Find two-digit numbers such that their ratio equals $0.254902$.

The rational number you give can be expressed as a fraction of two integers as $$0.254902=\frac{254902}{1000000}=\frac{127451}{500000},$$ where the latter cannot be simplified any further.

To find integers less than $100$ whose ratio is close to $0.254902$, note that $$0.254902>\frac{1}{4}\qquad\text{ and }\qquad0.004902<\frac{1}{200},$$ where $0.004902=.254902-\tfrac{1}{4}$. It follows that $$\frac{50}{200}<0.254902<\frac{51}{200}.$$ So for a fraction to approximate the given value, it must be of the form $\tfrac{x}{4x-1}$ for some $x\leq25$. This is a strictly decreasing function in $x$, allowing us to 'close in' on the desired value of $x$ by computing a few values. We find that for $x=13$ we obtain the repeating decimal expansion $$\frac{13}{51}=0.\overline{2549019607843137},$$ which rounds to the given value.

Feeding $0.254902$ to Wolfram Alpha also returns $\frac{13}{51}$ as a possible closed form.