Question 5
Alpha and Beta both took part in a two-day problem-solving competition. At the end of the second day, each had attempted questions worth a total of $500$ points. Alpha scored $160$ points out of $300$ points attempted on the first day, and scored $140$ points out of $200$ points attempted on the second day. Beta, who did not attempt $300$ points on the first day, had a positive integer score on each of the two days, and Beta’s daily success ratio (points scored divided by points attempted) on each day was less than Alpha’s on that day. Alpha’s two-day success ratio was $\frac{300}{500} = \frac{3}{5}.$ The largest possible two-day success ratio that Beta could have achieved is $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. What is $m + n?$