Percentage Change: How to Calculate Increase and Decrease
Master percentage change calculations for finance, business, data analysis, and everyday life. Learn the formulas, avoid common errors, and apply percent change with confidence.
What Is Percentage Change?
Percentage change is a mathematical measure that expresses the difference between two values as a percentage of the original value. It answers the question: "How much did this value change relative to where it started?" This simple calculation is one of the most widely used mathematical operations in business, finance, economics, science, and everyday decision-making. Stock market analysts measure percentage change in share prices. Retailers calculate percentage change in sales. Scientists measure percentage change in experimental results.
The power of percentage change is that it normalizes changes across different scales. A $1 change in a $5 stock (20%) is treated very differently from a $1 change in a $200 stock (0.5%). Percentage change makes these comparisons meaningful by relating the absolute change to the original value. This normalization — expressing change relative to the starting point — is what makes percentage change useful across all disciplines.
The Core Formula
Percentage Change = ((New Value — Original Value) ÷ Original Value) × 100. If a stock price rises from $50 to $60, the percentage change is (($60 — $50) ÷ $50) × 100 = ($10 ÷ $50) × 100 = 20%. If the price falls from $50 to $40, the change is (($40 — $50) ÷ $50) × 100 = (—$10 ÷ $50) × 100 = —20%. The sign of the result indicates the direction of the change — positive for increases, negative for decreases.
Percentage Increase vs Percentage Decrease
The same formula handles both increases and decreases. The sign of the result tells you the direction, and the absolute value tells you the magnitude. A result of +35% means a 35% increase. A result of —22% means a 22% decrease. In everyday language, we usually drop the negative sign and say "decreased by 22%," but the mathematical formula always produces a signed result.
Percentage Increase: Growth and Appreciation
Percentage increase measures how much a value has grown. Common applications include: investment returns (a stock that goes from $100 to $130 gained 30%), population growth (a town that grows from 10,000 to 11,500 residents increased by 15%), sales growth (quarterly revenue from $1M to $1.2M is a 20% increase), and salary increases (a raise from $50,000 to $55,000 is a 10% increase).
Percentage Decrease: Decline and Depreciation
Percentage decrease measures how much a value has declined. Common applications include: price discounts (a $200 item on sale for $150 is a 25% decrease), asset depreciation (a car worth $25,000 new that is now worth $18,000 depreciated by 28%), weight loss (from 200 lb to 170 lb is a 15% decrease), and budget cuts (from $500,000 to $425,000 is a 15% decrease).
Swipe sideways to compare columns.
| Scenario | Original Value | New Value | Absolute Change | Percentage Change | Interpretation |
|---|---|---|---|---|---|
| Stock price increase | $50.00 | $65.00 | +$15.00 | +30% | 30% gain on investment |
| Sale discount | $80.00 | $60.00 | —$20.00 | —25% | 25% off original price |
| Population growth | 25,000 | 27,500 | +2,500 | +10% | 10% population increase |
| Car depreciation | $30,000 | $21,000 | —$9,000 | —30% | 30% value loss over 3 years |
| Test score improvement | 72/100 | 90/100 | +18 | +25% | 25% improvement from baseline |
| Revenue decline | $800,000 | $640,000 | —$160,000 | —20% | 20% revenue drop |
Step-by-Step Calculation Guide
Calculating percentage change is straightforward but requires careful attention to the order of operations and identifying which value is the "original" or reference value.
- Identify the original (starting) value and the new (ending) value. The original value is the baseline against which the change is measured. For "month over month" sales, last month is the original. For "year over year," last year is the original.
- Subtract the original value from the new value. This gives the absolute change — the raw difference. A positive result means the new value is larger; a negative result means it is smaller.
- Divide the absolute change by the original value (NOT the new value). This converts the absolute change into a proportion relative to the starting point. This decimal expresses "how many times the original value the change represents."
- Multiply by 100 to convert the decimal to a percentage. Add a % sign. Include a + or — sign to indicate direction unless the context makes it obvious.
