"Percentage" covers at least five distinct questions, and mixing them up is why the subject feels harder than it is. Here is each one on its own.
1. What is P% of a number?
Turn the percentage into a decimal by dividing by 100, then multiply.
The working
15% of £80 15 ÷ 100 = 0.15 0.15 × £80 = £12.00
Worth knowing: percentages are reversible. 15% of 80 is the same as 80% of 15, and the second is often easier to do in your head.
2. What is X as a percentage of Y?
Divide the part by the whole, then multiply by 100.
The working
37 out of 50 37 ÷ 50 = 0.74 0.74 × 100 = 74%
3. Percentage increase and decrease
To increase by P%, multiply by 1 + (P ÷ 100). To decrease, multiply by 1 − (P ÷ 100). One multiplication does the whole job.
The working
Increase £250 by 12%: £250 × 1.12 = £280.00 Decrease £250 by 12%: £250 × 0.88 = £220.00
A rise then a fall of the same percentage does not return you to where you started. £100 up 20% is £120; £120 down 20% is £96. The second percentage is taken from a bigger number.
4. Percentage change between two numbers
Subtract the old from the new, divide by the old, multiply by 100. The sign tells you the direction.
The working
From 40 to 52 52 − 40 = 12 12 ÷ 40 = 0.3 0.3 × 100 = 30% increase
Dividing by the *old* value is the part that matters. Divide by the new one and you get 23%, which answers a different question nobody asked.
5. Working backwards to the original
If you know the figure after a percentage was applied, divide rather than subtract — the same trap as removing VAT.
The working
£90 is a price after a 25% discount. What was it before? £90 ÷ 0.75 = £120.00 Check: £120 × 0.75 = £90 ✓
Common questions
What is the difference between percentage change and percentage difference?
Percentage change has a direction — it compares a new value against an old one and divides by the old. Percentage difference is symmetrical: it compares two values with no before-and-after, and divides by their average. Use change for anything over time, difference for two independent measurements.
How do I work out a percentage in my head?
Find 10% by moving the decimal point one place left, then build from it. 10% of 240 is 24, so 5% is 12, 20% is 48, and 35% is 24 + 24 + 24 ÷ 2 + 12, which is 84. Almost every everyday percentage is a short sum away from 10%.
Why is a 20% rise followed by a 20% fall not back to the start?
Because each percentage is taken from a different base. The rise is 20% of the original; the fall is 20% of the larger figure that resulted, so it removes more than was added. £100 becomes £120, then £96. To undo a 20% rise you need a fall of 16.67%.
What is a percentage point?
The plain difference between two percentages. A rate going from 4% to 6% has risen by 2 percentage points, but by 50 per cent. Newspapers mix these up constantly and the distinction is usually where the real story is.
Check this before you rely on it. A free guide to everyday arithmetic — not tax, accounting, financial, legal or medical advice. The working is shown above so you can verify it against your own figures. See our terms of use.