If m men can do a work in r days, then the number of days taken by `(m+n)` men do it is :

Options

A. `(m+n)/(mn)`
B. `(m+n)/(mr)`
C. `(mr)/(m+n)`
D. `((m+n)r)/(mn)`

Answer

To solve the problem step by step, we can use the concept of work done, which is based on the relationship between the number of workers, the time taken, and the amount of work done.

### Step-by-Step Solution:

1. **Understand the Given Information:**
- Let \( m \) be the number of men.
- Let \( r \) be the number of days taken by \( m \) men to complete the work.
- We need to find the number of days taken by \( (m+n) \) men to complete the same work.

2. **Calculate the Total Work Done:**
- The total work done can be expressed as:
\[ \text{Total Work} = \text{Number of Men} \times \text{Number of Days} = m \times r \]

3. **Set Up the Equation for \( (m+n) \) Men:**
- Let \( d_2 \) be the number of days taken by \( (m+n) \) men to complete the same work. The work done by \( (m+n) \) men is:
\[ \text{Total Work} = (m+n) \times d_2 \]

4. **Equate the Total Work Done:**
- Since the total work done remains the same, we can set the two expressions for total work equal to each other:
\[ m \times r = (m+n) \times d_2 \]

5. **Solve for \( d_2 \):**
- Rearranging the equation to solve for \( d_2 \):
\[ d_2 = \frac{m \times r}{m+n} \]

6. **Conclusion:**
- Therefore, the number of days taken by \( (m+n) \) men to complete the work is:
\[ d_2 = \frac{m \times r}{m+n} \]

### Final Answer:
The number of days taken by \( (m+n) \) men to complete the work is \( \frac{m \times r}{m+n} \).

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