Question

Two pipes A and B can fill a cistern in $12$ minutes and $15$ minutes respectively while a third pipe C can empty the full tank in $6$ minutes. A and B kept open for $5$ minutes in the beginning and then C was also opened. In what time is the cistern emptied?
A) $30\min $
B) $33\min $
C) $37\dfrac{1}{2}\min $
D) \[45\]min

Answer 1

Hint: First, we shall note down the given information so that it will be easy for us to solve. It is given that two pipes A and B can fill a cistern in $12$ minutes and $15$ minutes. Also, it is given that the third pipe C can empty the full tank in $6$ minutes. Now, A and B are kept open for $5$ minutes in the beginning and then C is also opened. We are asked at what time the cistern is emptied.
In this question, first, we need to find the part filled by pipe A in one minute. Similarly, we need to find the part filled by pipe B in one minute. Also, we need to calculate the part emptied by pipe C in one minute. And then, we can calculate the part emptied when all pipes are opened in one minute. Then, we can calculate the part emptied by pipes A and B.

Complete step by step answer:
From the given two pipes A and B can fill a cistern in $12$ minutes and $15$ minutes.
First, we need to find the part filled by pipe A in one minute.
The part filled by pipe A in one minute $ = \dfrac{1}{{12}}$
 Similarly, the part filled by pipe B in one minute$ = \dfrac{1}{{15}}$
The part filled in $5$ minutes by both pipes$ = 5 \times \left( {\dfrac{1}{{12}} + \dfrac{1}{{15}}} \right)$
$ = 5 \times \dfrac{9}{{60}}$
$
   = \dfrac{9}{{12}} \\
   = \dfrac{3}{4} \\
 $
Since the third pipe C can empty the full tank in$6$minutes, we get,
The part emptied by pipe C in one minute$ = \dfrac{1}{6}$
The part emptied when all pipes are opened in one minute\[ = \dfrac{1}{6} - \left( {\dfrac{1}{{12}} + \dfrac{1}{{15}}} \right)\]
$ = \dfrac{1}{6} - \dfrac{9}{{60}}$
$ = \dfrac{1}{{60}}$
Hence, $\dfrac{1}{{60}}$ part is emptied in one minute.
One part is emptied in $60$ minutes.
Hence, $\dfrac{3}{4}$ the part can be emptied by pipes A and B in $\dfrac{3}{4} \times 60$$ = 45$minutes.

So, the correct answer is “Option D”.

Note: First, we need to find the part filled by pipe A in one minute. Similarly, we need to find the part filled by pipe B in one minute. Also, we need to calculate the part emptied by pipe C in one minute. And then, we can calculate the part emptied when all pipes are opened in one minute. Then, we can calculate the part emptied by pipes A and B.