Percentage change is one of the most frequently used calculations in everyday life — yet one of the most frequently done wrong. Whether you're tracking investment returns, comparing prices, reading economic statistics, or analysing business data, understanding how to calculate it correctly (and avoid the common traps) is essential.
The Basic Formula
Percentage change = ((New value − Old value) / Old value) × 100
This can also be written as:
Percentage change = ((New / Old) − 1) × 100
A positive result = an increase. A negative result = a decrease.
Worked Examples
Price increase: A product costs £80. It rises to £92.
% change = ((92 − 80) / 80) × 100 = (12 / 80) × 100 = 15%
Price decrease: A share falls from £5.40 to £4.86.
% change = ((4.86 − 5.40) / 5.40) × 100 = (−0.54 / 5.40) × 100 = −10%
Population change: A city grows from 340,000 to 389,100.
% change = ((389,100 − 340,000) / 340,000) × 100 = (49,100 / 340,000) × 100 = 14.44%
Increase vs Decrease: Why the Base Matters
A 20% increase followed by a 20% decrease does not return you to the start. This surprises many people.
Start: £100
After 20% increase: £100 × 1.20 = £120
After 20% decrease: £120 × 0.80 = £96
You end up 4% below where you started. This is because the percentage in the second step is calculated on a larger base (£120, not £100).
The same logic explains why a 50% drop requires a 100% gain to recover:
- £100 → £50 (−50%) → £100 (+100%)
Percentage Points vs Percentage Change
These are different things that are commonly confused.
Percentage points = the arithmetic difference between two percentages.
Percentage change = the relative change in a percentage value.
Example: Interest rates rise from 3% to 5%.
- Change in percentage points = 5 − 3 = 2 percentage points
- Percentage change = ((5 − 3) / 3) × 100 = 66.7%
Both statements are technically true. "Interest rates rose by 2 percentage points" and "interest rates rose by 66.7%" describe the same event from different angles. News articles sometimes conflate these — always check which is being used.
Reverse Percentage Change
If you know the result after a percentage change and want to find the original value:
Original = New value / (1 + percentage change/100)
Example: After a 15% increase, a price is £138. What was the original price?
Original = 138 / (1 + 0.15) = 138 / 1.15 = £120
Common mistake: subtracting 15% from £138 gives £117.30 — which is wrong. You'd be applying 15% to the wrong base.
Example: An item in a sale is £63 after a 30% discount. What was the original price?
Original = 63 / (1 − 0.30) = 63 / 0.70 = £90
Compound Percentage Change
When multiple percentage changes occur sequentially, multiply the multipliers:
Example: A salary increases by 5% in year 1, then 3% in year 2, then decreases by 2% in year 3. Starting from £40,000:
Final = £40,000 × 1.05 × 1.03 × 0.98
Final = £40,000 × 1.05969
Final = £42,388
Overall percentage change: ((42,388 − 40,000) / 40,000) × 100 = +5.97% over 3 years.
Not 5 + 3 − 2 = 6%. The order doesn't change the result, but the compounding does.
Average Annual Growth Rate (CAGR)
When something grows from a starting value to an ending value over multiple years, the Compound Annual Growth Rate gives the equivalent steady annual growth:
CAGR = (End / Start)^(1/years) − 1
Example: Revenue grows from £2m to £3.2m over 4 years.
CAGR = (3.2 / 2)^(1/4) − 1 = 1.6^0.25 − 1 = 1.1247 − 1 = 12.47%
This is the growth rate that, applied consistently for 4 years, would produce the observed result. It's more informative than saying "60% growth over 4 years."
Key Formulas at a Glance
| Calculation | Formula | |-------------|---------| | Percentage change | ((New − Old) / Old) × 100 | | New value after % increase | Old × (1 + %/100) | | New value after % decrease | Old × (1 − %/100) | | Original before % increase | New / (1 + %/100) | | Original before % decrease | New / (1 − %/100) | | CAGR | (End/Start)^(1/n) − 1 |
Use our Percentage Change Calculator for instant results, and our CAGR Calculator for compound growth rates.