if `(m% of m) + (n% of n) = 2%` of (mn), then what percentage of m is n?

Options

A. 50`%`
B. 75`%`
C. 100`%`
D. Cannot be determined due to insufficient data

Answer

To solve the problem, we need to analyze the given equation step by step.

### Step 1: Understand the given equation
The equation provided is:
\[ (m\% \text{ of } m) + (n\% \text{ of } n) = 2\% \text{ of } (mn) \]

### Step 2: Convert percentages to fractions
We can express the percentages in the equation as fractions:
- \( m\% \text{ of } m = \frac{m}{100} \times m = \frac{m^2}{100} \)
- \( n\% \text{ of } n = \frac{n}{100} \times n = \frac{n^2}{100} \)
- \( 2\% \text{ of } (mn) = \frac{2}{100} \times (mn) = \frac{2mn}{100} \)

### Step 3: Substitute these into the equation
Substituting the fractions back into the equation gives us:
\[ \frac{m^2}{100} + \frac{n^2}{100} = \frac{2mn}{100} \]

### Step 4: Eliminate the denominator
To eliminate the denominator (100), we can multiply the entire equation by 100:
\[ m^2 + n^2 = 2mn \]

### Step 5: Rearrange the equation
Rearranging the equation gives:
\[ m^2 - 2mn + n^2 = 0 \]

### Step 6: Recognize the perfect square
The left-hand side can be factored as:
\[ (m - n)^2 = 0 \]

### Step 7: Solve for m and n
From the equation \((m - n)^2 = 0\), we find:
\[ m - n = 0 \implies m = n \]

### Step 8: Determine the percentage of m that is n
Since \(m = n\), we can express \(n\) as a percentage of \(m\):
\[ \frac{n}{m} \times 100\% = \frac{m}{m} \times 100\% = 100\% \]

### Final Answer
Thus, the percentage of \(m\) that is \(n\) is:
\[ \boxed{100\%} \]