Pipe A can fill a tank in 16 minutes and pipe B can empty the same tank in 24 minutes. If both pipes are opened together, after how many minutes should pipe B be closed so that the tank is exactly full in a total time of 30 minutes?

Introduction / Context:
This problem mixes a filling pipe and a draining pipe that initially work together, but one pipe is later closed. We must choose the correct moment to close the outlet pipe so that the tank becomes exactly full after a specified total time. This needs careful handling of rates and time intervals.

Given Data / Assumptions:
- Pipe A fills the tank in 16 minutes
- Pipe B empties the tank in 24 minutes
- Both start together at time zero
- Total time for the tank to become full = 30 minutes
- Pipe B will be closed after some time t minutes, while A continues

Concept / Approach:
We divide the process into two phases: an initial phase where both A and B are open, and a second phase where only A runs. We then compute the fraction of the tank filled in each phase in terms of t. The sum of these fractions must equal 1. Solving this equation gives the required time t after which pipe B should be closed.

Step-by-Step Solution:
Step 1: Rate of pipe A = 1/16 tank per minute. Step 2: Rate of pipe B = -1/24 tank per minute (since it empties). Step 3: Combined rate with both open = 1/16 - 1/24. Step 4: Take LCM of 16 and 24 which is 48. So combined rate = (3 - 2) / 48 = 1/48 tank per minute. Step 5: Let pipe B be open for t minutes. Fraction filled in this period = t * 1/48 = t/48. Step 6: Remaining time with only pipe A = 30 - t minutes. Fraction filled during this time = (30 - t) * 1/16. Step 7: Total fraction filled = t/48 + (30 - t)/16 = 1. Step 8: Convert to common denominator 48: t/48 + 3(30 - t)/48 = 1, so t + 90 - 3t = 48. Step 9: Simplify: 90 - 2t = 48, so 2t = 42 and t = 21 minutes.

Verification / Alternative check:
For the first 21 minutes, both pipes run, filling 21 * 1/48 = 21/48 = 7/16 of the tank. Remaining fraction = 1 - 7/16 = 9/16. Only A runs for the last 9 minutes, and A fills 9 * 1/16 = 9/16 of the tank. Total = 7/16 + 9/16 = 1 full tank, matching the requirement.

Why Other Options Are Wrong:
24, 20 or 22 minutes: Each of these values for t fails to satisfy the equation t/48 + (30 - t)/16 = 1. They would either overfill or underfill the tank by the end of 30 minutes.

Common Pitfalls:
Learners sometimes forget to treat the outlet as a negative rate or incorrectly assume that the tank is half full at half time. Another frequent error is incorrect algebra when solving the linear equation for t. Writing out the equation clearly and checking the arithmetic avoids such mistakes.

Final Answer:
Pipe B should be closed after 21 minutes so that the tank is full in exactly 30 minutes.