A worked example: 100 units, half-life of 10, after 30 units of time
Starting with 100 units of something with a half-life of 10, after 30 units of elapsed time — exactly 3 half-lives — only 12.5 units remain: 100 → 50 → 25 → 12.5, halved three times in a row.
The formula: N(t) = N₀ × (1/2)^(t ÷ half-life)
Dividing elapsed time by the half-life gives the number of half-lives that have passed, and raising one-half to that power gives the fraction of the original quantity remaining. When that count is a whole number, this is just repeated halving; the exponent form handles any elapsed time, whole or fractional.
A fractional half-life count: 200 units, half-life of 8, after 20 units of time
20 units of elapsed time against an 8-unit half-life is 2.5 half-lives — not a whole number — leaving 35.355339 units remaining out of 200. The fractional exponent correctly interpolates between "2 half-lives" (50 units) and "3 half-lives" (25 units), landing between them rather than requiring a whole-number count.
What half-life is really describing
Half-life is exponential decay parameterized by a specific milestone — the time to lose half the quantity — rather than by a period-by-period percentage rate. Both describe the identical underlying decay curve, just from different starting information.