Understanding the Problem: How Much Time Does it Take for Tapes A and B to Fill the Tank?

To solve the problem of determining how long it will take for tapes A and B to fill the tank together, we need to understand the individual rates at which each tap fills the tank and how these rates combine to fill the tank in a collaborative effort.

Step-by-Step Analysis

Given:

Tape A takes 6 hours to fill the tank alone. Tape B takes 4 hours to fill the tank alone.

Let's translate these times into rates.

Individual Rates

Tape A:

In 1 hour, Tape A fills 1/6 of the tank.

Tape B:

In 1 hour, Tape B fills 1/4 of the tank.

Combined Rate

Working together, the combined rate of Tapes A and B can be calculated as follows:

In 1 hour, Tapes A and B together fill:

1/6   1/4  2/12   3/12  5/12 of the tank

Therefore, to fill the entire tank, it will take:

1 ÷ (5/12)  12/5 hours  2.4 hours  2 hours and 24 minutes

Mathematical Formulation

To generalize the solution using algebra, let:

x be the number of hours it takes Tapes A and B together to fill the tank.

The equation based on the combined rate is:

1/6   1/4  1/x

Multiplying every term by the least common denominator (12), we get:

2/12   3/12  12/12
5/12  12/12
Therefore, 5x  12, and solving for x:

x  12/5  2.4 hours

Conclusion

The time it takes for Tapes A and B to fill the tank together is 2.4 hours or 2 hours and 24 minutes. This solution adheres to the principles of work rate in connected systems, where the individual rates are combined to find the collective rate.

Key Points:

Understand the individual rates of each tape. Add the rates to find the combined rate. Solve the equation using the least common denominator.

To verify the correctness of such problems, always check the combined time by summing the individual contributions and ensuring the total work equals the complete job.