1. The problem is to convert the repeating decimal $0.0\overline{25}$ to a fraction. 2. Let $x = 0.0\overline{25}$, which means $x = 0.0252525\ldots$ 3. Since the repeating block "25" has 2 digits, multiply $x$ by $10^2 = 100$ to shift the decimal point two places: $$100x = 2.5252525\ldots$$ 4. Now subtract the original $x$ from this equation to eliminate the repeating part: $$100x - x = 2.5252525\ldots - 0.0252525\ldots$$ $$99x = 2.5$$ 5. Solve for $x$: $$x = \frac{2.5}{99}$$ 6. Convert $2.5$ to a fraction: $$2.5 = \frac{5}{2}$$ 7. Substitute back: $$x = \frac{\frac{5}{2}}{99} = \frac{5}{2 \times 99} = \frac{5}{198}$$ 8. The fraction $\frac{5}{198}$ is in simplest form because 5 and 198 share no common factors other than 1. Final answer: $$0.0\overline{25} = \frac{5}{198}$$