A and B can do a piece of work in 15 days. B and C can do the same work in 10 days and A and C can do the same in 12 days. Time taken by A, B and C together to do the job is

Options

A. 4 days
B. 9 days
C. 8 days
D. 5 days

Answer

To solve the problem, we need to determine the time taken by A, B, and C together to complete the work based on the information given about their individual and pair work rates.

### Step-by-Step Solution:

1. **Understanding the Work Rates**:
- Let the total work be represented in units. We can assume the total work is 1 unit for simplicity.
- A and B can complete the work in 15 days, so their combined work rate is:
\[ \text{Work rate of A and B} = \frac{1}{15} \text{ units/day} \]

- B and C can complete the work in 10 days, so their combined work rate is:
\[ \text{Work rate of B and C} = \frac{1}{10} \text{ units/day} \]

- A and C can complete the work in 12 days, so their combined work rate is:
\[ \text{Work rate of A and C} = \frac{1}{12} \text{ units/day} \]

2. **Setting Up the Equations**:
- Let the work rates of A, B, and C be \( a \), \( b \), and \( c \) respectively. We can set up the following equations based on the combined work rates:
\[ a + b = \frac{1}{15} \tag{1} \]
\[ b + c = \frac{1}{10} \tag{2} \]
\[ a + c = \frac{1}{12} \tag{3} \]

3. **Adding the Equations**:
- Adding equations (1), (2), and (3) gives:
\[ (a + b) + (b + c) + (a + c) = \frac{1}{15} + \frac{1}{10} + \frac{1}{12} \]
- This simplifies to:
\[ 2a + 2b + 2c = \frac{1}{15} + \frac{1}{10} + \frac{1}{12} \]
- Dividing everything by 2:
\[ a + b + c = \frac{1}{2} \left( \frac{1}{15} + \frac{1}{10} + \frac{1}{12} \right) \tag{4} \]

4. **Finding a Common Denominator**:
- The least common multiple (LCM) of 15, 10, and 12 is 60. We can convert the fractions:
\[ \frac{1}{15} = \frac{4}{60}, \quad \frac{1}{10} = \frac{6}{60}, \quad \frac{1}{12} = \frac{5}{60} \]
- Thus:
\[ \frac{1}{15} + \frac{1}{10} + \frac{1}{12} = \frac{4 + 6 + 5}{60} = \frac{15}{60} = \frac{1}{4} \]

5. **Substituting Back**:
- Substituting back into equation (4):
\[ a + b + c = \frac{1}{2} \cdot \frac{1}{4} = \frac{1}{8} \]

6. **Calculating Time Taken by A, B, and C Together**:
- The combined work rate of A, B, and C is \( \frac{1}{8} \) units/day. Therefore, the time taken by A, B, and C together to complete the work is:
\[ \text{Time} = \frac{\text{Total Work}}{\text{Combined Work Rate}} = \frac{1}{\frac{1}{8}} = 8 \text{ days} \]

### Final Answer:
The time taken by A, B, and C together to do the job is **8 days**.