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This question is similar to this one, but it does not help me, so I am asking it by myself.

I want to know which one is appropriate:

The number of coins in bottle A and bottle B is compared.

The numbers of coins in bottle A and bottle B are compared.

Here what I want to say is "The number of coins in bottle A and the number of coins in bottle B are compared."

I want to know the general rule for this kind of situations (this specific example is not important). In case, the example is strange, I give another one.

The weight of bottle A and bottle B is compared.

The weights bottle A and bottle B are compared.

3

You should use the plural.

Let's look at it without the center section:

The number is compared.
The numbers are compared.
The weight is compared.
The weights are compared.

For me, the answer is obvious. You can't compare a single thing with itself without getting equivalence. The plural is required here. So, when we add more information, the plural should be retained:

The numbers of coins are compared.
The weights of bottles are compared.

And this, too makes more sense than the singular. With the singular, I end up asking "to what???"... The number of coins is compared... to what?

So, again, the plural is correct. There are two weights and two numbers.

I also note that you don't have to repeat the nouns:

The numbers of coins in bottles A and B are compared.
The weights of bottles A and B are compared.

And I'd say that here, the plural is better sounding, now that you obviously have bottles - plural.

When you use the singular, I'm prepared for a single number/weight, so the second half of the sentence becomes confusing.

The number of coins in bottle A and bottle B is 484.
The weight of bottle A and bottle B [together] is 1.3 kilos.

0

In your specific examples, you'd best stick to the singular. There is one number that accounts for the coins in each bottle. Bottle A has a single number of coins, as does bottle B. Bottle A has a weight, not weights, and the same is true of bottle B. However, you could write that the weights of bottle A and bottle B are compared, as the plural will then refer to the weights of both bottle A and bottle B, but you can also write that Bottle A's weight is compared to that of Bottle B.

So, Bottle A has a number of coins. Bottle A has a weight. Bottle B has a number of coins and a weight. Bottle A and B each has a number of coins. They have numbers of coins, in the plural, unless you wish to refer to the coins collectively, in which case they are a number of coins distributed into two bottles. The same is true of the weights. Comparing the two weights can lead to use of the plural, but collectively measuring the weight of the two bottles together will use the singular.

0

As you have stated, what you want to say is:

The number of coins in bottle A and the number of coins in bottle B are compared.

This is grammatically correct.

In asking which of these is correct:

The number of coins in bottle A and bottle B is compared.

The numbers of coins in bottle A and bottle B are compared.

from your extended question you can construct the contraction by omitting the repetition, thus:

The number of coins in bottle A and [the number of coins in] bottle B are compared.
The number of coins in bottle A and bottle B are compared.*

This is both correct and would be understood to mean that you have compared the number of coins in each bottle (and found that the number of coins differs or is the same).

Because the number(s) of coins are properties of the bottles, and the bottles are named separately; the number of coins in bottle A where number (referring to quantity) is singular because there is only one bottle, and are compared is the plural form because there are two bottles. And you have quantity (s) of objects (pl) in objects (pl), which becomes cumbersome if it were plural-plural-plural.

The weight of bottle A and bottle B is compared.

is not correct; the case is slightly different here. The weight of bottle A is compared with the weight of bottle B., when rearranged, becomes The weight of bottle A and the weight of bottle B are compared.

The weights [of] bottle A and bottle B are compared.

Whether you say weight (s) or weights (pl) can be open to interpretation, but in either case it will be are compared.

Also, using the passive voice is often less clear than using the active voice.

* The numbers of coins in bottle A and bottle B are compared is also correct.

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