Ravi alone does a piece of work in 2 days and Rajesh does it in 6 days. In how many days will the two do it together?

Options

A. 1 day
B. `1/2` day
C. `1 1/2` day
D. None of these

Answer

To solve the problem of how many days Ravi and Rajesh will take to complete the work together, we can follow these steps:

### Step-by-Step Solution:

1. **Determine the work rate of Ravi:**
- Ravi can complete the work in 2 days.
- Therefore, in one day, Ravi completes \( \frac{1}{2} \) of the work.

2. **Determine the work rate of Rajesh:**
- Rajesh can complete the work in 6 days.
- Therefore, in one day, Rajesh completes \( \frac{1}{6} \) of the work.

3. **Combine their work rates:**
- When Ravi and Rajesh work together, their combined work rate is the sum of their individual work rates.
- Combined work rate = \( \frac{1}{2} + \frac{1}{6} \).

4. **Find a common denominator to add the fractions:**
- The least common multiple (LCM) of 2 and 6 is 6.
- Convert \( \frac{1}{2} \) to have a denominator of 6:
\[ \frac{1}{2} = \frac{3}{6} \]
- Now add the two fractions:
\[ \frac{3}{6} + \frac{1}{6} = \frac{4}{6} \]

5. **Simplify the combined work rate:**
- \( \frac{4}{6} \) can be simplified to \( \frac{2}{3} \).

6. **Calculate the time taken to complete the work together:**
- If together they complete \( \frac{2}{3} \) of the work in one day, we need to find out how many days it will take to complete 1 whole work.
- To find the number of days, we take the reciprocal of their combined work rate:
\[ \text{Days} = \frac{1}{\frac{2}{3}} = \frac{3}{2} \]

7. **Interpret the result:**
- \( \frac{3}{2} \) days means 1 day and a half day, or 1.5 days.

### Final Answer:
Ravi and Rajesh together will complete the work in \( \frac{3}{2} \) days, which is equivalent to 1 day and 6 hours.

---