If A can do a piece of work in 4 days, B can do the same work in 8 days and C can do the same work in 12 days , then working together , how many days will they take to complete the work ?

Answer

To solve the problem, we need to find out how long A, B, and C will take to complete the work when they work together. We can do this by calculating their individual work rates and then combining them.

### Step-by-Step Solution:

1. **Determine the Work Rates:**
- A can complete the work in 4 days, so A's work rate is \( \frac{1}{4} \) of the work per day.
- B can complete the work in 8 days, so B's work rate is \( \frac{1}{8} \) of the work per day.
- C can complete the work in 12 days, so C's work rate is \( \frac{1}{12} \) of the work per day.

2. **Combine the Work Rates:**
- To find the combined work rate of A, B, and C, we add their individual work rates:
\[ \text{Combined Work Rate} = \frac{1}{4} + \frac{1}{8} + \frac{1}{12} \]

3. **Find a Common Denominator:**
- The least common multiple (LCM) of 4, 8, and 12 is 24. We will convert each fraction to have a denominator of 24:
\[ \frac{1}{4} = \frac{6}{24}, \quad \frac{1}{8} = \frac{3}{24}, \quad \frac{1}{12} = \frac{2}{24} \]

4. **Add the Fractions:**
- Now we can add the fractions:
\[ \frac{6}{24} + \frac{3}{24} + \frac{2}{24} = \frac{6 + 3 + 2}{24} = \frac{11}{24} \]

5. **Calculate the Time Taken to Complete the Work:**
- The combined work rate of A, B, and C is \( \frac{11}{24} \) of the work per day. To find the total time taken to complete the work, we take the reciprocal of the combined work rate:
\[ \text{Time} = \frac{1}{\frac{11}{24}} = \frac{24}{11} \text{ days} \]

6. **Final Answer:**
- Thus, working together, A, B, and C will take \( \frac{24}{11} \) days to complete the work, which is approximately 2.18 days.