A and B can complete a piece of work in 8 days, B and C can do it in 12 days, C and A can do it in 8 days. A, B and C together can complete it in

Options

A. 4 days
B. 5 days
C. 6 days
D. 7 days

Answer

To solve the problem, we need to find out how many days A, B, and C together can complete the work. We are given the following information:

1. A and B can complete the work in 8 days.
2. B and C can complete the work in 12 days.
3. C and A can complete the work in 8 days.

Let's denote the work done by A, B, and C in one day as \( A \), \( B \), and \( C \) respectively.

### Step 1: Write the equations for the work done
From the information provided, we can write the following equations:

- For A and B:
\[ A + B = \frac{1}{8} \quad \text{(1)} \]

- For B and C:
\[ B + C = \frac{1}{12} \quad \text{(2)} \]

- For C and A:
\[ C + A = \frac{1}{8} \quad \text{(3)} \]

### Step 2: Add the three equations
Now, we can add equations (1), (2), and (3):
\[ (A + B) + (B + C) + (C + A) = \frac{1}{8} + \frac{1}{12} + \frac{1}{8} \]

This simplifies to:
\[ 2A + 2B + 2C = \frac{1}{8} + \frac{1}{12} + \frac{1}{8} \]

### Step 3: Find a common denominator and simplify
The common denominator for 8 and 12 is 24. We convert each fraction:
\[ \frac{1}{8} = \frac{3}{24}, \quad \frac{1}{12} = \frac{2}{24} \]
Thus,
\[ \frac{1}{8} + \frac{1}{12} + \frac{1}{8} = \frac{3}{24} + \frac{2}{24} + \frac{3}{24} = \frac{8}{24} = \frac{1}{3} \]

### Step 4: Substitute back to find A + B + C
Now we have:
\[ 2A + 2B + 2C = \frac{1}{3} \]
Dividing both sides by 2 gives:
\[ A + B + C = \frac{1}{6} \]

### Step 5: Calculate the time taken by A, B, and C together
If \( A + B + C = \frac{1}{6} \), it means that A, B, and C together can complete the work in 6 days.

### Final Answer
Thus, A, B, and C together can complete the work in **6 days**.

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