Example 7 The roots of the equation $x^{2}+p x+q=0$ are both positive integers, and $p+q=1992$. Then the ratio of the larger root to the smaller root is $\qquad$.
Given: $\because x_{1}+x_{2}=-p, x_{1} x_{2}=q$,
$$
\begin{array}{l}
\therefore x_{1} x_{2}-x_{1}-x_{2}=q+p=1992, \\
\left(x_{1}-1\right)\left(x_{2}-1\right)=1993 .
\end{array}
$$
$\because 1993$ is a prime number,
$$
\therefore\left\{\begin{array}{l}
x_{1}-1=1, \\
x_{2}-1=1993 .
\end{array}\right.
$$

Solving, we get $\left\{\begin{array}{l}x_{1}=2, \\ x_{2}=1994 .\end{array}\right.$
$$
\text { Therefore, } \frac{x_{2}}{x_{1}}=997 \text {. }
$$