Mike@Magoosh wrote: ↑

Tue Mar 18, 2014 9:20 am

At a certain high school, there are three sports: baseball, basketball, and football. Some athletes at this school play two of these three, but no athlete plays in all three. At this school, the ratio of (all baseball players) to (all basketball players) to (all football players) is 15:12:8. How many athletes at this school play baseball?
Statement (1): 40 athletes play both baseball and football, and 75 play football only and no other sport
Statement (2): 60 athletes play only baseball and no other sport

Solution:

Let’s show that even if we assume both statements, there are at least two scenarios with different numbers of baseball players.

Since baseball : basketball : football = 15 : 12 : 8, we can find a positive integer k such that 15k students play baseball, 12k students play basketball and 8k play football.

Notice that the number of football players is 40 + 75 + #football and basketball = 115 + #football and basketball. We know from the question stem that the number of football players is a multiple of 8, so let’s choose the number of football players to be the first multiple of 8 greater than 115, which is 120. This means that #football and basketball = 120 - 115 = 5. Also, 8k = 120; so k = 15. Then, there are 15k = 225 students who play baseball and 12k = 180 students who play basketball. Since 60 play baseball only and 40 play baseball and football, 225 - (60 + 40) = 125 students play baseball and basketball. Further, 180 - (125 + 5) = 50 play only basketball. Notice that the ratio of the baseball, basketball, and football players agrees with the information given to us, as well as the numbers for football only, baseball only, and the number for baseball and football. In this scenario, 225 students play baseball.

In another scenario, let’s choose the number of football players as 128. Then, #football and basketball = 128 - 115 = 13. Since the number of football players is represented by 8k, we have 8k = 128 and k = 16. So, there are 15k = 240 students who play baseball and 12k = 192 students who play basketball. Since 60 play baseball only and 40 play baseball and football, then 240 - (60 + 40) = 140 students play baseball and basketball. Further, 192 - (140 + 13) = 39 students play basketball only. Once again, the ratio of the number of players for the three sports, the number of students who play only football, only baseball and only baseball and football all agree with the information given to us. This shows that 240 is another possibility for the number of baseball players.

As we can see, even when we assume both statements, there is more than one possibility for the number of baseball players. So, statements one and two together are not sufficient to answer the question.

Answer: E