GMAT Club World Cup 2022 (DAY 1): A poll conducted in a football club

First make the Venn diagram, and consider the following regions in the overlapping circles.
a, b , c - Only Portugal(P), France(F) and Argentina(A) respectively
d, e, f - Both P&F, Both P&A, Both F&A respectively
g - All 3
T - Total
N - Neither

(i) T - N = a+b+c+d+e+f+g
(ii) a+d+e+g = 100 (Portugal)
(iii) b+d+g+f = 150(France)
(iv) e+g+f+c = 200(Argentina)

(v) i + ii + iii + iv = a + b + c + 2(d+e+f) + 3g = 450 ----- (d+e+f = 120, given)
This will be simplified to a+b+c +3g = 210, hence a+b+c = 210-3g (vi)
Putting (d+e+f = 120) & (vi) in eqn (i) we get
T = 330 - 2g - N, hence we need value of g & N to get the total

St 1 - Tells a = c, Not sufficient

St 2 - For every 1 g, there are 2 N's hence N/G is in ratio of 2:1, hence N = 2g
Put this in equation , T = 330 - 2g + 2g, T =330 Answer Hence: Sufficient

Overlapping set formula for 3 set:

Total-Neither= A+B+C-(Exactly 2)-2(Exactly 3)

Statement1: Equal number of members root for Portugal only and for Argentina only.- We still don't have any information of the number of members who root for none of the team in question.

Statement 2: For every 2 members of the club who root for none of the three teams, there is 1 member who roots for all three of the teams.
simplifying Statement 2:

none: all three= 2:1
let member who like all three be x, therefore member who doesnt support any of the team = 2x
using the overlapping set formula,

Total- 2x= 100+150+200-120-2(x)
Total= 330

Statement 2

I--- Only P+ Only A + Only F
II--- P&F together but not A + P&A together but not F + A&F together but not P
III--- all P, F, A together
O--- None of the P,F&A

II=120
I+2II+3III=450
I+3III=210

Total Football Fanclub members, T=O+I+II+III

S(1) Equal number of members root for Portugal only and for Argentina only.
Only P= Only A
does not help us to identify T

S(2) For every 2 members of the club who root for none of the three teams, there is 1 member who roots for all three of the teams.

If III=x, O will be 2x

T= I+II+x+2x
T=I+II+3x or I+II+3III

T=120+I+3III=120+210=330

S2 is sufficient
Therefore, B

Chose E as there was insufficient information based on the answers.

A poll conducted among the members of a football fan club, revealed that 100 of them root for Portugal, 150 of them root for France, and 200 of them root for Argentina. Also, 120 of them root for exactly two of the three teams. How many members does the fan club have ?

(1) Equal number of members root for Portugal only and for Argentina only.
we don't know the number that doesn't root any at all or root for all three teams.
insufficient
(2) For every 2 members of the club who root for none of the three teams, there is 1 member who roots for all three of the teams.
we still don't know the number of members who root for none and for all three. We only know the ratio.

Therefore E.

Given -
a + d + e + g = 100 --- 1
b + d + g + f = 150 --- 2
c + g + f + c = 200 --- 3

d + e + f = 120 -- 4

Adding the equations 1,2 & 3 we get -

(a+b+c) + 2(d+e+f) + 3g = 450

Using 4 we get

a + b + c + 3g = 450 - 240 = 210

Statement 1

Equal number of members root for Portugal only and for Argentina only.

We do not know the number of member who do not root for any of the teams.

Hence this statement is not sufficient.

Statement 2

For every 2 members of the club who root for none of the three teams, there is 1 member who roots for all three of the teams.

Number of members of the club = (a + b + c + d + e + f + g) + None

As we are not given the number of members who do not support or support all three teams the information is inconclusive.

Combining

We dont have sufficient information.

IMO E

As per the question we have

1) Even with this information we don't know about all three fans and people who are not fans
2) We don't know about the values hence we won't be able to find out the total number of people

Even with both statements 1 and 2 there is not sufficient information to find out the total number of people in the fan club

IMHO Option E

total = A+B+C - ( sum of exactly 2- group over laps ) - 2 * (ALL THREE ) + NEITHER


FROM given info we have
total = 100+150+200 - ( 120- group overlaps) - 2 * (ALL THREE ) + NEITHER

#1
(1) Equal number of members root for Portugal only and for Argentina only.

total = 450- ( 120- 2x + france ) - 2 * (ALL THREE ) + NEITHER
insufficient

#2
For every 2 members of the club who root for none of the three teams, there is 1 member who roots for all three of the teams.

