# SAT Math Multiple Choice Question 686: Answer and Explanation

### Test Information

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.

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.

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:

Simplify:

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

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:

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