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 classical pipes and cisterns question involving changing conditions during the filling process. Initially, two pipes work together, and then one pipe is closed. To solve such problems, we translate times into rates of work, compute how much of the tank is filled in each phase, and then determine the total time taken.

Given Data / Assumptions:

Pipe A fills the tank in 15 minutes, so its rate is 1/15 of the tank per minute.

Pipe B fills the tank in 20 minutes, so its rate is 1/20 of the tank per minute.

Both pipes are opened together for the first 4 minutes.

After 4 minutes, pipe A is closed and only pipe B continues to fill the tank.

Concept / Approach:
We use the idea that work done equals rate multiplied by time. We first find the combined filling rate of A and B, then compute the fraction of the tank filled in 4 minutes. From that, we find the remaining fraction and calculate how long pipe B alone needs to fill the rest. Finally we add the two time segments.

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 = (4 + 3) / 60 = 7 / 60 tank per minute.Step 4: In the first 4 minutes, amount filled = 4 * (7 / 60) = 28 / 60 = 7 / 15 of the tank.Step 5: Remaining fraction of the tank = 1 - 7 / 15 = 8 / 15.Step 6: Now only B works, with rate 1/20 tank per minute.Step 7: Time taken by B to fill the remaining 8 / 15 = (8 / 15) / (1 / 20) = (8 / 15) * 20 = 160 / 15 = 32 / 3 minutes.Step 8: 32 / 3 minutes equals 10 minutes 40 seconds.Step 9: Total time = initial 4 minutes + 10 minutes 40 seconds = 14 minutes 40 seconds.

Verification / Alternative check:
We can convert everything into seconds to double check. The total time is 14 minutes 40 seconds, which is 880 seconds. In 880 seconds, pipe B works for 880 seconds, and pipe A works only for the first 240 seconds. Converting to minutes and applying rates will again give exactly one full tank, confirming our answer.

Why Other Options Are Wrong:
10 minutes 20 seconds and 11 minutes 45 seconds come from mixing up the remaining fraction or miscomputing combined rates.

12 minutes 30 seconds underestimates the total time and usually indicates that the 4 minutes of joint work and the later solo work of B were not separated correctly.

Common Pitfalls:
Students often forget to compute the remaining fraction after the first phase and instead try to average the times. Another common error is to mis-handle fractions when converting into minutes and seconds, leading to small but important numerical mistakes. Always compute rates carefully and break the problem into clear stages.

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