We present a simple numerical method based on a Lagrangian-Eulerian framework for approximate solutions of nonlinear balance law problems. This framework has been used for numerically solving partial differential equations of several types, such as hyperbolic conservation laws [3, 8], balance laws problems [4]. As in [3, 5] the mass conservation takes place in the space-time volume Dn j , and this region in the form of [3] is used to define naturally a balance law. This balance law is the central idea to build a efficient numerical method to approximate solution to balance law problems. Verification of the technique is also made by comparison with analytical solutions when they are available.