A and B can do a work in 12 days, B and C in 15 days, C and A in 20 days. If A, B and C Work together, they will complete the work in :

Options

A. 5 days
B. `7 5/6` days
C. 10 days
D. `15 2/3 ` days

Answer

To solve the problem, we need to find out how long it will take for A, B, and C to complete the work together based on the information given about their pairs.

### Step-by-step Solution:

1. **Understanding the Work Rates**:
- A and B can complete the work in 12 days.
- B and C can complete the work in 15 days.
- C and A can complete the work in 20 days.

2. **Finding the Work Done in Units**:
- Let's assume the total work is 60 units (this is a common multiple of 12, 15, and 20).
- Therefore, the work rates for each pair can be calculated as follows:
- A + B = 60 units in 12 days → (60/12) = 5 units/day
- B + C = 60 units in 15 days → (60/15) = 4 units/day
- C + A = 60 units in 20 days → (60/20) = 3 units/day

3. **Setting Up the Equations**:
- From the above, we can set up the following equations based on the work rates:
- (1) A + B = 5 (units/day)
- (2) B + C = 4 (units/day)
- (3) C + A = 3 (units/day)

4. **Solving the Equations**:
- To find the individual work rates, we can add all three equations:
\[ (A + B) + (B + C) + (C + A) = 5 + 4 + 3 \]
This simplifies to:
\[ 2A + 2B + 2C = 12 \]
Dividing by 2:
\[ A + B + C = 6 \quad (4) \]

5. **Finding Individual Work Rates**:
- Now, we can use equation (4) to find the individual work rates:
- From (1): A + B = 5 → C = 6 - 5 = 1
- From (2): B + C = 4 → B = 4 - 1 = 3
- From (3): C + A = 3 → A = 3 - 1 = 2

6. **Final Work Rates**:
- A = 2 units/day
- B = 3 units/day
- C = 1 unit/day

7. **Combined Work Rate**:
- Now, we can find the combined work rate of A, B, and C:
\[ A + B + C = 2 + 3 + 1 = 6 \text{ units/day} \]

8. **Calculating Total Time**:
- Since the total work is 60 units and their combined work rate is 6 units/day, the time taken to complete the work together is:
\[ \text{Time} = \frac{\text{Total Work}}{\text{Combined Work Rate}} = \frac{60}{6} = 10 \text{ days} \]

### Conclusion:
A, B, and C will complete the work together in **10 days**.