Write and solve an equation of ratios: 6 books/11 people = x/13543 people.
Then 6 = 11x/13543, and x = 13543/11, or 6*13543=11x, and x = 7307 books.
The library already has 3500 books. Thus,to achieve this goal, the library would have to acquire (7307-3500) more books: 3887 books.