In response to the question "What time is it?", we can say something like
(1) It's a quarter to seven.
(2) It's half past eight.

Is there a logical reasoning behind why there is a determiner "a" before "quarter" and no determiner before "half"?

There are usage where "half" is used with "a".
(3) One and a half months have passed since.....
(4) a half pound of.....

Thank you.

apple

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Original Post

When the time is 15 minutes after the hour, British people say quarter past the hour (with out "a" in front) but American people say a quarter after the hour. When the time is 15 minutes before the hour, British people say quarter to the hour (again without "a" in front) but American people use a quarter of the hour.

Ex:
At quarter past four on an overcast day, more than 1,100 years of building work came to fruition.(Megastar News, UK)

At a quarter past 9 pm, the count was not even close 13,393 "no" votes to 4,077 "yes."(McMinnville News-Register, OR, USA)
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Half differs from the other fractional expressions in that it is used without an article. This is a property of the word itself. We say

The storm did a great deal of damage to the corn. Half (of) the crop had already been harvested, but at least a quarter of the crop was a total loss.

In time expressions, half is also used without an article:

It's half [an hour] past midnight. Why aren't you in bed?

I'll see you in half an hour

You can find more information on the usage of half and the other fractions in

Collins COBUILD English Grammar, Sections 2.258-2.266 (1990).

Marilyn Martin
It's interesting to know the following phenomenon.

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British people say quarter past the hour (with out "a" in front) but American people say a quarter after the hour.

But my corpus based dictionary states that Americans sometimes say "quarter" without "a".

My mind is still a bit murky, because when telling the time, one rarely says "It's a half past ten". One should almost always say "It's half past ten". A quarter means a one fourth of 60 minutes. Half is one half chunk of 60 minutes. Then why just half not "a half." "A half "is used on many occasions, such as two hours and a half, one and a half tons. I have a hard time explaining to the students. "This is how it is used and learn them individually" would be the solution?

apple
Yes, the students need to learn that half is different from the other expressions of fractions. In some uses it has an article ("a minute and a half," "three and a half hours"), while in others ("half past ten," "half an hour") it does not. Each use must be learned on its own.

Marilyn Martin