Summary: In this work we consider a mixed precision approach to accelerate the implementation of multi-stage methods. We show that Runge-Kutta methods can be designed so that certain costly intermediate computations can be performed as a lower-precision computation without adversely impacting the accuracy of the overall solution. In particular, a properly designed Runge-Kutta method will damp out the errors committed in the initial stages. This is of particular interest when we consider implicit Runge-Kutta methods. In such cases, the implicit computation of the stage values can be considerably faster if the solution can be of lower precision (or, equivalently, have a lower tolerance). We provide a general theoretical additive framework for designing mixed precision Runge-Kutta methods, and use this framework to derive order conditions for such methods. Next, we show how using this approach allows us to leverage low precision computation of the implicit solver while retaining high precision in the overall method. We present the behavior of some mixed-precision implicit Runge-Kutta methods through numerical studies, and demonstrate how the numerical results match with the theoretical framework. This novel mixed-precision implicit Runge-Kutta framework opens the door to the design of many such methods.