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Given total=90
all 90 must participate in atleast one of DC or MC or both

statement 1
Only Math=3/5 of total=3/5 * 90=54
and 54 is more than the half of 90 i.e 45. Which tells us that Math participants are greater, hence sufficient

statement 2
both DC and MC=2/9 of 90=20 and we have no info about only math or only debate, hence insufficient.

Therefore Option A is correct

Bunuel
A survey of 90 students found that each student participates in at least one of two activities: Debate Club or Math Club. Is the number of students who participate in Debate Club greater than the number who participate in Math Club?

(1) 3/5 of the students participate in only Math Club.
(2) 2/9 of the students participate in both Debate Club and Math Club.

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I know how well these questions can be solved via a table, but this one presents a logical challenge that might work even mentally. The goal, of course, would be to catch that logic in its tracks.

90 students participate in either debate club, math club, or both. Great. We need to find whether the debate club participants are more or fewer than the math club participants. No direct answers needed - we simply need to see if one of them is higher than the other. This can be done by determining if one of the clubs has more than 45 participating members, as 90 is the total count. The other one will naturally have fewer than 45 then.

Statement I: 3/5 of the students participate in only Math Club.

Well, here we go. If 3 / 5 or 54 / 90 students participate in the Math Club alone, at most 36 students will participate in the Debate Club. We have our answer. However, let's see if Statement II alone also helps solve this question.

Statement II: 2/9 students participated in both Debate Club and Math Club.


This means 18 out of the 90 students participate in both clubs, leaving us 72 who participate in either Math or Debate. This could mean something like 70 of the 72 are members of the Debate Club, making that the more popular club; or 70 of the 72 can be members of the Math Club. We have no way of knowing via this Statement alone.

In other words, Statement I alone is sufficient. A.



Bunuel
A survey of 90 students found that each student participates in at least one of two activities: Debate Club or Math Club. Is the number of students who participate in Debate Club greater than the number who participate in Math Club?

(1) 3/5 of the students participate in only Math Club.
(2) 2/9 of the students participate in both Debate Club and Math Club.

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The question: is Debate club participant > Math club participant?

(1) Since 3/5 participated in Math club (only) meaning, 2/5 participated in (Debate club + Both club)
so it must be debate club < math club

sufficient

(2) 2/9 participate in Debate & Math club, meaning 7/9 participate in (only debate + only math)
but we cannot justify how many from 7/9 contribute to either debate or math club

insufficient

so the answer is A.
Bunuel
A survey of 90 students found that each student participates in at least one of two activities: Debate Club or Math Club. Is the number of students who participate in Debate Club greater than the number who participate in Math Club?

(1) 3/5 of the students participate in only Math Club.
(2) 2/9 of the students participate in both Debate Club and Math Club.

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Total students=90
d for debate
m for math
b for both

d+m+b=90

Is d>m
3/5=math only
m-b=3/5x90=54
m=54+b
d and b not known

2/9=both clubs
b=2/9x90=20

m=54+b, b=20
m=54+20=74
d+m-b=90
d+74-20=90
d=90+20-74
d=36, m=74
d<m so
Together sufficient, individually insufficient
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Statement 1 :
3/5 students participate only in Math Clubs = Only M = 3/5 x 90 = 54
Remaining students = 90-54 = 36 . There are 36 , either Only Debate or both .(Not Sufficient)
We don't know how 36 splits between Only D and Both .
Statement 2 : 2/9 Participate in both = 2/9 x 90 = 20 , Remaining 70 . We don't know split of 70 between Only Math and Only Debate .(Not Sufficient)

St 1 & 2 Together
From 1 , Only M = 54 . Both = 20 . Only D = 90-(54+20)= 16 . Clearly Debate < Math . C is sufficient.✅
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Bunuel
A survey of 90 students found that each student participates in at least one of two activities: Debate Club or Math Club. Is the number of students who participate in Debate Club greater than the number who participate in Math Club?

(1) 3/5 of the students participate in only Math Club.
(2) 2/9 of the students participate in both Debate Club and Math Club.

