5. A.

Let the original number of members in the activity group be $x$, and the total sum of their ages be $y$.
From the problem, we have
$$\begin{array}{l} \left\{\begin{array} { l } { \frac { y - 65 } { x - 5 } = \frac { y } { x } + 1 , } \\ { \frac { y + 85 } { x + 5 } = \frac { y } { x } + 1 } \end{array} \Rightarrow \left\{\begin{array}{l} x^{2}+60 x=5 y, \\ x^{2}-80 x=-5 y \end{array}\right.\right. \\ \Rightarrow 2 x^{2}-20 x=0 \end{array}$$