It is the aim of this paper to investigate a suitable approach to compute solutions of the powerful Michaelis-Menten enzyme reaction equation with less computational effort. We obtain analytical-numerical solutions using piecewise finite series by means of the differential transformation method, DTM. In addition, we compute a global analytical solution by a modal series expansion. The Michaelis-Menten equation considered here describes the rate of depletion of the substrate of interest and in general is a powerful approach to describe enzyme processes. A comparison of the numerical solutions using DTM, Adomian decomposition and Runge-Kutta methods is presented. The numerical results show that the DTM is accurate, easy to apply and the obtained solutions retain the positivity property of the continuous model. It is concluded that the analytic form of the DTM and global modal series solutions are accurate, and require less computational effort than other approaches thus making them more convenient.