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

Introduction / Context:
This is a two-phase filling problem. Initially, one tap is running, and later additional identical taps are opened, increasing the total filling rate. We need to carefully handle the fraction of the tank filled in each phase and then add the times.

Given Data / Assumptions:

One tap alone fills the tank in 6 hours.

After half the tank is filled, three more taps identical to the first are opened.

All taps are assumed to have constant flow rates.

We consider the tank capacity as 1 full unit for easier calculation.

Concept / Approach:
We treat the filling process in two parts. In the first part, a single tap fills half the tank. In the second part, four identical taps (the original one plus three more) fill the remaining half. Using rate times time equals work done, we compute the time for each phase and then sum them.

Step-by-Step Solution:
Step 1: Let the tank capacity be 1 unit. One tap fills it in 6 hours, so its rate is 1 / 6 tank per hour.Step 2: Time to fill half the tank with one tap = (1 / 2) / (1 / 6) = 3 hours.Step 3: After this, three more identical taps are opened. There are now 4 taps in total.Step 4: Combined rate of 4 taps = 4 * (1 / 6) = 4 / 6 = 2 / 3 tank per hour.Step 5: Remaining volume to be filled = 1 - 1 / 2 = 1 / 2 tank.Step 6: Time to fill the remaining half with 4 taps = (1 / 2) / (2 / 3) = (1 / 2) * (3 / 2) = 3 / 4 hours.Step 7: 3 / 4 hours equals 45 minutes.Step 8: Total filling time = 3 hours + 45 minutes = 3 hours 45 minutes.

Verification / Alternative check:
We can compute how much work is done per quarter hour in the second phase. In 15 minutes (1 / 4 hour) with 4 taps, the amount filled is (2 / 3) * (1 / 4) = 1 / 6. Therefore, in 45 minutes (3 such intervals) the taps fill 3 * (1 / 6) = 1 / 2 of the tank, which confirms our earlier calculation that 45 minutes are enough to complete the remaining half.

Why Other Options Are Wrong:
3 hours 15 minutes assumes an incorrect combined rate or mistakenly uses only three taps in the second phase.

4 hours and 4 hours 15 minutes both overestimate the time, indicating that the contribution of multiple taps in the second phase was not properly accounted for.

Common Pitfalls:
Some students forget to distinguish between the two phases and incorrectly assume four taps work from the start. Others mis-handle the fractional calculations, such as adding rates incorrectly or forgetting that only half the tank remains when extra taps are opened. Always compute how much of the tank is filled in each phase and use the correct number of taps and rates for each period.

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