Question

Two pipes can fill a cistern in 30 and 15 h respectively. The pipes are opened simultaneously and it is found that due to leakage in the bottom 5 h extra are taken for cistern to be filled up. If the cistern in full in what time would the leak empty it?

Answer 1

Hint: Here in the given question based on pipe and cistern, it is basically a general aptitude question, this can be solved by using the concept of time and further simplified using the basic mathematical operation. We get the required time for the cistern filled up with leaks and empty it.

Complete step-by-step solution:
A pipe connected with a tank or a cistern or a reservoir that fills it, is known as an inlet and a pipe connected with a tank or cistern or reservoir, emptying it, is known as an outlet.
If a pipe can fill a tank in x hours, then:
part filled in 1 hour = \[\dfrac{1}{x}\].
If a pipe can empty a tank in y hours, then:
part emptied in 1 hour = \[\dfrac{1}{y}\].
If a pipe can fill a tank in x hours and another pipe can empty the full tank in y hours (where y > x), then on opening both the pipes, then
The net part filled in 1 hour = \[\left( {\dfrac{1}{x} - \dfrac{1}{y}} \right)\].
If a pipe can fill a tank in x hours and another pipe can empty the full tank in y hours (where x > y), then on opening both the pipes, then
The net part emptied in 1 hour = \[\left( {\dfrac{1}{y} - \dfrac{1}{x}} \right)\].
Consider the data from given question
Let a two pipes A and B can fill a cistern in 30 and 15 hour respectively
Work done by the pipe A in 1 hour = \[\dfrac{1}{{30}}\] and
Work done by the pipe B in 1 hour = \[\dfrac{1}{{15}}\].
Work done by two pipes A and B in 1 hour = \[\dfrac{1}{{30}} + \dfrac{1}{{15}}\] \[ = \dfrac{3}{{30}} = \dfrac{1}{{10}}\].
Time taken by these two pipes to fill the tank = 10 hours.
Due to leakage, time taken = 10 hrs + 5 hrs = 15 hours.
Therefore, work done by (two pipes + leak) in 1 hour = \[\dfrac{1}{{15}}\]
work done by leak in 1 hour = \[\dfrac{1}{{10}} - \dfrac{1}{{15}}\]= \[\dfrac{1}{{30}}\]
Hence, the leak will empty the cistern in 30 hours.

Note: Here in this question while solving the fractions we take LCM for the denominators. In a fraction if the denominators are the same then there is no need to take LCM. but if the denominators are different, we take LCM. This question is like an aptitude type of a question which can be solved.