Percentage Increase & Decrease Explained
Percentage increase and decrease measure how much a value has changed relative to its starting point. They answer: "By what fraction of the original did this change?" Both use a simple formula dividing the absolute change by the original value.
Understanding these formulas is essential for interpreting salary raises, price changes, interest rate movements, population growth, and investment returns — essentially any time a value changes over time.
Formula
Percentage Change = ((New − Old) / |Old|) × 100
Percentage Increase Formula
Formula: Percentage Increase = ((New − Old) / Old) × 100. Used when New > Old.
Example 1 — Salary raise: Annual salary goes from $52,000 to $57,200. Increase = ((57,200 − 52,000) / 52,000) × 100 = (5,200 / 52,000) × 100 = 10%.
Example 2 — House price: Property value rises from $280,000 to $336,000. Increase = ((336,000 − 280,000) / 280,000) × 100 = (56,000 / 280,000) × 100 = 20%.
New value from % increase: New = Old × (1 + Rate/100). 15% increase on $200 → $200 × 1.15 = $230.
Finding the rate when Old and New are known: Rate = ((New/Old) − 1) × 100. From $80 to $96: (96/80 − 1) × 100 = (1.20 − 1) × 100 = 20%.
| Old Value | New Value | Absolute Increase | % Increase |
|---|---|---|---|
| 100 | 110 | 10 | 10.0% |
| 200 | 230 | 30 | 15.0% |
| 50,000 | 57,500 | 7,500 | 15.0% |
| 1.20 (price) | 1.44 (price) | 0.24 | 20.0% |
| 8,000,000 | 8,400,000 | 400,000 | 5.0% |
Percentage Decrease Formula
Formula: Percentage Decrease = ((Old − New) / Old) × 100. Used when New < Old. Result is positive.
Example 1 — Price discount: Item price drops from $120 to $84. Decrease = ((120 − 84) / 120) × 100 = (36 / 120) × 100 = 30%.
Example 2 — Population decline: Town population falls from 45,000 to 40,500. Decrease = ((45,000 − 40,500) / 45,000) × 100 = (4,500 / 45,000) × 100 = 10%.
New value from % decrease: New = Old × (1 − Rate/100). 25% decrease on $160 → $160 × 0.75 = $120.
Maximum percentage decrease: 100% decrease means the value reached zero. Percentage decreases cannot exceed 100% (values cannot fall below zero in most real-world contexts).
Common Mistakes — % Change vs Percentage Points
Mistake 1 — Confusing % change with percentage points: If a tax rate rises from 20% to 25%, it rose by 5 percentage points (absolute) but by 25% relative (5/20 × 100). Both are correct — specify which you mean.
Mistake 2 — Adding percentages that should be multiplied: A 10% increase followed by a 10% decrease is NOT 0% change. Result: 1.10 × 0.90 = 0.99 = −1% overall.
Mistake 3 — Dividing by the wrong base: The denominator must always be the ORIGINAL (Old) value, not the new value. If price rises from $80 to $100, the increase is 25% (20/80), NOT 20% (20/100).
Mistake 4 — Treating percentage change as reversible: A 20% increase does not undo a 20% decrease. See the asymmetry section below.
Real-World Examples
Salary raise: $65,000 → $72,800. Increase = (7,800/65,000) × 100 = 12%. To find the new salary after any raise %: New = Old × (1 + Rate/100).
Store discount: $85.00 → $68.00. Decrease = (17/85) × 100 = 20%. The item is "20% off."
Population change: City grows from 1,200,000 to 1,380,000. Increase = (180,000/1,200,000) × 100 = 15% growth over the period.
Fuel price: Petrol rises from 142p/litre to 163.3p/litre. Increase = (21.3/142) × 100 = 15%.
Weight change: Person goes from 90 kg to 81 kg. Decrease = (9/90) × 100 = 10% weight loss.
Asymmetry — When Increase ≠ Reverse of Decrease
A percentage decrease and a percentage increase of the same rate applied sequentially do NOT cancel out. This is because the base changes between operations.
Example: Start at $100. Apply 30% decrease: $100 × 0.70 = $70. Apply 30% increase: $70 × 1.30 = $91. Net result: −9%, not 0%.
General rule: After a p% decrease, you need an increase of p/(1−p/100)% to restore the original. After a 30% decrease, recovery requires 30/0.70 = 42.86% increase.
Investing implication: A stock falls 50% (from $100 to $50). To return to $100, it must DOUBLE (+100% increase), not just rise 50%. This is why avoiding large losses is disproportionately important.
To correctly stack two percentage changes: Multiply the factors: (1 + r₁/100) × (1 + r₂/100). A 20% increase then a 15% decrease: 1.20 × 0.85 = 1.02 = +2% overall.