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                                                                   \(X\)        \(\to \)           \(Y\)

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.