Explanation

(a) As $\frac{N}{N_{0}} = \left( \frac{1}{2} \right)^{n}$; where, Number of half lives, $n = \frac{t}{T}$
$T$ is the half life period
For $X$ sample,
$\frac{1}{16} = \left( \frac{1}{2} \right)^{8/T_{X}}$ or $\left( \frac{1}{2} \right)^{4} = \left( \frac{1}{2} \right)^{8/T_{X}} \Rightarrow 4 = \frac{8}{T_{X}}$ …(i)
For $Y$ sample,
$\left( \frac{1}{256} \right) = \left( \frac{1}{2} \right)^{8/T_{Y}}$ or $\left( \frac{1}{2} \right)^{8} = \left( \frac{1}{2} \right)^{8/T_{Y}} \Rightarrow 8 = \frac{8}{T_{Y}}$ …(ii)
Divide (i) by (ii) we get
$\frac{4}{8} = \frac{8}{T_{X}} \times \frac{T_{Y}}{8} \Rightarrow \frac{1}{2} = \frac{T_{Y}}{T_{X}}\ or\ \frac{T_{X}}{T_{Y}} = \frac{2}{1}$