A can do a job in 10 days, B can do the same job in 12 days and C can do the same job in 15 days. In how many days they will finish the work together?

Answer

To solve the problem, we need to determine how many days A, B, and C will take to complete a job if they work together.

### Step-by-Step Solution:

1. **Determine Individual Work Rates**:
- A can complete the job in 10 days, so A's work rate is:
\[ \text{Work rate of A} = \frac{1}{10} \text{ (job per day)} \]
- B can complete the job in 12 days, so B's work rate is:
\[ \text{Work rate of B} = \frac{1}{12} \text{ (job per day)} \]
- C can complete the job in 15 days, so C's work rate is:
\[ \text{Work rate of C} = \frac{1}{15} \text{ (job per day)} \]

2. **Combine the Work Rates**:
- To find the combined work rate when A, B, and C work together, we add their individual work rates:
\[ \text{Combined work rate} = \frac{1}{10} + \frac{1}{12} + \frac{1}{15} \]

3. **Find the Least Common Multiple (LCM)**:
- The LCM of 10, 12, and 15 is 60. We will use this to add the fractions:
\[ \frac{1}{10} = \frac{6}{60}, \quad \frac{1}{12} = \frac{5}{60}, \quad \frac{1}{15} = \frac{4}{60} \]
- Now, add these fractions:
\[ \text{Combined work rate} = \frac{6}{60} + \frac{5}{60} + \frac{4}{60} = \frac{15}{60} = \frac{1}{4} \]

4. **Calculate the Total Time Taken**:
- The combined work rate of A, B, and C is \(\frac{1}{4}\) of the job per day. To find out how many days it takes to complete the entire job, we take the reciprocal of the combined work rate:
\[ \text{Total time} = \frac{1}{\frac{1}{4}} = 4 \text{ days} \]

### Final Answer:
A, B, and C together will finish the work in **4 days**.