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Let,
D= Debate only
M = Math only
B = both Debate and Math
T = 90

Since each student participates in at least one of the two clubs => Neither = N = 0

T - N = D + M + B

90 = D +M + B

We need, D + B > M +B ?

(1) 3/5 of the students participate in only Math club

=> M = (3/5) * 90 = 54
=> D + B = 90 - 54 = 36

D + B < M + B => 36 < 54 + B. YES

Statement (1) is SUFFICIENT

(2)
2/9 participate in both clubs

=> B = (2/9) * 90 = 20
=> 90 - 20 = M + D => M + D = 70. This does not tell us anything about M or D

Statement (2) is NOT Sufficient

Therefore, Answer A) Statement A alone is sufficient
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If we analyze the options

Considering statement 1 alone, if there are 3/5 of the members in just the Maths Dept alone & there are 90 people in total in at least one of the two clubs we have around 3/5*90 = 54 people in just the Maths Dept alone. if we consider the remaining members to be part of the Debate Club the number would be (90-54=36). In any case the Maths Dept would have more members than the Debate Club. Hence statement 1 alone is sufficient. Hence we can eliminate options C & E.

If we consider statement 2 alone, we find that 2/9 of the students were part of both the clubs, which would amount to 2/9*90 = 20. This statement does not help determine the number of members across the clubs individually, hence the statement is not suffiicient & so we can eliminate options B & D.

Hence the answer is option A
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DebateNo DebateTotal
Mathaby
No Mathc0c
Totalxb90

We need to know if x > y

(1) It is given that b = (3/5)(90)

=> b = 54

That means x = 90-b = 90-54 = 36

And y = a+b = a+54

With this info, it is obvious that y > x ----------> Sufficient

(2) It is given that a = (2/9)(90) = 20

But without knowing values for b & c, we cannot conclude the relationship between x & y ----------> Insufficient

Answer is (A)
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S1 : Only Math = 54 => Debate would be always lesser even if we make both = 0. => Definite answer (S)
S2 : Both is known. Still, we won't be able to say whether only debate is greater or only Math? (NS)

A
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Total = 90

|D |no D |
------------------------------
M |(2)20|(1)54| y
-----------------------------
No M | | |
-----------------------------
| x | | 90

1. M and No D = 54
ymin = 54
xmax = 36
x<y - Sufficient

2. With M and D to be 20, we can have y = 90, x=20 OR y=20, x=90. Therefore, we cannot say if x>y or y>x. Insufficient.

Answer A
Bunuel
A survey of 90 students found that each student participates in at least one of two activities: Debate Club or Math Club. Is the number of students who participate in Debate Club greater than the number who participate in Math Club?

(1) 3/5 of the students participate in only Math Club.
(2) 2/9 of the students participate in both Debate Club and Math Club.

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Let the number of only math=a
Number of both math and debate=b
only debate=c
given a+b+c=90
statement a.
given a=3/5*90=54
substitute a we get b+c=36
we get total no of debate students as 36.
total no of math students = 54+b
we know that b>=0
54+b>366
from this we can conclude that no of students in maths > debate

statement b:
given both math and debate as 2/9*90=20
from this we dont have any idea about the only maths or debate students.
so this statement is not sufficient.

therefore statement a alone is sufficient.
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(1) 3/5 of the students participate in only Math Club
=> 2/5 remaining students participate in only Debate or both Club
Since the maximum number of students participate in Debate Club is 2/5*a (a is the number of total students = 90) < 3/5*a, (1) is SUFFICIENT.

(2) Both Debate & Math = 2/9*a
We cannot find the number of students participate in Debate Club or Math Club. This is not SUFFICIENT.

Answer: A
Bunuel
A survey of 90 students found that each student participates in at least one of two activities: Debate Club or Math Club. Is the number of students who participate in Debate Club greater than the number who participate in Math Club?

(1) 3/5 of the students participate in only Math Club.
(2) 2/9 of the students participate in both Debate Club and Math Club.

