#### 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

\[\text{ Time taken by the tap to fill the cistern = 4 hours } \]

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

\[\text{ Time taken by the pipe to empty the cistern = 6 hours }\]

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

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

\[\text{ We have: } \]

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

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

\[\text{ Hence, the cistern will be filled in 12 hours } .\]