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GMAT Diagnostic Test Question 24 Field: word problems (overlapping sets) Difficulty: 750

Rating:

Among 60 members of a club, 6p players play soccer, 11p players play tennis, 8p players play golf and 2p players play none of the games. If p players play all of the games, how many players play only one game?

(1) The number of players who play soccer and golf but not tennis is half the number of players who play any other combination of two sports (2) p = 3

A. Statement (1) ALONE is sufficient, but Statement (2) ALONE is not sufficient B. Statement (2) ALONE is sufficient, but Statement (1) ALONE is not sufficient C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient D. EACH statement ALONE is sufficient E. Statements (1) and (2) TOGETHER are NOT sufficient
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suppose there are: s: players only play soccer t: only tennis g: only golf x: only play soccer and golf y: only play soccer and tennis z: only play tennis and golf p: play all the three.

we can have: a) s+t+g+x+y+z+p+2p=60 ==> s+t+g+x+y+z=60-3p ==>> a1) 2*(s+t+g)+2*(x+y+z)=120-6p b) s+x+y+p=6p c) t+y+z+p=11p d) g+x+z+p=8p

from b) c) d), we have bcd) s+t+g+2*(x+y+z)=22p

use equation a1)- bcd), we can get: s+t+g=120-28p=36

Answer is B for this question, but it seems dzyubam's calculation is not correct. Or am I wrong?

Each formula is a different way of expressing the same equality.

Let me reprhase Total = Soccer + Tennis + Golf - (Soccer&Tennis + Soccer&Golf + Tennis&Golf) - 2*Soccer&Tennis&Golf + None as Total = Soccer + Tennis + Golf - (ONLY Soccer&Tennis + Soccer&Golf + Tennis&Golf) - 2*ONLY Soccer&Tennis&Golf + None, Note that the expression between parenthesis does not include the intersections of Soccer&Tennis, Soccer&Golf, and Tennis&Golf. To make it easier, refer to the graph posted by flyingbunny above. Total = s + t + g - y - x - z - 2p + 2P

Meanwhile, P(A u B u C) : P(A) + P(B) + P(C) – P(A n B) – P(A n C) – P(B n C) + P(A n B n C) is the same as saying Total = Soccer + Tennis + Golf - (Soccer&Tennis + Soccer&Golf + Tennis&Golf) + ONLY Soccer&Tennis&Golf + None, In this case the intersections of "Soccer&Tennis + Soccer&Golf + Tennis&Golf" are included. This expression counts/substracts the intersections thrice, so it is necessary to add it once. Using the nomenclature of the graph, this one could be written as: Total = s + t + g - (y+p) - (x+p) - (z+p) + p + 2P

You can use either formula. It depends on personal preference. Data provided and question asked should tilt the balance for using one or the other.
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This is a badly-worded exercise, as well as logically incorrect in its phrasing. I had to skip this exercise during the diagnostic test.

Quote:

GMAT Diagnostic Test Question 24

Among 60 members of a club, 6p players play soccer, 11p players play tennis, 8p players play golf and 2p players play none of the games. If p players play all of the games, how many players play only one game?

(1) The number of players who play soccer and golf is half of the players who play each of the rest two games (2) p = 3

The OA is fine, but I had to read it to actually understand the question.

All the problems are in this statement "The number of players who play soccer and golf is half of the players who play each of the rest two games " which is completely meaningless.

1) Reading the OA, what you meant to say is this: "The number of players who only play soccer and golf..." Why? For the same reason that "6p players play soccer" means "6p players play soccer, or soccer and another sport, or all 3 sports". So the meaning of the word "only" is absolutely critical if you want the OA to actually be correct.

2) As other posters pointed out, "players who play each of the rest two games" has no meaning in English (or even if translated in any other language).

3) Grammar parallelism rules need to be followed. So compare a "number of a player" with another "number of players" not with just "players".

I would write statement as (1):

"The number of players who play soccer and golf but not tennis is half the number of players who play any other combination of two sports."

We need to know the number inside the parentheses. Statement 1 only provides Soccer&Golf to (Soccer&Tennis + Tennis&Golf) relationship, which is not sufficient. We have one equation and two variables. Insufficient.

Statement 2 is sufficient. Knowing the value of \(p\) we can find the exact values of each group from the formula above:

Thank you, there was a typo in the OE. I've corrected that. +1 for you!

First of all, we have to remember that we don't have to find the exact values in DS questions. We should save time and move on to the next question once we know that the info we have is sufficient.

There was a typo that I've fixed in the OE. The final value is different now, but the answer is still B.

arvs212 wrote:

can u plz elaborate a bit more..im confused with the explanation given..

leandrobrufman (or anyone else who might be reading this):

I have yet to see my specific query addressed regarding the first statement. Everyone on this thread seems to be in agreement that the statement p=3 is sufficient by itself.

However, the other statement, that "(1) The number of players who play soccer and golf but not tennis is half the number of players who play any other combination of two sports" should also be sufficient.

