Question

A water tank is \[{\dfrac{2}{5}^{th}}\] full. Pipe A can fill the tank in 10 minutes and pipe B can empty it in 6 minutes. If both the pipes are open, how long will it take to empty or fill the tank completely?

A) 6 minutes to empty
B) 6 minutes to fill
C) 9 minutes to empty
D) 9 minutes to fill

Answer 1

Hint:
In the given problem, the tank is given to be \[{\dfrac{1}{2}^{th}}\] filled. Pipe A can fill the tank and Pipe B can empty the tank while pipe A can do its job in 10 minutes, whereas pipe B can do its job in 6 minutes. So, it’s definite that the tank is going to be empty first. We are to find how much part of the tank gets emptied in 1 minute. Then by the further calculation, we can find the required result.

Complete step by step solution:
Now, the \[{\dfrac{2}{5}^{th}}\] part of the Tank is already filled as given in the problem. Also, it is given that Pipe A can fill the tank in 10 minute whereas pipe B can empty the tank in 6 minutes.
Hence, pipe B is faster than pipe A. So, the tank got emptied first.
Now, the part of the tank to be emptied = \[\dfrac{2}{5}\]. The part emptied by pipes A and B together
i.e. $\left( {A + B} \right)$ in $1$ minute \[ + \dfrac{1}{6} - \dfrac{1}{{10}} = \dfrac{1}{{15}}\]
By taking L.C.M.and solving it;
The part emptied by pipes \[(A + B) = \dfrac{2}{{30}}\] in \[1\] minute
\[ = \dfrac{1}{{15}}\]
Now, \[{\dfrac{1}{{15}}^{th}}\] part of the tank has been emptied in \[1\] minute.
So, suppose, \[{\dfrac{2}{5}^{th}}\] part of the tank has been emptied in ‘m’ minutes
Now, by using Ratio & proportions.
\[\dfrac{1}{{15}}:\dfrac{2}{5}::1:m\]
On solving these terms, we get
$\dfrac{{\dfrac{1}{{15}}}}{{\dfrac{2}{5}}} = \dfrac{1}{m}$
i.e. $m = \dfrac{{\dfrac{2}{5}}}{{\dfrac{1}{{15}}}} = 2 \times 3 = 6\min .$

Hence, the whole tank would be emptied in $6$ minutes.

Note:
In such problems we first need to calculate the unit measures and hence, we can get the required result by further tricks and calculations.
Now, as pipe B is given to be faster, both pipes A and B together will empty the tank only.
So, we calculated that If pipe A is filling the whole tank in $10$ minutes.
In \[1\] minute, it will fill \[{\dfrac{1}{{10}}^{th}}\] of the tank.
Similarly,
If pipe B is emptying the whole tank in \[6\] minutes. In \[1\] minute, it will fill \[{\dfrac{1}{6}^{th}}\] of the tank.
Hence, to find the part of the tank both of them will empty would be \[ + \dfrac{1}{6} - \dfrac{1}{{10}} = \dfrac{1}{{15}}\]
and, hence we solved the whole problem by unitary method. So, the correct option is
i.e. 6 minutes to empty.