 15 / 100 = £10.20.
Rule of thumb. 10% is always easy in your head (move the point). 15% = 10% + half of it.
68 → 6.80 + 3.40 = 10.20.
2. How many per cent is A of B?
percentage = A / B × 100
Example. 34 marks out of a maximum of 40 in a test:
34 / 40 × 100 = 85%.
Rule of thumb. The number after the word "of" or "out of" goes on the bottom as the divisor.
3. A is X% of what number?
total = A / (X / 100)
Example. A deposit of £4,500 is 15% of the sale price:
4,500 / 0.15 = £30,000.
Rule of thumb. When you're looking for a bigger number, you divide. If you notice
the answer is smaller than the starting number, you've done it the wrong way round.
4. By how much did the value change, in per cent?
change % = (new − old) / old × 100
Example. Rent £780 → £815: (815 − 780) / 780 × 100 = 4.49%.
Rule of thumb. The divisor is always the starting point, because that's the 100% you're
comparing to.
Derived formulas you'll often need
| Situation | Formula | Example |
|---|---|---|
| Add X% | value × (1 + X/100) | 200 + 15% = 230 |
| Subtract X% | value × (1 − X/100) | 200 − 15% = 170 |
| Original before an increase | new / (1 + X/100) | 230 / 1.15 = 200 |
| Original before a discount | new / (1 − X/100) | 170 / 0.85 = 200 |
| VAT onto a net price | price × (1 + rate/100) | 100 × 1.20 = 120 |
| VAT off a gross price | price / (1 + rate/100) | 120 / 1.20 = 100 |
| Two successive discounts | price × (1−a) × (1−b) | 200 × 0.7 × 0.8 = 112 |
| Difference in percentage points | new % − old % | 3% − 2% = 1 pp |
| Share in per mille | part / whole × 1000 | 2/1000 = 2 ‰ |
Practice exercises with solutions
1. A coat costs £149 with 35% off. What do you pay?
You pay 65% of the price: 149 × 0.65 = £96.85. The saving is £52.15.
2. 7 of a class of 28 pupils are absent. What percentage?
7 / 28 × 100 = 25%. So 75% of the class is present.
3. A product costs £84 after a discount. The discount was 30%. What was the normal price?
84 / 0.70 = £120. Check: 120 × 0.7 = 84 ✓
4. Pay rose from £2,400 to £2,520. What percentage?
(2,520 − 2,400) / 2,400 × 100 = 5%.
5. A share rose from 12% to 15%. By how much did it rise?
Three percentage points. In relative terms the rise is 3 / 12 × 100 = 25%.
Both answers are correct – always say which one you mean.
6. A price rose 10% and then fell 10%. Is it back where it started?
No. 100 × 1.1 × 0.9 = £99. The price fell one per cent, because the fall was
calculated from a bigger number.
The most common mistakes in percentage maths
- The wrong reference number. A change is always calculated from the starting value, not the end value.
- Adding percentages together. Successive discounts and increases are multiplied, not added.
- Adding a percentage back. Undoing a discount needs a division, not adding the same percentage back.
- Mixing up per cent and percentage point. Especially common when talking about interest rates and tax rates.
- Rounding too early. Round only the final result, not the intermediate steps – otherwise the error compounds.
Applying the percentage formulas, and the pitfalls
Four basic formulas cover almost everything, but mistakes come from which number the percentage is taken from and when percentages must not be added. Below are the most common applied formulas, the concepts and the calculator's limits.
Formulas and examples
Share of two numbers
share = part / whole × 100Example
87 marks out of 240: 87 / 240 × 100 = 36.25%.
Note
The part is always divided by the whole, not the other way round – otherwise the percentage flips.
Reverse calculation: of what number
whole = part / (percentage / 100)Example
15% of some number is 45: 45 / 0.15 = 300.
Note
This comes up in questions like "45 is 15% of what?" – you divide, not multiply.
Per cent or percentage point
change % = (new − old) / old × 100Example
An interest rate 2% → 3%: the difference is 1 percentage point but 1 / 2 × 100 = 50% more.
Note
A percentage point compares percentages directly; a per cent compares the relative change.
Successive changes
end = start × (1 + c₁/100) × (1 + c₂/100)Example
+10% and −10%: 100 × 1.1 × 0.9 = 99, not 100.
Note
Up and down by the same percentage doesn't return you to the start, because the divisor changes.
Concepts and classification
| Base value | The number the percentage is calculated from (the divisor in a reverse calculation). |
|---|---|
| Share | How many per cent a part is of the whole. |
| Percentage point | The difference between two percentages, e.g. 2% to 3% = 1 point. |
| Multiplier | A percentage as a decimal: 25% = 0.25; a 25% increase = a multiplier of 1.25. |
| Change percentage | The relative change from the starting value to the new value. |
Limitations of the calculator
- The formulas assume the base value (what the percentage is taken from) is known and correct.
- Rounding is best done to the final result, not intermediate steps.
- Successive percentages must not be added – the multipliers are multiplied.
- The calculator does not interpret a word problem for you; choose the right formula yourself.
How up to date the information is
When it was checked
Content and the percentages used were checked on 12 August 2026.
Disclaimer
The calculator gives a mathematical result from the numbers you enter. It is not an official decision, an offer or professional advice. See the terms of use.
Topics
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Formulas in the practical calculators
Frequently asked questions about percentage formulas
What is the basic idea of percentage calculation?
A per cent is a hundredth, so a percentage is a multiplier. 25% is the same as 0.25. Once you convert a percentage to a decimal, every percentage calculation becomes ordinary multiplication or division.
Which formula is for which task?
The basic rule: to find a part, you multiply. To find a percentage, you divide the part by the whole. To find the whole, you divide the part by the percentage. To find a change, you divide the difference by the starting value.
Is it 25% or 25 %?
In UK English the percent sign is written closed up to the number: 25%. There is no space between the number and the sign.
Can a percentage be over 100?
Yes. Over 100% means the part is larger than the reference number. If sales double, the new sales are 200% of the old and the growth is 100%. Those two figures are easily confused.
How do I do a percentage in my head?
Start from ten per cent: move the decimal point one step to the left. Build the others from that: 5% is half of ten, 20% is twice ten, and 15% is 10% + 5%.