Vivek, Dev and Lalit alone can complete a work in 10 days, 15 days and 20 days respectively. In how many days Vivek and Dev together can complete half of the same work?

Options

A. 9 days
B. 5 days
C. 3 days
D. 6 days

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Determine the work rates of Vivek, Dev, and Lalit
- Vivek can complete the work in 10 days. Therefore, his work rate is:
\[ \text{Work rate of Vivek} = \frac{1 \text{ work}}{10 \text{ days}} = \frac{1}{10} \text{ work per day} \]

- Dev can complete the work in 15 days. Therefore, his work rate is:
\[ \text{Work rate of Dev} = \frac{1 \text{ work}}{15 \text{ days}} = \frac{1}{15} \text{ work per day} \]

- Lalit can complete the work in 20 days. Therefore, his work rate is:
\[ \text{Work rate of Lalit} = \frac{1 \text{ work}}{20 \text{ days}} = \frac{1}{20} \text{ work per day} \]

### Step 2: Calculate the total work done by Vivek and Dev together in one day
To find the combined work rate of Vivek and Dev, we add their individual work rates:
\[ \text{Combined work rate of Vivek and Dev} = \frac{1}{10} + \frac{1}{15} \]

To add these fractions, we need a common denominator. The least common multiple (LCM) of 10 and 15 is 30. Thus, we convert the fractions:
\[ \frac{1}{10} = \frac{3}{30}, \quad \frac{1}{15} = \frac{2}{30} \]
Now, adding these:
\[ \text{Combined work rate} = \frac{3}{30} + \frac{2}{30} = \frac{5}{30} = \frac{1}{6} \text{ work per day} \]

### Step 3: Determine how much work Vivek and Dev need to complete
Since we need to find out how many days it takes for them to complete half of the work, we calculate half of the total work:
\[ \text{Half of the work} = \frac{1}{2} \text{ work} \]

### Step 4: Calculate the time taken to complete half of the work
Now, we need to find out how many days it will take for Vivek and Dev to complete half of the work at their combined rate:
\[ \text{Time} = \frac{\text{Work}}{\text{Rate}} = \frac{\frac{1}{2}}{\frac{1}{6}} = \frac{1}{2} \times \frac{6}{1} = 3 \text{ days} \]

### Final Answer
Thus, Vivek and Dev together can complete half of the work in **3 days**.

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