The hyperbolic 0-1 knapsack problem (HKP) consists to obtain a 0-1 solution that maximizes a linear fractional objective function under the constraint of one linear inequality. We propose in this paper a linear programming-based method to obtain in polynomial time a 1/2-approximate solution to (HKP). The method is based on the solution of the continuous relaxation of (HKP) obtained by removing the integrality constraints on the variables. Moreover we report some computational results which show that the method allows approximate solutions to be found quickly with a small relative gap compared to the optimum (between 0.05% and 2.5%) for instances comprising up to 10000 variables.