The Formula
Percentage Change = ((New Value − Old Value) ÷ Old Value) × 100
- Positive result = increase
- Negative result = decrease
Example: Price went from $80 to $100
- (100 − 80) ÷ 80 × 100 = 25% increase
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Percentage Increase Step by Step
Example: Your rent went from $1,200 to $1,380
- Find the difference: $1,380 − $1,200 = $180
- Divide by the original: $180 ÷ $1,200 = 0.15
- Multiply by 100: 0.15 × 100 = 15%
Your rent increased by 15%.
Percentage Decrease Step by Step
Example: A TV dropped from $800 to $600
- Find the difference: $600 − $800 = −$200
- Divide by the original: −$200 ÷ $800 = −0.25
- Multiply by 100: −0.25 × 100 = −25%
The price decreased by 25%.
Common Percentage Changes
Money & Finance
| Scenario | From | To | Change |
|---|---|---|---|
| 10% raise on $50K salary | $50,000 | $55,000 | +$5,000 |
| 20% off $80 item | $80.00 | $64.00 | −$16.00 |
| 3% annual inflation | $100 | $103 | +$3.00 |
| 8% investment return on $10K | $10,000 | $10,800 | +$800 |
| 15% tip on $60 bill | $60.00 | $69.00 | +$9.00 |
Year-Over-Year Growth
| Company Revenue | Year 1 | Year 2 | Growth |
|---|---|---|---|
| Small startup | $100K | $250K | 150% |
| Growing business | $1M | $1.3M | 30% |
| Mature company | $50M | $52.5M | 5% |
Percentage Change vs Percentage Difference
These are two different calculations:
Percentage Change (from A to B)
- Has a direction (old → new)
- Formula: ((B − A) ÷ A) × 100
- Example: 100 → 150 = 50% increase
Percentage Difference (between A and B)
- No direction (symmetric)
- Formula: (|A − B| ÷ ((A + B) ÷ 2)) × 100
- Example: 100 vs 150 = 40% difference
| Values | % Change (A→B) | % Difference |
|---|---|---|
| 100 → 150 | +50% | 40% |
| 150 → 100 | −33.3% | 40% |
| 200 → 250 | +25% | 22.2% |
| 50 → 75 | +50% | 40% |
When to use which:
- Percentage change: Comparing a before and after (price increase, salary raise, weight loss)
- Percentage difference: Comparing two independent values (your salary vs national average)
Reverse Percentage Calculations
"What was the original price before a 25% discount?"
If the sale price is $60 after 25% off:
Original = Sale Price ÷ (1 − discount rate)
- $60 ÷ (1 − 0.25) = $60 ÷ 0.75 = $80
"What price after a 20% increase?"
Starting at $50 with a 20% increase:
New Price = Original × (1 + increase rate)
- $50 × 1.20 = $60
The Compounding Trap
A common mistake: a 50% decrease followed by a 50% increase does NOT get you back to the original.
| Step | Value |
|---|---|
| Start | $100 |
| After 50% decrease | $50 |
| After 50% increase | $75 (not $100!) |
To recover from a loss, you need a larger percentage gain:
| Loss | Gain Needed to Recover |
|---|---|
| 10% | 11.1% |
| 20% | 25% |
| 30% | 42.9% |
| 50% | 100% |
| 75% | 300% |
This is why protecting against losses matters more than chasing gains in investing.
Key Takeaways
- Percentage change = (new − old) ÷ old × 100
- Positive result = increase, negative = decrease
- Percentage change has direction; percentage difference is symmetric
- A 50% drop needs a 100% gain to recover (not 50%)
- Always divide by the original value, not the new value