All: a singular noun?

I am struggling with the verb at the end of this sentence.

"I will make this a quick meeting, since all of you are busy."

Why is it "are"? I thought "all" was a singular noun, like "each". If the sentence used "each", it would clearly be "each of you is busy", so why does "all" change to verb from "is" to "all"?

"all of", "a lot of", and "the rest of" do not agree with the the verb. It is the word which follows them which agrees with the verb (in number).

All of the students are busy.
All of the milk is spoiled.

A lot of us are tired.
A lot of it is still to be determined.

The rest of the men were asked to return the next day.
The rest of the butter was used in another recipe.

CJ

A noun in the singular is not in the plural, that is you're speaking about only one thing or person - or of none -.

a girl, a dog, a table: these are nouns (in the singular). If you say "there's a girl in the room", girl is singular (only 1)/ "there's no girl in the room": girl is singular again (0 girl) / "there are 3, 102, many etc girls in the room", girls is plural (more than one)

a dog, no dog: singular/ 2,3, 250 dogs: plural

a table, no table: singular / many, 10 etc... tables: plural.