13.313. The denominator of the fraction is less than the square of its numerator by 1. If 2 is added to both the numerator and the denominator, the value of the fraction will be greater than $1 / 4$; if 3 is subtracted from both the numerator and the denominator of the original fraction, the value of the fraction will be equal to $1 / 12$. Find this fraction.
Solution.

Let $\frac{x}{y}$ be the desired fraction. According to the condition, we have the system $\left\{\begin{array}{l}y=x^{2}-1, \\ \frac{x-3}{y-3}=\frac{1}{12}\end{array}\right.$, from which the desired fraction is $\frac{4}{15}$, and the fraction $\frac{8}{63}$ does not satisfy the inequality $\frac{x+2}{y+2}>\frac{1}{4}$.

Answer: $\frac{4}{15}$.