Question

A cistern which could be filled in 9 hours takes one hour more to be filled owing to a leak in its bottom. If the cistern is full, in what time will the leak empty it?
A. 19 hours
B.1 hour
C.90 hours
D.\[\dfrac{3}{5}\] hours

Answer 1

Hint: Here we will firstly find the part of cistern which will be filled in one hour without any leak. Then we will find part of the cistern which will be filled in one hour with the leak at the bottom. Further, we will find the difference between them to get the total leakage in 1 hour. Then we will reciprocate it to get the value of the total time in hours to empty the cistern by that leak.

Complete step-by-step answer:
It is given that the cistern can be filled in 9 hours without any leak in it.
Therefore we will find the part of cistern which will be filled in one hour without any leak, we get
So in 1 hour, \[\dfrac{1}{9}\] part of the cistern without leak is filled.
It is also given that cistern takes one hour more to be filled owing to a leak in its bottom.
So the time taken to fill the cistern when it has a leakage in the bottom is 10 hours.
Therefore we will find the part of cistern which will be filled in one hour with the leak at the bottom, we get
So in 1 hour, \[\dfrac{1}{{10}}\] part of the cistern with a leak is filled.
So, the difference between the part of filled cistern without leak and the part of filled cistern with leak will give us the total leakage in 1 hour. Therefore, we get
Total leakage in 1 hour \[ = \dfrac{1}{9} - \dfrac{1}{{10}}\]
\[ \Rightarrow \] Total leakage in 1 hour \[ = \dfrac{{10 - 9}}{{90}} = \dfrac{1}{{90}}\]
Hence the total time taken by the full cistern to be emptied by the leak from the bottom of the cistern will be equal to \[ = \dfrac{1}{{\dfrac{1}{{90}}}} = 90hours\]
Hence, if the cistern is full, 90 hours will be taken for the leak to empty it.
So, option C is the correct option.

Note: Here we should note that while calculating the part of cistern filled in 1 hour we simply have to reciprocate to get the part of cistern. A cistern is a simple container which is generally used to store the water and has a closed structure. The difference between them will give us the total leakage in 1 hour not the addition of them will give us the leakage in 1 hour. In the calculation part, we need to keep the time in hours only otherwise it will lead to the wrong solution.