• # question_answer Half lives of two isotopes X and Y of a material are known to be $2\times {{10}^{9}}$years and $4\times {{10}^{9}}$ respectively. If a planet was formed with equal number of these isotopes, then the current age of planet, given that currently the material has 20% of X and 80% of Y by number, will be: A) $2\times {{10}^{9}}$years B) $4\times {{10}^{9}}$years C) $6\times {{10}^{9}}$years D) $8\times {{10}^{9}}$years

 ${{N}_{x}}=\frac{{{N}_{0}}}{2}\cdot {{e}^{-{{\lambda }_{1}}t}}=0.2{{N}_{0}}$ ${{N}_{y}}=\frac{{{N}_{0}}}{2}\cdot {{e}^{-{{\lambda }_{2}}t}}=0.8{{N}_{0}}$ ${{e}^{({{\lambda }_{1}}-{{\lambda }_{2}})}}=4$    $\Rightarrow$ $t=8\times {{10}^{9}}$years