2 * ( all three) = 1 Neither
or say
total = A+B+C - ( sum of exactly 2- group over laps ) - 2 NEITHER

insufficient as values of neither is not known and other also

from 1 &2

total = 450- ( 120- 2x + france ) - 2 * neither

we have 2 unkowns only france and neither
insufficient

OPTION E

A poll conducted among the members of a football fan club, revealed that 100 of them root for Portugal, 150 of them root for France, and 200 of them root for Argentina. Also, 120 of them root for exactly two of the three teams. How many members does the fan club have ?



Statement 1:

(1) Equal number of members root for Portugal only and for Argentina only.


Not sufficient as it does not give us any idea about people rooting for all 3 and how the members are distributed.

(2) For every 2 members of the club who root for none of the three teams, there is 1 member who roots for all three of the teams.

Not sufficient as it does not give us any idea about people rooting for all 3 and how the members are distributed.


IMO Option E

3 overlapping sets - Set P = 100, Set F = 150, Set A = 200

exactly 2 overlapping = 120
then total members in club = P +F + A - sum of exactly 2 overlaps - 2* (all 3) + N (None of 3 teams)
St 1. Only P = Only A.. but no info about all 3 and N..
Not Suff

st 2. N = 2 (all 3).

substituting in above equation the given values
total members in club = 100 +150+200 -120 = 230

Answer B

Given: A poll conducted among the members of a football fan club, revealed that 100 of them root for Portugal, 150 of them root for France, and 200 of them root for Argentina. Also, 120 of them root for exactly two of the three teams.

Asked: How many members does the fan club have ?



(1) Equal number of members root for Portugal only and for Argentina only.
We require value of X & Y to find total members
NOT SUFFICIENT


(2) For every 2 members of the club who root for none of the three teams, there is 1 member who roots for all three of the teams.
Y = 2X
Total = 100 + 150 +200 - 120 - 2X + Y = 100 + 150 + 200 - 120 = 230
SUFFICIENT

IMO B

Let the people who like only Portugal be 'p'
People like both Portugal and France be 'c'
People like both Portugal and Argentina be 'r'
People who like only France be 'q'
People who like all the 3 teams be 'x'
People who like none of the 3 teams be 'm'.

p+c+b+x=100 -i People who like Portugal
q+c+a+x=150-ii People who like France
r+b+a+x=200- iii People who like Argentina

a+b+c=120 , People who like exactly two of the three teams
On adding i, ii and iii
p+q+r+2(a+b+c)+3x=450
p+q+r+3x=240
Total member of the fan club are p+q+r+a+b+c+x+m
From statement 1, we get p=r,
then 2r+q+3x=240 but no value about m and x can be determined.

From statement 2 we have m=2x
but p+r+q+3x=240 no solution.

On combining statement 1 and statement 2,
2r+q+3x=240 and m=2x no solution combining these 2, Hence option E.

P=100
F=150
A=200
Exactly 2 =120
let total be x
let people in none of the groups be n
I+II+III=450---------------------------------------EQU 1
I+2(II)+3(III)+N=X--------------------------------EQU 2

(1) Equal number of members root for Portugal only and for Argentina only.
NOT sufficient

(2) For every 2 members of the club who root for none of the three teams, there is 1 member who roots for all three of the teams

n/III=2:1
n=2l
III=l
solving equ 1 and 2 and substituting the value of n and iii we will get
120+2l=x-450
not sufficient

st 1 and 2 are also not sufficient
oa:e

Let k be root for all three teams

A is not sufficient because we don't know what is the value of k and None.

Statement B-
we know, Total-None=100+150+200-120-2k
Option B states that None = 2k
So, Total-2k=100+150+200-120-2k
2k will cancel from both sides and we can find the answer for Total.
Hence, answer should be B.

Let's say that :
P= number of members that root for Portugal
F= number of members that root for France
A= number of members that root for Argentine

So Total of members = P+F+A -(Sum of exactly 2-group overlaps) -2.(members rootinf to all three) +N * / such as N=Members who root for none of the three teams (France, Portugal or Argentine)

the first statement says that :(1) Equal number of members root for Portugal only and for Argentina only
this statment gives nothing about N, so the total number cannot be determined, so insuffisiant

Second statement says that For every 2 members of the club who root for none of the three teams, there is 1 member who roots for all three of the teams.
And that simply means that N = 2x(members rooting to all three)

From * we conclude that : Total = 100+150+200 -120 =330 so Suffisant => Answer is B

We know from 3 set overlapping sets:
Total=p+f+a-#(Exactly 2)-2*#(All 3)+#(Neither)

Question : Total = ?

1) Let p=f=x
Thus Total = 2x+f
Therefore,
2x+f=450-120-2#(All 3)+#(Neither)

INSUFFICIENT

2) Total=450-120-0
Since as per problem :
"For every 2 members of the club who root for none of the three teams, there is 1 member who roots for all three of the teams."
So if
#(All 3)=a
#(Neither)=2*a

SUFFICIENT

ANS: B