This paper focuses on the numerical approximation of the solutions of a class of non-local systems in one space dimension, arising in traffic modeling. We propose alternative simple schemes by splitting the non-local conservation laws into two different equations, namely, the Lagrangian and the remap steps. We provide some properties and estimates recovered by approximating the problem with the L-AR scheme, and we prove the convergence to weak solutions in the scalar case. Finally, we show some numerical simulations illustrating the efficiency of the L-AR schemes in comparison with classical first and second order numerical schemes.