The problem is called fractional programming. It's quite well studied, first (I think) by Béla Martos (he has a book on it from 1975), but you can find texts in many places about this well-known problem. I suggest, for example, a reading in Murty's book on linear programming - which includes the standard transformation into a standard linear program. "Linear-fractional programming" is even a Wikipedia page, with some information, as well as more links.