I would further argue that the conceptual use of half-days is inappropriately imprecise, because that is itself ambiguous.

  12h       = 50% 24-hour calendar day
   6h       = 50% global mean 12 hours of daylight
   0h - 12h = 50% actual local hours of daylight
   4h       = 50% standard 8-hour workday
   5h       = 50% 8 AM to 6 PM business hours
   ???      = opening until midday break
   ???      = midday break until closing

Resolving ambiguity is a great job for a hyphen:

  2-1/2 days = two-and-a-half days
  2 1/2-days = two half-days

The rule to always hyphenate mixed fractions makes that explicit, although the hyphen is visually indistinguishable from a subtraction sign. Is it two minus a half? Which is why my personal rule is to never use mixed fractions, ever, for any reason, and to never write anything that looks like a mixed fraction. No style guide I have seen suggests 2+1/2, probably because some people might confuse (2+1)/2 with 2+(1/2). Just write 5/2 or 2.5 . (And if your number is at the end of a sentence, put a space between it and the period, so it won't be confused with a decimal point.)