A pump can fill a tank with water in $3$ hours. Due to leakage in the bottom, it takes $3\frac{1}{2}$ hours to fill it. In what time will the leak empty the fully filled tank?

Let the inlet pump be $A$ and the bottom leak be $B$.
* Work rate of pump $A=\frac{1}{3}$ of the tank per hour.
* Combined work rate of pump and leak $(A-B)=\frac{1}{3\frac{1}{2}}=\frac{1}{\frac{7}{2}}=\frac{2}{7}$ of the tank per hour.

We can solve for the leak rate ($B$):

$$\frac{1}{3}-B=\frac{2}{7}$$

$$B=\frac{1}{3}-\frac{2}{7}$$

$$B=\frac{7-6}{21}=\frac{1}{21}$$

The leak empties $\frac{1}{21}$ of the tank in one hour. Therefore, it will take $21\text{ hours}$ to empty the fully filled tank.