# Definiteness of fractions

I was tutoring a French student the other day. We were doing some math homework on fractions, and one question read:

Les trois huitièmes de la solution sont de ... Le quart est de ... Quelle partie reste de ... ?

Naturally, the English equivalent would be:

Three eighths of the solution consist of ... A quarter consists of ... What is the remaining fraction that consists of ... ?

Does anyone have a good answer for why this would be definite in French? (After all, there's more than set of "three eighths"!) Or is it just one of those arbitrary quirks?

• I didn't understand your question ; could you be more precise ? (especially the 'there's more than set of "three eighths" ' part) I'm a mathematic student, so I should be able to help :) – m.raynal Feb 10 '17 at 21:07
• What I meant was that if you say, in English, "The three eighths of the solution consist of ..." then the response would be, "The three eighths? Which three eighths?" The only similar case that jumps to mind might be for comparing and contrasting: "(The) one half is made of sugar, and the other of flour." – Luke Sawczak Feb 10 '17 at 22:52
• instead of "Les trois huitiemes", the author may have written "37.5%". For French people, it's totally equivalent in a mathematical context. – Graffito Feb 10 '17 at 22:59
• @Graffito - yes :-) – Frank Feb 10 '17 at 23:05
• Then the answer from Frank definitly explains well the whole thing. – m.raynal Feb 11 '17 at 9:39

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.

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.

• If the "le quart" part of the problem is expressed as "Un quart, could the "les 3/8" part of it be expressed as simply "Trois huitièmes" (without an additional determiner/quantifier)? If the answer is that French cardinal numbers cannot serve as quantitative determiners, wouldn't "un quart" (where "un" is a cardinal number) require le/l' in front of it? ("Les trois huitièmes" donc "L'un quart")? – Papa Poule Feb 10 '17 at 22:36
• Ah oui! Vous avez raison, je change ma réponse. – Frank Feb 10 '17 at 22:46
• But l'un quart is unheard of/doesn't exist in this context. – Frank Feb 10 '17 at 22:48
• @LukeSawczak - I don't think it's a contextual issue. It's just mathematics. In the problem, there is probably no interest in finding out which 3/8 is under consideration, as long as it is a 3/8. Any 3/8 will do (I'm guessing it's an exercise about fractions). – Frank Feb 10 '17 at 23:00
• Yes - blame it on usage :-) We can use both indifferently in French. – Frank Feb 10 '17 at 23:07

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.

• On the contrary, there are many more than just one! After these introductions one could certainly refer to the groups that had been named using definite articles, but at their first mention they aren't unique yet. Also, this explanation and Frank's leave it a mystery as to why in English you can't possible say: "There is some solution. The three eighths of it consist of ... the quarter of it consists of ..." – Luke Sawczak Feb 10 '17 at 23:04
• Ahhh, I think I understand your point on closer consideration. You mean that the fraction "3/8 of x" has only one solution. Thanks; that's certainly correct (even if 3/8 of a real substance could be many different real slices of that substance). It still doesn't explain the difference between the English and the French, but I don't know if there is any logic in that choice between the two languages. – Luke Sawczak Feb 10 '17 at 23:09
• @LukeSawczak Yes, I was writing an explanation but it seems you got it. Three quarters of 16 will always be 12, no matter "which one" you choose. Of course it's different if we talk about divisors, or any thing else that can have multiple solutions. – Teleporting Goat Feb 10 '17 at 23:12