The hyperbolic 0-1 knapsack problem (HKP) to obtain a 0-1 solution that maximizes a linear fractional objective function under the constraint of one linear inequality is considered. First it is shown how to find approximate solutions of this problem with a given accuracy by solving successively two mixed integer linear programs. Then, six mixed integer programming-based strategies are compared for finding exact solutions of HKP. Some of these strategies exploit the knowledge of an approximate solution. The computational results that are reported give a comparison of the different strategies and show the efficiency of one of them since it allows instances comprising up to 10,000 variables to be solved. Keywords: Fractional 0-1 programming, Hyperbolic 0-1 knapsack problem, Mixed integer programming, Approximate solutions, Exact solutions, Computational experiments.