Discussion
1600 atoms of a radioactive substance with a half life period T start decaying at t=0. The time during which 1550 atoms would have decayed is
Answer: D
It takes t = 5T.
When 1550 of the 1600 atoms have decayed, 50 remain.
\[ \begin{aligned} \frac{N}{N_{0}} &= \frac{50}{1600} = \frac{1}{32} \\ &= \left(\frac{1}{2}\right)^{5} \end{aligned} \]
So five half-lives have passed: 1600 → 800 → 400 → 200 → 100 → 50. 4T leaves 100 atoms and 6T leaves 25.
A quick count: each half-life halves the number left, so after n half-lives the fraction remaining is \(\frac{1}{2^{n}}\). Finding the n with \(2^{n} = \frac{1600}{50} = 32\) gives n = 5 directly.
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