Arjun started a task and left it after working for 2 days. Then, Bharath was called and he completed the task in 9 days. If Arjun alone had worked for 3 days, Bharath alone would have finished the remaining work in 6 days. In how many days can Arjun alone complete the task?

Options

A. 8 days
B. 12 days
C. 5 days
D. 10 days

Answer

To solve the problem, let's denote the total work as 1 unit. We will also denote the work done by Arjun in one day as \( A \) and the work done by Bharath in one day as \( B \).

### Step 1: Work done by Arjun in 2 days
Arjun works for 2 days, so the work done by Arjun in 2 days is:
\[ \text{Work done by Arjun} = 2A \]

### Step 2: Work done by Bharath in 9 days
After Arjun leaves, Bharath completes the remaining work in 9 days. Therefore, the work done by Bharath in 9 days is:
\[ \text{Work done by Bharath} = 9B \]

### Step 3: Total work equation
Since the total work is 1 unit, we can write the equation:
\[ 2A + 9B = 1 \quad \text{(Equation 1)} \]

### Step 4: Work done if Arjun worked for 3 days
If Arjun had worked for 3 days, the work done by him would be:
\[ \text{Work done by Arjun in 3 days} = 3A \]
Then, Bharath would finish the remaining work in 6 days, so:
\[ \text{Work done by Bharath in 6 days} = 6B \]

### Step 5: Total work equation for this scenario
The total work in this scenario can also be expressed as:
\[ 3A + 6B = 1 \quad \text{(Equation 2)} \]

### Step 6: Solve the system of equations
Now we have two equations:
1. \( 2A + 9B = 1 \)
2. \( 3A + 6B = 1 \)

We can solve these equations simultaneously. Let's multiply Equation 2 by 3 to eliminate \( B \):
\[ 9A + 18B = 3 \quad \text{(Equation 3)} \]

Now, we can subtract Equation 1 from Equation 3:
\[ (9A + 18B) - (2A + 9B) = 3 - 1 \]
\[ 7A + 9B = 2 \quad \text{(Equation 4)} \]

### Step 7: Substitute to find \( B \)
Now we can express \( B \) in terms of \( A \) using Equation 1:
\[ 9B = 1 - 2A \implies B = \frac{1 - 2A}{9} \]

Substituting \( B \) into Equation 4:
\[ 7A + 9\left(\frac{1 - 2A}{9}\right) = 2 \]
\[ 7A + 1 - 2A = 2 \]
\[ 5A = 1 \implies A = \frac{1}{5} \]

### Step 8: Find \( B \)
Now substitute \( A \) back to find \( B \):
\[ B = \frac{1 - 2\left(\frac{1}{5}\right)}{9} = \frac{1 - \frac{2}{5}}{9} = \frac{\frac{3}{5}}{9} = \frac{1}{15} \]

### Step 9: Calculate days for Arjun to complete the task alone
If Arjun can do \( A = \frac{1}{5} \) of the work in one day, then the number of days Arjun would take to complete the task alone is:
\[ \text{Days for Arjun} = \frac{1}{A} = \frac{1}{\frac{1}{5}} = 5 \text{ days} \]

### Final Answer
Arjun alone can complete the task in **5 days**.