🔢 Math
📈 📈 Percentage Change Calculator: Increase and Decrease Formula
Learn how to calculate percentage increase and decrease with the correct formula. Covers common mistakes, worked examples from finance and everyday life, and the CAGR formula.
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Percentage change is one of the most useful calculations in everyday life — you encounter it in salary negotiations, investment returns, sale prices, inflation reports, and scientific measurements. It converts a raw number change into a relative proportion, making it possible to compare changes across very different quantities. A $10 increase means very different things on a $20 item (50% increase) versus a $1,000 item (1% increase).
The Percentage Change Formula
Percentage Change = ((New Value − Old Value) ÷ |Old Value|) × 100
The absolute value in the denominator matters when the starting value is negative (more on this below).
- Positive result = percentage increase
- Negative result = percentage decrease (report as a positive number with "decrease" label)
Percentage Increase Formula
% Increase = ((New − Original) ÷ Original) × 100
Example: Salary raised from $55,000 to $60,500
- Change = $60,500 − $55,000 = $5,500
- % Increase = ($5,500 ÷ $55,000) × 100 = 10%
Percentage Decrease Formula
% Decrease = ((Original − New) ÷ Original) × 100
Example: Price reduced from $80 to $68
- Change = $80 − $68 = $12
- % Decrease = ($12 ÷ $80) × 100 = 15%
Real-World Worked Examples
Investment Return
You bought a stock at $45 per share; it's now $58.50. What's your return?
- % Change = (($58.50 − $45) ÷ $45) × 100 = ($13.50 ÷ $45) × 100 = 30% increase
Inflation / Price Comparison
A grocery basket cost $120 last year; it costs $129.60 today. Inflation rate?
- % Change = (($129.60 − $120) ÷ $120) × 100 = ($9.60 ÷ $120) × 100 = 8% increase
Science / Measurement Error
A measured length is 9.5 cm; the actual length is 10 cm. Percentage error?
- % Error = (|9.5 − 10| ÷ 10) × 100 = (0.5 ÷ 10) × 100 = 5% error
Negative Starting Value
A company's losses went from −$25,000 to −$15,000. What's the change?
- % Change = ((−15,000 − (−25,000)) ÷ |−25,000|) × 100 = (10,000 ÷ 25,000) × 100 = 40% decrease in losses
- The absolute value in the denominator prevents a confusing double-negative result
The Biggest Mistake: Using the Wrong Base
Always divide by the original (starting) value, not the final value. This is the most common error:
- Wrong: Stock went from $90 to $72. Change = 18 ÷ 72 × 100 = 25% decrease ❌
- Right: Change = (90 − 72) ÷ 90 × 100 = 18 ÷ 90 × 100 = 20% decrease ✅
The base always reflects "compared to where you started," not "compared to where you ended up."
The Asymmetry of Percentage Changes
A percentage increase and the same percentage decrease do NOT cancel out. This counterintuitive fact trips up many people:
- Start with $100. Increase 10%: $100 × 1.10 = $110
- Decrease 10% from $110: $110 × 0.90 = $99 (not $100)
A 10% increase followed by a 10% decrease leaves you 1% below where you started. To exactly reverse a 10% decrease, you need an 11.11% increase:
Recovery % = (1 ÷ (1 − decrease%)) − 1 × 100
Recovery from 10% loss = (1 ÷ 0.9 − 1) × 100 = 11.11%
This asymmetry is why investment losses are so painful — a 50% loss requires a 100% gain just to break even.
Chaining Multiple Percentage Changes
When multiple percentage changes apply sequentially, you multiply — not add — the multipliers:
Price increases 10%, then 20%: final = original × 1.10 × 1.20 = original × 1.32 = 32% total increase (not 30%)
This matters for salary negotiations: two 5% raises in two years gives 10.25% total, not 10%.
Compound Annual Growth Rate (CAGR)
When measuring change over multiple years, CAGR annualizes the growth rate correctly:
CAGR = (Final Value ÷ Beginning Value)^(1/years) − 1
Example: Investment grows from $10,000 to $16,105 over 5 years:
- CAGR = ($16,105 ÷ $10,000)^(1/5) − 1 = 1.6105^0.2 − 1 = 1.10 − 1 = 10% per year
CAGR smooths out year-to-year volatility to give the equivalent steady annual growth rate. It's the standard metric for comparing investment performance over different time periods.
Quick Reference: Common Percentage Shortcuts
- To increase by 15%: multiply by 1.15
- To decrease by 20%: multiply by 0.80
- To find what % A is of B: (A ÷ B) × 100
- To find original value after X% increase: Final ÷ (1 + X/100)
- To find original value after X% decrease: Final ÷ (1 − X/100)
❓ Frequently Asked Questions
What is the percentage change formula?▼
Percentage Change = ((New Value − Old Value) ÷ Old Value) × 100. A positive result is an increase; negative is a decrease. Example: price rises from $50 to $65 → ($15 ÷ $50) × 100 = 30% increase. Always divide by the original (starting) value, not the new value.
Why does a 10% increase followed by a 10% decrease not return to the original?▼
Because each percentage applies to a different base. Starting at $100: +10% gives $110. Then −10% of $110 = $11 reduction, leaving $99. The second percentage is calculated on the new (higher) amount. To exactly reverse any percentage decrease, you need a larger percentage increase: recovery from a 10% loss requires an 11.11% gain.
How do I calculate percentage change when the starting value is negative?▼
Use the absolute value in the denominator: % Change = ((New − Old) ÷ |Old|) × 100. Example: losses improve from −$50,000 to −$30,000. Change = (−30,000 − (−50,000)) ÷ |−50,000| × 100 = 20,000 ÷ 50,000 × 100 = 40% improvement. The absolute value prevents confusing sign errors.
What is CAGR and when do I use it?▼
CAGR (Compound Annual Growth Rate) measures the annualized growth rate over multiple years: (Final ÷ Beginning)^(1/years) − 1. Use it when comparing investments or business metrics over different time periods. A company that grew 60% over 5 years has a CAGR of (1.60)^0.2 − 1 = 9.86%/year — more informative than the raw 5-year total.
What is the most common mistake when calculating percentage change?▼
Dividing by the wrong number. Percentage change must always be divided by the original (starting) value, not the final value. If a stock price fell from $90 to $72: the correct denominator is $90 (original), giving an 18 ÷ 90 = 20% decrease — not 18 ÷ 72 = 25%, which uses the wrong base.