I would say: because there is only one “trois huitièmes” from the mathematical point of view, even if there are several subsets with a cardinality of 3 from the set of 8 things under consideration.
Interestingly, in this case, whether you write a “definite” or “indefinite” problem description changes nothing. In other words, this problem is about fractions (I’m assuming), and it is unimportant which 3/8th of the pie, hour, distance … this is about, only that it is some 3/8th. Any 3/8th will do to complete the problem.
The “indeterminate” form of the problem statement would be:
Trois huitièmes de la solution sont de … Un quart est de … Quelle partie reste de … ?
Same solution from the mathematical point of view. Same understanding of what this means in this context. Usage allows both styles without any perceptible difference.
Why wouldn’t it ?
For each number there only one number that is “the” quarter, or “the” remaining fraction of that number. It’s consistent with “the square”, “the opposite”, “the square root”, etc. It doesn’t feel like a quirk at all to me.
After all, you say “the half”, so why would you say “a quarter” ?
Note that when it’s not related to mathematics, we’ll say “un quart“, or “trois huitième“. Moitié is almost often definite though.
It’s a linguistic quirk, not a mathematical one. French prefers to use an article with a noun much more often than English. In English we say “last summer” but in French it’s “l’été passé” or “the last summer.” The profusion of articles helps identify genders and clarifies some other relationships, but mostly it’s just the way it is.