Worked example: A company's revenue was $2.5 million last year and $3.0 million this year. Original value = $2.5M. New value = $3.0M. Absolute change = $3.0M — $2.5M = +$0.5M. Proportion = $0.5M ÷ $2.5M = 0.20. Percentage = 0.20 × 100 = 20%. Revenue increased by 20% year over year.
The Reversibility Trap: Why 50% Up Then 50% Down Is Not Even
One of the most counterintuitive properties of percentage change is that it is not reversible. A 50% increase followed by a 50% decrease does not return to the original value — it results in a net 25% loss. Start with $100. Increase by 50%: $100 × 1.50 = $150. Then decrease by 50%: $150 × 0.50 = $75. The net change is —25%, not 0%.
This asymmetry exists because the two percentage changes are applied to different bases. The 50% increase is applied to the original $100 base. The 50% decrease is applied to the new $150 base, which is 50% larger. The 50% of $150 ($75) is larger than the 50% of $100 ($50), so the decrease outweighs the previous increase.
The mathematical consequence: to reverse a p% increase, you need a decrease of p/(1+p) × 100%. To reverse a 50% increase: 0.50/1.50 × 100 = 33.33%. A 33.33% decrease from $150 returns to $100. To reverse a 25% decrease: 0.25/0.75 × 100 = 33.33%. A 33.33% increase from $75 returns to $100.
Swipe sideways to compare columns.
| Initial Change | Multiplier | Result After Increase | Reverse Change Needed | Multiplier | Final Value |
|---|---|---|---|---|---|
| +10% | 1.10 | $110 | —9.09% | 0.9091 | $100 |
| +25% | 1.25 | $125 | —20.00% | 0.8000 | $100 |
| +50% | 1.50 | $150 | —33.33% | 0.6667 | $100 |
| +100% | 2.00 | $200 | —50.00% | 0.5000 | $100 |
| +200% | 3.00 | $300 | —66.67% | 0.3333 | $100 |
Percentage Change vs Percentage Difference
Percentage change measures change over time — from an original value to a new value. Percentage difference compares two values without designating one as the "original." The formula for percentage difference is: |V₁ — V₂| ÷ ((V₁ + V₂) ÷ 2) × 100. Note the denominator uses the AVERAGE of the two values, not the original value.
Percentage difference is used when there is no clear chronological or reference relationship between the values. For example, comparing the GDP of two countries, or comparing prices between two stores. The absolute value in the numerator ensures the result is always positive — percentage difference is nondirectional.
Swipe sideways to compare columns.
| Values | Percentage Change (V₁ → V₂) | Percentage Change (V₂ → V₁) | Percentage Difference (Nondirectional) |
|---|---|---|---|
| V₁ = 100, V₂ = 120 | +20.0% | —16.7% | 18.2% |
| V₁ = 100, V₂ = 80 | —20.0% | +25.0% | 22.2% |
| V₁ = 50, V₂ = 100 | +100% | —50.0% | 66.7% |
| V₁ = 100, V₂ = 100 | 0% | 0% | 0% |
| V₁ = 100, V₂ = 200 | +100% | —50.0% | 66.7% |
The table illustrates that percentage change depends entirely on which value you designate as the original, while percentage difference produces a single symmetric result. When reporting statistics, be precise about which metric you are using — "our sales increased 20% year over year" (percentage change) is very different from "our sales are 18% different from our competitor" (percentage difference).
Try the Percentage Change CalculatorCalculate percentage increase, decrease, and difference with step-by-step breakdowns.Real-World Applications of Percentage Change
Finance and Investing: Returns and Drawdowns
Investment returns are universally reported as percentage changes. A mutual fund that gains 12% in one year and loses 5% the next has a cumulative return of (1.12 × 0.95) — 1 = 1.064 — 1 = 6.4%, not 7%. The average annual return (arithmetic mean: (12% + (—5%)) ÷ 2 = 3.5%) differs from the compound annual growth rate (CAGR: approximately 3.15%). Financial professionals use CAGR because it accounts for the compounding effect and the asymmetry of percentage changes.
