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605-655 (Medium)|   Overlapping Sets|         
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GMAT Club Olympics 2025 (Day 2): At a tech seminar with a total 01 Jul 2025, 15:32
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CybCyb'
Compb-(a-c-d) c
Comp'a-c-dda-c
ba-bTotal=a

Fractions of participants who attended both = b-(a-c-d)/a = b-a+c+d/a. Option C.

Bunuel
At a tech seminar with a total participants, b attended a session on cybersecurity, and c attended a session on cloud computing. If exactly d participants attended neither session, then in terms of a, b, c, and d, what fraction of the participants attended both sessions?

A. \(\frac{a - b - c + d}{a}\)

B. \(\frac{a - b - c - d}{a}\)

C. \(\frac{b + c + d - a}{a}\)

D. \(\frac{a - b - c + 2d}{a}\)

E. \(\frac{b + c - a + d}{a-d}\)


 


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GMAT Club Olympics 2025 (Day 2): At a tech seminar with a total 01 Jul 2025, 15:52
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Total participants: a
Let number of participants who attended both sessions: x
number who attended only cybersecurity: b−x
number who attended only cloud computing: c−x
number who attended neither: d

Total participants=(b−x)+(c−x)+x+d

=>a=b+c−x+d⇒x=b+c+d−a

No. of participants who attended both sessions = x=b+c+d−a

Required fraction: x=b+c+d−a/a (option C)

Bunuel
At a tech seminar with a total participants, b attended a session on cybersecurity, and c attended a session on cloud computing. If exactly d participants attended neither session, then in terms of a, b, c, and d, what fraction of the participants attended both sessions?

A. \(\frac{a - b - c + d}{a}\)

B. \(\frac{a - b - c - d}{a}\)

C. \(\frac{b + c + d - a}{a}\)

D. \(\frac{a - b - c + 2d}{a}\)

E. \(\frac{b + c - a + d}{a-d}\)


 


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GMAT Club Olympics 2025 (Day 2): At a tech seminar with a total 01 Jul 2025, 15:57
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Bunuel
At a tech seminar with a total participants, b attended a session on cybersecurity, and c attended a session on cloud computing. If exactly d participants attended neither session, then in terms of a, b, c, and d, what fraction of the participants attended both sessions?

A. \(\frac{a - b - c + d}{a}\)

B. \(\frac{a - b - c - d}{a}\)

C. \(\frac{b + c + d - a}{a}\)

D. \(\frac{a - b - c + 2d}{a}\)

E. \(\frac{b + c - a + d}{a-d}\)


 


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Let us write the total participants a = b+c+d-(b overlap c) ---> (b overlap c) = the participants who attended both sessions = b+c+d-a
Thus, the fraction is (b+c+d-a)/a
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GMAT Club Olympics 2025 (Day 2): At a tech seminar with a total 01 Jul 2025, 16:22
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To see if someone attended both session we need to see the total people that attended to at least one session (a-d) and compare with the total people we know that attended some sessions (b+c).

1) At least one sesssion is a-d;
2) Attended some session is b+c;
3) Comparing than is b+c-(a-d) = b+c+d-a, is this number is more than 0 some people attended both sessions;
4) The fraction is (b+c+d-a)/a

Letter C
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GMAT Club Olympics 2025 (Day 2): At a tech seminar with a total 01 Jul 2025, 16:23
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At a tech seminar with a total participants, b attended a session on cybersecurity, and c attended a session on cloud computing. If exactly d participants attended neither session, then in terms of a, b, c, and d, what fraction of the participants attended both sessions?

The correct answer is (B).

To solve this question I first wrote out the various letters with what they represent. Then I wrote an equation representing the "fraction of the participants" that attended both sessions.

a = total participants
b = cybersecurity participants
c = cloud computing participants
d = participants that attended neither

a (the total) = b + c + d
The total participants all either attended one of the sessions or fall into the group that attended neither. Using this equation, if we want to find the amount that attended both sessions, we take a, then subtract d (those who attended neither). Then to find the fraction, we find the answer that has the denominator = a, as we need to divide by the total to understand the fraction. The only answer with a - d is (b), then we ensure it is over the right denominator.

a-b-c-d / a
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GMAT Club Olympics 2025 (Day 2): At a tech seminar with a total 01 Jul 2025, 16:43
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Bunuel
At a tech seminar with a total participants, b attended a session on cybersecurity, and c attended a session on cloud computing. If exactly d participants attended neither session, then in terms of a, b, c, and d, what fraction of the participants attended both sessions?

