If p men working p hours per day for p days produce p units of work, then the units of work produced by n men working n hours a day for n days is

Options

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

Answer

To solve the problem step by step, we will use the concept of work done, which can be expressed using the formula:

\[ \text{Work} = \text{Number of Men} \times \text{Number of Hours per Day} \times \text{Number of Days} \]

### Step 1: Establish the Work Done by p Men
Given that \( p \) men working \( p \) hours per day for \( p \) days produce \( p \) units of work, we can express this as:

\[ \text{Work} = p \times p \times p = p^3 \]

So, we have:

\[ p^3 = p \quad \text{(1)} \]

### Step 2: Establish the Work Done by n Men
Now, we need to find the work done by \( n \) men working \( n \) hours a day for \( n \) days. Using the same formula for work, we have:

\[ \text{Work} = n \times n \times n = n^3 \]

### Step 3: Relate the Two Scenarios
From the first scenario, we know that \( p^3 \) units of work corresponds to \( p \) units of work. Therefore, we can set up a ratio of the work done:

\[ \frac{m_1 t_1 d_1}{w_1} = \frac{m_2 t_2 d_2}{w_2} \]

Where:
- \( m_1 = p \) (number of men in the first scenario)
- \( t_1 = p \) (hours per day in the first scenario)
- \( d_1 = p \) (days in the first scenario)
- \( w_1 = p \) (work done in the first scenario)
- \( m_2 = n \) (number of men in the second scenario)
- \( t_2 = n \) (hours per day in the second scenario)
- \( d_2 = n \) (days in the second scenario)
- \( w_2 \) is the work done in the second scenario which we need to find.

### Step 4: Substitute Values into the Equation
Substituting the values into the equation gives us:

\[ \frac{p \times p \times p}{p} = \frac{n \times n \times n}{w_2} \]

This simplifies to:

\[ p^2 = \frac{n^3}{w_2} \]

### Step 5: Solve for w2
Rearranging the equation to solve for \( w_2 \):

\[ w_2 = \frac{n^3}{p^2} \]

### Final Result
Thus, the units of work produced by \( n \) men working \( n \) hours a day for \( n \) days is:

\[ w_2 = \frac{n^3}{p^2} \]