SAT Math Multiple Choice Question 686: Answer and Explanation
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
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.
Since one-third of the soccer players also run track, we must put in the overlapping region between soccer and track, and therefore the number who play only soccer is . Likewise, since one-half of the athletes who run track also play soccer, we must put in the overlapping region, and therefore the number of athletes who only run track is .
Sincer there are 60 athletes in total:
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:
Multiply by 6 (the common denominator):
2s = 3t
Substitute 2s = 3t into the previous equation:
3t + 3t = 180
6t = 180
Divide by 6:
t = 30
Substitute t = 30 into the other
equation to solve for s:
2s = 3(30)
2s = 90
Divide by 2:
s = 45
Now we can use these values to complete the Venn diagram:
From this diagram, we can see that the only true statement among the choices is (D).