The main result of the paper is that, for every positive integer \(k\), every \(d\)-dimensional simplex can be decomposed into \(k^d\) simplices of dimension \(d\), all of the same volume and shape characteristics. The approach is based on an algebraic description of a simplex. The elementary model is very effective in the detailed analysis of the algorithm employed to construct a subdivision of a simplex, and is very convenient for computer representation of simplices and subdivisions. It is worth mentioning that the same model can be used to decompose a simplex into Cartesian product of simplices [see \textit{D. R. Grayson}, K-Theory 3, No. 3, 247-260 (1989; Zbl 0701.18007)]. The construction has been also used by \textit{T. N. T. Goodman} and \textit{J. Peters} [Comput. Aided Geom. Des. 12, No. 1, 53-65 (1995; Zbl 0875.68829)] to design smooth manifolds.