Business: Markup and Margin
A 25% markup on cost is a 20% margin on revenue. The percentage change framework explains this: cost ($100) → selling price ($125) is a 25% increase from cost. But profit ($25) as a percentage of price ($125) is 20%. The difference is the base — cost vs price. This is covered in depth in our Markup vs Margin article, but the core insight is that percentage change interpretation depends entirely on which value you designate as the base.
Economics: Inflation and Real Values
Inflation is a percentage change in the Consumer Price Index over time. If CPI rises from 250 to 257, the inflation rate is (257 — 250) ÷ 250 × 100 = 2.8%. To adjust a $50,000 salary for inflation: $50,000 × (1 + 0.028) = $51,400. The distinction between nominal changes (before inflation) and real changes (after inflation) is essential — a 3% salary increase when inflation is 4% is actually a —1% real decrease.
Data Analysis: Year-over-Year and Month-over-Month
Businesses frequently calculate YoY and MoM percentage changes to smooth out seasonal patterns. A retailer with $2M in sales this January and $1.8M last January has 11.1% YoY growth. Comparing this January to this December ($2.5M) would show a —20% MoM decline, but this is seasonal — December is always higher. YoY comparisons eliminate seasonality and reveal the underlying trend.
Advanced Percentage Change Concepts
Logarithmic (Continuous) Returns
In finance, logarithmic returns are often preferred over simple percentage returns because they are time-additive. Log return = ln(V₂ / V₁). For a $100 to $150 increase, log return = ln(1.5) = 0.405 = 40.5%, compared to the simple return of 50%. For small changes (< 10%), log returns and simple returns are nearly identical. For large changes, they diverge. Log returns have the property that a +50% log return followed by a —50% log return DOES cancel out — which simple returns do not.
Percentage Points vs Percent Change
A common source of confusion: percentage points vs percent change. If an interest rate rises from 4% to 6%, it increased by 2 percentage points (absolute change) but by 50% (relative change: (6—4) ÷ 4 × 100 = 50%). Both statements are correct but mean different things. In finance and economics, "basis points" (100 basis points = 1 percentage point) provide additional precision. A 50-basis-point rate increase means a 0.50 percentage point increase.
Swipe sideways to compare columns.
| Scenario | From | To | Change in Percentage Points | Percent Change |
|---|---|---|---|---|
| Interest rate hike | 4.00% | 4.50% | +0.50 pp | +12.5% |
| Unemployment drop | 8.0% | 6.0% | —2.0 pp | —25.0% |
| Tax rate increase | 22.0% | 25.0% | +3.0 pp | +13.6% |
| Market share gain | 15.0% | 20.0% | +5.0 pp | +33.3% |
| Bond yield increase | 3.0% | 3.5% | +0.5 pp | +16.7% |
What is the formula for percentage decrease?
It is the same formula as percentage change: ((New — Original) ÷ Original) × 100. The result will be negative, indicating a decrease. The magnitude of the negative number tells you the percentage decrease.
How do I calculate percentage change when the original value is zero?
Percentage change is undefined when the original value is zero because division by zero is not mathematically defined. In this case, report the absolute change instead. For example, "revenue went from $0 to $1,000" rather than "revenue increased by infinity percent."
How do I calculate percentage change between negative numbers?
The standard formula still works but interpretation requires care. A change from —$100 to —$50: ((—50) — (—100)) ÷ (—100) × 100 = (50) ÷ (—100) × 100 = —50%. This means the loss decreased by 50% — which is an improvement despite the negative percentage.
What is the difference between percentage change and percentage point?
Percentage change measures relative change (new — old) ÷ old × 100. Percentage point measures absolute change (new — old). If a tax rate goes from 20% to 25%, the percentage point change is +5 pp, and the percent change is +25%. Always specify which you mean.
How do I calculate cumulative percentage change over multiple periods?
Convert each percentage change to a multiplier (1 + decimal change), multiply all multipliers, subtract 1, and convert back to percentage. For +10%, —5%, +8%: cumulative = (1.10 × 0.95 × 1.08) — 1 = 1.1286 — 1 = 0.1286 = 12.86%. Do NOT simply add the percentages (10 — 5 + 8 = 13% — close but not exact).