Scope and assumptions
Growth % compares end to start; compound form applies a fixed annual rate. Illustrative.
A city, a bank balance, and a bacterial culture share the same arithmetic: a starting number, an ending number, and time between them. The subtlety isn't the division — it's that a growth rate applied repeatedly behaves nothing like a growth rate applied once.
That distinction separates a back-of-envelope estimate from a projection that holds up.
One period: the percent
Growth % = (End − Start) ÷ Start × 100
| Start | End | Growth |
|---|---|---|
| 1,000 | 1,030 | 3% |
| 500 | 600 | 20% |
| 2,000 | 1,800 | −10% |
The base is the start. Dividing by the end instead gives a smaller number for any increase, because the end is the larger figure.
Many periods: the power
Final = Start × (1 + r)^t
A thousand people growing at 2% a year for a decade: 1,000 × 1.02¹⁰ ≈ 1,219. The naive estimate — 2% × 10 = 20%, so 1,200 — is close here but drifts further with every year or every increase in the rate.
| Start | Rate | Years | Final |
|---|---|---|---|
| 1,000 | 2% | 10 | 1,219 |
| 1,000 | 2% | 20 | 1,486 |
| 1,000 | 3% | 10 | 1,344 |
| 1,000 | 7% | 10 | 1,967 |
At 7% the linear guess (1,700) is off by nearly 270 people in a decade.
Why the compounding matters
Each year's growth is added to the base, so the next year grows from a bigger number. Two percent of 1,000 is 20; two percent of 1,219 is 24.4. The rate stayed constant while the absolute gain grew — which is why "2% a year" produces much more than "20% over ten years" once the horizon stretches.
Negative growth is the same formula
A fall from 2,000 to 1,800 is −10%. Nothing special happens to the maths; the sign carries the direction. The same compound formula works with a negative r: 1,000 × 0.98¹⁰ ≈ 817, a population losing 2% a year.
Projection Assumptions That Drift
Dividing by the end instead of the start; adding rates linearly across years; confusing a growth rate with a growth amount; dropping the sign on a decline; annualising a single year's change by squaring it; and extending a straight line over decades when compounding dominates.
Where to go deeper
This page is the entry point for the growth-and-decay cluster. Follow a spoke for the case you need:
- Decay calculation — Percentage Decay Calculation
- APR vs APY — Annual Percentage Rate Explained
- Compound growth — Compound Growth Percentage
- Inflation — Inflation Percentage Calculation
- ROI — Percentage Return on Investment
- Depreciation — Depreciation Percentage
- Doubling time — Doubling Time Percentage
- Half-life — Half-Life Percentage Decay
- Change over time — Percentage Change Over Time
FAQ
What's the difference between total growth and annual growth rate? Total growth is the whole change across the period; the annual rate is the per-year pace that compounds to it. A 10-year total of 22% corresponds to about 2% a year, not 2.2.
How do I compare growth across periods of different lengths? Annualise both. CAGR = (End ÷ Start)^(1 ÷ years) − 1 puts every series on a per-year footing so the comparison is fair.
Can a growth rate exceed 100%? Yes. Doubling is 100% growth; tripling is 200%. Rates above 100% are unusual but perfectly valid — the number describes the change, not a ceiling.
Why does 1% growth take about 70 years to double? The rule of 70: doubling time ≈ 70 ÷ rate. At 1% that's 70 years, at 7% it's 10, at 10% it's 7. It's an approximation from the logarithm, accurate enough for planning.
Sources: Khan Academy, Exponential growth; Investopedia, CAGR.
Disclaimer: Illustrative. Real populations use age-specific rates; this is a simplified model. Informational only.
Sources and references
- Khan Academy — Exponential growthhttps://www.khanacademy.org/math
- Investopedia — Compound Annual Growth Ratehttps://www.investopedia.com/terms/c/cagr.asp
Maintenance and corrections
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