GMAT Diagnostic Test Question 24

Question

GMAT Diagnostic Test Question 24 
Field: word problems (overlapping sets)
Difficulty: 750
 Rating:  yellow/79353

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

dzyubam · Answer

Explanation: 
 Rating:  yellow/793532 
 Official Answer: B

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

arvs212 · Answer

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..

dzyubam · Answer

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.

powerka · Answer

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.

flyingbunny · Answer

got the same problem here. The description is weird.

flyingbunny · Answer

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?

powerka · Answer

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.

blue4rain · Answer

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.

powerka · Answer

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.

defeatgmat · Answer

I always draw a diagram to help understanding this kinda question.

thb · Answer

What would that look like for this question?

LUGO · Answer

Agree with you - there are 33 and not 36 players who only played one game - Good to see probabilistic people here

ajonas · Answer

Venn diagrams are the best way to go for this one... Though it eats up time doing so.

vasyl27 · Answer

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)

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

BarneyStinson · Answer

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

Both statements are insufficient to solve the above equation. what's the OA?

GMAT TIGER · Answer

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

sojafon · Answer

I agree with "each of rest two games" being confusing. What exactly does that mean? Please rephrase this if you can. Thanks.

bangalorian2000 · Answer

OA is B.
we need to find the number of people who play exactly two games so from the equation 60 = n(S) + n(T) + n(G) -   T) + n(S   G) + n(T   G)] + n(S   G   T) + n(None) reduces to 
n(S   T) + n(S   G) + n(T   G) = 6p+11p+8p+2p+p-60
if p=3 then we can solve this hence B

gmatbull · Answer

Total = S+T+G - double -2(x) + N:
S=soccer; T=tennis; double=no of those playing 2 games; x = No of those playing all 3 games; N=none
60 = 6p+11p+8p - double -2(x) + 2p
60 = 25p - double - 2p + 2p

(1) 60 = 25p - double..........
Notwithstanding its interpretation, (1) introduces another variable relating 2-games players
INSUFF

(2) 60 = 25(3) - double
=> double = 15
1 game: total - double - triple(x)   - N 
 = 60 - 15 - 3 - 2(3)
=> 60 - 24 = 36

Hence, B

GMAT Diagnostic Test Question 24

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