A sample contains ${10}^{-2} \mathrm{kg}$ each of two substances $A$ and $B$ with half lives $4 \mathrm{s}$ and $8 \mathrm{s}$ respectively. The ratio of their atomic weights is $1:2$. The ratio of the amounts of $A$ and $B$ after $16 \mathrm{s}$ is $\frac{x}{100}$. The value of $x$ is _____ .

Ratio of number of moles of substance in the beginning will be, $\frac{{\left(N_0\right)}_A}{{\left(N_0\right)}_B}=\frac{{\displaystyle \frac{{10}^{-2} \mathrm{kg}}{M_A}}}{{\displaystyle \frac{{10}^{-2} \mathrm{kg}}{M_B}}}=\frac{M_B}{M_A}=\frac{1}{2}$.

Number of half-life in $16 \mathrm{s}$ for element $A$ will be, $n_A=\frac{16}{4}=4$

Number of half-life in $16 \mathrm{s}$ for element $B$ will be, $n_B=\frac{16}{8}=2$

Hence, ratio of number of nuclei after $16 \mathrm{s}$ will be, $=\frac{{\left(N_0\right)}_A\times {\left({\displaystyle \frac{1}{2}}\right)}^{n_A}}{{\left(N_0\right)}_B\times {\left({\displaystyle \frac{1}{2}}\right)}^{n_B}}=\frac{1}{2}\times {\left(\frac{1}{2}\right)}^{4-2}=\frac{1}{8}=\frac{25}{100}$

Thus, value of $x$ is $25$.