“Per cent” means “per hundred”: 25% is 25 out of every 100, or 0.25 as a decimal. That conversion, dividing by 100, is the key to every formula below. Once a percentage is a decimal (multiplier), most problems become a single multiplication or division. You can check any answer with our percentage calculator.
Quick reference
| Question | Formula | Example |
|---|---|---|
| What is X% of Y? | Y × X ÷ 100 | 15% of 240 = 36 |
| X is what % of Y? | X ÷ Y × 100 | 36 of 240 = 15% |
| Percentage change from A to B | (B − A) ÷ A × 100 | 80 → 92 = +15% |
| Increase Y by X% | Y × (1 + X/100) | 200 + 12% = 224 |
| Decrease Y by X% | Y × (1 − X/100) | 200 − 12% = 176 |
| Original value before an X% change | New ÷ (1 ± X/100) | 64 after 20% off → 80 |
The six core formulas
1. X% of a number
Formula: result = number × percentage ÷ 100
Example: A restaurant bill is $86 and you want to leave an 18% tip. 86 × 18 ÷ 100 = $15.48.
Mental shortcut: 10% is the number with the decimal point moved one place left. For 15%, take 10% and add half of it: 10% of 240 = 24, half is 12, total 36. For 5%, halve 10%. For 1%, move the decimal two places.
Percentages are also reversible: 8% of 50 equals 50% of 8, which is 4. Swap whichever is easier.
2. What percentage is X of Y?
Formula: percentage = part ÷ whole × 100
Example: You scored 42 out of 60 in an exam. 42 ÷ 60 = 0.7, × 100 = 70%.
Example: 1,350 of 4,500 website visitors signed up. 1,350 ÷ 4,500 = 0.3 → 30% conversion rate.
3. Percentage change (increase or decrease)
Formula: change % = (new − old) ÷ old × 100
Always divide by the old (starting) value.
Increase: A monthly rent goes from £1,200 to £1,290. (1,290 − 1,200) ÷ 1,200 × 100 = 90 ÷ 1,200 × 100 = +7.5%.
Decrease: A share price falls from $92 to $80. (80 − 92) ÷ 92 × 100 = −12 ÷ 92 × 100 ≈ −13.04%.
Notice the asymmetry: going from 80 to 92 is +15%, but going from 92 back to 80 is about −13.04%. The base is different each time.
4. Adding or removing a percentage
Increase by X%: multiply by (1 + X/100).
- £200 increased by 12%: 200 × 1.12 = £224.
- Salary of $52,000 with a 4% raise: 52,000 × 1.04 = $54,080.
Decrease by X%: multiply by (1 − X/100).
- A €150 jacket with 30% off: 150 × 0.70 = €105.
Successive changes multiply, they don’t add. A 10% rise followed by a 10% fall is 1.10 × 0.90 = 0.99, a net 1% decrease, not zero. Two discounts of 20% and then 10% give 0.8 × 0.9 = 0.72, a total of 28% off, not 30%.
5. Reverse percentages (finding the original)
This is where most mistakes happen. If a value already includes a percentage change, divide by the multiplier; don’t subtract the percentage from the new value.
Example (discount): A sale price is £64 after 20% off. Original = 64 ÷ 0.80 = £80. (Wrong method: 64 + 20% of 64 = £76.80.)
Example (increase): After a 5% pay rise, your salary is $63,000. Previous salary = 63,000 ÷ 1.05 = $60,000.
Example (tax): A price of £120 includes 20% UK VAT. Net price = 120 ÷ 1.20 = £100, so the VAT is £20, not £24. Our VAT calculation guide covers this in detail for UK, EU and US rates.
6. Percentage points vs percent
When comparing two percentages, be precise about what changed:
- A mortgage rate rises from 4% to 5%. That’s 1 percentage point higher, but 25% higher in relative terms (1 ÷ 4 × 100).
- Unemployment falls from 5.0% to 4.5%: down 0.5 percentage points, or 10% in relative terms.
News headlines often mix these up. “Interest rates up 50%” and “interest rates up 1 point” can describe the same move from 2% to 3%.
Other everyday uses
Margin vs markup
Both use cost and price, but different bases:
- Markup = profit ÷ cost × 100. Buy at $40, sell at $50: markup = 10 ÷ 40 = 25%.
- Margin = profit ÷ selling price × 100. Same item: margin = 10 ÷ 50 = 20%.
Confusing them leads to under-pricing: if you want a 25% margin, you need a markup of 33.3%.
Percentage of a total (shares)
Budget of £3,200: rent £1,100, food £480. Rent = 1,100 ÷ 3,200 × 100 = 34.375%; food = 480 ÷ 3,200 × 100 = 15%.
Compound growth
Growth that repeats on a growing base compounds. £1,000 growing 5% a year for 3 years: 1,000 × 1.05³ = £1,157.63, not £1,150. For loans and savings, see how loan interest works.
Common mistakes
- Dividing by the new value instead of the old one when calculating change.
- Adding successive percentage changes instead of multiplying them.
- Removing tax or a discount by subtracting the percentage from the final price.
- Mixing up percentage points and percent.
- Rounding too early in multi-step calculations. Keep full precision until the final answer.
Checklist
- Convert the percentage to a decimal (÷ 100).
- Identify the base: the original, the whole, or the final figure?
- Multiply for “X% of” and increases/decreases; divide for reverse problems.
- Multiply successive changes.
- Sense-check: a discount should make things smaller, an increase larger.