Between means non-inclusive, so 100 and 300 are not included in the list of multiples of 5.
There are 20 multiples of 5 between 100 and 200 including 200. There are 19 multiples of 5 between 200 and 300 not including 200 (we already included 200) and not including 300 from the problem stating "between 100 and 300".
There are 39 multiples of 5 altogether. Now look for the multiples of 5 that are divisible by 3. In other words, we want to determine which of these 39 multiples of 5 are also multiples of 15. A multiple of 15 is divisible by 5 and also by 3 !
105, 120, 135, 150, 165, 180, 195, 210, 225, 240, 255, 270, and 285 for a total of 13 out of 39, which simplifies to 13/39=1/3