The lowest common multiple of 30 and 45 is 90.

Suppose the book shelf can contain 90 units of thickness of books. It can fit exactly 30 large books or exactly 45 small books.

How many units is the thickness of each book?

90/30 = 3, so each large book is 3 units thick.
90/45 = 2, so each small book is 2 units thick.

There are 3 large books and 23 small books on the shelf. How much space is occupied by all these books?

For the large books, 3 x 3 = 9 units.
For the small books, 23 x 2 = 46 units

How much space is there left?

90 units - 9 units - 46 units = 35 units.

How many large books can fit the remaining space of 35 units at most?

35/3 = 11 (ignore the R2, no large book can fit the small gap)

So, only another 11 large books can be added to the shelf.

Thank you very much.