Question

A tank can be filled by Pipe A in 4 hours and by Pipe B in 6 hours. If both pipes are opened together, how long will it take to fill the tank?

• A. 2 hours
• B. 2.4 hours
• C. 3 hours
• D. 3.5 hours
• E.

Hint:

For problems involving two workers (or pipes, etc.), you can use the shortcut formula: \(\text{Time together} = \frac{\text{Product of individual times}}{\text{Sum of individual times}}\). In this case, \(\frac{4 \times 6}{4 + 6} = \frac{24}{10} = 2.4\) hours. This is a quick way to verify your answer.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This is a classic work-rate problem. The key is to determine the rate at which each pipe works and then add the rates together to find their combined rate. The total time is the reciprocal of the combined rate.
Step 2: Key Formula or Approach:

Rate of work = \(\frac{1}{\text{Time taken to complete the work}}\)
Combined Rate = Rate of A + Rate of B
Time taken together = \(\frac{1}{\text{Combined Rate}}\)
Step 3: Detailed Explanation:
1. Calculate the individual rates:
- Pipe A can fill the tank in 4 hours. So, the rate of Pipe A (\(R_A\)) is \(\frac{1}{4}\) of the tank per hour.
- Pipe B can fill the tank in 6 hours. So, the rate of Pipe B (\(R_B\)) is \(\frac{1}{6}\) of the tank per hour.
2. Calculate the combined rate:
When both pipes are open, their rates add up.
Combined Rate (\(R_C\)) = \(R_A + R_B\) \[ R_C = \frac{1}{4} + \frac{1}{6} \] To add these fractions, find a common denominator, which is 12.
\[ R_C = \frac{3}{12} + \frac{2}{12} = \frac{5}{12} \] So, together they fill \(\frac{5}{12}\) of the tank per hour.
3. Calculate the total time:
Time = \(\frac{\text{Total Work}}{\text{Combined Rate}}\). The total work is filling 1 tank.
\[ \text{Time} = \frac{1}{R_C} = \frac{1}{\frac{5}{12}} = \frac{12}{5} \] \[ \text{Time} = 2.4 \text{ hours} \] Step 4: Final Answer:
It will take 2.4 hours for both pipes together to fill the tank.