The school soccer team consists entirely of freshmen and sophomores. If there are more sophomores than freshmen on the

BTGmoderatorDC wrote: ↑

Sun Mar 08, 2020 6:00 pm

The school soccer team consists entirely of freshmen and sophomores. If there are more sophomores than freshmen on the team, which of the following must be true?

A. If there were two more freshmen on the team, there would be more freshmen than sophomores on the team.
B. If there were two fewer sophomores on the team, there would be more sophomores than freshmen on the team.
C. If all the freshmen were to quit the team, the team would have fewer than half the players it currently has.
D. If all the sophomores were to quit the team, the team would have fewer than half the players it current has.
E. If all the freshmen and the sophomores were to quit the team, the team would still have players.

OA D

Source: Veritas Prep

Say there are F no. of freshmen and S no. of sophomores in the team.

Let's analyze each option one by one.

A. If there were two more freshmen on the team, there would be more freshmen than sophomores on the team.

Not MUST be true:

Say S = 10 and F = 2, then still S > F + 2.

B. If there were two fewer sophomores on the team, there would be more sophomores than freshmen on the team.

Not MUST be true:

Say S = 3 and F = 2, then S – 2 < F.

C. If all the freshmen were to quit the team, the team would have fewer than half the players it currently has.

MUST be false:

Since S > F, and if all the freshmen were to quit the team, the team would rather have more than half the players it currently has.

D. If all the sophomores were to quit the team, the team would have fewer than half the players it currently has.

MUST be true:

Since S > F, and if all the sophomores were to quit the team, the team would certainly have fewer than half the players it currently has.

Let's see how:

Say S = F + a, where a is a positive integer. Total no. of players = S + F = 2F + a

When all sophomores quit, the no. of players left = F

Ratio of F to 2F + a = F/(2F + a) = 1/(2 + a/F). Since a/F is a positive value, 1/(2 + a/F) < 1/2. Correct answer.

E. If all the freshmen and the sophomores were to quit the team, the team would still have players.

MUST be false:

Since the school soccer team consists entirely of freshmen and sophomores, there won't be any player left when all the freshmen and the sophomores were to quit the team.

The correct answer: D

Hope this helps!

-Jay
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