Pipes A and B can fill a tank in 18 hours and 27 hours, respectively, whereas pipe C can empty the full tank in 54 hours. All three pipes are opened together, but pipe C is closed after 12 hours. In how much time (in minutes) will the one-third of the remaining part of the tank be filled by A and B together?

Let the total capacity of the tank be the LCM of 18, 27, and 54, which is 54 units.
Efficiency of pipe A =
18
54
​
=3 units/hour

Efficiency of pipe B =
27
54
​
=2 units/hour

Efficiency of pipe C =−
54
54
​
=−1 units/hour

When all three pipes are opened together, the combined efficiency is:

Combined efficiency=3+2−1=4 units/hour
In 12 hours, the part of the tank filled is:

Work done in 12 hours=4×12=48 units
The remaining part of the tank is:

Remaining part=54−48=6 units
One-third of the remaining part is:

Required part to fill=
3
1
​
×6=2 units
Now, pipe C is closed, so only A and B are working together with a combined efficiency of 3+2=5 units/hour.

Time taken to fill 2 units:

Time=
5
2
​
 hours
Converting hours into minutes:

Time in minutes=
5
2
​
×60=24 minutes