Umang can build a house alone in 16 days but Raj alone can build 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 ?

Question

Accepted Answer

To determine how many days it will take for Umang and Raj to build the house when working on alternate days, with Umang starting on the first day, we need to calculate the amount of work each can complete in a day and then determine how many days it will take to complete the entire house.

1. **Calculate the work done by each person in one day:**
   - Umang can build the house alone in 16 days. Therefore, in one day, Umang completes $\frac{1}{16}$ of the house.
   - Raj can build the house alone in 12 days. Therefore, in one day, Raj completes $\frac{1}{12}$ of the house.

2. **Calculate the work done in two days:**
   - On the first day, Umang works and completes $\frac{1}{16}$ of the house.
   - On the second day, Raj works and completes $\frac{1}{12}$ of the house.
   - In two days, the total work completed is $\frac{1}{16} + \frac{1}{12}$.

3. **Find a common denominator and add the fractions:**
   - The least common multiple of 16 and 12 is 48.
   - Convert the fractions: $\frac{1}{16} = \frac{3}{48}$ and $\frac{1}{12} = \frac{4}{48}$.
   - Add the fractions: $\frac{3}{48} + \frac{4}{48} = \frac{7}{48}$.

4. **Determine how many such two-day cycles are needed:**
   - To complete the house, we need the total work to equal 1 (the whole house).
   - Let $n$ be the number of two-day cycles needed. Then, $n \times \frac{7}{48} = 1$.
   - Solving for $n$, we get $n = \frac{48}{7}$.

5. **Calculate the total number of days:**
   - Each cycle consists of 2 days, so $n$ cycles take $2n = 2 \times \frac{48}{7} = \frac{96}{7}$ days.
   - However, since the house is completed in the middle of a cycle, we need to find the exact day count.
   - After 6 full cycles (12 days), $\frac{6 \times 7}{48} = \frac{42}{48}$ of the house is completed.
   - On the 13th day, Umang works and completes $\frac{3}{48}$ more, totaling $\frac{45}{48}$.
   - On the 14th day, Raj works and completes $\frac{4}{48}$, but only $\frac{3}{48}$ is needed to finish the house.
   - Therefore, the house is completed in $13 + \frac{3}{4} = 13\frac{3}{4}$ days.

Thus, the correct answer is Option 2: $13\frac{3}{4} \ \text{days}$.

**Explanation for why other options are incorrect:**

- **Option 1: $\frac{25}{2} \ \text{days}$** is incorrect because it suggests a longer time than necessary. The calculations show that the house is completed in less than 14 days.
  
- **Option 3: $\frac{48}{7} \ \text{days}$** is incorrect because it represents the number of two-day cycles needed, not the actual number of days. The house is completed in the middle of the 7th cycle.

- **Option 4: $\frac{24}{7} \ \text{days}$** is incorrect because it underestimates the time required. The calculations show that more than 12 days are needed to complete the house.

Therefore, the correct answer is indeed Option 2: $13\frac{3}{4} \ \text{days}$.