Question

A tank can be filled two pipes A and B separately in 3 hours and 3 hours 45 minutes respectively. A third pipe C can empty the full tank in 1 hour when the tank is exactly filled, all the pipes are opened. The tank will become empty in
(a) 1 hour 15 min
(a) 2 hours 30 min
(a) 3 hours 15 min
(d) 4 hours 10 min

Answer 1

Hint: In order to solve this problem, we need to find the part of the tank that will be emptied in one hour and then add all the parts as all the pipes are open according to the condition. And then, find the time required to empty the tank. We are asked to find the hours taken to empty the tank when half-filled. So we need to divide the answer by half.

Complete step-by-step answer:
We have been given three pipes from which two are used to fill the water and the third one is used to empty the tank.
The pipe A fills the tank in 3 hours individually,
And pipe B fills the tank in 3 hours and 45 minutes individually.
Whereas pipe C is used to empty the tank and it can empty the tank in 1 hour.
We are asked the time to empty the tank when its half-filled and all the taps the opened
We will first calculate the time required to empty the tank when fully with the three taps open and then by dividing by two we can get when half-filled.
Pipe A takes3 hrs to fill the tank fully so part of the tank filled in one hour is $\dfrac{1}{3}$ .
For pipe B first, we need to convert 3 hours 45 minutes in fraction form.
3 hours 45 minutes = $3\dfrac{45}{60}=\dfrac{15}{4}$ hrs. (Because one hour has 60 minutes).
Pipe B takes $\dfrac{15}{4}$ hours to fill the tank fully so part of the tank filled in one hour is $\dfrac{4}{15}$ .
Pipe C takes 1 hour to empty the tank so part of the tank emptying in one hour is fully that is 1.
When all the three pipes are open part of tank emptied in one hour is to add all the ratios taken
Therefore, $\text{part of tank emptied in 1 hour=1-}\left( \dfrac{\text{1}}{\text{3}}\text{+}\dfrac{\text{4}}{\text{15}} \right)$
Our action is to empty the tank but pipe A and B do the opposite action hence the negative sign.
Solving it further, we get,
\[\text{Part of tank emptied in 1 hour=1-}\dfrac{\text{9}}{\text{15}}\text{=}\dfrac{\text{6}}{\text{15}}\text{=}\dfrac{\text{2}}{\text{5}}.......(i)\]
If $\dfrac{2}{5}$ part of the tank is empty in one hour then the time taken to empty the full tank is $\dfrac{5}{2}$ hours.
But we have been asked to find the time taken to empty exactly half tank so dividing by 2 we get,
Time taken to the empty half-filled tank is = $\dfrac{1}{2}\times \dfrac{5}{2}=\dfrac{5}{4}$ hours.
Converting into hours and minutes format we get $\dfrac{5}{4}=1.25$hours.
Converting 0.25 hours to minutes we get, 0.25 x 60 = 15 minutes.
Therefore, the time taken to the empty half-filled tank is 1 hour 15 minutes.
Hence, the correct option is (a).

Note: As we just have to take the reciprocal of the part of the tank filled in one hour to get the number of hours taken to fill the tank. This is possible only to fill the tank full. While converting the decimal form into an hour-minutes clock we need to multiply only the decimal part by 60 to get the extra minutes because one hour has 60 minutes.