IMAT 2013 Q14 [Chairperson Elections]

At a society meeting, 1000 people are entitled to vote in the elections for Chairperson with a one-person-­one-­vote system. The election rules state if no candidate obtains more than 50% of the votes cast in the first ballot, a second ballot must be held between the top two candidates. 350 votes were cast for a particular candidate in the first ballot. Then a second ballot took place.

Under these circumstances which one of the following is possible?

A. The candidate won the election, came second, or came third.
B. The candidate either won the election or came second.
C. The candidate came second or third, but did not win.
D. The candidate came third.
E. The candidate definitely won the election.

Simple steps to solve word problems:

  • Underline key information
  • Determine what they are trying to ask, and what you will need to solve it
    • Eliminate any non-essential information
  • Draw a picture, graph, or equation
    • In moments of high stress like exam taking, always work with the paper they give you to avoid careless mistakes.
  • Solve.

This is a fairly straightfoward word problem we can solve using logic. We know that there are 1000 votes total, in which 350 went to one candidate. Since there was less than 50% votes for a single candidate, there is a second ballot. The top 2 candidates will participate in the second ballot. So, to be in the second ballot you have to have less than 50% of total votes (or else there is no need for a second ballot), and you must be in the top 2 candidates. The candidate with 350 votes must be in the second ballot because there cannot be more than one person who has more votes than him:

1000 total votes - 350 = 650

650 votes left does not allow for 2 candidates to have more than 350 votes each, therefore the candidate with 350 must proceed to the second ballot.

After the second ballot, we do not know the outcome of the trial. Instead, we know that there is only the top 2 candidates who make it here, meaning that the candidates must finish second or first.

\fcolorbox{red}{grey!30}{So B must be the correct answer.}
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