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It seems correct to me to say

These 5 apples weigh 100g each.

However it also seems correct to say

Each of these 5 apples weighs 100g.

What is the reasoning behind using the plural in the first, and the singular in the second? They seem to be saying the same thing about an individual apple.

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    That's right. They're both correct. And so is These 5 apples each weigh 100g. The quantifier each (like all; but unike every or any) can undergo "Quantifier-shift", which moves each from its original position, quantifying these 5 apples, to other positions that are in construction with its focussed constituent (100g here), or immediately before a constituent (like weigh 100g here) that contains the focussed constituent, losing a preposition along the way. – John Lawler Sep 18 '15 at 18:41
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If you parse the sentences into parts to discover the subject and the predicate, and remembering that the predicate has to correspond to the subject in number and person, you would see that in the sentence

These 5 apples weigh 100g each.

the subject is "apples", plural, and the predicate is "weigh" (plural): all fine.

In the other sentence

Each of these 5 apples weighs 100g.

the subject is "each", singular, and the predicate is "weighs" (3rd person singular): all fine.

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