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# GMAT Diagnostic Test Question 24

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GMAT Diagnostic Test Question 24 [#permalink]

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06 Jun 2009, 23:03
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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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Re: GMAT Diagnostic Test Question 24 [#permalink]

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23 Aug 2009, 09:32
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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?
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Re: 60 members of a club [#permalink]

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04 Aug 2010, 11:24
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Since I love Venn Diagrams, I am going to use one to solve this. (Hussain15 - Bear with me :D)

Here's a Venn Diagram of the situation where I've represented unknown quantities with variables a, b, c, x, y, and z respectively.

Attachment:
File comment: Venn Diagram

Club Sports.jpg [ 40.41 KiB | Viewed 8664 times ]

To find: $$a+b+c$$

Given Information:

Total$$= a+b+c+x+y+z+p+2p = 60$$

$$a+b+c+x+y+z = 60 - 3p$$ (1)

Total soccer = a + x + y + p = 6p

$$x+y = 5p - a$$ (2)

Total tennis$$= b + x + z + p = 11 p$$

$$x+z = 10p - b$$ (3)

Total golf $$= c + y + z + p = 8p$$

$$y+z = 7p - c$$ (4)

Adding (2), (3) and (4) and substituting into (1) we get:

$$22p - 2(x+y+z) = (a+b+c)$$

$$22p - 2(x+y+z) + (x+y+z) = 60 - 3p$$

$$(x+y+z) = 25p - 60$$

$$(a+b+c) = 22p - 25p + 60 = 60 - 3p$$

Statement 1:

$$y = \frac{(x+z)}{2}$$

Insufficient, because this doesn't tell us anything.

Statement 2

p = 3

Substituting p = 3 in the equation obtained for (a+b+c) we get a numeric answer. Hence sufficient. So, the answer is B.

Hope this helps.
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Re: GMAT Diagnostic Test Question 24 [#permalink]

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23 Aug 2009, 22:22
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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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Re: GMAT Diagnostic Test Question 24 [#permalink]

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30 Apr 2010, 10:07
2
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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."
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Re: GMAT Diagnostic Test Question 24 [#permalink]

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02 Jul 2009, 07:53
1
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Explanation:
 Rating:

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

Total = Soccer + Tennis + Golf - (Soccer&Tennis + Soccer&Golf + Tennis&Golf) - 2*Soccer&Tennis&Golf + None

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:

60 = 18 + 24 + 33 - (Soccer&Tennis + Soccer&Golf + Tennis&Golf) - 2*3 + 2*3
60 = 75 - 6 + 6 - (Soccer&Tennis + Soccer&Golf + Tennis&Golf)
60 = 75 - (Soccer&Tennis + Soccer&Golf + Tennis&Golf)
(Soccer&Tennis + Soccer&Golf + Tennis&Golf) = 15

Now that we know the number of club members playing exactly two games, we can find the number of club members playing only one game:

Total - (Soccer&Tennis + Soccer&Golf + Tennis&Golf) - Soccer&Tennis&Golf - None =
$$60 - 15 - 3 - 2*3 = 36$$
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Re: GMAT Diagnostic Test Question 24 [#permalink]

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10 Aug 2009, 22:18
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can u plz elaborate a bit more..im confused with the explanation given..

Total = Soccer + Tennis + Golf - (Soccer&Tennis + Soccer&Golf + Tennis&Golf) - 2*Soccer&Tennis&Golf + None

uve written this equation fr S1

then u hav used

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

60 = 18 + 24 + 33 - (Soccer&Tennis + Soccer&Golf + Tennis&Golf) - 2*3 - 2*3

this for S2..

it wuld b grt if u culd throw sum light n this topic..
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Re: GMAT Diagnostic Test Question 24 [#permalink]

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11 Aug 2009, 01:07
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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..

Total = Soccer + Tennis + Golf - (Soccer&Tennis + Soccer&Golf + Tennis&Golf) - 2*Soccer&Tennis&Golf + None

uve written this equation fr S1

then u hav used

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

60 = 18 + 24 + 33 - (Soccer&Tennis + Soccer&Golf + Tennis&Golf) - 2*3 - 2*3

this for S2..

it wuld b grt if u culd throw sum light n this topic..

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Re: GMAT Diagnostic Test Question 24 [#permalink]

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27 Nov 2011, 07:02
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alonbrodie wrote:
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.

