Scope and assumptions
Decay % compares the drop to the start; repeated decay compounds the loss. Illustrative.
A single 10% loss is easy to shake off. Ten percent a year for three years is not: you don't lose 30%, you lose about 27% — and the reason it's less than 30% is the same reason it feels worse than it looks. Each loss is taken from a smaller base, so the absolute damage shrinks while the relative damage accumulates.
One step, then many
Single step: Decay % = (Start − End) ÷ Start × 100 Repeated: Final = Start × (1 − r)^t
| Start | End | Decay |
|---|---|---|
| 500 | 450 | 10% |
| 1,000 | 850 | 15% |
| 200 | 180 | 10% |
Repeated: 500 at 10% a year for three years → 500 × 0.9³ = 364.5.
Watching it year by year
| Year | Start of year | Lost (10%) | End of year |
|---|---|---|---|
| 1 | 500 | 50.0 | 450.0 |
| 2 | 450 | 45.0 | 405.0 |
| 3 | 405 | 40.5 | 364.5 |
Three years of 10% removes 135.5 in total, which is 27.1% of the original 500 — not 30%. The lost amount falls every year while the rate stays fixed. That's the signature of a multiplicative process, and it's why decay curves flatten out instead of hitting zero on a schedule.
Decayed versus remaining
The most common reading error: "10% decay" leaves 90%, not 10%. The two phrases describe opposite ends of the same process, and confusing them inverts the conclusion. A process described as "10% remaining" has already lost 90% — a near-total collapse, not a minor dip.
Name the figure you're quoting: decay rate, or share remaining.
Compounding in reverse
The same logic applies to losses as to gains, which is why recovering from a big drop takes more than the drop itself. A 50% loss needs a 100% gain just to return to level. A 20% loss needs a 25% gain. Decay is asymmetric: going down and coming back up are not mirror images.
Decay Errors That Inflate the Result
Dividing by the end value instead of the start; adding the yearly rates across years; forgetting the exponent on repeated decay; conflating decay rate with remaining share; assuming a constant annual loss in dollars rather than a constant percentage; and quoting "10% decay" with no time period attached, which makes the figure meaningless.
Related growth & decay guides
For the rest of the growth-and-decay cluster, start at the hub: Population Growth Percentage. Adjacent: Half-Life Percentage Decay and Percentage Change Over Time.
FAQ
How do I find the value before a decay happened? Divide by what survived. If a 20% loss left 400, the original was 400 ÷ 0.8 = 500. Adding 20% to 400 instead gives 480, which is wrong.
Is decay the same as negative growth? Yes. The formulas are identical apart from the sign of the rate. Naming it "decay" usually signals an exponential process rather than a one-off change.
How is half-life related to the decay rate? Half-life is the time to lose 50%. It equals ln(2) ÷ ln(1 ÷ (1 − r)), which for small rates is roughly 70 ÷ (rate in percent).
Why do the dollar losses shrink each year even at a constant rate? Because the base shrinks. Ten percent of 500 is 50, but ten percent of 450 is 45. A constant percentage produces a declining absolute loss.
Sources: Khan Academy, Exponential decay; Investopedia.
Disclaimer: Illustrative. Real decay (radioactive, biological) follows specific models. Informational only.
Sources and references
- Khan Academy — Exponential decayhttps://www.khanacademy.org/math
- Investopedia — Exponential decayhttps://www.investopedia.com/
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