Let M = (E, I)be a matroid, and let S be a family of subsets of size p of E. A subfamily Ŝ⊆S represents if for every pair of sets X ε S and Y ⊆ E \X such that X ∪ Y ε I, there is a set X̂ε Ŝ disjoint from Y such that X̂ ∪ Y ε I.. Fomin et al. (Proc. ACM-SIAM Symposium on Discrete Algorithms, 2014) introduced a powerful technique for fast computation of representative families for uniform matroids. In this paper, we show that this technique leads to a unified approach for substantially improving the running times of parameterized algorithms for some classic problems. This includes, among others, k -Partial Cover, k -Internal Out-Branching, and Long Directed Cycle. Our approach exploits an interesting tradeoff between running time and the size of the representative families.