SAT Math Multiple Choice Question 686: Answer and Explanation

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Question: 686

11. Every athlete in a group of 60 female varsity athletes at Greenwich High School either runs track, plays soccer, or does both. If one-third of the athletes in this group who play on the soccer team also run on the track team, and one-half of the athletes in this group who run on the track team also play on the soccer team, which of the following statements must be true?

  • A. This group contains 40 soccer players.
  • B. This group contains 20 athletes who play soccer but do not run track.
  • C. This group contains 20 athletes who play both track and soccer.
  • D. The number of soccer players in this group is 15 greater than the number of track team members in this group.

Correct Answer: D

Explanation:

D

Data Analysis (set relations) HARD

Let's let s = the total number of athletes in the group who play soccer, and t = the number of athletes in the group who run track. We can set up a Venn diagram to show the relationship between these two overlapping sets.

Images

Since one-third of the soccer players also run track, we must put Images in the overlapping region between soccer and track, and therefore the number who play only soccer is Images. Likewise, since one-half of the athletes who run track also play soccer, we must put Images in the overlapping region, and therefore the number of athletes who only run track is Images.

Sincer there are 60 athletes in total:

Images

Simplify:

Images

Multiply by 3 to simplify:

2s + 3t = 180

The number of soccer players who run track must equal the number of track athletes who play soccer:

Images

Multiply by 6 (the common denominator):

2s = 3t

Substitute 2s = 3t into the previous equation:

3t + 3t = 180

Simplify:

6t = 180

Divide by 6:

t = 30

Substitute t = 30 into the other

equation to solve for s:

2s = 3(30)

Simplify:

2s = 90

Divide by 2:

s = 45

Now we can use these values to complete the Venn diagram:

Images

From this diagram, we can see that the only true statement among the choices is (D).

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