A. \(\frac{a - b - c + d}{a}\)

B. \(\frac{a - b - c - d}{a}\)

C. \(\frac{b + c + d - a}{a}\)

D. \(\frac{a - b - c + 2d}{a}\)

E. \(\frac{b + c - a + d}{a-d}\)


 


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Answer: C
Participants attending at least one session = total participants minus participants who participated in neither session = a-d
But total participants attending at least one session can also be derived as: attendees of each session (b+c) minus participants who participated in BOTH sessions (Because we don't know this number, let's call it X)

Thus: a-d=b+c - X, which can be rearranged as: X = b+c+d-a
To find the fraction, we divide by the total participants (a): (b+c+d-a)/a
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GMAT Club Olympics 2025 (Day 2): At a tech seminar with a total 01 Jul 2025, 17:41
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Let e be the number of participants that attended both sessions. We have: b + c - e + d = a --> e = b + c + d - a --> e/a = (b+c+d-a)/a
Bunuel
At a tech seminar with a total participants, b attended a session on cybersecurity, and c attended a session on cloud computing. If exactly d participants attended neither session, then in terms of a, b, c, and d, what fraction of the participants attended both sessions?

A. \(\frac{a - b - c + d}{a}\)

B. \(\frac{a - b - c - d}{a}\)

C. \(\frac{b + c + d - a}{a}\)

D. \(\frac{a - b - c + 2d}{a}\)

E. \(\frac{b + c - a + d}{a-d}\)


 


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GMAT Club Olympics 2025 (Day 2): At a tech seminar with a total 01 Jul 2025, 18:14
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divide the group

G1= CS
G2= NO CS
G3= CC
G4= NO CC

G1_G3=X
G1_G4=Y
G2_G3=Z
G2_G4=U


SO GIVEN IS G1=b=X+Y ; G3=c=X+Z, U=d TOTAL=a
THEN G2=a-b ; G4= a-c, Y=a-c-d, X=b+c+d-a

need to calculate X/total

so reqd fraction= b+c+d-a/a


Bunuel
At a tech seminar with a total participants, b attended a session on cybersecurity, and c attended a session on cloud computing. If exactly d participants attended neither session, then in terms of a, b, c, and d, what fraction of the participants attended both sessions?

A. \(\frac{a - b - c + d}{a}\)

B. \(\frac{a - b - c - d}{a}\)

C. \(\frac{b + c + d - a}{a}\)

D. \(\frac{a - b - c + 2d}{a}\)

E. \(\frac{b + c - a + d}{a-d}\)


 


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GMAT Club Olympics 2025 (Day 2): At a tech seminar with a total 01 Jul 2025, 18:28
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Total = b+c-x+d. So a = b+c-x+d

Then both equals: x=b+c+d-a

And to get both as a fraction of all participants: \(\frac{b + c + d - a}{a}\)


Bunuel
At a tech seminar with a total participants, b attended a session on cybersecurity, and c attended a session on cloud computing. If exactly d participants attended neither session, then in terms of a, b, c, and d, what fraction of the participants attended both sessions?

A. \(\frac{a - b - c + d}{a}\)

B. \(\frac{a - b - c - d}{a}\)

C. \(\frac{b + c + d - a}{a}\)

D. \(\frac{a - b - c + 2d}{a}\)

E. \(\frac{b + c - a + d}{a-d}\)


 


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GMAT Club Olympics 2025 (Day 2): At a tech seminar with a total 01 Jul 2025, 18:39
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Well, since I am a lawyer and my math is rusty this took me a little longer. As i see it the goal is to calculate X. So, even thou in the exercice only A, B, C and D are mentioned, the goal is to calculate the non mentioned variable: X (those who attended both sessions). So it would be wrong to think that A (total) =B +C +D, because the exercice is assuming there is an additional variable.

Then the right way would be A= B +C+D-X or in other words to determine correctly A, you should add B+C+D and sustract X because those numbers are repeated.

After that it gets easy: X= B+C+D-A/A

So right answer is C.

Bunuel
Quote:
At a tech seminar with a total participants, b attended a session on cybersecurity, and c attended a session on cloud computing. If exactly d participants attended neither session, then in terms of a, b, c, and d, what fraction of the participants attended both sessions?

A. \(\frac{a - b - c + d}{a}\)

B. \(\frac{a - b - c - d}{a}\)

C. \(\frac{b + c + d - a}{a}\)

D. \(\frac{a - b - c + 2d}{a}\)

E. \(\frac{b + c - a + d}{a-d}\)


 


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GMAT Club Olympics 2025 (Day 2): At a tech seminar with a total 01 Jul 2025, 18:40
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Bunuel
At a tech seminar with a total participants, b attended a session on cybersecurity, and c attended a session on cloud computing. If exactly d participants attended neither session, then in terms of a, b, c, and d, what fraction of the participants attended both sessions?

