Four everyday percentage questions answered instantly — percent of, what percent, percentage change and reverse percentage.
Percentage change is everywhere: salary rises, price hikes, stock moves, population growth. It’s also one of the most commonly misunderstood calculations, because the answer depends entirely on which number you start from.
The formula
Percentage change = (New value − Old value) ÷ |Old value| × 100
A positive result is an increase; a negative result is a decrease.
Rent goes from $1,200 to $1,320: (1,320 − 1,200) ÷ 1,200 × 100 = +10%
A laptop falls from $900 to $720: (720 − 900) ÷ 900 × 100 = −20%
The percentage calculator has a dedicated row for this, and it labels the result as an increase or decrease.
Increases and decreases aren’t symmetric
Go from 100 to 150: that’s +50%. Now go back from 150 to 100: that’s −33.3%, not −50%. The base changed from 100 to 150.
This asymmetry has a painful consequence for investments:
| Loss | Gain needed to recover |
|---|---|
| −10% | +11.1% |
| −20% | +25% |
| −33% | +50% |
| −50% | +100% |
| −75% | +300% |
| −90% | +900% |
Recovery gain = 1 ÷ (1 − loss) − 1
A portfolio that halves needs to double just to break even.
Successive percentage changes
Changes multiply; they don’t add.
A price rises 10%, then rises another 10%: 1.1 × 1.1 = 1.21, a 21% total increase, not 20%.
A price rises 20%, then falls 20%: 1.2 × 0.8 = 0.96, a 4% net loss, not zero.
The same logic explains why stacked store discounts are worth less than they look; see Stacked Discounts Explained: Why 20% + 10% Isn't 30% Off.
Average annual growth (CAGR)
If something grows from A to B over several years, the compound annual growth rate is:
CAGR = (B ÷ A)^(1 ÷ years) − 1
Revenue grows from $200k to $320k over 4 years: (320 ÷ 200)^(0.25) − 1 ≈ 12.5% a year. Simply dividing the 60% total by 4 (15%) overstates the yearly rate, because it ignores compounding. More on compounding in Compound vs Simple Interest: The Difference, With Examples.
Applying a percentage change
To increase a number by p%: multiply by (1 + p/100). To decrease it by p%: multiply by (1 − p/100).
- $50 + 15% = 50 × 1.15 = $57.50
- $50 − 15% = 50 × 0.85 = $42.50
Common mistakes
- Using the new value as the base. Always divide by the original.
- Confusing percentage change with percentage-point change. A rate going from 2% to 3% is +1 percentage point but a +50% change. See Percent vs Percentage Points: What's the Difference?.
- Averaging percentages directly. +50% then −50% doesn’t average to zero; you lose 25%.
- Percentage change from zero. It’s undefined. You can’t grow by a percentage from nothing.
Quick practice
- 40 → 50? +25%
- 50 → 40? −20%
- 80 → 200? +150%
- What’s 250 after a 12% fall? 220
Check your answers with the calculator; the small print under each result shows the working.
Four everyday percentage questions answered instantly — percent of, what percent, percentage change and reverse percentage.