In an election between two candidates, 12% of the votes polled were rejected as invalid. The winner took 55% of the valid votes and the loser received 9900 votes. How many votes were polled in all?

The loser's 9900 votes are 45% of the valid votes, so the valid votes number 9900/0.45 = 22000. Those are the 88% that survived rejection, so the votes polled = 22000/0.88 = 25000. (Stopping at 22000 answers a different question, and dividing 9900 by 0.55 uses the winner's share by mistake.)