A cistern can be filled by a tap in 4 hours and emptied by an outlet pipe in 6 hours.
Question:

A cistern can be filled by a tap in 4 hours and emptied by an outlet pipe in 6 hours. How long will it take to fill the cistern if both the tap and the pipe are opened together?

Solution:

Time taken by the tap to fill the cistern $=4$ hours

$\therefore$ Tap fills $\frac{1}{4}$ th part of the cistern in 1 hour.

Time taken by the pipe to empty the cistern $=6$ hours

$\therefore$ Pipe empties out $\frac{1}{6}$ th part of the cistern in 1 hour.

Thus, in 1 hour, $\left(\frac{1}{4}-\frac{1}{6}\right)$ th part of the cistern is filled.

We have:

$\frac{1}{4}-\frac{1}{6}=\frac{6-4}{24}=\frac{2}{24}=\frac{1}{12}$

Thus, in 1 hour, $\frac{1}{12}$ th part of the cistern is filled.

Hence, the cistern will be filled in 12 hours.

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