Ponder This Challenge - December 2010 - Counting votes from percentage

Ponder This Challenge:

In an election, Charles came in last and Bob received 24.8% of the votes.
After counting two additional votes, he overtook Bob with 25.1% of the votes.
Assuming there were no ties and all the results are rounded to the nearest promille (one-tenth of a percent), how many votes did Alice get?

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Solution

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  Let's denote the number of votes Alice, Bob, and Charles received at the beginning as A, B,and C.
  Since there were no ties, C=B-1, and after counting the two additional votes, C'=C+2=B+1.
  Since Charles was last and there were no ties, all the candidates got more votes than Bob.
  Careful checking proves there should be exactly one more candidate (Alice); furthermore, we can know not only the percentages, but there is actually a unique solution for the number of votes she received.
  The unique solution is A=84, B=41, and C=40, where 41/165=24.8% and 42/167=25.1%.

  We made December's challenge relatively easy to compensate for the difficulty of November's challenge. If it was too easy for you, you can find other numbers and conditions leading to the unique solution (truncating instead of rounding, using the nearest percentile instead of promille, etc.).

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  If you have any problems you think we might enjoy, please send them in. All replies should be sent to: ponder@il.ibm.com

Solvers

• Arthur Breitman (12/01/2010 12:58 PM EDT)
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