Samarth can complete a work in 6 h. Soumik can complete same work in 8 h. How long should it take for both Samarth and Soumnik working together but independently, to do the same work?

Options

A. `1(2)/7 d`
B. `3(1)/7 d`
C. `3(3)/7 d`
D. `4(3)/7` d

Answer

To solve the problem of how long it will take for Samarth and Soumik to complete the work together, we can follow these steps:

### Step 1: Determine the work rates of Samarth and Soumik.
- Samarth can complete the work in 6 hours, so his work rate is:
\[ \text{Work rate of Samarth} = \frac{1 \text{ work}}{6 \text{ hours}} = \frac{1}{6} \text{ work/hour} \]
- Soumik can complete the work in 8 hours, so his work rate is:
\[ \text{Work rate of Soumik} = \frac{1 \text{ work}}{8 \text{ hours}} = \frac{1}{8} \text{ work/hour} \]

### Step 2: Combine their work rates.
When working together, their combined work rate is the sum of their individual work rates:
\[ \text{Combined work rate} = \frac{1}{6} + \frac{1}{8} \]

### Step 3: Find a common denominator and add the fractions.
The least common multiple (LCM) of 6 and 8 is 24. We can rewrite the fractions:
\[ \frac{1}{6} = \frac{4}{24}, \quad \frac{1}{8} = \frac{3}{24} \]
Now, add the two fractions:
\[ \text{Combined work rate} = \frac{4}{24} + \frac{3}{24} = \frac{7}{24} \text{ work/hour} \]

### Step 4: Determine the time taken to complete the work together.
Let \( x \) be the time taken to complete the work together. Since their combined work rate is \( \frac{7}{24} \) work/hour, we can set up the equation:
\[ \frac{7}{24} \cdot x = 1 \text{ work} \]
To find \( x \), we rearrange the equation:
\[ x = \frac{1}{\frac{7}{24}} = \frac{24}{7} \text{ hours} \]

### Step 5: Convert the time into a mixed number.
To convert \( \frac{24}{7} \) into a mixed number:
- Divide 24 by 7, which gives 3 with a remainder of 3.
Thus, \( \frac{24}{7} = 3 \frac{3}{7} \) hours.

### Final Answer:
It will take Samarth and Soumik \( 3 \frac{3}{7} \) hours to complete the work together.

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