We give an analysis of the solutions of the n-dimensional vacuum Einstein equations with a metric in the form of a direct sum of a Friedmann-Robertson-Walker (FRW) metric and a Kasner-type Euclidean metric. The solutions are interpreted as four-dimensional perfect-fluid cosmological FRW models, using the simple ansatz proposed by Ib\'an\ifmmode \tilde{}\else \~{}\fi{}ez and Verdaguer. We first obtain the general solution for flat models. These are perfect-fluid solutions that can be made compatible with contraction of all the extra dimensions. The general compatibility of the field equations is then discussed. It is found that for ng5 both open and closed models admit a range of perfect-fluid solutions whose qualitative behavior is analyzed.