Percentage change is the difference between a new and an old value, divided by the old value, times 100. If a price goes from £80 to £100, that is (100 − 80) ÷ 80 × 100 = a 25% increase. Discounts work the same way in reverse: take the percentage off the original price to get the sale price.
The percentage change formula
Percentage change = (new − old) ÷ old × 100.
A positive answer is an increase and a negative one is a decrease. The key point is that you always divide by the old (starting) value, not the new one. If you are new to the basics, start with How to calculate percentages.
Why increases and decreases are not symmetrical
Here is a common trap. Going from £80 to £100 is a 25% increase. Going from £100 back to £80 is only a 20% decrease. The change in pounds is the same (£20), but the starting point is different.
| From | To | Change |
|---|---|---|
| £80 | £100 | +25% |
| £100 | £80 | −20% |
This is why a 50% fall needs a 100% rise to get back to where you began. The Percentage change tool shows the difference and the direction.
How to calculate a discount
Sale price = original price × (1 − discount ÷ 100).
Example: a £120 jacket with 20% off. 20% of £120 is £24, so the sale price is £96. Quick check: £120 × 0.8 = £96.
What happens with two discounts in a row
A second discount applies to the already reduced price, so two discounts do not simply add up. Take £120 with 20% off, then an extra 10% off:
- After 20% off: £96.00
- After a further 10% off: £86.40
- Total saved: £33.60, which is a 28% overall discount rather than 30%
The Discount tool handles one or two discounts and shows the total saving. Remember too that VAT is a percentage added on top of a price, covered in VAT explained: adding and removing VAT, so check whether a displayed price includes it.
Splitting a bill with a tip
A tip is a percentage of the bill, then divided between the group. For an £84 bill with a 12.5% tip across 4 people, the tip is £10.50, the total is £94.50 and each person pays £23.63 (rounded). Tipping customs and any service charge already on the bill vary, so check the menu before adding more. The Tip splitter tool does the sums.
Note: These are general maths methods and estimates, not financial advice. Check the price, terms and any exclusions with the retailer.
Useful percentage change examples
- Pay rise: from £28,000 to £29,400 is (1,400 ÷ 28,000) × 100 = 5%.
- Price drop: from £250 to £200 is a 20% decrease.
- Weight or measurement changes: use the same formula with kilograms or centimetres.
For pay conversions, see Hourly rate to salary, and how to work out overtime.
Percentage change in daily life
The same formula helps with many everyday questions. Comparing this month’s energy use with last month’s, checking how much a savings pot has grown, or seeing how far a train fare has risen over a few years all come down to the old value, the new value and a division by the old one.
When reading headlines, check what the percentage is based on. “Up 50%” from a very small starting figure may still be a small amount. It helps to look at both the percentage and the actual difference before drawing conclusions.
Reversing a percentage change
Sometimes you know the result and need the starting point. If a price after a 20% rise is £60, the original was £60 ÷ 1.2 = £50, not £60 less 20%. The rule is to divide by the multiplier rather than subtract the percentage from the new figure. The same applies to sale prices: divide by 0.8 for a 20% discount, or by 0.9 for 10%. This avoids the asymmetry described above and is a handy check when a shop quotes only the discounted price.
Try the calculator
Use Percentage change for increases and decreases, Discount for sale prices and Tip splitter to split a bill. Results are estimates based on your entries.
Open the free Percentage change
Step-by-step: percentage change with a worked example
Take a train season ticket that rose from £2,400 to £2,520.
- Find the difference: £2,520 − £2,400 = £120.
- Divide by the old value: £120 ÷ £2,400 = 0.05.
- Multiply by 100: 0.05 × 100 = 5%.
The ticket rose by 5%. If it later fell back to £2,400, step 1 would give −£120, but step 2 would divide by £2,520, giving 0.0476, which is a 4.76% decrease. The fall is smaller in percentage terms for exactly the reason covered above: the starting point is larger.
Worked example 2: stacking discounts and a coupon
A coat costs £150. The shop offers 30% off, and you have a further 15% off voucher.
| Step | Calculation | Price |
|---|---|---|
| Original | £150.00 | |
| After 30% off | £150 × 0.70 | £105.00 |
| After a further 15% off | £105 × 0.85 | £89.25 |
| Total saved | £150 − £89.25 | £60.75 |
| Overall discount | £60.75 ÷ £150 | 40.5% |
Adding 30% and 15% would suggest 45% off, or £82.50. The real total is 40.5% because the voucher applies to the lower price. A quick method is to multiply the two multipliers together: 0.70 × 0.85 = 0.595, so you pay 59.5% of the original and save 40.5%.
