The half-life of a radioactive nucleus is 5 years. The fraction of the original sample that would decay in 15 years is:

Solution

Physics sir · 3 steps

15 years is three half-lives, so find what is left first, then subtract it from the whole.

1. Count half-lives. The time elapsed is 15 years and each half-life is 5 years, so n = 15/5 = 3 half-lives pass.

   N/N₀ = (1/2)ⁿ = (1/2)³ = 1/8

2. Remaining, not decayed. This 1/8 is the fraction of nuclei still undecayed. The sample goes 1 → 1/2 → 1/4 → 1/8 over the three half-lives.
3. So. The fraction that has decayed is 1 - 1/8 = 7/8, which is option (c).

The idea to keep. Always check whether the question asks for the fraction left or the fraction decayed. Get the fraction left from (1/2)ⁿ, and the decayed part is 1 minus that.

Answer from the NTA final answer key. Solution by PrepWiser.