A pipe can fill a tank in $16$ hours. Due to a leak in the bottom, it is instead filled in $24$ hours. If the tank is completely full, how much time will the leak take to empty it?

Let the filling pipe be $A$ and the leak be $B$.
* Rate of pipe $A$ working alone = $+\frac{1}{16}$ of the tank per hour.
* Combined rate of pipe $A$ and leak $B$ together = $+\frac{1}{24}$ of the tank per hour.

Let the emptying rate of the leak $B$ be $-\frac{1}{x}$ per hour.

$$\frac{1}{16}-\frac{1}{x}=\frac{1}{24}$$

$$\frac{1}{x}=\frac{1}{16}-\frac{1}{24}$$

Finding a common denominator (which is $48$):

$$\frac{1}{x}=\frac{3-2}{48}=\frac{1}{48}$$

$$x=48\text{ hours}$$

Thus, the leak will empty the full tank in $48$ hours.