B and C can complete a piece of work in 12 days, C and A can do it in 8 days. All the three can do it in 6 days. A and B together can complete It In

Options

A. 4 days
B. 6 days
C. 8 days
D. 10 days

Answer

To solve the problem, we need to determine how long A and B together can complete the work based on the information given about the work rates of A, B, and C.

### Step-by-Step Solution:

1. **Understanding the Work Rates**:
- Let the total work be represented as 1 unit of work.
- B and C can complete the work in 12 days, so their combined work rate is:
\[ \text{Work rate of B and C} = \frac{1 \text{ unit}}{12 \text{ days}} = \frac{1}{12} \text{ units/day} \]

2. **C and A's Work Rate**:
- C and A can complete the work in 8 days, so their combined work rate is:
\[ \text{Work rate of C and A} = \frac{1 \text{ unit}}{8 \text{ days}} = \frac{1}{8} \text{ units/day} \]

3. **All Three Together**:
- A, B, and C can complete the work in 6 days, so their combined work rate is:
\[ \text{Work rate of A, B, and C} = \frac{1 \text{ unit}}{6 \text{ days}} = \frac{1}{6} \text{ units/day} \]

4. **Setting Up the Equations**:
- Let the work rates of A, B, and C be \(a\), \(b\), and \(c\) respectively.
- From the information we have:
\[ b + c = \frac{1}{12} \quad \text{(1)} \]
\[ c + a = \frac{1}{8} \quad \text{(2)} \]
\[ a + b + c = \frac{1}{6} \quad \text{(3)} \]

5. **Solving the Equations**:
- From equation (3), we can express \(a\) in terms of \(b\) and \(c\):
\[ a = \frac{1}{6} - (b + c) \]
- Substitute \(b + c\) from equation (1) into this:
\[ a = \frac{1}{6} - \frac{1}{12} \]
- To perform the subtraction, convert \(\frac{1}{6}\) to a fraction with a denominator of 12:
\[ a = \frac{2}{12} - \frac{1}{12} = \frac{1}{12} \]

6. **Finding \(b\) and \(c\)**:
- Substitute \(a\) back into equation (2):
\[ c + \frac{1}{12} = \frac{1}{8} \]
- Rearranging gives:
\[ c = \frac{1}{8} - \frac{1}{12} \]
- Convert both fractions to a common denominator (24):
\[ c = \frac{3}{24} - \frac{2}{24} = \frac{1}{24} \]
- Now substitute \(c\) back into equation (1) to find \(b\):
\[ b + \frac{1}{24} = \frac{1}{12} \]
- Rearranging gives:
\[ b = \frac{1}{12} - \frac{1}{24} = \frac{2}{24} - \frac{1}{24} = \frac{1}{24} \]

7. **Final Work Rates**:
- We have:
\[ a = \frac{1}{12}, \quad b = \frac{1}{24}, \quad c = \frac{1}{24} \]

8. **Finding A and B's Combined Work Rate**:
- The combined work rate of A and B is:
\[ a + b = \frac{1}{12} + \frac{1}{24} \]
- Convert \(\frac{1}{12}\) to a fraction with a denominator of 24:
\[ a + b = \frac{2}{24} + \frac{1}{24} = \frac{3}{24} = \frac{1}{8} \]

9. **Calculating Time for A and B**:
- The time taken by A and B together to complete the work is:
\[ \text{Time} = \frac{1 \text{ unit}}{\frac{1}{8} \text{ units/day}} = 8 \text{ days} \]

### Final Answer:
A and B together can complete the work in **8 days**.