Aditya, Vedus and Yuvraj alone can do a job for 6 weeks, 9 weeks and 12 weeks respectively. They work together for 2 weeks. Then, Aditya leaves the job. Vedus leaves the job a week earlier to the completion of the work. The job would be completed in:

Options

A. 4 weeks
B. 5 weeks
C. 7 weeks
D. None of these

Answer

To solve the problem step-by-step, let's break it down:

### Step 1: Determine the work done by each person in one week.
- **Aditya's work rate:** He can complete the job in 6 weeks, so his work rate is \( \frac{1}{6} \) of the job per week.
- **Vedus's work rate:** He can complete the job in 9 weeks, so his work rate is \( \frac{1}{9} \) of the job per week.
- **Yuvraj's work rate:** He can complete the job in 12 weeks, so his work rate is \( \frac{1}{12} \) of the job per week.

### Step 2: Calculate the combined work rate when all three work together.
- Combined work rate = \( \frac{1}{6} + \frac{1}{9} + \frac{1}{12} \)

To add these fractions, we need a common denominator. The least common multiple (LCM) of 6, 9, and 12 is 36.

- Convert each fraction:
- \( \frac{1}{6} = \frac{6}{36} \)
- \( \frac{1}{9} = \frac{4}{36} \)
- \( \frac{1}{12} = \frac{3}{36} \)

Now, add them together:
\[ \frac{6}{36} + \frac{4}{36} + \frac{3}{36} = \frac{13}{36} \]

### Step 3: Calculate the total work done in 2 weeks.
- Work done in 2 weeks = \( 2 \times \frac{13}{36} = \frac{26}{36} = \frac{13}{18} \) of the job.

### Step 4: Determine the remaining work.
- Total work = 1 (the whole job).
- Remaining work = \( 1 - \frac{13}{18} = \frac{5}{18} \) of the job.

### Step 5: Determine who works next and their work rates.
- After 2 weeks, Aditya leaves. Now, Vedus and Yuvraj will work together.
- Combined work rate of Vedus and Yuvraj = \( \frac{1}{9} + \frac{1}{12} \)

Convert these fractions to a common denominator:
- LCM of 9 and 12 is 36.
- Convert:
- \( \frac{1}{9} = \frac{4}{36} \)
- \( \frac{1}{12} = \frac{3}{36} \)

Combined work rate = \( \frac{4}{36} + \frac{3}{36} = \frac{7}{36} \)

### Step 6: Calculate the work done by Vedus and Yuvraj in 1 week.
- Work done in 1 week = \( \frac{7}{36} \).

### Step 7: Determine the remaining work after Vedus leaves.
- Vedus leaves a week earlier, so they work together for 1 week.
- Work done in 1 week = \( \frac{7}{36} \).
- Remaining work = \( \frac{5}{18} - \frac{7}{36} \).

Convert \( \frac{5}{18} \) to a fraction with a denominator of 36:
- \( \frac{5}{18} = \frac{10}{36} \).

Now, calculate remaining work:
\[ \frac{10}{36} - \frac{7}{36} = \frac{3}{36} = \frac{1}{12} \]

### Step 8: Determine how long Yuvraj will take to finish the remaining work.
- Yuvraj's work rate is \( \frac{1}{12} \).
- Time taken to finish \( \frac{1}{12} \) of the job = \( \frac{1/12}{1/12} = 1 \) week.

### Step 9: Calculate the total time taken to complete the job.
- Total time = 2 weeks (initial work) + 1 week (Vedus and Yuvraj) + 1 week (Yuvraj alone) = 4 weeks.

### Final Answer:
The job would be completed in **4 weeks**.

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