5. If the quadratic equation with positive integer coefficients
$$
4 x^{2}+m x+n=0
$$

has two distinct rational roots $p$ and $q$ ($p<q$), and the equations
$$
x^{2}-p x+2 q=0 \text { and } x^{2}-q x+2 p=0
$$

have a common root, find the other root of the equation
$$
x^{2}-p x+2 q=0
$$
Let the common root be $x_{0}$. Then
$$
\begin{array}{l}
x_{0}^{2}-p x_{0}+2 q=0, x_{0}^{2}-q x_{0}+2 p=0 \\
\Rightarrow x_{0}=-2 \Rightarrow p+q=-2 .
\end{array}
$$

Given $p+q=-\frac{m}{4}$, then $m=-8$.
From the equation $4 x^{2}+8 x+n=0$, we know
$$
\Delta=64-16 n>0 \text {. }
$$

Thus, $n<4$, and $n$ is a positive integer, $\Delta$ must be a perfect square.
Therefore, $n=3$.
Hence, $p=-\frac{3}{2}, q=-\frac{1}{2}$.
So the equation $x^{2}-p x+2 q=0$, which is $2 x^{2}+3 x-2=0$, has another root $x=\frac{1}{2}$.