The solution of simple kinetic equations is analyzed without referencing any topic from differential equations or integral calculus. Guided by the physical meaning of the rate equation, a systematic procedure is used to generate an approximate solution that converges uniformly to the exact solution in the case of zero, first, and second order kinetics. The approximate solution is shown to have unequivocally the same form as the exact solution, except for an effective parametric rate constant that turns into the actual rate constant of the process at the appropriate limit. The use of simple algebraic methods makes the analysis elementary and its generalization to processes with fractional order kinetics straightforward. We discuss the educational value of this approach while highlighting its use for introducing chemical kinetics and related topics to high school and first-year students. Furthermore, instructors and students already familiar with the main results derived here will also benefit from this presentation as it offers a different perspective on the solution of routinely found differential equations.