Lokesh starts a work and after 6 days he left and remaining work finished by Rakesh in 12 days. Had Lokesh worked for 9 days, Rakesh would have finished the remaining work in 8 days. Find the time taken by Lokesh to complete the work alone.

Options

A. 12 days
B. 9 days
C. 15 days
D. 18 days

Answer

To solve the problem step by step, we will denote Lokesh's efficiency as \( L \) and Rakesh's efficiency as \( R \).

### Step 1: Understand the Work Done
Lokesh works for 6 days and then Rakesh completes the remaining work in 12 days. We can express the total work in terms of their efficiencies.

### Step 2: Set Up the Equations
From the first condition:
- Work done by Lokesh in 6 days = \( 6L \)
- Work done by Rakesh in 12 days = \( 12R \)

The total work can be expressed as:
\[ 6L + 12R = \text{Total Work} \quad (1) \]

### Step 3: Analyze the Second Condition
If Lokesh had worked for 9 days, Rakesh would have finished the remaining work in 8 days.
- Work done by Lokesh in 9 days = \( 9L \)
- Remaining work done by Rakesh in 8 days = \( 8R \)

The total work can also be expressed as:
\[ 9L + 8R = \text{Total Work} \quad (2) \]

### Step 4: Equate the Total Work from Both Conditions
From equations (1) and (2), we can set them equal to each other:
\[ 6L + 12R = 9L + 8R \]

### Step 5: Rearranging the Equation
Rearranging gives:
\[ 12R - 8R = 9L - 6L \]
\[ 4R = 3L \]

### Step 6: Find the Ratio of Efficiencies
From the equation \( 4R = 3L \), we can express the ratio of Lokesh's efficiency to Rakesh's efficiency:
\[ \frac{L}{R} = \frac{4}{3} \]

### Step 7: Express Rakesh's Efficiency in Terms of Lokesh's
Let \( R = \frac{3}{4}L \).

### Step 8: Substitute Back to Find Total Work
Substituting \( R \) in either equation (1) or (2) to find the total work. Let's use equation (1):
\[ 6L + 12\left(\frac{3}{4}L\right) = \text{Total Work} \]
\[ 6L + 9L = \text{Total Work} \]
\[ 15L = \text{Total Work} \]

### Step 9: Calculate the Time Taken by Lokesh Alone
To find the time taken by Lokesh to complete the work alone, we use:
\[ \text{Time} = \frac{\text{Total Work}}{\text{Efficiency of Lokesh}} = \frac{15L}{L} = 15 \text{ days} \]

### Final Answer
Thus, the time taken by Lokesh to complete the work alone is **15 days**.