AIME Math Question Help

The members of a distinguished committee were choosing a president, and each member gave one vote to one of the 27 candidates. For each candidate, the exact percentage of votes the candidate got was smaller by at least 1 than the number of votes for that candidate. What was the smallest possible number of members of the committee?
The solution is 134.
Please explain VERY SIMPLY. (as if you were explaining this to a 6th grader)!

Thanks!

First, some comments.

The percentages are bound to be 100 or less, so the voters must be above 100 for the votes to be more than the percentages.

For the percentages to be at least one point less than the votes, the number of voters has to be at least 127.

Now, why not just 127? Let's see if we come up with the same answer.

With 127 voters, if everyone voted for one, then there would be 127 votes and 100 % of the vote, easily accomplishing the goal. (If all voted for one, then 101 voters would be enough.) The way to challenge the goal is to have the votes spread-out.

So, with 127 voters, each candidate could get 4 votes, and 19 could get 5.

How about 4 votes per candidate, would that work? With 108 voters?
4 / 108 = 3.7037... %. Compared to 4 votes, that is /not/ at least 1 less.

So, 5 votes per candidate?
5 * 27 = 135.
5 / 135 = 3.7037...%. 5 votes is at least 1 more than the percentage. Yay!

But 134? If 134, then, one candidate would have gotten 4 votes. Now, 4/134 = 2.985...%, which is at least 1 less than 4 votes.

Maybe there is some wiggle room /below 134/. To find it, we can find out a number of votes that just makes 5 votes be 4 % of the vote.

5 / x = 4%
5 / x = 4 / 100
4x = 500
x = 125

If 125, then some candidates could have 5 votes and some could have 4 votes. Let's test each.
5 / 125 = 4 %. Compared to 5 votes -- check!
4 / 125 = 3.2 %. Compared to 5 votes - uh, no.