Two pipes A and B can fill a tank in 15 minutes and 20 minutes respectively. Both pipes are opened together, but after 4 minutes pipe A is turned off. What is the total time required to fill the tank completely?

Introduction / Context:
This is a staged filling question where two pipes initially work together, and then one pipe is shut off while the other continues. We must compute how much of the tank is filled in the first stage, determine the remaining fraction, and then calculate how long the remaining pipe needs to fill that remainder.

Given Data / Assumptions:
- Pipe A fills the tank in 15 minutes
- Pipe B fills the same tank in 20 minutes
- Both start together and run for 4 minutes
- After 4 minutes, pipe A is closed and only pipe B continues
- The tank is initially empty and there are no leaks

Concept / Approach:
We consider two phases: phase one with both pipes open, and phase two with only pipe B open. For each phase we calculate the fraction of the tank filled using rate multiplied by time. The sum of the fractions from both phases must be exactly equal to 1. This gives us the total time, which we convert into minutes and seconds.

Step-by-Step Solution:
Step 1: Rate of A = 1/15 tank per minute. Step 2: Rate of B = 1/20 tank per minute. Step 3: Combined rate of A and B = 1/15 + 1/20. Step 4: LCM of 15 and 20 is 60, so combined rate = 4/60 + 3/60 = 7/60 tank per minute. Step 5: In the first 4 minutes, fraction filled = 4 * 7/60 = 28/60 = 7/15. Step 6: Fraction of tank remaining after 4 minutes = 1 - 7/15 = 8/15. Step 7: In phase two, only B works with rate 1/20 tank per minute. Step 8: Let t be the additional time needed by B alone. Then t * 1/20 = 8/15. Step 9: Solve for t: t = 8/15 * 20 = 160/15 minutes = 10 2/3 minutes. Step 10: 10 2/3 minutes = 10 minutes 40 seconds. Step 11: Total time = initial 4 minutes + 10 minutes 40 seconds = 14 minutes 40 seconds.

Verification / Alternative check:
We can verify by computing the total filled fraction. Phase one fills 7/15 of the tank. In the next 10 2/3 minutes, B fills (10 2/3) / 20 = 32/60 = 8/15 of the tank. Sum = 7/15 + 8/15 = 1, confirming that the tank becomes full exactly at 14 minutes 40 seconds.

Why Other Options Are Wrong:
15 minutes 20 seconds, 16 minutes 40 seconds and 13 minutes 10 seconds: Each of these totals leads to either more or less than one full tank when the filled fractions from both phases are computed. They do not satisfy the fractional equation precisely.

Common Pitfalls:
One common mistake is to continue using the combined rate for the whole time, forgetting that pipe A is closed after 4 minutes. Another error is mishandling the conversion between fractional minutes and seconds. Keep track of each phase separately and always check that the total fraction filled equals exactly one.

Final Answer:
The tank will be completely filled in 14 minutes 40 seconds.