A and B working together can complete the work in 12 days. A can complete the work in 20 days. If B does work for half day each day, then in what time can the work be completed.

Answer

To solve the problem, we need to determine how much work A and B can do together and how B's half-day work affects the total time taken to complete the work.

### Step-by-Step Solution:

1. **Determine the work done by A and B together:**
- A and B together can complete the work in 12 days.
- Therefore, the work done by A and B together in one day is:
\[ \text{Work rate of A and B together} = \frac{1}{12} \text{ (work/day)} \]

2. **Determine the work done by A alone:**
- A can complete the work in 20 days.
- Therefore, the work done by A in one day is:
\[ \text{Work rate of A} = \frac{1}{20} \text{ (work/day)} \]

3. **Determine the work done by B alone:**
- Let the work done by B in one day be \( b \).
- Since A and B together can complete the work in 12 days, we have:
\[ \frac{1}{20} + b = \frac{1}{12} \]
- To find \( b \), we solve the equation:
\[ b = \frac{1}{12} - \frac{1}{20} \]
- Finding a common denominator (60):
\[ b = \frac{5}{60} - \frac{3}{60} = \frac{2}{60} = \frac{1}{30} \text{ (work/day)} \]
- Thus, B can complete the work in 30 days.

4. **Determine the effective work done by B when he works for half a day:**
- If B works for half a day, the work done by B in half a day is:
\[ \text{Work done by B in half a day} = \frac{1}{2} \times \frac{1}{30} = \frac{1}{60} \text{ (work)} \]

5. **Determine the total work done in one day when A works full day and B works half day:**
- In one day, A works for a full day and B works for half a day, so the total work done in one day is:
\[ \text{Total work done in one day} = \frac{1}{20} + \frac{1}{60} \]
- Finding a common denominator (60):
\[ \text{Total work done in one day} = \frac{3}{60} + \frac{1}{60} = \frac{4}{60} = \frac{1}{15} \text{ (work/day)} \]

6. **Calculate the total time to complete the work:**
- Since the total work is 1 unit, the time taken to complete the work is:
\[ \text{Time} = \frac{1 \text{ (total work)}}{\frac{1}{15} \text{ (work/day)}} = 15 \text{ days} \]

### Final Answer:
The work can be completed in **15 days**.