A. \(\frac{a - b - c + d}{a}\)

B. \(\frac{a - b - c - d}{a}\)

C. \(\frac{b + c + d - a}{a}\)

D. \(\frac{a - b - c + 2d}{a}\)

E. \(\frac{b + c - a + d}{a-d}\)

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Total participants = a
Total participants on cybersecurity = b
Total participants on cloud computing = c
Attended neither = d

|Cyber\(\cup\)Cloud| = a-d = Cyber + Cloud - |Cyber\(\cap\) Cloud| =b+c-|Cyber\(\cap\) Cloud|

|Cyber\(\cap\) Cloud|=b+c+d-a

Therefore, the fraction attending both = \(\frac{b + c + d - a}{a}\)
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GMAT Club Olympics 2025 (Day 2): At a tech seminar with a total 01 Jul 2025, 18:44
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Answer: C

Cybersecurity + Cloud computing -Both + Neither= Total participants
=> b + c - both + d = a
=> both = b + c + d - a

Fraction of the participants attended both sessions = both/total participants = (b+c+d−a)/a
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GMAT Club Olympics 2025 (Day 2): At a tech seminar with a total 01 Jul 2025, 18:53
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Total = A
Cyber = B
Cloud Computing = C
Cyber and cloud = X
No course = D
Look for : X/A
So: A= B + C - (X) + D --> X = B + C + D - A
Answer is: (B+C + D - A)/A ... (D)
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GMAT Club Olympics 2025 (Day 2): At a tech seminar with a total 01 Jul 2025, 18:55
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Bunuel
At a tech seminar with a total participants, b attended a session on cybersecurity, and c attended a session on cloud computing. If exactly d participants attended neither session, then in terms of a, b, c, and d, what fraction of the participants attended both sessions?

A. \(\frac{a - b - c + d}{a}\)

B. \(\frac{a - b - c - d}{a}\)

C. \(\frac{b + c + d - a}{a}\)

D. \(\frac{a - b - c + 2d}{a}\)

E. \(\frac{b + c - a + d}{a-d}\)


 


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From the statement we have the equation as below

Total = Cybersecurity + Cloud computing - Both + Neither
=> a = b + c - Both + d
=> Both = b + c + d - a

Required fraction = \(\frac{Both}{a}\) = \(\frac{b + c + d - a}{a}\)

IMHO Option C
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GMAT Club Olympics 2025 (Day 2): At a tech seminar with a total 01 Jul 2025, 19:02
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Total no of participants = a
Cyber security(cs) = b
Cloud computing(cc) = c
Neither = d

Let's take attending both cs and cc = x

Based on the overlapping sets formula OR we can draw a basic venn diagram as well to derive the following:
a=b+c-x+d => x=b+c+d-a

Fraction of the total attending both cs and cc = x/a = (b+c+d-a)/a

Answer choice D
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GMAT Club Olympics 2025 (Day 2): At a tech seminar with a total 01 Jul 2025, 19:25
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Total = A + B - Both + Neither

a = b + c - x + d

x = b+c+d - a

Fraction = b+c+d - a/a

C is the answer
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GMAT Club Olympics 2025 (Day 2): At a tech seminar with a total 01 Jul 2025, 19:26
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Basically it is a sets question and asks overlap.

a=b+c-bc+d
=> bc=b+c+d-a

therefore fraction= b+c+d-a/a
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GMAT Club Olympics 2025 (Day 2): At a tech seminar with a total 01 Jul 2025, 19:26
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\(b\) = those who attended cybersecurity
\(c \) = those who attended cloud computing
\(d\) = those who attended none
let \( x\) = those who attended both cybersecurity and cloud computing
We need to find \(\frac{x}{a}\)

from the overlapping sets formula, \(b(cyber) + c (cloud) - x (both) + d (none) = a (all)\)
\(x= b+c+d - a\)

answer = \( \frac{x}{a} = \frac{b + c + d - a}{a} \)

C is the answer
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GMAT Club Olympics 2025 (Day 2): At a tech seminar with a total 01 Jul 2025, 19:39
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Let X be number of people who were in both session
At least one session=b+c-X
b+c-X+d=a
=>X=b+c+d-a
X/a=b+c+d-a/a( Answer C)
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GMAT Club Olympics 2025 (Day 2): At a tech seminar with a total 01 Jul 2025, 19:55
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Total participants = a
Number of participants who attended cybersecurity = b
Number of participants who attended cloud computing = c
Number of participants who attended neither session = d
Number of participants who attended both sessions = x

=> Total number of participants => a = b+c-x+d
Rearranging this we get, x = b+c+d-a
Now to find the fraction of participants who attended both sessions which is the solution required, we divide by total number of partipants
=>
x/a = (b+c+d-a) / a
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Tuck School Moderator
853 posts