Understanding Pipe Filling Rates

First, let's determine the rate at which each pipe fills the tank. The rate is the fraction of the tank filled per hour.

• Rate of Pipe A = $ \frac{1}{8} $ tank per hour
• Rate of Pipe B = $ \frac{1}{12} $ tank per hour
• Rate of Pipe C = $ \frac{1}{24} $ tank per hour

Calculating Work Done in Stages

The filling process occurs in three distinct stages.

Stage 1: Pipe A Alone (1 hour)

Pipe A works alone for the first hour.

• Work done by A in 1 hour = Rate of A $ \times $ Time = $ \frac{1}{8} \times 1 = \frac{1}{8} $ tank.
• Tank capacity remaining = $ 1 - \frac{1}{8} = \frac{7}{8} $ tank.

Stage 2: Pipes A and B Together (2 hours)

Next, pipes A and B work together for 2 hours.

• Combined rate of A and B = $ \frac{1}{8} + \frac{1}{12} $.
• To add these fractions, find a common denominator (24): $ \frac{3}{24} + \frac{2}{24} = \frac{5}{24} $ tank per hour.
• Work done by A and B in 2 hours = Combined rate $ \times $ Time = $ \frac{5}{24} \times 2 = \frac{10}{24} = \frac{5}{12} $ tank.
• Total work done after 2 hours = Work from Stage 1 + Work from Stage 2 = $ \frac{1}{8} + \frac{5}{12} $.
• Common denominator (24): $ \frac{3}{24} + \frac{10}{24} = \frac{13}{24} $ tank.
• Tank capacity remaining = $ 1 - \frac{13}{24} = \frac{11}{24} $ tank.

Stage 3: Pipes A, B, and C Together

Finally, all three pipes work together until the tank is full.

• Combined rate of A, B, and C = $ \frac{1}{8} + \frac{1}{12} + \frac{1}{24} $.
• Common denominator (24): $ \frac{3}{24} + \frac{2}{24} + \frac{1}{24} = \frac{6}{24} = \frac{1}{4} $ tank per hour.

Determining Final Filling Time

Now, calculate the time needed for Stage 3 and the total time.

• Time required for Stage 3 = Remaining capacity / Combined rate of A, B, C
• Time for Stage 3 = $ \frac{11/24}{1/4} = \frac{11}{24} \times 4 = \frac{44}{24} = \frac{11}{6} $ hours.
• Total time to fill the tank = Time (Stage 1) + Time (Stage 2) + Time (Stage 3)
• Total time = $ 1 \text{ hour} + 2 \text{ hours} + \frac{11}{6} \text{ hours} = 3 + \frac{11}{6} $ hours.
• Total time = $ \frac{18}{6} + \frac{11}{6} = \frac{29}{6} $ hours.

Convert the total time into hours and minutes:

• $ \frac{29}{6} $ hours = 4 whole hours and $ \frac{5}{6} $ of an hour.
• $ \frac{5}{6} $ hour = $ \frac{5}{6} \times 60 $ minutes = 50 minutes.
• Therefore, the total time is 4 hours and 50 minutes.