One tap can fill an empty water tank completely in 50 minutes, while another tap can empty a completely filled tank in 75 minutes. If both taps are opened together when the tank is already half full, in how many minutes will the tank become completely full?

Introduction / Context:
This is a pipes and cisterns problem, which is conceptually identical to a time and work problem. One tap fills the tank and another empties it. We are given the individual times for filling and emptying, and the tank is initially half full. The task is to determine how long it will take for the tank to become completely full when both taps are open simultaneously.

Given Data / Assumptions:

• Inlet tap (filling tap) alone fills the tank in 50 minutes.
• Outlet tap (emptying tap) alone empties a full tank in 75 minutes.
• Initially, the tank is half full.
• Both taps are opened together.
• The rates of filling and emptying are constant.
• Total capacity of the tank is considered as 1 unit.

Concept / Approach:
We first compute the filling and emptying rates in units per minute, then find the net rate when both taps are open simultaneously. Since the tank is already half full, only half of the tank's capacity remains to be filled. Dividing the remaining volume by the net rate gives the time to fill the tank completely.

Step-by-Step Solution:
Step 1: Let the total capacity of the tank be 1 unit. Step 2: Inlet tap fills 1 unit in 50 minutes, so filling rate = 1/50 units per minute. Step 3: Outlet tap empties 1 unit in 75 minutes, so emptying rate = 1/75 units per minute. Step 4: When both taps are open, the net rate = filling rate - emptying rate = 1/50 - 1/75. Step 5: Find a common denominator for 1/50 and 1/75. The least common multiple of 50 and 75 is 150. Step 6: Convert rates: 1/50 = 3/150 and 1/75 = 2/150. Step 7: Net rate = 3/150 - 2/150 = 1/150 units per minute. Step 8: Initially, the tank is half full, so remaining volume to be filled = 1/2 unit. Step 9: Time required = remaining volume / net rate = (1/2) / (1/150) = 1/2 * 150 = 75 minutes.

Verification / Alternative check:
We can reason about the result qualitatively. If there were no outlet, the remaining half would be filled in 25 minutes (since full tank is 50 minutes). But because water is also draining, it should take more than 25 minutes. Our result is 75 minutes, which seems plausible because the outlet is slowing down the filling significantly. The exact rate calculation confirms the answer.

Why Other Options Are Wrong:

• 60 minutes: Implies a higher net filling rate than 1/150 units per minute, which does not match the given individual rates.
• 125 minutes: This is too long, and at that rate the outlet would almost cancel the inlet more than our calculation allows.
• 150 minutes: This is the time required to fill the entire tank at the net rate starting from empty, not from half full.

Common Pitfalls:
Common mistakes include adding the rates instead of subtracting them, or forgetting that the tank is initially half full and treating it as empty. Another pitfall is mishandling the fractions when computing 1/50 - 1/75. Always find the correct common denominator and subtract accurately to avoid arithmetic errors.

Final Answer:
The half full tank will become completely full in 75 minutes when both taps are opened together.