Solution:

At any time, let $a$ and $b$ represent the respective number of rounds won by Aerith and Bob. Now, after $n$ games, $a-3b = n(85\% - 3 \cdot 15\%) > 0$. However, at some point in the past $b$ was greater than $a$, so $a-3b$ was negative. Consider the first subsequent moment when $a-3b$ was non-negative. At that moment,
$$(a-1)-3b < 0 \leq a-3b,$$
so as $a$ and $b$ were integers, $a-3b=0$ and $a=75\% \cdot (a+b)$, as desired.