Law of radioactivity : $$N = {N_0}{e^{ - \lambda t}}$$ where $$\lambda$$ = decay constant

Activity $$\left| A \right| = \left| {{{ - dN} \over {dt}}} \right| = {N_0}\lambda {e^{ - \lambda t}}$$

Rate of activity $$R = {{d|A|} \over {dt}} = {N_0}{\lambda ^2}{e^{ - \lambda t}}$$

At t = 0, A₁ = A₂. Therefore,

$${N_{OP}}{\lambda _P} = {N_{OQ}}{\lambda _Q}$$

At $$t = 2\tau ,\,{{{R_P}} \over {{R_Q}}} = {\left( {{{{\lambda _P}} \over {{\lambda _Q}}}} \right)^2}\left( {{{{N_{OP}}} \over {{N_{OQ}}}}} \right){{{e^{ - \lambda P(25)}}} \over {{e^{ - \lambda Q(25)}}}} = {{{\lambda _P}} \over {{\lambda _Q}}}{e^{({\lambda _Q} - {\lambda _P})25}}$$

Since mean life is given by $$\tau = {1 \over \lambda }$$.

Therefore,

$${{{R_P}} \over {{R_Q}}} = {{{\lambda _P}} \over {{\lambda _Q}}}{e^{{{\left( {{1 \over {25}} - {1 \over 5}} \right)}^{25}}}} = {{{\lambda _P}} \over {{\lambda _Q}}}{e^{ - 1}}$$

$${{{R_P}} \over {{R_Q}}} = {{{\lambda _\{ }} \over {{\lambda _Q}}}{1 \over e} = {n \over e}$$

$$n = {{{\lambda _P}} \over {{\lambda _Q}}} = {{2\tau } \over \tau } = 2$$