A water tank is two-fifths full. Pipe A can fill the whole tank in 10 minutes and pipe B can empty the whole tank in 6 minutes. If both pipes are opened together, will the tank be emptied or filled, and how long will it take for the process to complete?

Introduction / Context:
This problem combines an inlet pipe and an outlet pipe starting from a partially filled tank. We must determine whether the net effect is filling or emptying and then compute how long it takes to either reach full or become empty. Such questions are solved by comparing the rates of filling and emptying and then applying them to the current volume.

Given Data / Assumptions:

The tank is initially two-fifths full.

Pipe A fills the entire tank in 10 minutes.

Pipe B empties the entire tank in 6 minutes.

Both pipes are opened at the same time.

Tank capacity is taken as 1 unit.

Concept / Approach:
We convert the times for A and B into rates. Then we compute the net rate when both A and B are open together. If the net rate is negative, the tank is being emptied. Knowing the initial content, we divide that volume by the magnitude of the net rate to find the time required for the tank to become empty.

Step-by-Step Solution:
Step 1: Let tank capacity be 1 unit. Initially, volume of water = 2 / 5 unit.Step 2: Rate of pipe A (filling) = 1 / 10 tank per minute.Step 3: Rate of pipe B (emptying) = 1 / 6 tank per minute.Step 4: Net rate when both are open = 1 / 10 - 1 / 6.Step 5: Compute 1 / 10 - 1 / 6 = (3 / 30 - 5 / 30) = -2 / 30 = -1 / 15 tank per minute.Step 6: The negative sign shows the tank is being emptied at a net rate of 1 / 15 tank per minute.Step 7: Time to empty the tank from 2 / 5 full to empty = (2 / 5) / (1 / 15) = (2 / 5) * 15 = 6 minutes.

Verification / Alternative check:
We can check the net effect after 6 minutes: net emptying rate is 1 / 15 per minute, so in 6 minutes, total emptied volume is 6 * (1 / 15) = 6 / 15 = 2 / 5, which equals the initial volume of water. That means the tank just becomes empty after 6 minutes, matching our calculation.

Why Other Options Are Wrong:
6 minutes to fill would require a positive net rate and an initial content lower than final, which is not the case.

9 minutes to empty or 9 minutes to fill both correspond to using incorrect net rates or miscalculating the initial fraction of water.

Common Pitfalls:
Some students mistakenly add the rates, as if both pipes were filling, which gives the wrong net effect. Others forget to account for the initial partial filling and instead compute the time to empty a full tank. Always first determine whether the net flow is into or out of the tank, and then apply that net rate to the actual starting volume.

Final Answer:
The tank will be emptied, and it will take 6 minutes to empty completely.