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✔1. There are many health problems to over come, each is of vital importance.
❌2. There are many health problems to over come, each are of vital importance.

Why is the word "are" incorrect in this sentence (no.2)? Can you please explain this to me? Thank you in advance 🙏

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Each is a pronoun and determiner used to refer to every one of two or more people or things, regarded and identified separately.

You should use is because you are referring to one thing.

  • You can make it even simpler: "There are many health problems to overcome, each of vital importance." overcome is one word. – user3169 Oct 31 '18 at 1:06
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Neither of your sentences are grammatically correct.

It would be better to say:

There are many health problems to overcome. Each (one) is of vital importance.

There are many health problems to overcome; each (one) is of vital importance.

Overcome is one word (not two), and a comma is not correct here because a comma cannot be used to connect two independent clauses. You can either use a period/full stop or a semicolon (see this lovely infographic on semicolon rules).

You can also better understand why we would not use "are" in this context when we add "one" to the phrase. "One" thing is singular, so it uses the singular "is". Since we are talking about each health problem as individual problems, we think of them separately. Each one is vital, not the group of them all as a whole.

However, if we use an example from the Oxford Living Dictionary (as linked to in a previous answer by gusmog):

they each have their own personality

We would not add "one" here, so we would not use the singular "has". Try to pronounce "they each one has"; it is very strange to even read.

(This "adding one" trick may not be universal, but hopefully it will help you to understand.)

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