A and B can do a given piece of work in 8 days, B and C can do the same work in 12 days and A, B, C complete it in 6 days. Number of days required to finish the work by A and C is

Options

A. 16
B. 8
C. 12
D. 24

Answer

To solve the problem step by step, we will first determine the work efficiencies of A, B, and C based on the information provided.

### Step 1: Determine the efficiencies of A + B, B + C, and A + B + C

1. **A + B can complete the work in 8 days.**
- Efficiency of A + B = 1/8 (work done per day)

2. **B + C can complete the work in 12 days.**
- Efficiency of B + C = 1/12 (work done per day)

3. **A + B + C can complete the work in 6 days.**
- Efficiency of A + B + C = 1/6 (work done per day)

### Step 2: Convert efficiencies to a common base

To find the individual efficiencies, we can use the least common multiple (LCM) of the denominators (8, 12, and 6). The LCM of these numbers is 24.

- **Efficiency of A + B:**
\[ \text{Efficiency} = \frac{24}{8} = 3 \text{ (units of work per day)} \]

- **Efficiency of B + C:**
\[ \text{Efficiency} = \frac{24}{12} = 2 \text{ (units of work per day)} \]

- **Efficiency of A + B + C:**
\[ \text{Efficiency} = \frac{24}{6} = 4 \text{ (units of work per day)} \]

### Step 3: Find the individual efficiencies of A, B, and C

Now we can set up equations based on the efficiencies we calculated:

1. Let the efficiency of A be \( a \), B be \( b \), and C be \( c \).
- From A + B: \( a + b = 3 \) (1)
- From B + C: \( b + c = 2 \) (2)
- From A + B + C: \( a + b + c = 4 \) (3)

### Step 4: Solve the equations

From equation (3), we can express \( c \) in terms of \( a \) and \( b \):
\[ c = 4 - (a + b) = 4 - 3 = 1 \]

Now substitute \( c \) back into equation (2):
\[ b + 1 = 2 \implies b = 1 \]

Now substitute \( b \) back into equation (1):
\[ a + 1 = 3 \implies a = 2 \]

### Step 5: Summary of efficiencies

- Efficiency of A = 2
- Efficiency of B = 1
- Efficiency of C = 1

### Step 6: Find the efficiency of A + C

Now we need to find the efficiency of A + C:
\[ \text{Efficiency of A + C} = a + c = 2 + 1 = 3 \]

### Step 7: Calculate the number of days required for A and C to finish the work

The total work is represented by the LCM we calculated earlier, which is 24 units. Therefore, the number of days required for A and C to finish the work is:
\[ \text{Days} = \frac{\text{Total Work}}{\text{Efficiency of A + C}} = \frac{24}{3} = 8 \text{ days} \]

### Final Answer
The number of days required to finish the work by A and C is **8 days**.

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