Two identical conducting rods are first connected independently to two vessels, one containing water at 100$^{\circ}$C and the other containing ice at 0$^{\circ}$C. In the second case, the rods are joined end to end and connected to the same vessels. Let q$_1$ and q$_2$ gram per second be the rate of melting of ice in the two cases respectively. The ratio $\frac{q_1}{q_2}$ is