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.
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Re: GMAT Diagnostic Test Question 24 [#permalink]

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22 Aug 2009, 13:57
I found confusing the wording of statement 1.
"...is half of the players who play each of the rest two games"
Wonder if any high scorer would find it confusing.
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Re: GMAT Diagnostic Test Question 24 [#permalink]

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23 Aug 2009, 09:18
powerka wrote:
I found confusing the wording of statement 1.
"...is half of the players who play each of the rest two games"
Wonder if any high scorer would find it confusing.

got the same problem here. The description is weird.
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Re: GMAT Diagnostic Test Question 24 [#permalink]

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23 Aug 2009, 10:50
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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Re: GMAT Diagnostic Test Question 24 [#permalink]

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23 Aug 2009, 21:53
can someone please explain why the formula is:

Total = Soccer + Tennis + Golf - (Soccer&Tennis + Soccer&Golf + Tennis&Golf) - 2*Soccer&Tennis&Golf + None,

in particular, why it is - 2*Soccer&Tennis&Golf

aren't we using the formula that

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)

and so Total = P(A u B u C) + complement (P(A u B u C) )

= Soccer + Tennis + Golf - (Soccer&Tennis + Soccer&Golf + Tennis&Golf) + Soccer&Tennis&Golf + None,

thanks.
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Re: GMAT Diagnostic Test Question 24 [#permalink]

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23 Aug 2009, 23:12
I always draw a diagram to help understanding this kinda question.
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Re: GMAT Diagnostic Test Question 24 [#permalink]

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06 Sep 2009, 01:46
defeatgmat wrote:
I always draw a diagram to help understanding this kinda question.

What would that look like for this question?
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Re: GMAT Diagnostic Test Question 24 [#permalink]

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18 Oct 2009, 13:40
blue4rain wrote:
can someone please explain why the formula is:

Total = Soccer + Tennis + Golf - (Soccer&Tennis + Soccer&Golf + Tennis&Golf) - 2*Soccer&Tennis&Golf + None,

in particular, why it is - 2*Soccer&Tennis&Golf

aren't we using the formula that

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)

and so Total = P(A u B u C) + complement (P(A u B u C) )

= Soccer + Tennis + Golf - (Soccer&Tennis + Soccer&Golf + Tennis&Golf) + Soccer&Tennis&Golf + None,

thanks.

Agree with you - there are 33 and not 36 players who only played one game - Good to see probabilistic people here
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Re: GMAT Diagnostic Test Question 24 [#permalink]

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12 Nov 2009, 19:15
Venn diagrams are the best way to go for this one... Though it eats up time doing so.
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Re: GMAT Diagnostic Test Question 24 [#permalink]

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14 Dec 2009, 07:34
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)
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Re: GMAT Diagnostic Test Question 24 [#permalink]

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14 Dec 2009, 08:44
vasyl27 wrote:
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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Re: GMAT Diagnostic Test Question 24 [#permalink]

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16 Dec 2009, 00:14
bb wrote:
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.

Rephrasing the OE:

p = 3
Soccer = 6p = 18
Tennis = 11p = 33
Golf = 8p = 24
Soccer&Tennis&Golf = p = 3
None = 2p = 6

Total = Soccer + Tennis + Golf - (Soccer&Tennis + Soccer&Golf + Tennis&Golf) - 2 (Soccer&Tennis&Golf) + None
Total = 6p + 11P + 8p - (Soccer&Tennis + Soccer&Golf + Tennis&Golf) - 2p + 2p
60 = 18 + 24 + 33 - (Soccer&Tennis + Soccer&Golf + Tennis&Golf) - 2*3 + 2*3
60 = 75 - 6 + 6 - (Soccer&Tennis + Soccer&Golf + Tennis&Golf)
60 = 75 - (Soccer&Tennis + Soccer&Golf + Tennis&Golf)
(Soccer&Tennis + Soccer&Golf + Tennis&Golf) = 15

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:

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

Total = Soccer + Tennis + Golf - (Soccer&Tennis + Soccer&Golf + Tennis&Golf) - 2*Soccer&Tennis&Golf + None

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:

60 = 18 + 24 + 33 - (Soccer&Tennis + Soccer&Golf + Tennis&Golf) - 2*3 + 2*3
60 = 75 - 6 + 6 - (Soccer&Tennis + Soccer&Golf + Tennis&Golf)
60 = 75 - (Soccer&Tennis + Soccer&Golf + Tennis&Golf)
(Soccer&Tennis + Soccer&Golf + Tennis&Golf) = 15

Now that we know the number of club members playing exactly two games, we can find the number of club members playing only one game:

Total - (Soccer&Tennis + Soccer&Golf + Tennis&Golf) - Soccer&Tennis&Golf - None =
$$60 - 15 - 3 - 2*3 = 36$$

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Re: GMAT Diagnostic Test Question 24   [#permalink] 16 Dec 2009, 00:14

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# GMAT Diagnostic Test Question 24

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