$X$        $\to $           $Y$

Initial number of atoms,                            $N_0$                     $N$

Number of atoms after time $t$                $N$                         $N_0-N$

As per question

$\frac{N}{{{N_0} – N}} = \frac{1}{7} \Rightarrow 7N = {N_0} – N$  or  $8 N=N_{0}$

${\frac{N}{N_{0}}=\frac{1}{8}}$

As $\frac{N}{N_{0}}=\left(\frac{1}{2}\right)^{n}$ where $n$ is the no. of half lives

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

$\therefore n=3$

$n=\frac{t}{T_{1 / 2}} \text { or } t=n T_{1 / 2}=3 \times 20 \text { years }=60 \text { years }$

Hence, the age of rock is $60$ years.