Shivam alone and Deepak alone can finish a task in 6 hours and 10 hours respectively. Find the time taken by both of them together to do twice of the work?

Options

A. 8 hr
B. `7(1)/(2)` hr
C. `7(3)/(4)` hr
D. 6 hr

Answer

To solve the problem step by step, we will find the time taken by Shivam and Deepak together to complete twice the work.

### Step 1: Determine the individual work rates of Shivam and Deepak.
- Shivam can complete the task in 6 hours. Therefore, his work rate is:
\[ \text{Shivam's work rate} = \frac{1 \text{ task}}{6 \text{ hours}} = \frac{1}{6} \text{ tasks per hour} \]
- Deepak can complete the task in 10 hours. Therefore, his work rate is:
\[ \text{Deepak's work rate} = \frac{1 \text{ task}}{10 \text{ hours}} = \frac{1}{10} \text{ tasks per hour} \]

### Step 2: Calculate the combined work rate of Shivam and Deepak.
- To find their combined work rate, we add their individual work rates:
\[ \text{Combined work rate} = \text{Shivam's work rate} + \text{Deepak's work rate} = \frac{1}{6} + \frac{1}{10} \]
- To add these fractions, we need a common denominator. The least common multiple of 6 and 10 is 30:
\[ \frac{1}{6} = \frac{5}{30} \quad \text{and} \quad \frac{1}{10} = \frac{3}{30} \]
- Thus,
\[ \text{Combined work rate} = \frac{5}{30} + \frac{3}{30} = \frac{8}{30} = \frac{4}{15} \text{ tasks per hour} \]

### Step 3: Calculate the total work for twice the task.
- Since we need to complete twice the work, we multiply the work of one task by 2:
\[ \text{Total work} = 2 \text{ tasks} \]

### Step 4: Calculate the time taken to complete twice the work.
- To find the time taken to complete 2 tasks at the combined work rate, we use the formula:
\[ \text{Time} = \frac{\text{Total work}}{\text{Combined work rate}} = \frac{2 \text{ tasks}}{\frac{4}{15} \text{ tasks per hour}} \]
- This simplifies to:
\[ \text{Time} = 2 \times \frac{15}{4} = \frac{30}{4} = 7.5 \text{ hours} \]

### Final Answer:
The time taken by both Shivam and Deepak together to do twice the work is **7.5 hours**.

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