A water tank is initially 2/5 full. Inlet pipe A can fill the entire tank in 12 minutes, while outlet pipe B can empty the full tank in 6 minutes. If both pipes are opened together when the tank is 2/5 full, will the tank be filled or emptied, and how long will it take for this to happen completely?

Introduction / Context:
This problem involves one inlet and one outlet pipe acting on a partly full tank. The core concept tested is the net rate of change of water level when both filling and emptying happen simultaneously. Students must determine both the direction of water level change (filling or emptying) and the time required for the tank to reach either full or empty state from the given initial level.

Given Data / Assumptions:

The tank is initially 2/5 full.

Inlet pipe A can fill an empty tank in 12 minutes.

Outlet pipe B can empty a full tank in 6 minutes.

Both pipes are opened at the same time.

Flow rates are constant and only these two pipes affect the water level.

Concept / Approach:
First, compute the individual rates of A and B in terms of tanks per minute. The inlet contributes a positive rate and the outlet contributes a negative rate. The net rate indicates whether the tank is filling or emptying overall. Then, starting from the initial level 2/5 of capacity, we determine how long it takes at this net rate to reach a water level of zero or one full tank. Since the net rate here is negative, the water level will decrease and the tank will eventually empty before it can fill.

Step-by-Step Solution:
Let the tank capacity be 1 unit.Rate of inlet A = 1/12 tank per minute (filling).Rate of outlet B = 1/6 tank per minute (emptying).When both pipes are open, net rate = 1/12 - 1/6.Compute 1/6 as 2/12, so net rate = 1/12 - 2/12 = -1/12 tank per minute.The negative sign shows that the tank is being emptied overall.Initial water volume = 2/5 of capacity.We want the time taken for the volume to decrease from 2/5 to 0 at a rate of 1/12 tank per minute in magnitude.Time = initial volume / magnitude of net rate = (2/5) / (1/12).This equals (2/5) * 12 = 24/5 = 4.8 minutes.So the tank empties completely in 4.8 minutes.
Verification / Alternative check:
In 4.8 minutes, the net volume change is net rate * time = (-1/12) * 4.8.4.8 minutes is 24/5, so net change = (-1/12) * (24/5) = -24 / 60 = -2/5 of a tank.Since we started at 2/5 full, the final volume is 2/5 - 2/5 = 0, confirming that the tank is exactly empty after 4.8 minutes.
Why Other Options Are Wrong:
The option saying it fills in 4.8 minutes contradicts the negative net rate. The options involving 5.6 minutes are based on incorrect calculations of net rate or initial volume. The claim that the level remains constant would require equal filling and emptying rates, which is not the case here because 1/12 and 1/6 are not equal.

Common Pitfalls:
Students often incorrectly add the rates without considering direction, treating both pipes as filling. Others may mistakenly think that because there is an inlet and an outlet, the net effect is zero, which happens only when the magnitudes of the rates match. Another frequent mistake is to compute time to fill the whole tank instead of considering the initial 2/5 level. Always check the sign of the net rate and the starting volume before deciding whether the tank will fill or empty.

Final Answer:
The tank will be emptied completely in 4.8 minutes when both pipes are open.