word choice - “Are” Vs “Is” in the following sentences

I'm confused what verb would concord in the following couple of sentences:

The number of students and the number of teachers are/is decreasing
day by day in the school.

Either a number of teachers or a number of students is/are
participating in the college's festival.

According to my understanding the number of is always followed by a singular verb even two the number of are joined with and. Also, a number is always followed by a plural verb. So I think the former will take is while the latter concord with are. But I'm not sure.

Answer

In the first sentence, you are talking about two things, the number of students and the number of teachers. Let's call them A and B as a shorthand.

• A and B is decreasing


• A and B are decreasing

The second form is obviously correct:

The number of students and the number of teachers are decreasing day by day in the school.

In your second sentence, you are again talking of two things, and choosing between them:

• Either A or B is decreasing


• Either A or B are decreasing

In this case the first is correct. A potential difficulty arises because both A and B refer to a number of particular people which could be construed as a plural. So either is potentially correct. Most people would match the verb to the word immediately preceding it and use "students are".

Either a number of teachers or a number of students is participating in the college's festival.
Either a number of teachers or a number of students are participating in the college's festival.