The ratio of fiction to non-fiction books in a library is 7:3.
60% of the fiction books are hardback, and 20% of the non-fiction books are hardback.
What percentage of all the books in the library are hardback?
Let's say there are 7x fiction books in the library, where x represents the common ratio.
There are also 3x non-fiction books in the library.
The number of hardback fiction books is 0.6 * 7x = 4.2x.
The number of hardback non-fiction books is 0.2 * 3x = 0.6x.
The total number of hardback books in the library is 4.2x + 0.6x = 4.8x.
The total number of books in the library is 7x + 3x = 10x.
Therefore, the percentage of hardback books in the library is (4.8x / 10x) * 100% = 48%. Answer: \boxed{48}.