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The first and the last chapter are tough.

i guess last chapter is correct but is should be used in place of are and chapters should be used in place of chapter

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This is the same as saying:

The first (chapter) and the last chapter are tough.

(chapter) is implied. Since you are referring to each as named items (because of the definite article the), each is a single chapter, so use "chapter".

If you wrote:

The first and last chapters are tough.

now you are just referring to two chapters, so using "chapters" is OK.

In either case, the subject has two items, so being plural use are.

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Are you asking this question based on a correction you received concerning this sentence? If so, I disagree with it.

The first and the last chapter are tough.

Using are here is entirely correct. You are referring to a plural subject, since you are using and to present both possible subjects (the first chapter and the last chapter).

On the other hand, using is would be correct if your subject was singular or if the multiple subjects were used in the exclusive sense, meaning using or rather than and.

Either the first chapter or the last chapter is the one he finds difficult.

As for the question of chapter vs. chapters...

The pluralization of chapters follows the same logic of is vs. are. The sentence could be rephrased in this way:

The first chapter and the last chapter are tough.

This is why it becomes:

The first and the last chapters are tough.

Using the plural is important because you have two adjectives (first and last) applied to two different instances of the noun (chapter in both cases). See, hypothetically, you could cause confusion in cases where the two adjectives could apply to the same noun. For example:

  1. The tallest and fastest athlete is sitting at the table by the stage.

  2. The tallest and fastest athletes are sitting at the table by the stage.

  3. The tallest and the fastest athletes are sitting at the table by the stage.

  4. The tallest athlete and the fastest athlete are sitting at the table by the stage.

In the first example, the sentence indicates that there is only one athlete, and that they are both the tallest and the fastest.

In examples 2 and 3, the phrasing is a bit ambiguous. It's not entirely clear whether you're talking about two individuals (one of whom is the tallest and one of whom is the fastest), or any number of individuals (which would be described by the adjectives tallest and fastest in a vague, less specific sense).

In the final example, it is clear from the usage of are that you are referring to multiple individuals, and the duplication of the singular noun ("athlete") indicate fairly clearly that we are talking about one athlete who is the tallest, and one athlete who is the fastest. Using the definite article "the" twice is the most grammatically clear, but you could probably drop the second "the" and it would still transmit the meaning accurately. You could also insert the word both (as in, "are both sitting") for emphasis.

One final note...

Occasionally, native speakers will use a singular noun form in a case where it should be plural, as it does not always "sound" wrong, although this isn't technically proper grammar.

In your original example ("The first and the last chapter are tough."), it doesn't exactly sound wrong to use chapter rather than chapters. Some examples:

  • The first and the last sandwich on the menu are especially good.

  • The white and the black cat get along pretty well.

  • The short and the tall bottle are both Mexican beers.

All of these examples are technically incorrect, but in everyday communication they don't sound incorrect.

I hope you'll find my answer helpful.

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