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Answer: A

This can be answered in the form of a Venn diagram,
Let M be the number of students participating in Math only
Let D be the number of students participating in Debate only
Let B be the number of students participating in Both Math and Debate

We have to find if (D+B) > (M+B)?
or
if D> M?

(1) tells us that M=54, thus, D+B= 90-54 =36
Now, there can be 3 scenarios:
a) D=36, B=0
b) D=36-B, 0<B<36
c) D=0, B=36

In all 3 scenarios, M>D, thus the answer to the above question is No. Sufficient.

(2) tells us that B= 20, this info in itself does not help us reach any conclusion. Thus, Not Sufficient.


Bunuel
A survey of 90 students found that each student participates in at least one of two activities: Debate Club or Math Club. Is the number of students who participate in Debate Club greater than the number who participate in Math Club?

(1) 3/5 of the students participate in only Math Club.
(2) 2/9 of the students participate in both Debate Club and Math Club.

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A survey of 90 students found that each student participates in at least one of two activities: Debate Club (D) or Math Club (M). Is the number of students who participate in Debate Club greater than the number who participate in Math Club?
Total students= 90
Student in debate club= D
Student in Math club= M
Student in Both clubs = B

(1) 3/5 of the students participate in only Math Club.
Only M= (3/5)*90=54
We do not know anything about the number of students in the debate club. It is insufficient

(2) 2/9 of the students participate in both Debate Club and Math Club.
Both = (2/9)90=20
We do not know about students in only debate club or the math club. It is insufficient

(1)&(2)
Only M= 54 and B=20
Total number of students in Math Club= 54+20=74
Total students in debate club= 90-54=36
D=36 and M=74 and D>M is False
Sufficient

C
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The main point of info is that we have 90 students, and all of them participate in at least something (there's no student who doesn't belong to any club).

(1) If 60% of students are only part of Maths club, it leaves 40%, 36 students, to attend only Debate and both. Therefore, even if none go to both, and all 36 go only to Debate, there's still no way for Debate participation to be higher than that of Maths.
Hence, the answer is a definite 'no', and condition 1 is sufficient.

(2) If 2/9, or 20 people, participate in both clubs, then the remaining 70 can go in any direction. That is, all 70 can go to Maths, or to Debate, so there's no clarity. This is insufficient.

Therefore, the answer is A.
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Question Extraction:

Assuming, D - Number of debate club members & M - Number of math club members

Total = 90

All students participated in atleast 1 club, so number of students participating in neither club is 0.

Is D > M?

Soln:

D + M - Both = 90

i) 3/5 of 90 participated only in Math club.

3/5 (90) = 54; Insufficient, as we don't know how many participated in the other club.

Rule out D & A

ii) 2/9 participated in both.

2/9 (90) = 20; Insufficient. This value is helpless to know separate values to compare.

So, rule out B

Evaluating both statements,

M = 54 + 20 = 74
D = 20 + (90 - M + Both) = 56

Hence, M > D.

Answer C.
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Bunuel
A survey of 90 students found that each student participates in at least one of two activities: Debate Club or Math Club. Is the number of students who participate in Debate Club greater than the number who participate in Math Club?

(1) 3/5 of the students participate in only Math Club.
(2) 2/9 of the students participate in both Debate Club and Math Club.

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90 = M + D - M&D + Neither
Now neither = 0 hence this becomes
90 = M + D - M&D -- (a)

Statement (1)
Only Math's = 3/5 of total students
=> Only Math's = 90* (3/5) = 54
=> M - M&D = 54
Putting this in equation (a) we get

90 = 54 + D
=> D = 36
Since D is 36, Only D cannot be greater than 36.
Since Only M = 54, we know that M > D

Sufficient

Statement (2)
Both Math's and Debate = 2/9 of Total students
=> M&D = 2/9 * 90 = 20
Putting this is eq(a) we get

90 = M + D - 20
=> M + D = 70

We cannot say if M or D is greater.
Insufficient

Option A
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