Quantity I: Two pipes, X and Y, have different filling rates for a tank. Pipe X can fill the tank in 20 minutes, while pipe Y can do it in 30 minutes. They are both activated simultaneously, but after 6 minutes, pipe Y is shut off. Calculate the total time it takes to completely fill the tank.

Quantity II: There are two taps, A and B, with distinct filling rates for a cistern. Tap A can fill the cistern in 24 minutes, and tap B can do it in 16 minutes. Vijay initiates both taps at precisely 11:00 a.m. and leaves. Upon his return, he notices that an outlet pipe has been opened, so he promptly closes it. Interestingly, the cistern is completely filled by 11:15 a.m. Can you determine the duration required for the outlet pipe, operating independently, to empty the entire cistern?

Solution

ATQ, Quantity I: Let the total capacity of the tank be LCM of (20 and 30) = 60 units Number of units of water filled by pipe X in one minute = 60 ÷ 20 = 3 units Number of units of water filled by pipe Y in one minute = 60 ÷ 30 = 2 units Since, both the pipes are opened for 6 minutes. So, number of units of water filled by both in 6 minutes = (3 + 2) × 6 = 30 units Remaining units of water i.e. 60 – 30 = 30 units is filled by pipe X alone. So, time taken by pipe X alone to fill the remaining water = 30 ÷ 3 = 10 minutes Therefore, the total time taken to fill the tank = 6 + 10 = 16 minute Quantity II: Let the capacity of the tank be LCM of (24 and 16) = 48 units Number of units of water filled by pipe A in one minute = 2 units Number of units of water filled by pipe B in one minute = 3 units Number of units of water filled by both pipes together in one minute = 5 units So, time taken by both pipes together to fill the tank = 48 ÷ 5 = 9 minutes 36 secs But due to outlet pipe opened, the total time taken to fill the tank = 15 minutes Number of units of water emptied by outlet pipe in 9 minutes 36 secs = Number of units of water filled by both pipes together in 5 minute 24 secs Number of units of water emptied by outlet pipe in 9 minutes 36 secs = 5 × 5.4 = 27units Number of units of water emptied by outlet pipe in 1 minute = 27/9.6 Total time taken by outlet pipe to empty the full tank = 48/(27/9.6) = 17.06 minutes