BTGmoderatorDC wrote: ↑

Wed Oct 07, 2020 5:45 pm

In a class election, 25 students each cast one vote for one of four candidates for student council. If Jill received the third-highest number of votes, and no two candidates received the same number of votes, what is the greatest number of votes she could have received?

A. 7
B. 8
C. 9
D. 10
E. 11



OA A

Solution:

We can let the number of votes Jill received = n. Since we want to maximize the number of her votes, we want to minimize the number of votes the other three candidates received. Since Jill received the third-highest number of votes and no two candidates received the same number of votes, we can let n + 1 = the number of votes the second place candidate received and n + 2 = the number of votes the first place candidate received, while the last place candidate received 0 votes. Thus, we have:

0 + n + (n + 1) + (n + 2) = 25

3n + 3 = 25

3n = 22

n = 7⅓

Of course, n has to be an integer; thus, n must be 7, and it’s the largest integer value n could be. It’s possible that the first place candidate could have received 9 votes, the second place candidate 8 votes, Jill 7 votes, and last place candidate 1 vote, or the first place candidate could have received 10 votes, the second place candidate 8 votes, Jill 7 votes, and last place candidate 0 votes.

We note that Jill could not have received 8 or more votes, because in that case, the second place candidate would have received at least 9 votes and the first place candidate would have received at least 10 votes, but those votes alone exceed 25 votes, even if the last place candidate received 0 votes.

Answer: A