After an election, the candidates were ranked in decreasing order of votes. Every citizen voted for a single candidate. The three highest ranked candidates together got 30% of the votes. The six lowest ranked candidates together got 48% of the votes. How many candidates were neither in the top two or the bottom five?
Explanation:
The average percentage of votes for the top three candidates is 30/3 = 10%. One of them must, therefore, have a percentage of votes less than or equal to 10%.
The average percentage of votes for the bottom six candidates is 45/6 = 7.5%. At least one of them must, therefore, have a percentage of votes greater than or equal to 7.5%.
Thus, the average percentage of the other candidates must lie between 10% and 7.5%, otherwise one of the other candidates would be either in the top three or the bottom six.
The total percentage of votes got by the other candidates is 100% - 30% - 45% = 25%. If there are N other candidates, their average percentage of votes is 25/N.
We have to solve the inequality 7.5 < 25/N < 10, for integral N.
This is possible only if N = 3.
Hence, option (b).
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