On May first, in order to reduce the number of overdue books, a childr

On 1st May, a children’s library instituted a policy of forgiving fines and giving bookmarks to children returning all of their overdue books (so that children return the books they have)
On July 1st, there were twice as many overdue books as there had been on May 1st, although a record number of books had been returned during the interim.

So record number of books were returned in the 2 months but there were twice as many overdue books!

What will explain this?

(A) The librarians did not keep accurate records of how many children took advantage of the grace period, and some of the children returning overdue books did not return all of their overdue books.

Doesn't explain anything. What is grace period, we don't know so no accurate records of that doesn't help us at all. Why were there more overdue books on 1st July, we don't know.

(B) Although the grace period enticed some children to return all of their overdue books, it did not convince all of the children with overdue books to return all of their books.

Why were there more overdue books on 1st July, we don't know.

(C) The bookmarks became popular among the children, so in order to collect the bookmarks, many children borrowed many more books than they usually did and kept them past their due date.

Now this makes sense. Children borrowed many more books (and returned) in the 2 months. They did keep the books past their due dates to collect the bookmarks. This explains why twice as many overdue books were there on 1st July.

(D) The children were allowed to borrow a maximum of five books for a two-week period, and hence each child could keep a maximum of fifteen books beyond their due date within a two-month period.

The actual number of books each child is allowed to keep doesn't help.
varotkorn
Since 5 books were allowed for 2 weeks, in 2 months (about 8 weeks) each child could borrow 5*3 = 15 books and keep beyond due date. In the last 2 weeks, the 2 month period would be over so the child would not be able to keep the books overdue. The child could effectively take out 20 books but keep only 15 overdue.

(E) Although the library forgave overdue fines during the grace period, the amount previously charged the children was minimal; hence, the forgiveness of the fines did not provide enough incentive for them to return their overdue books.

Why were there more overdue books on 1st July, we don't know.

Answer (C)