Three pipes A, B and C can fill a cistern in 8 minutes,12 minutes and 16 minutes respectively. What is the time taken by three pipes to fill the cistern when they are opened together?

Options

A. 5 minutes
B. 4 minutes
C. 3.7 minutes
D. 4.5 minutes

Answer

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

### Step 1: Determine the rate of each pipe
- Pipe A fills the cistern in 8 minutes. Therefore, in 1 minute, Pipe A fills \( \frac{1}{8} \) of the cistern.
- Pipe B fills the cistern in 12 minutes. Therefore, in 1 minute, Pipe B fills \( \frac{1}{12} \) of the cistern.
- Pipe C fills the cistern in 16 minutes. Therefore, in 1 minute, Pipe C fills \( \frac{1}{16} \) of the cistern.

### Step 2: Calculate the combined rate of all three pipes
To find out how much of the cistern is filled by all three pipes in one minute, we add their rates together:
\[ \text{Combined rate} = \frac{1}{8} + \frac{1}{12} + \frac{1}{16} \]

### Step 3: Find the least common multiple (LCM)
To add these fractions, we need to find the LCM of the denominators (8, 12, and 16). The LCM of 8, 12, and 16 is 48.

### Step 4: Convert each fraction to have a common denominator
Now we convert each fraction:
- \( \frac{1}{8} = \frac{6}{48} \)
- \( \frac{1}{12} = \frac{4}{48} \)
- \( \frac{1}{16} = \frac{3}{48} \)

### Step 5: Add the fractions
Now, we can add these fractions:
\[ \frac{6}{48} + \frac{4}{48} + \frac{3}{48} = \frac{13}{48} \]

### Step 6: Determine the time to fill the cistern
The combined rate of the three pipes is \( \frac{13}{48} \) of the cistern per minute. To find the total time taken to fill the cistern, we take the reciprocal of the combined rate:
\[ \text{Time} = \frac{1}{\frac{13}{48}} = \frac{48}{13} \text{ minutes} \]

### Step 7: Convert to a more understandable format
Calculating \( \frac{48}{13} \) gives approximately 3.69 minutes, which can be rounded to about 3 minutes and 42 seconds.

### Final Answer
Thus, the time taken by the three pipes to fill the cistern when opened together is approximately \( \frac{48}{13} \) minutes or about 3 minutes and 42 seconds.

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