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Why is it acceptable to say "It is half as much" but not "It is a half as much"

When you can say: "It is a third as much" but not "It is third as much"

What is different about "half" here compared to other things like quarter or third?

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    There's no "meaning" to this. It's just that it's a little bit awkward to articulate that "neutral vowel" immediately before a word starting with /h/, and it's only a meaningless "function word" here anyway. Note that you'll still sometimes see one used instead of the indefinite article with half (and third, quarter, etc.), but that usage is declining too.. Commented Dec 20, 2020 at 18:17
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    Also note that half is by far the most common "fraction" in the real world - and unlike third, fifth, tenth,... there's no danger of confusing fractions and ordinals with half and second. So it's easier to run a half-marathon than a whole one, but if you were told you had to run a third marathon, you might find that a bit intimidating (maybe you're only supposed to run one third of the distance, but maybe you'd have to run a total of three marathons to complete the third [one]! :) Commented Dec 20, 2020 at 18:24

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There are times when you can say 'a half'... he ate 'a half of an orange'; where the 'half' ends up being a quanity. But in your example the difference is this: when you cut something in half you end up with two halves. You 'halve' it. And taking 'half' of the result is the natural end of that operation. But when you cut something in quarters you take one of the quarters and leave three of the quarters behind. The same with thirds etc.

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