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 two pipes A and B taking times \(t_A\) and \(t_B\) respectively, the combined time \(T\) can be quickly calculated using the formula: \( T = \frac{t_A \times t_B}{t_A + t_B} \). In this case, \( T = \frac{4 \times 6}{4 + 6} = \frac{24}{10} = 2.4 \) hours.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
This is a classic "Work and Rate" problem. The key is to determine the rate at which each pipe works and then combine their rates to find the total time taken when they work together. The work to be done is filling 1 full tank.
Step 2: Key Formula or Approach
Rate of work is the reciprocal of the time taken to complete the work. \[ \text{Rate} = \frac{1}{\text{Time}} \] When working together, their rates add up: \[ \text{Combined Rate} = \text{Rate of A} + \text{Rate of B} \] The time taken to complete the work together is the reciprocal of the combined rate: \[ \text{Time} = \frac{1}{\text{Combined Rate}} \] Step 3: Detailed Explanation
Calculate the individual rates:
Rate of Pipe A = \(\frac{1}{4}\) of the tank per hour.
Rate of Pipe B = \(\frac{1}{6}\) of the tank per hour.
Calculate the combined rate:
\[ \text{Combined Rate} = \frac{1}{4} + \frac{1}{6} \] To add these fractions, find a common denominator, which is 12. \[ \text{Combined Rate} = \frac{3}{12} + \frac{2}{12} = \frac{5}{12} \] This means together they fill \(\frac{5}{12}\) of the tank every hour.
Calculate the total time taken together:
\[ \text{Time} = \frac{1}{\text{Combined Rate}} = \frac{1}{\frac{5}{12}} = \frac{12}{5} \text{ hours} \] To convert this to a decimal: \[ \text{Time} = 12 \div 5 = 2.4 \text{ hours} \] Step 4: Final Answer
It will take 2.4 hours to fill the tank if both pipes are opened together. This corresponds to option (B).