Umang can built a house alone in 16 days but Raj alone can built it in 12 days. Umang and Raj work on alternate days. If Umang works on first day, the house will be built in how many days?

Options

A. `25/2 days`
B. `13 3/4 days`
C. `48/7 days`
D. `24/7 days`

Answer

To solve the problem, we need to determine how many days it will take for Umang and Raj to build the house when they work on alternate days, starting with Umang.

### Step-by-Step Solution:

1. **Determine the work done by Umang and Raj in one day:**
- Umang can complete the house in 16 days, so his work rate is:
\[ \text{Umang's work rate} = \frac{1}{16} \text{ of the house per day} \]
- Raj can complete the house in 12 days, so his work rate is:
\[ \text{Raj's work rate} = \frac{1}{12} \text{ of the house per day} \]

2. **Calculate the total work done in two days (one cycle of work):**
- On the first day, Umang works:
\[ \text{Work done by Umang on Day 1} = \frac{1}{16} \]
- On the second day, Raj works:
\[ \text{Work done by Raj on Day 2} = \frac{1}{12} \]
- Therefore, the total work done in two days is:
\[ \text{Total work in 2 days} = \frac{1}{16} + \frac{1}{12} \]
- To add these fractions, we need a common denominator. The least common multiple of 16 and 12 is 48:
\[ \frac{1}{16} = \frac{3}{48}, \quad \frac{1}{12} = \frac{4}{48} \]
- Thus,
\[ \text{Total work in 2 days} = \frac{3}{48} + \frac{4}{48} = \frac{7}{48} \]

3. **Determine how many cycles are needed to complete the house:**
- Let \( x \) be the number of complete 2-day cycles needed to complete the house. The total work done after \( x \) cycles is:
\[ \text{Total work after } x \text{ cycles} = x \cdot \frac{7}{48} \]
- We need this to equal 1 (the whole house):
\[ x \cdot \frac{7}{48} = 1 \implies x = \frac{48}{7} \]

4. **Calculate the total time taken:**
- Each cycle takes 2 days, so the total time for \( x \) cycles is:
\[ \text{Total time for } x \text{ cycles} = 2x = 2 \cdot \frac{48}{7} = \frac{96}{7} \text{ days} \]
- This is approximately 13.71 days.

5. **Calculate the remaining work after \( x \) cycles:**
- After \( \frac{48}{7} \) cycles, the work done is:
\[ \text{Work done} = \frac{48}{7} \cdot \frac{7}{48} = 1 \text{ (the house is complete)} \]
- However, since Umang starts the next day, we need to check how much work is left after the complete cycles.

6. **Final adjustments:**
- After \( 6 \) complete cycles (12 days), the work done is:
\[ 6 \cdot \frac{7}{48} = \frac{42}{48} = \frac{7}{8} \]
- Remaining work:
\[ 1 - \frac{7}{8} = \frac{1}{8} \]
- On the next day, Umang works and completes:
\[ \text{Work done by Umang on Day 13} = \frac{1}{16} \]
- Since \( \frac{1}{16} \) is more than \( \frac{1}{8} \), Umang will finish the house on Day 13.

### Conclusion:
The total time taken to build the house is:
\[ \text{Total time} = 12 + \frac{1}{2} = 13 \frac{1}{2} \text{ days} = 13 \frac{3}{4} \text{ days} \]

### Final Answer:
**B) 13 and 3/4 days**