A and B can do a work in 8 days, B and C in 12 days, and A and C in 16 days. In what time can they do it, all working together ?

Answer

To solve the problem step by step, we will first determine the work done by each pair of workers (A and B, B and C, A and C) and then find out how long it will take for all three (A, B, and C) to complete the work together.

### Step-by-Step Solution:

1. **Determine the work done by each pair:**
- A and B can complete the work in 8 days. Therefore, their one day work is:
\[ A + B = \frac{1}{8} \]
- B and C can complete the work in 12 days. Therefore, their one day work is:
\[ B + C = \frac{1}{12} \]
- A and C can complete the work in 16 days. Therefore, their one day work is:
\[ A + C = \frac{1}{16} \]

2. **Add the equations:**
- Now, we will add all three equations:
\[ (A + B) + (B + C) + (A + C) = \frac{1}{8} + \frac{1}{12} + \frac{1}{16} \]
- This simplifies to:
\[ 2A + 2B + 2C = \frac{1}{8} + \frac{1}{12} + \frac{1}{16} \]

3. **Calculate the right side:**
- To add the fractions, we need a common denominator. The least common multiple (LCM) of 8, 12, and 16 is 48.
- Convert each fraction:
\[ \frac{1}{8} = \frac{6}{48}, \quad \frac{1}{12} = \frac{4}{48}, \quad \frac{1}{16} = \frac{3}{48} \]
- Now add them:
\[ \frac{6}{48} + \frac{4}{48} + \frac{3}{48} = \frac{13}{48} \]

4. **Substitute back into the equation:**
- Now we have:
\[ 2A + 2B + 2C = \frac{13}{48} \]
- Dividing both sides by 2 gives:
\[ A + B + C = \frac{13}{96} \]

5. **Find the time taken by A, B, and C together:**
- The time taken to complete the work when A, B, and C work together is the reciprocal of their one day work:
\[ \text{Time} = \frac{1}{A + B + C} = \frac{1}{\frac{13}{96}} = \frac{96}{13} \]
- This can be simplified to:
\[ 96 \div 13 = 7 \frac{5}{13} \text{ days} \]

### Final Answer:
A, B, and C together can complete the work in \( 7 \frac{5}{13} \) days.