Always check the terms. Some vouchers apply to the original price rather than the sale price, and some exclude sale items altogether.
Percentage points are not percentages
A common mix-up in news and reports is percentage points versus per cent. If an interest rate moves from 4% to 5%, it has risen by one percentage point, but it has risen by 25% in relative terms (1 ÷ 4 = 0.25).
- Use percentage points for the simple gap between two percentages.
- Use per cent change for the relative change, using the formula above.
When a figure is described as “up 20%”, check whether the writer means 20 points or 20 per cent. They can be very different.
Comparing offers and prices fairly
Percentages help you compare, but only if the bases match. Two examples:
- Different starting prices. A 10% saving on a £400 item is £40, while 25% off a £90 item is £22.50. The bigger percentage is the smaller saving.
- Unit prices. A pack that is “20% bigger” for the same price is a different deal from one that is “20% off”, because the first divides the price over more units. 20% more product costs about 16.7% less per unit (1 ÷ 1.2 = 0.833).
Where you can, convert to pounds saved or price per unit. For VAT and how it is added or removed from displayed prices, see VAT explained: adding and removing VAT.
Common mistakes to avoid
- Dividing by the new value. Always divide by the starting figure.
- Adding successive percentages. Two discounts or two rises multiply, they do not add.
- Taking the percentage off the wrong total. To find a price before a rise, divide by the multiplier rather than subtracting.
- Mixing percentage points and per cent. Say which one you mean.
- Rounding too early. Keep several decimal places through the working and round only the final answer.
- Ignoring the base. A rise from £2 to £4 is 100%, but the pound difference is small. Look at both.
Glossary of terms
- Base (old value): the starting number you divide by.
- Multiplier: a single number, such as 1.2 or 0.8, that applies a percentage change in one step.
- Percentage points: the arithmetic gap between two percentages.
- Reverse percentage: working back from the result to the original figure.
- Compound change: a change applied repeatedly, each time to the new total. The guide on Percentage increase and decrease, and reverse percentages covers this in more detail.
Worked example 3: a price rise, then a sale
A monthly subscription costs £12.50. It rises by 8%, then the provider offers 8% off as a loyalty deal. Many people expect to land back on £12.50.
| Step | Calculation | Price |
|---|---|---|
| Original | £12.50 | |
| After 8% rise | £12.50 × 1.08 | £13.50 |
| After 8% loyalty discount | £13.50 × 0.92 | £12.42 |
You finish 8p lower than where you began, because the discount is taken from the higher price. Over a year that is a small gap, but the principle matters on larger amounts: a 10% rise followed by 10% off on £1,000 ends at £990, not £1,000. To see how the same logic runs through repeated changes, look at Percentage increase and decrease, and reverse percentages.
Quick checks before you trust a percentage
- Is the percentage of the old or the new figure?
- Is it a change in per cent or in percentage points?
- Does the price include VAT, delivery or a service charge?
- Is the “was” price a real previous price? Check how long it was charged for before relying on the saving.
- Does the maths still work if you convert it to pounds?
Frequently asked questions
How do I work out a percentage increase?
Subtract the old value from the new one, divide by the old value and multiply by 100. If the answer is positive it is an increase.
Why is a 20% discount followed by 10% not 30% off?
The second discount applies to the lower, already discounted price, so it takes off less. The total on £120 is 28%, not 30%.
How do I add a percentage to a price?
Multiply by 1 plus the percentage as a decimal. For 20% on £50, multiply by 1.2 to get £60.
How do I find the original price before a discount?
Divide the sale price by 1 minus the discount. If £96 is after 20% off, £96 ÷ 0.8 = £120.
What is the difference between percentage change and percentage difference?
Percentage change compares a new value with an old one, so order matters. Percentage difference is used when neither value is the starting point and typically divides by the average of the two. Check which one a source is using.
How do I calculate a percentage increase in my salary?
Subtract your old salary from the new one, divide by the old salary and multiply by 100. A rise from £32,000 to £33,600 is £1,600 ÷ £32,000 = 5%.
Can a percentage decrease be more than 100%?
No, a fall cannot exceed 100% because that takes the value to zero. A percentage increase has no upper limit, so a value can rise by 200% or more.
How do I split a discount across several items?
Work out the sale price for each item and add them, or apply the same multiplier to the combined total if the percentage is the same for all of them.