We develop a new dimensionless representation of the time-dependent mass balances for the Michaelis-Menten (MM) reaction mechanism; we identify several dimensionless parameters that control the fundamental nature of the solution; and we solve the scaled equations with a combined regular and singular perturbation expansion. Unlike several approximate solutions to the MM problem offered previously in the literature, each of which is valid only for some limited range of conditions, the new solution converges accurately for any combination of initial substrate concentration, initial enzyme concentration, and kinetic rate constants. We discuss the physical significance and interdependence of the dimensionless parameters that emerge from our scaling analysis; we use these parameters to categorize previous approximations for the MM problem and to delimit their accuracy; and we verify the accuracy of our solution via comparisons to an exact numerical solution and various approximations offered previously by others.