If three pipes A, B and C can fill the tank alone in 5, 6 and 8 hrs, then when all the three pipes are opened together, find the time to fill the tank completely.

Answer

To solve the problem of how long it takes for pipes A, B, and C to fill a tank together, we can follow these steps:

### Step 1: Determine the rate of work for each pipe.
- Pipe A can fill the tank in 5 hours, so its rate is \( \frac{1}{5} \) of the tank per hour.
- Pipe B can fill the tank in 6 hours, so its rate is \( \frac{1}{6} \) of the tank per hour.
- Pipe C can fill the tank in 8 hours, so its rate is \( \frac{1}{8} \) of the tank per hour.

### Step 2: Add the rates of all three pipes.
To find the combined rate of all three pipes working together, we add their individual rates:
\[ \text{Combined rate} = \frac{1}{5} + \frac{1}{6} + \frac{1}{8} \]

### Step 3: Find a common denominator.
The least common multiple (LCM) of 5, 6, and 8 is 120. We will convert each rate to have a denominator of 120:
- For \( \frac{1}{5} \):
\[ \frac{1}{5} = \frac{24}{120} \]
- For \( \frac{1}{6} \):
\[ \frac{1}{6} = \frac{20}{120} \]
- For \( \frac{1}{8} \):
\[ \frac{1}{8} = \frac{15}{120} \]

### Step 4: Add the converted rates.
Now we can add the rates:
\[ \frac{24}{120} + \frac{20}{120} + \frac{15}{120} = \frac{24 + 20 + 15}{120} = \frac{59}{120} \]

### Step 5: Find the time to fill the tank.
The combined rate of the three pipes is \( \frac{59}{120} \) of the tank per hour. To find the time \( T \) taken to fill the tank completely, we take the reciprocal of the combined rate:
\[ T = \frac{120}{59} \text{ hours} \]

### Step 6: Convert to a mixed fraction.
To convert \( \frac{120}{59} \) into a mixed fraction:
- Divide 120 by 59, which gives 2 with a remainder of 2.
- Thus, \( \frac{120}{59} = 2 \frac{2}{59} \) hours.

### Final Answer:
The time taken to fill the tank completely when all three pipes are opened together is \( 2 \frac{2}{59} \) hours.

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