One eighth of a radioactive sample remains after 420 days. Find its half-life.

Solution & explanation

1. Given: Elapsed time = 420 days; remaining fraction = 1/8.

2. Halving once gives 1/2, twice 1/4 and three times 1/8.

3. Identify the number of half-lives from the remaining fraction.

   $$\frac{1}{8}={\left(\frac{1}{2}\right)}^3$$

4. Divide elapsed time by the number of half-lives.

   $$T_h=\frac{420}{3}=140$$

5. State the result with units: T½ = 140 days.

140 days