A radioactive isotope decays with a half-life of $t_{1/2}=100\text{ years}$. How many years will it take for $87.5\%$ of the sample to decay?

Solution

1. If $87.5\%$ decays, $12.5\%$ remains.
2. $12.5\%=1/8$ of the initial amount.
3. Since $(1/2{)}^3=1/8$, the time taken is 3 half-lives.
4. Time $$\begin{array}{rl}& =3\times 100\text{ years}\\ & =300\text{ years}\end{array}$$.
5. Hence the answer is (B).