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?
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
Every vote counts (spoiler)