Two radioactive materials $X_1$ and $X_2$ have decay constants  $5\lambda$ and $\lambda$ respectively. If initially they have the same number of nuclei, then the ratio of the number of nuclei of $X_1$ to that of $X_2$ will be $\frac{1}{e}$ after a time 

Let the required time bet. Then

${\mathrm{N}}_1={\mathrm{N}}_0{\mathrm{e}}^{-{\lambda }_1\mathrm{t}};{\mathrm{N}}_2={\mathrm{N}}_0{\mathrm{e}}^{-{\lambda }_2\mathrm{t}}$

Where  

$N_1=$number of nuclei of $X_1$ after time t

$N_2=$ number of nuclei of $X_2$ after time t 

$N_0=$ initial number of nuclei of $X_1$ and $X_2$ each.

Now, $\frac{{\mathrm{N}}_1}{{\mathrm{N}}_2}=\frac{{\mathrm{N}}_0{\mathrm{e}}^{-{\lambda }_1\mathrm{t}}}{{\mathrm{N}}_0{\mathrm{e}}^{-{\lambda }_2\mathrm{t}}}\text{ Here }\frac{{\mathrm{N}}_1}{{\mathrm{N}}_2}=\frac{1}{\mathrm{e}}$

$\begin{array}{l}{\lambda }_1=5\lambda ;{\lambda }_2=\lambda \\ \therefore \frac{1}{\mathrm{e}}=\frac{{\mathrm{e}}^{-5\lambda \mathrm{t}}}{{\mathrm{e}}^{-\lambda \mathrm{t}}}\Rightarrow {\mathrm{e}}^{-1}={\mathrm{e}}^{-4\lambda t}\Rightarrow 4\lambda \mathrm{t}=1\\ \therefore \mathrm{t}=\frac{1}{4\lambda }\end{array}$