Half-Life Calculator
Find how much of a decaying substance remains after a given time.
3 half-lives elapsed · 12.5% remaining
With a half-life of 5 units, a 100-unit sample loses half its amount every 5 units of time. After 15 units, that’s 15 ÷ 5 = 3 half-lives, so the amount halves three times: 100 → 50 → 25 → 12.5. So 12.5 units (12.5%) remain.
What a half-life is
A half-life (t½) is the time it takes for half of a decaying substance to disappear. After one half-life half remains, after two half-lives a quarter remains, after three an eighth — the amount falls by the same factor of ½ over each equal interval, which is what makes the decay exponential rather than linear.
N₀ = initial amount · t = elapsed time · t½ = half-life (t and t½ in the same units)
Worked example
How much of a 100-unit sample remains after 15 units of time, given a half-life of 5 units?
- 1 Count the half-lives elapsed. Divide elapsed time by the half-life: n = t ÷ t½ = 15 ÷ 5 = 3.
- 2 Apply the decay law. N = N₀ × (½)ⁿ = 100 × (½)³ = 100 × ⅛.
- 3 Compute the remaining amount. N = 100 × 0.125 = 12.5 units — that’s 12.5% of the original sample left.
Fraction remaining after each half-life
Each half-life multiplies the remaining amount by ½.
| Half-lives elapsed | Fraction remaining | Percent remaining |
|---|---|---|
| 1 | ½ | 50% |
| 2 | ¼ | 25% |
| 3 | ⅛ | 12.5% |
| 4 | 1⁄16 | 6.25% |
| 5 | 1⁄32 | 3.125% |
Reading the result
It never truly reaches zero. Exponential decay always leaves some fraction behind — after 10 half-lives roughly 0.1% remains, after 20 about a millionth. In practice the amount becomes undetectable, but mathematically it only approaches zero.
Units must match. The half-life and the elapsed time must use the same time units. Carbon-14, for instance, has a half-life of about 5730 years, so a 11,460-year-old sample (two half-lives) holds about a quarter of its original carbon-14 — the basis of radiocarbon dating.
The elapsed time need not be a whole number of half-lives. The exponent t ÷ t½ can be fractional; (½) raised to a non-integer power still gives a valid fraction remaining.