To solve the expression \( \frac{5 \, \text{dm}^2}{25} = \frac{\text{mm} \cdot 25 \, \text{dm}^2}{25} = \text{mm}^2 \), we can break it down step-by-step.

1. **Convert decimeters to millimeters**:
- We know that \( 1 \, \text{dm} = 100 \, \text{mm} \).
- Therefore, \( 1 \, \text{dm}^2 = (100 \, \text{mm})^2 = 10000 \, \text{mm}^2 \).

2. **Convert \( 5 \, \text{dm}^2 \) to mm²**:
- \( 5 \, \text{dm}^2 = 5 \times 10000 \, \text{mm}^2 = 50000 \, \text{mm}^2 \).

3. **Now, substitute this into the left-hand side**:
- \( \frac{5 \, \text{dm}^2}{25} = \frac{50000 \, \text{mm}^2}{25} = 2000 \, \text{mm}^2 \).

4. **Now handle the right side**:
- The expression \( \frac{\text{mm} \cdot 25 \, \text{dm}^2}{25} \) simplifies to just \( \text{mm} \cdot \text{dm}^2 \) because the \( 25 \) cancels out.
- Since \( 1 \, \text{dm}^2 \) is equal to \( 10000 \, \text{mm}^2 \), the expression becomes \( \text{mm} \cdot 10000 \, \text{mm}^2 = 10000 \, \text{mm}^3 \).

This means the interpretations on both sides do not equate.

If you intended a different operation or a different equality, please provide specific details for clarification.