Let's break down the problem step by step.

Step 1: Define variables for efficiencies

Let the efficiency of pipe B be $b$, which represents the amount of the tank filled per minute by pipe B.

• Efficiency of pipe A is $0.60b$, since A is 60% efficient compared to B.
• Efficiency of pipe C is $0.50b$, since C is 50% efficient compared to B.

Step 2: Determine the total work done in one cycle

The pipes are opened alternatively, each for 1 minute. This gives a repeating cycle of 3 minutes. In each cycle, the work done is:

• Pipe A runs for 1 minute, so it fills $0.60b$ of the tank.
• Pipe B runs for 1 minute, so it fills $b$ of the tank.
• Pipe C runs for 1 minute, so it fills $0.50b$ of the tank.

Thus, the total amount of tank filled in one cycle of 3 minutes is:

$0.60b + b + 0.50b = 2.10b$

Step 3: Determine the number of cycles

We are told that the three pipes together fill the tank in $25 \frac{5}{7}$ minutes. Convert this to an improper fraction:

$25 \frac{5}{7} = \frac{180}{7} \text{ minutes.}$

Since each cycle lasts 3 minutes, the number of cycles required to fill the tank is:

$\frac{180}{7} \div 3 = \frac{180}{7} \times \frac{1}{3} = \frac{60}{7} \text{ cycles.}$

Step 4: Set up an equation for the total work done

In $\frac{60}{7}$ cycles, the total amount of tank filled is:

$\text{Total work} = \left( \frac{60}{7} \right) \times 2.10b = 1 \text{ (since the tank is completely filled).}$

Thus, we have:

$\frac{60}{7} \times 2.10b = 1$

Solving for $b$:

$\frac{60}{7} \times 2.10b = 1$ $2.10 \times \frac{60}{7} = 1 / b$ $b = \frac{60 \times 2.10}{7} = \frac{126}{7} = 18 \text{ per minute.}$

Step 5: Time taken for A and B together to fill the tank

Now that we know the efficiency of pipe B is 18, the efficiency of pipe A is $0.60 \times 18 = 10.8$. The combined efficiency of pipes A and B is:

$\text{Efficiency of A and B together} = 10.8 + 18 = 28.8$

To find the time taken for A and B together to fill the tank, we use the formula:

$\text{Time} = \frac{1}{\text{Efficiency of A and B together}} = \frac{1}{28.8}$

Calculating the time:

$\text{Time} = \frac{1}{28.8} = \frac{10}{288} = \frac{5}{144} \text{ minutes.}$

Thus, the time taken for A and B together to fill the tank is:

$\boxed{ \text{Time taken} = \frac{5}{144} \text{ minutes.} }$