A pipe can fill a cistern in 9 hours.

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.

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