Chandni and Divakar can do a piece of work in 9 days and 12 days respectively. If they work for a day alternatively, Chandni beginning, in how many days, the work will be completed?

Options

A. `10(1)/(4)`
B. `9(1)/(4)`
C. 11.11
D. 10

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Determine the work done by Chandni and Divakar in one day.
- **Chandni** can complete the work in **9 days**. Therefore, in one day, she can do:
\[ \text{Work done by Chandni in one day} = \frac{1}{9} \text{ of the work} \]

- **Divakar** can complete the work in **12 days**. Therefore, in one day, he can do:
\[ \text{Work done by Divakar in one day} = \frac{1}{12} \text{ of the work} \]

### Step 2: Find the total work done in two days when they work alternatively.
- On the **first day**, Chandni works and completes:
\[ \text{Work done on Day 1} = \frac{1}{9} \]

- On the **second day**, Divakar works and completes:
\[ \text{Work done on Day 2} = \frac{1}{12} \]

- Therefore, the total work done in **two days** is:
\[ \text{Total work in 2 days} = \frac{1}{9} + \frac{1}{12} \]

### Step 3: Calculate the total work done in two days.
To add \(\frac{1}{9}\) and \(\frac{1}{12}\), we need a common denominator. The least common multiple (LCM) of 9 and 12 is 36.

- Convert \(\frac{1}{9}\) to a fraction with a denominator of 36:
\[ \frac{1}{9} = \frac{4}{36} \]

- Convert \(\frac{1}{12}\) to a fraction with a denominator of 36:
\[ \frac{1}{12} = \frac{3}{36} \]

- Now add the two fractions:
\[ \text{Total work in 2 days} = \frac{4}{36} + \frac{3}{36} = \frac{7}{36} \]

### Step 4: Determine how many such cycles are needed to complete the work.
- In **two days**, they complete \(\frac{7}{36}\) of the work.
- To find out how many complete cycles (2 days) are needed to finish the work, we set up the equation:
\[ n \cdot \frac{7}{36} = 1 \quad \text{(where n is the number of 2-day cycles)} \]
\[ n = \frac{36}{7} \approx 5.14 \]

### Step 5: Calculate the total days taken.
- Since \(n\) is approximately 5.14, this means they can complete 5 full cycles (10 days) and will have some work left.
- In 10 days, they will complete:
\[ 5 \cdot \frac{7}{36} = \frac{35}{36} \]
- This means that after 10 days, \(\frac{1}{36}\) of the work is still remaining.

### Step 6: Determine how long it takes to complete the remaining work.
- Now, it is Chandni's turn to work again. She can complete \(\frac{1}{9}\) of the work in one day.
- To complete the remaining \(\frac{1}{36}\) of the work, we find out how much time she will take:
\[ \text{Time taken by Chandni} = \frac{\text{Remaining work}}{\text{Work done by Chandni in one day}} = \frac{\frac{1}{36}}{\frac{1}{9}} = \frac{1}{36} \times 9 = \frac{1}{4} \text{ days} \]

### Step 7: Calculate the total time taken.
- Total time taken = 10 days + \(\frac{1}{4}\) days:
\[ \text{Total time} = 10 + \frac{1}{4} = 10 \frac{1}{4} \text{ days} \]

Thus, the work will be completed in **10 \(\frac{1}{4}\) days**.