If m men working m hr per day, can do m units of work in m days, then n men working n hr per day would be able to complete how many units of work in n days?

Options

A. `(n^3)/(m^2) `
B. `(m^3)/(n^2)`
C. `(m^4)/(n^2)`
D. `(n^4)/(m^2)`

Answer

To solve the problem step by step, we will use the relationship between men, days, hours, and work done.

### Step-by-Step Solution:

1. **Understanding the Given Information:**
- We know that \( m \) men working \( m \) hours per day can complete \( m \) units of work in \( m \) days.
- We need to find out how many units of work \( n \) men working \( n \) hours per day can complete in \( n \) days.

2. **Setting Up the Equation:**
- We can use the formula based on the relationship of men, days, hours, and work:
\[ \frac{m_1 \cdot d_1 \cdot h_1}{w_1} = \frac{m_2 \cdot d_2 \cdot h_2}{w_2} \]
- Here:
- \( m_1 = m \)
- \( d_1 = m \)
- \( h_1 = m \)
- \( w_1 = m \)
- \( m_2 = n \)
- \( d_2 = n \)
- \( h_2 = n \)
- \( w_2 \) is what we need to find.

3. **Substituting the Values:**
- Substitute the values into the equation:
\[ \frac{m \cdot m \cdot m}{m} = \frac{n \cdot n \cdot n}{w_2} \]
- This simplifies to:
\[ m^2 = \frac{n^3}{w_2} \]

4. **Rearranging the Equation:**
- Rearranging the equation to find \( w_2 \):
\[ w_2 = \frac{n^3}{m^2} \]

5. **Conclusion:**
- Therefore, the number of units of work that \( n \) men working \( n \) hours per day can complete in \( n \) days is:
\[ w_2 = \frac{n^3}{m^2} \]

### Final Answer:
The answer is \( \frac{n^3}{m^2} \) units of work.

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