High school percentage question that I got wrong.

Question

Calvin and Susie are running for class president. Of the first 80% of the ballots that are counted, Susie receives 53% of the votes and Calvin receives 47%. At least what percentage of the remaining votes must Calvin receive to catch up to Susie in the election?

Let V be the total number of votes. 80% = 0.8, so 0.8V votes have been counted.

Susie has 53% of this, which is (0.53)(0.8V).

Calvin has 47% of this, which is (0.47)(0.8V).

The difference between their counts is (0.53)(0.8V) - (0.47)(0.8V) = (0.06)(0.8V) = 0.048V.

The remaining votes make up 20% of V, or 0.2V.

Divide the amount Calvin needs by the remaining amount: 0.048V / 0.2V = 0.24, or 24%.

My answer was wrong, it's actually 62% can anyone please help me understand what I did wrong and how to do this problem?

Answer 1

Calvin has received $0.47\cdot 0.8V = 0.376V$ votes, but he needs $0.5V$ total votes to catch up. Thus of the remaining $0.2V$, he needs $0.124V$ votes, so the percentage is $0.124/0.2 = 0.62$.

Your mistake was assuming that if Calvin receives a number of votes that allows him to catch up to Susie's current total, he will win. But surely Susie gets some of those $0.2V$ votes as well.

Answer 2

You neglected her extra votes. Among the new ones, he needs to beat her by $24$ points, i.e. get $62$ to $38$.

Answer 3

Under a certain interpretation, your answer is correct. Among the remaining 20% of ballots, Calvin only needs 24% of them to tie Susie's current vote total. The book answer assumes that any vote that does not go to Calvin goes to Susie, however, so if Susie gets 76% of the remaining 20% of ballots, she will still win. If Calvin and Susie are the only options on the ballot, Calvin can't catch up by performing worse than he did in the already-counted 80% - he'll have to do better than the 47% support he already has in order to win.

If, however, ballots have a third party or an "abstention" option, it's not true that every ballot is won by either Calvin or Susie. If Calvin gets at least 24% of the remaining 20% of ballots, and the other 76% are not votes for Susie, Calvin wins the election, capturing a plurality (but not a majority) of votes, with just over 42% of the votes.