In your approach, it should be:
$91$ trips - $2.5$ trips (10 days he won't work) = 88.5 trips.
So, from your approach, since the year has $91$ trips + 1 day, he can make $88.5$ trips $+ 1$ day = $88.75$ trips.
Since he can't make a fractional part of a trip, then, based on your approach, he is limited to $88$ trips.
Edit
Note that if the problem was altered so that he gets $9$ days off instead of $10$ days off, then your approach, modified as I have suggested, would get the correct answer of $89$ trips.
That is $91 - 2.25 = 88.75$
So, $91$ trips + $1$ day = $88.75$ trips + $1$ day $ = 89$ trips.
Addendum
Actually, both my modification of your approach, and the first resolution are careless. In the original problem, a critical question is, how many days are there between January 1 and June 30, inclusive.
It turns out that there are $181 = [(4 \times 45) + 1]$ days. So, by forcing the vacation to start on June 1, only $1$ day is wasted, between between January 1 and June 30, inclusive.
As it turns out, in the 1st resolution, which computes $88$ trips, there are $3$ extra days. Therefore, regardless of how many days are wasted between January 1 and June 30, inclusive, you can still make $88$ trips.
As an example where my modified approach leads to the wrong answer, suppose the vacation is June 1 through June 9.
Then, there are $(4 \times 89) = 356$ work days. However, these work days are split into $2$ sections: $181$ days before June 1, and $175$ days after June 9.
Therefore, if the vacation starts on June 1, and ends on June 9, my original conclusion of $89$ trips is wrong.
