Question

There are two pipes A and B connected to a tank. It is known that A can fill the tank in $4$ hours while B can empty the tank in $6$ hours. If both the pipes are opened together, how much time will it take to fill up the tank?
(A) $12\text{ hours}$
(B) $10\text{ hours}$
(C) $2\text{ hours}$
(D) None of the above

Answer 1

Hint:In this type of question we find out the rate at which pipes can fill and empty and tank by combining the work of both the pipes we can get the time at which the tank will be filled. Take the work of one pipe as positive, another pipe work as negative because the second pipe is making the tank empty.

Complete step-by-step answer:
We have given information which, A can fill the tank in $4$ hours And B can empty the tank in $6$ hours.
Now we assume a tank of capacity as $24L$ tank, then the calculation would be easier
So according to the question ‘A’ takes $4$ hours of inflow gives us a full $24L$.
 Thus for getting the rate of inflow we divide the capacity of the tank by number of hours, we get
\[ \Rightarrow Rate{\text{ }}of{\text{ }}inflow = \dfrac{{24}}{4}\; = 6L/hr - - - - (1)\]
Now, according to the question ‘B’ empty the tank in $6$ hours
So, $6$ hours of outflow of empty a full $24L$ gives us rate of outflow of ‘B’
Thus for getting the rate of outflow we divide the capacity of the tank by number of hours, we get
\[ \Rightarrow Rate{\text{ }}of{\text{ out}}flow = \dfrac{{24}}{6}\; = 4L/hr - - - - - (2)\]
Now Both pipes are open, and we find the combined work of ‘A’, ‘B’, we get
\[ \Rightarrow Work{\text{ }}of{\text{ }}A{\text{ }}-{\text{ }}Work{\text{ }}of{\text{ }}B\]
We take the work of B negative because it makes the tank empty.
We take the work as the part of the whole so we write the work done by both in fraction.
Now we put the values of work of ‘A’ and ‘B’, we get
$ \Rightarrow \dfrac{1}{4} - \dfrac{1}{6}$
Now we take the L.C.M of the denominator we get
$ = \dfrac{{3 - 2}}{{12}}$
Now by solving we get
$ = \dfrac{1}{{12}}$
Thus, the time taken to fill the tank is 12 hours.

So, the correct answer is “Option A”.

Note:This type of question is of time and work. So we should consider the work done by both and consider them as a fraction because they are the part of the whole. And check whether the work done is positive or negative then find the value and at the end take the reciprocal of the answer because work done is inversely proportional to the time.