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Percentage Change (Before → After)

Percentage change measures increase or decrease from an original value: (New − Original) / Original × 100. It's asymmetric—the baseline matters. $100→$125 is +25%; $125→$100 is −20%. CAGR extends this to multi-period growth.

Concept Fundamentals
+25%
$100→$125
−20%
$125→$100
(End/Start)^(1/n)−1
CAGR
New/Original
Multiplier

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Asymmetric: +25% then −20% does not return to start. $100→$125→$100. CAGR assumes constant annual growth; actual returns may vary year to year. Log percentage change ln(New/Orig) is additive across periods.

Key quantities
+25%
$100→$125
Key relation
−20%
$125→$100
Key relation
(End/Start)^(1/n)−1
CAGR
Key relation
New/Original
Multiplier
Key relation

Ready to run the numbers?

Why: Stock prices, salaries, GDP, inflation—all use percentage change to express growth or decline. CAGR (compound annual growth rate) smooths multi-year changes into a single annual rate.

How: Formula: (New − Original) / |Original| × 100. For CAGR over n periods: (New/Original)^(1/n) − 1, then ×100 for percent. Handle negative originals with absolute value in denominator.

Asymmetric: +25% then −20% does not return to start. $100→$125→$100.CAGR assumes constant annual growth; actual returns may vary year to year.
Sources:Percentage ChangeCAGR

Run the calculator when you are ready.

Understanding ChangePercentage change is relative to the starting value—always divide by the original.

Quick Examples — Click to Load

Input Values

pct_change
CALCULATED
$ calc --original=100 --new=125
Percentage Change
+25.00%
Absolute Change
25.00
Direction
increase
Multiplier
×1.250
results.cagr !== null || results.logChange !== null || results.symmetricPercentageChange !== null&& (
Log Change
22.31%
Symmetric % Change
22.22%
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Before vs After

Step-by-Step Breakdown

CALCULATION
Formula
% Change = ((New - Old) / |Old|) × 100
Absolute change
125 - 100 = 25.00
ext{New} - ext{Original}
Divide by original
25.00 ÷ 100 = 0.250000
Multiply by 100
0.250000 × 100
ext{Convert} ext{to} ext{percentage}
ANSWER
RESULT
+25.00%
Log change
ln(125/100) × 100 = 22.31%
ext{Useful} ext{for} ext{multiplicative} ext{processes}
Symmetric % change
200 × (125-100)/(125+100) = 22.22%
ext{Order}- ext{independent} ext{measure}

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

— Asymmetry

— CAGR

Key Takeaways

  • • Percentage change measures relative change — direction matters (increase vs decrease)
  • • A 50% increase followed by a 50% decrease does not return you to the original value
  • • Use CAGR when comparing growth over multiple periods (smooths volatility)
  • • Symmetric percentage change is order-independent — useful when comparing two values without a clear "before/after"
  • • The multiplier (new/old) tells you how many times larger or smaller the new value is

Did You Know?

📈A 100% increase doubles the value; a 100% decrease reduces it to zero — the asymmetry is why percentage changes don't "cancel out"Source: Finance
📊Economists often use log returns (ln(new/old)) because they are additive: two consecutive log returns sum to the total log returnSource: Economics
🏦CAGR smooths volatile returns: if a stock goes +50% then -30%, the 2-year CAGR is about 5.8%, not the arithmetic averageSource: Investing
🌍Population growth rates are typically small percentages (1–3%) but compound dramatically — 2% for 35 years doubles the populationSource: Demographics
💡Symmetric percentage change: going from 100 to 150 gives +40%, but 150 to 100 gives -33.3% — symmetric formula gives ±40% for bothSource: Statistics
📉A 50% drop requires a 100% gain to recover — percentages are not symmetric around zeroSource: Mathematics

How Percentage Change Works

Percentage change measures the relative difference between an original (or "before") value and a new (or "after") value. It answers: "By what percent did the value change?"

The Formula

% Change = ((New − Old) / |Old|) × 100

We use the absolute value of the old value in the denominator so that the sign of the result correctly indicates direction: positive = increase, negative = decrease.

CAGR (Compound Annual Growth Rate)

When growth happens over multiple periods, CAGR gives the constant rate that would produce the same final value: CAGR = (Final/Initial)^(1/n) − 1, where n is the number of periods.

Symmetric Percentage Change

For comparing two values without a clear order: 200 × (A − B) / (A + B). This gives the same magnitude whether you go from A to B or B to A.

Expert Tips

Recovery Asymmetry

After a 50% loss, you need a 100% gain to break even. After a 25% loss, you need ~33% gain. The required recovery is always larger than the loss.

Base Effect

A change from 1% to 2% is a 100% relative increase but only 1 percentage point. Always clarify: "percentage change" vs "percentage point change."

Log Returns

For multiplicative processes (stock returns, population), log change = ln(new/old)×100. Small changes approximate percentage change; large changes differ.

Quick Mental Check

Doubling = +100%. Halving = -50%. 1.5× = +50%. 0.75× = -25%. Use the multiplier (new/old) for quick sanity checks.

Comparison: Percentage Change vs Percentage Difference vs Relative Change

MeasureFormulaUse Case
Percentage Change(New−Old)/|Old|×100Before→After (direction matters)
Percentage Difference|A−B|/avg(A,B)×100Comparing two values (order doesn't matter)
Relative Change(New−Old)/Old (as decimal)Same as % change, just not ×100

Frequently Asked Questions

How do I calculate percentage change?

Subtract the old value from the new value, divide by the absolute value of the old value, and multiply by 100. Formula: ((New − Old) / |Old|) × 100.

What is the difference between percentage change and percentage difference?

Percentage change has a clear before and after — direction matters. Percentage difference is symmetric and measures how far apart two values are relative to their average, regardless of order.

Why can't the original value be zero?

Division by zero is undefined. Percentage change compares the change to the starting value; if you started at zero, any change is infinite in relative terms.

When should I use CAGR instead of simple percentage change?

Use CAGR when you have growth over multiple periods (years, quarters) and want a single smoothed rate. Simple percentage change is for two points only.

What is symmetric percentage change?

It's 200×(A−B)/(A+B). It gives the same magnitude whether you go from A to B or B to A, useful when there's no natural "before" and "after."

How do I find the new value after a percentage increase?

Multiply the original by (1 + rate/100). For example, 100 after +25% = 100 × 1.25 = 125. For a decrease, use (1 − rate/100).

Quick Reference Numbers

+100%
Doubling
-50%
Halving
×1.5
+50% Multiplier
×0.8
-20% Multiplier

Disclaimer: This calculator provides mathematical results for educational and practical purposes. For financial decisions, tax calculations, or investment analysis, always verify with a qualified professional. Past performance does not guarantee future results.

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