Two coaching centres each teach 200 students in a morning batch and an evening batch. Centre P: morning 40 students of whom 80 percent passed, evening 160 students of whom 50 percent passed. Centre Q: morning 160 students of whom 75 percent passed, evening 40 students of whom 45 percent passed. Which statement is correct?

Batch by batch, P is ahead: 80 percent against 75 percent in the morning and 50 percent against 45 percent in the evening. But the overall rates must be built from COUNTS, not from an average of the two percentages. Centre P passes $0.80\times 40+0.50\times 160=32+80=112$ of 200, that is 56 percent. Centre Q passes $0.75\times 160+0.45\times 40=120+18=138$ of 200, that is 69 percent. So Q is ahead overall. The reversal happens because P puts 160 of its 200 students into its weak evening batch while Q puts 160 into its strong morning batch, and the batch SIZES weight the rates. Options that argue from being 'ahead in both batches' assume the overall rate is inherited from the parts, which it is not, and equal totals of 200 students guarantee nothing, since it is the split between the batches that decides the outcome.