Efficient Tank Filling Techniques: Combining Pipe A and B

In many real-world scenarios, we often need to determine the efficiency of multiple pipes working together to fill a tank. For instance, consider two pipes, Pipe A and Pipe B, that individually can fill a tank in 20 minutes and 30 minutes respectively. This problem requires us to calculate the time it would take for both pipes to fill the tank when they are opened simultaneously.

Problem Statement

The problem is to find the time taken to fill the tank when both pipes A and B are opened together. Let's break down the process to solve this problem step by step.

Step-by-Step Calculation

To find the combined rate of filling:

Determine the individual rates: Pipe A fills the tank in 20 minutes, so the rate is (frac{1}{20}) of the tank per minute. Pipe B fills the tank in 30 minutes, so the rate is (frac{1}{30}) of the tank per minute. Combine the rates: The combined rate of Filling is given by: (text{Combined rate} frac{1}{20} frac{1}{30}) To add these fractions, find a common denominator. The least common multiple (LCM) of 20 and 30 is 60. (frac{1}{20} frac{3}{60} quad text{and} quad frac{1}{30} frac{2}{60}) (text{Combined rate} frac{3}{60} frac{2}{60} frac{5}{60} frac{1}{12})

Calculation of Time to Fill the Tank

The combined rate is (frac{1}{12}) of the tank per minute. This means that together, pipes A and B can fill (frac{1}{12}) of the tank in one minute.

Therefore, the time taken to fill the tank is:

(text{Time} frac{1}{frac{1}{12}} 12 text{ minutes})

Conclusion

When both pipes A and B are opened together, the time taken to fill the tank is 12 minutes.

Alternative Methods and Variations

Using X as Volume: If the volume of the tank is (X), the discharge rates are: Pipe A: (frac{X}{20}) Pipe B: (frac{X}{30}) Combined: (frac{X}{20} frac{X}{30} frac{X}{6}) Time required: (6 text{ minutes}) Part Filled per Minute: Pipe A: (frac{1}{20}) Pipe B: (frac{1}{45}) Combined: (frac{1}{20} frac{1}{45} frac{1}{13.85}) Time required: Approximately (14 text{ minutes}) Another Approach: Pipe A: (frac{1}{20}) Pipe B: (frac{1}{90}) Combined: (frac{1}{20} frac{1}{90} frac{92}{180} frac{11}{180}) Time required: (frac{180}{11} approx 16.36 text{ minutes} text{ or } 16 text{ minutes 22 seconds})

These different methods provide multiple perspectives on the problem and can be used to verify the solution.