At a certain high school, there are three sports: baseball,

Question

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

For a discussion of ratios & proportions on the GMAT, further practice problems, and a solution of this problem, see:
https://magoosh.com/gmat/2013/gmat-quant ... oportions/

Mike

Smallwonder · Answer

Given, ratio of Base B: Basket B: Foot B = 15:12:8 and non of them play all three games.

Each form of game will have players who plays only that game and two games.

((Only Base B)+(Base B+Basket B)+(Base B+Foot B)) : ((Only Basket B)+(Basket B+Base B)+(Basket B+Foot B)) : ((Only Foot B)+(Foot B+Base B)+(Foot B+Basket B)) = 15:12:8

1) ((Only Base B)+(Base B+Basket B)+ 40 ) : ( 75 + 40 +(Foot B+Basket B)) = 15:8 
from the provided information, we can't deduct the total number of base ball players. Hence, insufficient.(Options A and D can be eliminated)

2) from the provided information, we can't deduct the total number of base ball players. Hence, insufficient. (Option B can be eliminated)

Both 1 & 2 : (60 +(Base B+Basket B)+ 40 ) : ( 75 + 40 +(Foot B+Basket B)) = 15:8 
From the provided information, we still can't deduct the total number of base ball players. Hence, insufficient. (Option C also can be eliminated)

Hence the answer is   E

/SW

fskilnik · Answer

Obs.: my solution is for the "15:12: 18 " version, slightly different that the original (first) post presented in this topic. Obviously the reasoning is identical.

Nice example of the  Venn diagrams  ("overlapping sets") and the  k technique  together!

15:12:18 = 5:4:6\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left\{ \matrix{
 \,{\rm{Baseball}} = 5k \hfill \cr 
 \,{\rm{Basketball}} = 4k \hfill \cr 
 \,{\rm{Football}} = 6k \hfill \cr} \right.\,\,\,\,\,\,\,\left( {k > 0} \right)

? = 5k\,

We go straight to (1+2): a  BIFURCATION  will guarantee that the correct answer is (E).

https://preview.ibb.co/kWjvTL/31-Oct18-7k.gif

This solution follows the notations and rationale taught in the GMATH method.

Regards, 
Fabio.

ScottTargetTestPrep · Answer

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

bumpbot · Answer

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At a certain high school, there are three sports: baseball,

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