Rajan can finish a certain task in 6 days Bhavesh takes 8 days to complete the same task. Charan takes as long as Rajan and Bhavesh would take working together. How long will it take Bhavesh and Charan to complete the work together ?

Options

A. `2(2)/(5)` days
B. `2(7)/(5)` days
C. `2(3)/(5)` days
D. `2(4)/(5)` days

Answer

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

### Step 1: Determine the work done by Rajan and Bhavesh in one day.

- Rajan can finish the task in 6 days. Therefore, the work done by Rajan in one day is:
\[ \text{Work by Rajan in one day} = \frac{1}{6} \text{ of the task} \]

- Bhavesh can finish the task in 8 days. Therefore, the work done by Bhavesh in one day is:
\[ \text{Work by Bhavesh in one day} = \frac{1}{8} \text{ of the task} \]

### Step 2: Calculate the combined work done by Rajan and Bhavesh in one day.

To find the combined work done by Rajan and Bhavesh in one day, we add their individual work rates:
\[ \text{Combined work in one day} = \frac{1}{6} + \frac{1}{8} \]

To add these fractions, we need a common denominator. The least common multiple of 6 and 8 is 24. Therefore:
\[ \frac{1}{6} = \frac{4}{24} \quad \text{and} \quad \frac{1}{8} = \frac{3}{24} \]
\[ \text{Combined work in one day} = \frac{4}{24} + \frac{3}{24} = \frac{7}{24} \]

### Step 3: Determine how long it takes for Rajan and Bhavesh to complete the task together.

If they can complete \(\frac{7}{24}\) of the task in one day, the total time taken to complete the entire task is:
\[ \text{Time taken} = \frac{1}{\frac{7}{24}} = \frac{24}{7} \text{ days} \]

### Step 4: Determine how long it takes Charan to complete the task.

Charan takes as long as Rajan and Bhavesh would take working together, which we calculated to be \(\frac{24}{7}\) days.

### Step 5: Calculate the work done by Charan in one day.

The work done by Charan in one day is:
\[ \text{Work by Charan in one day} = \frac{1}{\frac{24}{7}} = \frac{7}{24} \text{ of the task} \]

### Step 6: Calculate the combined work done by Bhavesh and Charan in one day.

Now, we need to find the combined work done by Bhavesh and Charan:
\[ \text{Combined work in one day} = \frac{1}{8} + \frac{7}{24} \]

Again, we convert \(\frac{1}{8}\) to a fraction with a denominator of 24:
\[ \frac{1}{8} = \frac{3}{24} \]
\[ \text{Combined work in one day} = \frac{3}{24} + \frac{7}{24} = \frac{10}{24} = \frac{5}{12} \]

### Step 7: Determine how long it will take Bhavesh and Charan to complete the task together.

To find the time taken for them to complete the entire task together:
\[ \text{Time taken} = \frac{1}{\frac{5}{12}} = \frac{12}{5} \text{ days} = 2.4 \text{ days} \]

Thus, Bhavesh and Charan will take **2.4 days** to complete the work together.

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