Question

Pipe X can fill a tank in 12 hours and pipe Y can empty the tank in 18 hours. Both pipes are opened at 8 am and after some time Y is closed, and the tank is full at 10 pm on the same day. At what time was pipe Y closed?

• A. 1 pm
• B. 10 am
• C. 11 am
• D. 12 noon

Answer 1

The Correct Option is C

To solve this problem, we need to calculate the time at which pipe Y was closed. We know the following:

• Pipe X fills the tank in 12 hours.
• Pipe Y empties the tank in 18 hours.
• Both pipes are opened at 8 am and the tank is fully filled by 10 pm. 

The question is asking at what time pipe Y was closed. Let's break down the problem using step-by-step reasoning:

1. Determine the rates of work of Pipe X and Pipe Y:
   • Rate of filling by Pipe X = \(\frac{1}{12}\) of the tank per hour.
   • Rate of emptying by Pipe Y = \(\frac{1}{18}\) of the tank per hour.
2. Find the net rate when both pipes are open:
   • Net rate = Rate of Pipe X - Rate of Pipe Y
   • Net rate = \(\frac{1}{12} - \frac{1}{18} = \frac{1}{36}\) of the tank per hour.
3. Calculate the total time when both pipes are open:
   • Both pipes are open from 8 am to the time when Y is closed. Suppose Y is closed after 't' hours.
   • So, the effective time with both pipes open is 't' hours.
   • Therefore, after 't' hours, fraction of tank filled = \(\frac{t}{36}\).
4. Calculate the remaining time for full tank:
   • After pipe Y is closed, pipe X fills the tank alone.
   • If total time from 8 am to 10 pm is 14 hours, then remaining time with only Pipe X open is 14 - t hours.
   • During this time, Pipe X fills \(\frac{14-t}{12}\) of the tank.
5. Equate the two expressions to find 't':
   • For the tank to be full, the sum of the filled parts should be 1 (full tank).
   • So, \(\frac{t}{36} + \frac{14-t}{12} = 1\)
   • Solve for 't':
   • \(\frac{t}{36} + \frac{14-t}{12} = 1\)
   • \(\frac{t}{36} + \frac{14 \times 3 - t \times 3}{36} = 1\)
   • Simplifying: \(\frac{t + 42 - 3t}{36} = 1\)
   • \(\frac{42 - 2t}{36} = 1\)
   • Simplify and solve for 't':
   • \(42 - 2t = 36\)
   • \(2t = 6\)
   • \(t = 3\)

Therefore, pipe Y was closed after 3 hours from 8 am, which is 11 am.