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

Question

Accepted Answer

To solve the problem, we need to analyze the given equation:

$$
(m\% \text{ of } m) + (n\% \text{ of } n) = 2\% \text{ of } (mn)
$$

This can be rewritten using the percentage formula:

$$
\left(\frac{m}{100} \times m\right) + \left(\frac{n}{100} \times n\right) = \frac{2}{100} \times (mn)
$$

Simplifying each term, we get:

$$
\frac{m^2}{100} + \frac{n^2}{100} = \frac{2mn}{100}
$$

Multiply the entire equation by 100 to eliminate the denominators:

$$
m^2 + n^2 = 2mn
$$

Rearrange the equation:

$$
m^2 - 2mn + n^2 = 0
$$

This equation can be recognized as a perfect square:

$$
(m - n)^2 = 0
$$

The only solution to this equation is:

$$
m - n = 0 \quad \Rightarrow \quad m = n
$$

Since $m = n$, $n$ is 100% of $m$. Therefore, the correct answer is Option 3: 100%.

**Explanation for Other Options:**

1. **Option 1 (50%)**: This would imply that $n$ is half of $m$, which does not satisfy the equation $(m - n)^2 = 0$. Therefore, this option is incorrect.

2. **Option 2 (75%)**: This would imply that $n$ is 75% of $m$, which also does not satisfy the equation $(m - n)^2 = 0$. Therefore, this option is incorrect.

4. **Option 4 (Cannot be determined due to insufficient data)**: The equation $(m - n)^2 = 0$ provides a clear solution that $m = n$, making the data sufficient to determine that $n$ is 100% of $m$. Therefore, this option is incorrect.

Thus, the correct answer is indeed Option 3: 100%.