Vincen wrote: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 currently has.
E. If all the freshmen and the sophomores were to quit the team, the team would still have players.

[spoiler]OA=D[/spoiler]

Source: Veritas Prep

Say there are f no. of freshmen and s no. of sophomores.

We are given that s > f.

Let's take each option one by one. Note that this is a 'must be true' question; thus, for all situations, the correct option must be valid.

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

Not necessarily true. Say s = 10 and f = 2, then as per the option f = 2 + 2 = 4. We see that f is not greater than s. Incorrect.

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

Not necessarily true. Say s = 3 and f = 2, then as per the option s = 3 - 2 = 1. We see that s is not greater than f. Incorrect.

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

Not necessarily true. Say s = 10 and f = 3, so currently the team has 10 + 3 = 13 players. Now as per the option f = 3 - 3 = 0, and there are 10 players. We see that 10 is not less than 13/2. Incorrect.

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

Say s = 2 and f = 1, then as per the option f = 2 + 1 = 3. Now as per the option s = 0, and there is only 1 player. We see that 1 is less than 3/2. Correct.

The above scenario was taken such that s and f have minimum values. Let's take a couple of more cases.

Say, s = 21 and f = 20, then as per the option f = 21 + 20 = 41. Now as per the option s = 0, and there are 0 + 20 = 20 players. We see that 20 is less than 41/2. Correct.

Another case...

Say, s = 21 and f = 3, then as per the option f = 21 + 3 = 24. Now as per the option, s = 0, and there are 0 + 3 = 3 players. We see that 3 is less than 24/2. Correct.

Correct answer.

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

This is must be false option. So, we have s = f = 0. Since the team has only freshmen and sophomores, there would not be any player in the team.

The correct answer: D

Hope this helps!

-Jay
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