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**