imc

Let $P$ be the probability Amelia wins. Note that $P=\text{chance she wins on her first turn}+\text{chance she gets to her turn again}\cdot P$, since if she gets to her turn again, she is back where she started with probability of winning $P$. The chance she wins on her first turn is $\frac{1}{3}$. The chance she makes it to her turn again is a combination of her failing to win the first turn - $\frac{2}{3}$ and Blaine failing to win - $\frac{3}{5}$. Multiplying gives us $\frac{2}{5}$. Thus, $$P=\frac{1}{3}+\frac{2}{5}P$$ Therefore, $P=\frac{5}{9}$, so the answer is $9-5=\boxed{4}$.