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Counting votes from percentages (Posted on 2019-02-07) Difficulty: 2 of 5
In an election among three candidates, 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 one-tenth of a percent, how many votes did Alice get?

No Solution Yet Submitted by Danish Ahmed Khan    
Rating: 4.0000 (1 votes)

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Solution Every vote counts (spoiler) Comment 1 of 1
Since there are no ties, Charles got both extra votes.  
He started one vote behind Bob, and finished one vote ahead.

Let b = Bob's votes and t = total votes before counting the extra 2.
Ignoring rounding, b/t = .248  and (b+1)/(t+2) = .251

Solving gives t = 166 and b = 41.168

This suggest a solution (before the two extra votes) of

   (b,c,t,a) = (41, 40, 166, 85).
   
This does not work, however, as 41/166 (rounded) = .247, not .248

I played around with solutions near this, and the one that works is 

   (b,c,t,a) = (41, 40, 165, 84).
   
41/165 (rounded) = .248
42/167 (rounded) = .251 

So Alice started with 84 votes out of 165.

To my surprise, I could not find other solutions.

Edited on February 7, 2019, 8:25 pm
  Posted by Steve Herman on 2019-02-07 11:42:38

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