Question:
A pipe can fill a cistern in 9 hours. Due to a leak in its bottom, the cistern fills up in 10 hours. If the cistern is full, in how much time will it be emptied by the leak?
Solution:
Time taken by the pipe to fill the cistern $=9$ hours
Part of the cistern filled in one hour $=\frac{1}{9}$
Suppose the leak empties the full cistern in x hours.
Part of the cistern emptied in one hour $=-\frac{1}{x}$ (negative sign implies a leak)
Time taken by the cistern to fill completely due to the leak $=10$ hours
Part of the cistern filled in one hour due to the leak $=\frac{1}{10}$
$\therefore \frac{1}{10}=\frac{1}{9}-\frac{1}{x}$
$\Rightarrow \frac{1}{x}=\frac{1}{9}-\frac{1}{10}=\frac{1}{90}$
$x=90$ hours
Therefore, the leak will empty a full cistern in 90 hours.