A radioactive isotope X with a half-life of 1.37 $\times$ 10⁹ yr decays to Y which is stable. A sample
of rock from the moon was found to contain both the elements X and Y which were in the ratio of 1 : 7.  The age of the rock is

If in the rock there is no Y element, then the time taken by element X to reduce to $\left(\frac{1}{8}\right)$th the initial value will be equal to

$\frac{1}{8}={\left(\frac{1}{2}\right)}^n\text{ or }n=3$

Therefore, from the beginning three half-life time is spent.
Hence, the age of the rock is

$=3\times 1.37\times {10}^9=4.11\times {10}^9\mathrm{yr}$