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I've heard native speakers use "half" but never "halves" in such sentences. If we cut an apple into two identical pieces, we get two halves. It shouldn't, therefore, sound wrong if one used "halves". Would it be wrong to use the plural here?

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It would be "please cut the apple into halves", which is perfectly correct.

But you are right, in practice it is rarer in everyday usage than "please cut the apple in half".

"To cut in half" is a fixed expression which stands for the verb "to halve" (which is more formal or archaic) and describes the action of dividing into equal parts. There is no equivalent phrase for other proportions. For example a budget or workforce can be "cut in half": it has a similar function to decimate (though a different proportion is involved).

By contrast, "to cut into halves" focuses on the end product, -- the halves -- and the means (to cut). It is a living phrase and other proportions and mechanisms can be substituted at will ("cut into quarters", etc).

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The subscription-only LDOCE mentions this in its collocations section:

divide/split something in half
Divide the dough in half.

reduce/cut something by half (=make something 50% smaller or 50% less)
The company has reduced the number of staff by half.
a plan to cut European forces by half

The other alternative you can use, is:

Please halve the apples.

And interestingly, as mentioned here, "to cut into halves" would be used when you're splitting a single object into two pieces, creating two separately definable items or quantities, each being half-size of the original, whereas it won't be correct to say: "The man was cut into halves". while "The man was cut in half." makes sense.

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