How are you building your hash?
If I was doing this I might use a hash of hashes such as:
$book{author}{book_name} = $borrower; which you could output all in one swoop with

for my $author (keys %book) {
  print FILE $author;
  while (my ($book,$borrow) = each %{$book{$author}}) {
     print FILE ':' . $book;
     print FILE '/' . $borrow if $borrow;
  }
  print FILE "\n";
}
[download]

The input is of course comparable.
If you are using (as you seem to be) keys which are the author's name joined to the book title I would say you are making trouble for yourself and making the output code harder than it should be.

Note: since this is homework I probably should not have given you that code, but well I did anyway....