Vijay can do a work in 8 hours. Vijay and Puneet together can do the same work in 6 hours. Puneet and Sachin together can do the same work in 4 hours. Sachin alone can complete the same work in how many hours ?

Options

A. `22//5`
B. `28//5`
C. `24//5`
D. `26//5`

Answer

To solve the problem step by step, we will first determine the work done by each individual based on the information provided.

### Step 1: Determine the work done by Vijay
Vijay can complete the work in 8 hours. Therefore, the work done by Vijay in one hour is:
\[ \text{Work done by Vijay in 1 hour} = \frac{1}{8} \text{ of the work} \]

### Step 2: Determine the combined work done by Vijay and Puneet
Vijay and Puneet together can complete the work in 6 hours. Therefore, the work done by both in one hour is:
\[ \text{Work done by Vijay and Puneet in 1 hour} = \frac{1}{6} \text{ of the work} \]

### Step 3: Set up the equation for Puneet's work
Let Puneet's work in one hour be denoted as \( P \). Thus, we can write:
\[ \frac{1}{8} + P = \frac{1}{6} \]
To solve for \( P \), we first find a common denominator, which is 24:
\[ \frac{3}{24} + P = \frac{4}{24} \]
Subtracting \( \frac{3}{24} \) from both sides gives:
\[ P = \frac{4}{24} - \frac{3}{24} = \frac{1}{24} \]
This means Puneet can complete \( \frac{1}{24} \) of the work in one hour.

### Step 4: Determine the combined work done by Puneet and Sachin
Puneet and Sachin together can complete the work in 4 hours. Therefore, the work done by both in one hour is:
\[ \text{Work done by Puneet and Sachin in 1 hour} = \frac{1}{4} \text{ of the work} \]
Using Puneet's work rate, we can set up the equation:
\[ P + S = \frac{1}{4} \]
Substituting \( P = \frac{1}{24} \):
\[ \frac{1}{24} + S = \frac{1}{4} \]
Again, using a common denominator of 24:
\[ \frac{1}{24} + S = \frac{6}{24} \]
Subtracting \( \frac{1}{24} \) from both sides gives:
\[ S = \frac{6}{24} - \frac{1}{24} = \frac{5}{24} \]
This means Sachin can complete \( \frac{5}{24} \) of the work in one hour.

### Step 5: Calculate the time taken by Sachin to complete the work alone
To find out how long it will take Sachin to complete the entire work alone, we take the reciprocal of his work rate:
\[ \text{Time taken by Sachin} = \frac{1}{S} = \frac{1}{\frac{5}{24}} = \frac{24}{5} \text{ hours} \]

### Final Answer
Sachin alone can complete the work in \( \frac{24}{5} \) hours.

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