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Radioactive material 'A' has decay constant '$8\lambda$' and material 'B' has decay constant '$\lambda$'. Initially they have same number of nuclei. After what time, the ratio of number of nuclei of material 'B' to that 'A' will be $\frac{1}{e}$?
Explanation
$N_A/N_B = e^{-7\lambda t}$; setting it to $1/e$ gives $t = 1/7\lambda$.
Detailed Solution
As framed ($N_B/N_A = 1/e$), no option is correct, since B decays more slowly and $N_B/N_A$ only increases.
If we take $\frac{N_A}{N_B} = \frac{1}{e}$, then
$\frac{N_A}{N_B} = \frac{e^{-8\lambda t}}{e^{-\lambda t}} = e^{-7\lambda t}$
$\frac{1}{e} = e^{-7\lambda t} \Rightarrow -1 = -7\lambda t$
$t = \frac{1}{7\lambda}$
Note: the source states that no option is correct as the question is worded; the key accepts $\frac{1}{7\lambda}$ by taking the ratio $N_A/N_B = 1/e$.