Percentage Formulas
Every percentage formula worth memorising, each with its inverse and a worked example you can check line by line. One page to bookmark instead of eight.
The four you will use most
Almost every percentage question in ordinary life is one of these four, or an inverse of one. They all come from the same definition: a percentage is a fraction over 100.
1. A percentage of a number
part = whole × (percent ÷ 100)
18% of 250: 250 × 0.18 = 45. Inverses: whole = part ÷ (percent ÷ 100), and percent = (part ÷ whole) × 100.
2. One number as a percentage of another
percent = (part ÷ whole) × 100
34 out of 40: (34 ÷ 40) × 100 = 85%. The number you are measuring goes on top; the total goes underneath. Reversing them gives 117.6%, which is the answer to a different question.
3. Increase or decrease by a percentage
result = start × (1 ± percent ÷ 100)
£640 plus 12.5%: 640 × 1.125 = £720. Minus 12.5%: 640 × 0.875 = £560. Inverse: start = result ÷ the same multiplier.
4. Percentage change between two numbers
change = ((new − old) ÷ old) × 100
£640 to £720: (80 ÷ 640) × 100 = +12.5%. Going back from £720 to £640 is −11.1%, not −12.5%, because the denominator changed.
Four more that come up regularly
5. Removing a percentage that is already included
net = gross ÷ (1 + rate ÷ 100)
Stripping 20% VAT from £300: 300 ÷ 1.20 = £250. Subtracting 20% from £300 gives £240 and is wrong, because the tax was charged on the smaller net figure.
6. Percentage points versus relative change
points = second − first
relative change = ((second − first) ÷ first) × 100
4% to 5% is one percentage point and a 25% relative change. When both quantities are already percentages, saying which of the two you mean is not optional.
7. Compound growth over several periods
total factor = (1 + rate ÷ 100)ⁿ
5% a year for three years is 1.05³ = 1.1576, or 15.76% overall, not 15%. To find the annual rate from a total, take the nth root: growing 40% over five years is 1.40^(1/5) = 1.0696, about 6.96% a year.
8. Margin and markup
margin = ((price − cost) ÷ price) × 100
markup = ((price − cost) ÷ cost) × 100
Same profit, different denominator. A £52 cost sold at £80 is a 35% margin and a 53.8% markup.
Every formula and its inverse
The most common source of a wrong answer is not the formula but its direction — reaching for an addition where the situation calls for a division. This table pairs each operation with the one that undoes it.
| You know | You want | Formula | Example |
|---|---|---|---|
| Whole and percent | The part | whole × (pct ÷ 100) | 18% of 250 = 45 |
| Part and percent | The whole | part ÷ (pct ÷ 100) | 45 is 18% of 250 |
| Part and whole | The percent | (part ÷ whole) × 100 | 45 of 250 = 18% |
| Start and increase | The result | start × (1 + pct ÷ 100) | 250 + 18% = 295 |
| Result and increase | The start | result ÷ (1 + pct ÷ 100) | 295 ÷ 1.18 = 250 |
| Start and decrease | The result | start × (1 − pct ÷ 100) | 250 − 18% = 205 |
| Result and decrease | The start | result ÷ (1 − pct ÷ 100) | 205 ÷ 0.82 = 250 |
| Old and new | The change | ((new − old) ÷ old) × 100 | 250 → 295 = +18% |
| Gross and rate | The net | gross ÷ (1 + rate ÷ 100) | 300 ÷ 1.20 = 250 |
| Gross and rate | The tax | gross × rate ÷ (100 + rate) | 300 × 20 ÷ 120 = 50 |
| Cost and margin | The price | cost ÷ (1 − margin ÷ 100) | 52 ÷ 0.65 = 80 |
| Cost and markup | The price | cost × (1 + markup ÷ 100) | 52 × 1.538 = 80 |
| Markup | The margin | markup ÷ (100 + markup) × 100 | 50% markup = 33.3% margin |
| Margin | The markup | margin ÷ (100 − margin) × 100 | 33.3% margin = 50% markup |
| Rate and periods | Total growth | (1 + rate ÷ 100)ⁿ − 1 | 1.05³ − 1 = 15.76% |
| Total growth and periods | Annual rate | (1 + total)^(1÷n) − 1 | 1.40^0.2 − 1 = 6.96% |
| A drop of p% | The gain to recover | p ÷ (100 − p) × 100 | A 50% drop needs +100% |
Two habits that prevent most errors
Name the base out loud
Every percentage is a percentage of something. Before writing anything down, say which number the percentage is being taken of. Most wrong answers are the right formula applied to the wrong base — a discount taken from the post-tax total, a raise divided by the new salary, a tax rate subtracted from a gross price.
Prefer multipliers to two-step arithmetic
Working with 1.18 and 0.82 rather than "add 18%" and "subtract 18%" makes the inverse obvious, makes chains of changes a single product, and removes the temptation to undo a change by applying it in reverse. Once you are thinking in multipliers, the fact that 1.18 × 0.82 = 0.9676 rather than 1 stops being surprising.
250 × 1.18 × 0.82 = 241.90, not 250
Frequently Asked Questions
- What is the basic percentage formula?
- part = whole × (percent ÷ 100). Everything else on this page is that identity rearranged for a different unknown.
- How do I reverse a percentage increase?
- Divide by the multiplier you would have multiplied by. To undo a 25% increase, divide by 1.25 — subtracting 25% leaves you 6.25% short.
- What is the formula for percentage change?
- ((new − old) ÷ old) × 100. The old value is always the denominator, which is why the same absolute move gives different percentages in each direction.
- How do I calculate compound growth?
- Raise the multiplier to the power of the number of periods: (1 + rate ÷ 100)ⁿ. Three years at 5% is 1.05³ = 1.1576, or 15.76% in total.
- How do I get the tax out of a price that includes it?
- Multiply the gross by rate ÷ (100 + rate). At 20% that is 1/6 of the gross, so a £300 price contains £50 of tax.
- What gain undoes a 40% loss?
- A gain of 40 ÷ 60 × 100 = 66.7%. The general form is p ÷ (100 − p) × 100.
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