A worked example: 1000 decaying at 10% for 5 periods
A quantity of 1,000 declining by 10% each period is down to 590.49 after 5 periods — a total decline of 409.51, noticeably less than the naive "50% off after 5×10%" guess would suggest, since each period's 10% applies to an already-shrunken amount.
The formula: final value = initial value × (1 − rate)^periods
Each period multiplies the current value by (1 − decay rate) — 0.90 in the example above — and that same multiplication happens again on the new, smaller value the next period. This compounding-in-reverse is why exponential decay always slows down over time in absolute terms, even at a constant percentage rate.
A slower decay: 500 at 5% for 8 periods
A gentler 5% decay rate on a starting value of 500, over 8 periods, leaves 331.710216 — a total decline of 168.289784, proportionally smaller than the first example's decline despite running for more periods, since the per-period rate itself is much lower.
Why a decaying quantity never fully reaches zero
Multiplying by (1 − rate) repeatedly always leaves a nonzero (if eventually tiny) result — mathematically, exponential decay approaches zero but never technically reaches it, which is why real-world applications like radioactive decay or drug elimination talk in terms of "negligible amount remaining" rather than "completely gone."
A special, widely-used case of this same formula
Half-life describes decay defined by a specific milestone — how long it takes to lose exactly half the quantity — rather than a period-by-period percentage. It's the identical underlying math, just parameterized differently.