This paper investigates the effect of grouping students by prior achievement into different classes (or schools) in settings where students are competing for admission to programs offering only a limited number of places. We first develop a model that identifies the conditions under which the practice of tracking students by prior achievement increases inequalities between students that do not initially have the same academic background, such as may exist between students with different social backgrounds. We then test our model using new data on the competitive entrance exams to elite scientific higher education programs in France. We find that 70% of the inequality in success in these exams between students from different social backgrounds can be explained by the practice of tracking students by prior achievement that prevails during the years of preparation for these exams.