$Q = \frac{{KA\left( {{T_1} - {T_2}} \right)t}}{L}$                          $,,,(i)$

Where $K$ is the thermal conductivity of the material of the rod.

Area of $cross-section$ of new rod

$A' = \pi {\left( {\frac{R}{2}} \right)^2} = \frac{{\pi {R^2}}}{4} = \frac{A}{4}$                 $...(ii)$

As the volume of the rod remains unchanged

$\therefore AL = A'L'$

Where $L'$ is the length the new rod

$or\,\,\,\,L' = L\frac{A}{{A'}}$                           $,,,(iii)$

$ = 4L$                                                     $(Using (ii))$

Now, the amount of heat flows in same time $t$ in the new rod with its ends maintained at the same temperatures $T_1$ and $T_2$ is given by

$Q' = \frac{{KA'\left( {{T_1} - {T_2}} \right)t}}{{L'}}$                   $...(iv)$

Substituting the values of $A'$ and $L'$ from equations $(ii)$ and $(iii)$ in the above equation, we get

$Q' = \frac{{K\left( {A/4} \right)\left( {{T_1} - {T_2}} \right)t}}{4L}$

$ = \frac{1}{{16}}\frac{{KA\left( {{T_1} - {T_2}} \right)t}}{L} = \frac{1}{{16}}Q$