In this case, I will cite the work of leandrobrufman. The conclusion that most of us come to (some of us use different variables, but the idea is identical every time) is:

" 1) 2A=B+C Therefore we have 60 = 2p + X + (3A) + p and 22p = X+ 6A

NOT SUFFICIENT"

I disagree that that information is insufficient. Here is why: let's take the two equations within the quotation marks above, and simplify as much as possible.

We end up with 60=3p+X+3A=3p+(X+6A)-3A=25p-3A. In other words, 60+3A=25p. Therefore, p MUST BE A MULTIPLE OF 3.

As I mentioned in an earlier post, p has to be less than 6, because there are 60 people in total, and 11p of them play tennis. So p MUST BE 3.

What we have here are two statements, the first of which, if true, necessarily implies the that the second is true.

Since we all agree that the second statement is sufficient, we now can conclude that the first one is.

Statement 1) is actually meaningful because it tells us that p is a multiple of 3, which narrows down the value of p to one possibility.

Will somebody please address this, perhaps by finding a counterexample?

You are right. Well done! OA should be changed to reflect that. Both statements are sufficient. Indeed, even if statement 1 provides one equation with two unknowns, we still have other information in the useful in the question stem (p is an integer, 0<p<6), and as you showed, statement 1 requires p to be a multiple of 3.

I found confusing the wording of statement 1. "...is half of the players who play each of the rest two games" I had to read the explanation above to understand it. Wonder if any high scorer would find it confusing.

I found confusing the wording of statement 1. "...is half of the players who play each of the rest two games" I had to read the explanation above to understand it. Wonder if any high scorer would find it confusing.

got the same problem here. The description is weird.
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I agree with flyingbunny. Number of players that play only one game is 36, not 33.

Demonstration using the first explanation as base.

Total = Soccer + Tennis + Golf - (Soccer&Tennis + Soccer&Golf + Tennis&Golf) - 2*Soccer&Tennis&Golf + None => 60 = 6*3 + 8*3 + 11*3 - (Soccer&Tennis + Soccer&Golf + Tennis&Golf) - 2*3 + 2*3 => (Soccer&Tennis + Soccer&Golf + Tennis&Golf) = 15 => Number of players playing two games only is 15.

Therefore, number of players playing one game only = Total - 2only - 3only - none = Total - (Soccer&Tennis + Soccer&Golf + Tennis&Golf) - Soccer&Tennis&Golf - None = 60 - 15 - 3 - 6 = 36

First explanation makes a mistake on the last step by counting "3only" twice, when it should be counted once.

Overlapping sets exercises are so simple yet is so easy to get confused with them.
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People, it never says in the wording of the problem that "6p people play ONLY soccer", "11p people play ONLY tennis" etc. To me this wording sounds like 6p includes people who play solely soccer, as well as others who play soccer in a combination (e.g. soccer + tennis).

Given the above, I would say that we can't find the solution, given the information provided (E)

People, it never says in the wording of the problem that "6p people play ONLY soccer", "11p people play ONLY tennis" etc. To me this wording sounds like 6p includes people who play solely soccer, as well as others who play soccer in a combination (e.g. soccer + tennis).

Given the above, I would say that we can't find the solution, given the information provided (E)

Exactly, I'd go with E as well. In my understanding of the question, where [INT] represents Intersection. n(S) = 6p, n(T) = 11p, n(G) = 8p, n(S [INT] G [INT] T) = p, n(None) = 2p. The equation for total should be 60 = n(S) + n(T) + n(G) - [ n(S [INT] T) + n(S [INT] G) + n(T [INT] G)] + n(S [INT] G [INT] T) + n(None)

Both statements are insufficient to solve the above equation. what's the OA?
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GMAT Diagnostic Test Question 24 Field: word problems (overlapping sets) Difficulty: 750

Rating:

Among 60 members of a club, 6p players play soccer, 11p players play tennis, 8p players play golf and 2p players play none of the games. If p players play all of the games, how many players play only one game?

(1) The number of players who play soccer and golf is half of the players who play each of the rest two games (2) p = 3

I did not get what does mean by "rest of the two games" in st. 1?

I agree with the OA ( i.e B) assuming that statement 1 is insufficient however the number should be 33 not 36.

Members playing only one game = Total - (Soccer&Tennis + Soccer&Golf + Tennis&Golf) - 2 (Soccer&Tennis&Golf) - None Members playing only one game = 60 - 15 - 2(3) - 2*3 = 33

dzyubam wrote:

Explanation:

Rating:

Official Answer: B

Statement 1 is insufficient. For simplicity's sake we will write down the formula for three overlapping sets:

We need to know the number inside the parentheses. Statement 1 only provides Soccer&Golf to (Soccer&Tennis + Tennis&Golf) relationship, which is not sufficient. We have one equation and two variables. Insufficient.

Statement 2 is sufficient. Knowing the value of \(p\) we can find the exact values of each group from the formula above: