Which of the following sentences is grammatically correct in english language. a. the book is their's but it's very old. b. the book is theirs but its very old. c. the book is theirs' but its' very old. d. the book is theirs but it's very old.

d. The book is theirs but it's very old.

The word their never uses an apostrophe. Therefore options A and C are wrong. "Its" shows ownership. In the sentence, "it" is not owning anything. "It's" (with an apostrophe) is a contraction meaning it is. D is correct because you need the contraction for it is very old. Just a sidenote - technically, none of these sentences are grammatically correct because there should be a comma after theirs. "The book is theirs" is a complete sentence. "it's very old" is a complete sentence. When combining two complete sentences with a conjunction, there needs to be a comma before the conjunction: The book is theirs, but it's very old.