Understanding the Problem: Tank Filling Pipes

This problem involves calculating the time taken by individual pipes to fill a tank, given the combined time taken by pairs or groups of pipes. The core concept we will use is the work rate, which is the inverse of the time taken to complete the work (fill the tank in this case).

Work Rate Concept Explained

If a pipe can fill a tank in \(T\) hours, its work rate is \(1/T\) of the tank filled per hour. When multiple pipes work together, their individual work rates add up to give the combined work rate.

Given Information

We are given the following times for different combinations of pipes:

• Pipes A and B together fill the tank in 30 hours.
• Pipes B and C together fill the tank in 24 hours.
• Pipes A, B, and C together fill the tank in 20 hours.

Calculating Combined Work Rates

Based on the given times, we can find the combined work rates:

• Combined work rate of A and B = \(\frac{1}{30}\) tanks/hour.
• Combined work rate of B and C = \(\frac{1}{24}\) tanks/hour.
• Combined work rate of A, B, and C = \(\frac{1}{20}\) tanks/hour.

Finding Individual Pipe Work Rates

We can use the combined work rates to find the individual work rates of pipes A, B, and C.

The work rate of pipe C alone can be found by subtracting the work rate of (A + B) from the work rate of (A + B + C):

Work rate of C = Work rate of (A + B + C) - Work rate of (A + B)

Work rate of C = \(\frac{1}{20} - \frac{1}{30}\)

To subtract these fractions, we find a common denominator, which is 60:

Work rate of C = \(\frac{3}{60} - \frac{2}{60} = \frac{3-2}{60} = \frac{1}{60}\) tanks/hour.

Similarly, the work rate of pipe A alone can be found by subtracting the work rate of (B + C) from the work rate of (A + B + C):

Work rate of A = Work rate of (A + B + C) - Work rate of (B + C)

Work rate of A = \(\frac{1}{20} - \frac{1}{24}\)

To subtract these fractions, we find a common denominator, which is 120:

Work rate of A = \(\frac{6}{120} - \frac{5}{120} = \frac{6-5}{120} = \frac{1}{120}\) tanks/hour.

Now we can find the work rate of pipe B alone. We can do this by subtracting the work rate of A from the combined work rate of (A + B):

Work rate of B = Work rate of (A + B) - Work rate of A

Work rate of B = \(\frac{1}{30} - \frac{1}{120}\)

Using a common denominator of 120:

Work rate of B = \(\frac{4}{120} - \frac{1}{120} = \frac{4-1}{120} = \frac{3}{120} = \frac{1}{40}\) tanks/hour.

Alternatively, we could subtract the work rate of C from the combined work rate of (B + C):

Work rate of B = Work rate of (B + C) - Work rate of C

Work rate of B = \(\frac{1}{24} - \frac{1}{60}\)

Using a common denominator of 120:

Work rate of B = \(\frac{5}{120} - \frac{2}{120} = \frac{5-2}{120} = \frac{3}{120} = \frac{1}{40}\) tanks/hour.

Both methods give the same work rate for pipe B.

Calculating Time for Pipe B to Fill the Entire Tank

Since the work rate of pipe B is \(\frac{1}{40}\) tanks/hour, pipe B alone can fill the entire tank in the inverse of this rate:

Time for B alone to fill 100% tank = \(\frac{1}{\text{Work rate of B}} = \frac{1}{\frac{1}{40}} = 40\) hours.

Calculating Time for Pipe B to Fill 35% of the Tank

We need to find the time pipe B takes to fill only 35% of the tank. 35% as a decimal is 0.35.

Time = (Fraction of tank to fill) \(\times\) (Time to fill whole tank)

Time = \(0.35 \times 40\) hours

Time = \(\frac{35}{100} \times 40 = \frac{35 \times 40}{100} = \frac{1400}{100} = 14\) hours.

Therefore, pipe B alone can fill 35% of the tank in 14 hours.

Revision Table: Key Concepts in Tank Filling Problems

Additional Information: Solving Time and Work Problems

Problems involving pipes filling or emptying tanks are a common type of 'Time and Work' problem. Here are some key points:

• Identify the "Work": In tank problems, the work is usually filling (or emptying) one full tank.
• Consistent Units: Ensure all times are in the same unit (hours in this case).
• Emptying Pipes: If a pipe is emptying the tank, its work rate is negative. Subtract its work rate from the filling pipes' rates.
• Partially Filled/Empty Tanks: If the tank is initially partially filled or the question asks about filling only a fraction, adjust the total 'work' accordingly (e.g., filling 35% means the work is 0.35 instead of 1).
• Efficiency: Sometimes problems mention efficiency. Efficiency relates directly to the work rate – a more efficient pipe has a higher work rate.