The half life period of radioactive element x is same as the mean life time of another radioactive element y. Initially they have the same number of atoms. Then :

Given, ($\tau$_1/2)_x = ($\tau$)_y<br/><br/>Here, $\tau$_1/2 = half-life period of radioactive element and $\tau$ = mean life period of radioactive element.<br/><br/>As we know the expression,<br/><br/>Half-life of the radioactive element x,<br/><br/>${\tau _{1/2}} = {{\ln (2)} \over {{\lambda _x}}}$<br/><br/>Mean life of the radioactive element y,<br/><br/>$\tau = {1 \over {{\lambda _y}}}$<br/><br/>Substituting the values in Eq. (i), we get<br/><br/>$${{\ln 2} \over {{\lambda _x}}} = {1 \over {{\lambda _y}}} \Rightarrow {\lambda _x} = 0.693{\lambda _y}$$<br/><br/>Initially they have same number of atoms,<br/><br/>${N_x} = {N_y} = {N_0}$<br/><br/>As we know that,<br/><br/>Activity, $A = \lambda N$<br/><br/>As, ${\lambda _x} < {\lambda _y} \Rightarrow {A_x} < {A_y}$<br/><br/>Therefore, y will decay faster than x.