Question

A cistern can be filled by two pipes a and b in \[10\] and \[15\] hours respectively and is then emptied by a tap in \[8\] hours.If all the taps are opened,then cistern will be fill in how many hours?

Answer 1

Hint: The individual time needed for each pipe for filling or emptying the cistern is given. So we can calculate the part of the cistern filled in one hour for both inlet pipes, add them and subtract the part emptied by outlet pipe in one hour. Now we found part of the cistern filled by all three pipes in one hour.Using this we found the time required to fill the whole cistern.

Complete step-by-step answer:
It is given that pipes a and b can fill a cistern in \[10\] hours and \[15\] hours respectively and another pipe c can empty the cistern in \[8\] hours.
We are asked to find the time needed to fill the tank if the three pipes are opened together.
Since the pipe needs \[10\] hours to fill the cistern, we can say that \[\dfrac{1}{{10}}\] of the cistern is filled in one hour by a.
Since the pipe b needs \[15\] hours to fill the cistern, we can say that \[\dfrac{1}{{15}}\] of the cistern is filled in one hour by b.
So in one hour, part of the cistern filled by a and b is \[\dfrac{1}{{10}} + \dfrac{1}{{15}}\] .
Also since the pipe c needs \[8\] hours to empty the cistern, we can say that \[\dfrac{1}{8}\] of the tank is emptied in one hour by c.
Considering all these we can say that the part of tank filled in one hour when three pipes are open is \[\dfrac{1}{{10}} + \dfrac{1}{{15}} - \dfrac{1}{8}\].
This gives,
cistern filled in one hour = \[\dfrac{1}{{10}} + \dfrac{1}{{15}} - \dfrac{1}{8}\]
Simplifying we get,
cistern filled in one hour \[ = \dfrac{{12 + 8 - 15}}{{120}} = \dfrac{5}{{120}} = \dfrac{1}{{24}}\]
Now we want to find the time needed to fill the cistern.
Since the number of hours times the part of cistern filled in one hour gives one, we have,
\[t \times \dfrac{1}{{24}} = 1\] , where t is the number of hours.
So we get,
\[t = 24\] hours .
So the time needed to fill the tank if a,b and c are open is \[t = 24\] hours.
 The answer is \[24\] hours.
So, the correct answer is “\[24\] hours”.

Note: This problem can also be done in an efficient method. Efficiency is the work done in unit time (here one hour).
Let us consider the capacity of the cistern as \[120\] (LCM of \[10,15,8\]). The efficiency of the pipes can be considered as \[12,8\] and \[ - 15\] respectively. So, total efficiency is \[12 + 8 - 15 = 5\] .
We know Work done is the product of efficiency and time.
So time is gained by dividing the capacity (work done) by the total efficiency.
This gives, time \[ = \dfrac{{120}}{5} = 24\] hours