A single tap can fill a tank in 6 hours. After this tap alone has filled half of the tank, three more identical taps are opened. What is the total time taken to fill the tank completely from empty?

Introduction / Context:
This is a typical staged filling problem where initially one tap works alone and later more identical taps are opened. Because the taps are identical, their combined rate is a simple multiple of the single tap rate. We split the process into two phases and sum the times for each phase.

Given Data / Assumptions:
- One tap alone fills the tank in 6 hours
- This tap is used alone until half the tank is full
- After half the tank is filled, three more similar taps are opened, making four taps in total
- All taps are identical and there are no leaks

Concept / Approach:
For a single tap with filling time 6 hours, its rate is 1/6 of the tank per hour. In the second phase, four identical taps work, so their combined rate is 4 times that of the single tap, that is 4/6 = 2/3 of the tank per hour. The first phase fills half the tank, and the second phase fills the remaining half. We compute time for each phase separately and then add them.

Step-by-Step Solution:
Step 1: Rate of one tap = 1/6 tank per hour. Step 2: Time to fill half the tank with one tap = (1/2) / (1/6) = 1/2 * 6 = 3 hours. Step 3: After this, 1/2 of the tank remains. Step 4: In the second phase, there are 4 identical taps, so combined rate = 4 * 1/6 = 4/6 = 2/3 tank per hour. Step 5: Time to fill the remaining half = (1/2) / (2/3) = 1/2 * 3/2 = 3/4 hour. Step 6: 3/4 hour equals 45 minutes. Step 7: Total time = 3 hours + 45 minutes = 3 hours 45 minutes.

Verification / Alternative check:
We can verify by computing total work. Phase one: 3 hours * 1/6 tank per hour = 1/2 tank. Phase two: 45 minutes is 0.75 hours and the rate is 2/3 tank per hour, so filled amount = 0.75 * 2/3 = 1/2 tank. Total = 1/2 + 1/2 = 1 tank, confirming that the times are correct.

Why Other Options Are Wrong:
4 hours 15 minutes, 3 hours 24 minutes, 4 hours 51 minutes: Each of these totals would imply either too much or too little work done when multiplied by the appropriate rates. They do not satisfy both phase requirements simultaneously.

Common Pitfalls:
A common error is to think that adding more taps simply divides the initial time by the number of taps without considering that this only applies if they all operate for the entire duration. Another mistake is miscalculating the time for the second phase when converting between fractions of an hour and minutes.

Final Answer:
The tank will be completely filled in a total of 3 hours 45 minutes.