Solving Pipe Filling Problem: A, B, and C Pipes

This problem involves calculating the time taken to fill a portion of a tank when multiple pipes with different filling and emptying rates are involved. We need to determine the time in minutes for pipes A and B to fill one-third of the remaining tank capacity after pipe C has been closed.

Understanding Pipe Rates

First, let's determine the rate at which each pipe works:

• Pipe A fills the tank in 18 hours. Its filling rate is $ \frac{1}{18} $ of the tank per hour.
• Pipe B fills the tank in 27 hours. Its filling rate is $ \frac{1}{27} $ of the tank per hour.
• Pipe C empties the tank in 54 hours. Its emptying rate is $ \frac{1}{54} $ of the tank per hour.

Calculating Combined Rate (A, B, and C Together)

When all three pipes are opened together, the combined rate is the sum of filling rates minus the emptying rate:

Combined Rate (A, B, C) = Rate(A) + Rate(B) - Rate(C)

Combined Rate = $ \frac{1}{18} + \frac{1}{27} - \frac{1}{54} $

To add/subtract these fractions, we find the least common multiple (LCM) of 18, 27, and 54, which is 54.

Combined Rate = $ \frac{3}{54} + \frac{2}{54} - \frac{1}{54} = \frac{3 + 2 - 1}{54} = \frac{4}{54} = \frac{2}{27} $ tank per hour.

Work Done in the First 12 Hours

All three pipes work together for 12 hours. The amount of tank filled during this time is:

Work Done = Combined Rate $ \times $ Time

Work Done = $ \frac{2}{27} \times 12 = \frac{24}{27} = \frac{8}{9} $ of the tank.

Calculating the Remaining Part of the Tank

After 12 hours, the fraction of the tank that remains empty is:

Remaining Part = Total Tank Capacity - Work Done

Remaining Part = $ 1 - \frac{8}{9} = \frac{1}{9} $ of the tank.

Target Work: One-Third of the Remaining Part

The question asks for the time to fill one-third of the *remaining* part. So, the target work is:

Target Work = $ \frac{1}{3} \times $ Remaining Part

Target Work = $ \frac{1}{3} \times \frac{1}{9} = \frac{1}{27} $ of the tank.

Calculating Combined Rate of A and B Only

After pipe C is closed, only pipes A and B are working together to fill the remaining part. Their combined rate is:

Combined Rate (A + B) = Rate(A) + Rate(B)

Combined Rate (A + B) = $ \frac{1}{18} + \frac{1}{27} $

The LCM of 18 and 27 is 54.

Combined Rate (A + B) = $ \frac{3}{54} + \frac{2}{54} = \frac{5}{54} $ tank per hour.

Time Taken by A and B to Fill the Target Work

Now, we calculate the time needed for pipes A and B together to fill $ \frac{1}{27} $ of the tank:

Time = Target Work $ \div $ Combined Rate (A + B)

Time = $ \frac{1}{27} \div \frac{5}{54} $

Time = $ \frac{1}{27} \times \frac{54}{5} = \frac{54}{27 \times 5} = \frac{2}{5} $ hours.

Converting Time to Minutes

The question asks for the time in minutes. To convert hours to minutes, we multiply by 60:

Time in minutes = $ \frac{2}{5} \times 60 $

Time in minutes = $ 2 \times 12 = 24 $ minutes.

Final Answer

Therefore, it will take 24 minutes for pipes A and B together to fill one-third of the remaining part of the tank.