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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, 08:00
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45% (medium)

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75% (01:46) correct 25% (02:00) wrong based on 828 sessions

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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?

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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C
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GMAT Club Olympics 2025 (Day 2): At a tech seminar with a total 01 Jul 2025, 08:30
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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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­

GMAT Club Official Explanation:



Total = cybersecurity + cloud computing - both + neither

So:

a = b + c - both + d

Solving for both gives: both = b + c + d - a

Therefore, the required fraction both/total is (b + c + d - a)/a

Answer: C.
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The formula for two overlapping sets is:
Total = Group 1 + Group 2 - Both + Neither

Substitute the variables from the problem:
a = b + c - Both + d

Solve for the number of participants who attended both sessions:
Both = b + c + d - a

To find the fraction of participants who attended both, divide this by the total number of participants, 'a':
Fraction = (b + c + d - a) / a

The correct answer is (C).
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Total = CS + CC - both + None
a=b+c-both+d
both = b+c+d-a

Fraction of the participants attended both sessions = b+c+d-a/a

Answer 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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Answer
(C) (b + c + d - a) / a

Explanation

This is a classic overlapping sets problem. The easiest way to solve it is with a formula.

1. The Formula:

The formula for any two overlapping groups is:
Total = Group A + Group B - Both + Neither

2. Plug in the Variables:

Let's use the variables from the problem:

  • a = Total
  • b = Attended Cybersecurity (Group A)
  • c = Attended Cloud Computing (Group B)
  • d = Attended Neither

Plugging these into the formula gives us:
a = b + c - Both + d

3. Solve for "Both":

The question asks for the number of people who attended both sessions. We just need to rearrange the formula to solve for "Both":
Both = b + c + d - a

4. Find the Fraction:

The question asks for the fraction of participants who attended both. To get the fraction, we divide the number who attended "Both" by the "Total" number of participants, which is a.
Fraction = (b + c + d - a) / a
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a = Total participants
d = attend neither

a-d = attended one or both


answer e : (cyber total) + ( cloud total) - (total) + (didnt attend) / (attended)

Im not so sure on this one. Guessing E


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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We can say that - people who attended at least one session = a − d (Total - neither session)

also - people who attended at least one session = b + c − both (because we are taking union here)

People who attended both = b + c + d − a

therefore fraction would be (b + c + d − a) / a
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Participants who attended at least one of the two b & c sessions: b+c-x(participants who attended both b & c). And a = b+c-x+d => x=b+c+d-a => Fraction of participants who attended both b & c : (b+c+d-a)/a => 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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a = total number of participants
b = Attended cybersecurity
c = Attended cloud computing

No of participants not attended cloud computing = a-b
No of participants not attended cybersecurity = a-c

d = Attended neither cloud computing nor cybersecurity

No of participants attended cybersecurity but not cloud computing = a-c-d
No of participants attended both cybersecurity and cloud computing = b-(a-c-d) = b+c+d-a

Hence required fraction is \(\frac{b + c + d - a}{a}\)

Option C is correct
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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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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Union Formula: a = b + c - both(b&c) + d;

Than b&c = b+c+d-a;

Than Fraction b&c/a = (b+c+d-a)/a; Answer
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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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We can plot the information on a 2*2 matrix and get the following table

Cybersecurity Not Cybersecurity
Cloud Computingc
Not Cloud computing
ba

Adding the missing values we get

CybersecurityNot Cybersecurity
Cloud Computingb - (a - c - d) = b - a + c + da - b - dc
Not Cloud computinga - c - dd
ba-ba

Both Cloud Computing and Cybersecurity = b - a + c + d

Ratio= (b - a + c + d) / a

Option C
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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?

Basic information give in the question: -
Cyber securityNot cyber securityTotal
Cloud computingc
Not cloud computingd
Totalba


After filling up the blank fields: -
Cyber securityNot cyber securityTotal
Cloud computingb + c + d - aa - b - dc
Not cloud computinga - c -d da - c
Totalba - ba

The fraction of the participants attended both sessions = \(\frac{b+c+d-a}{a}\)

IMO C
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Total participants = a
Attended cybersecurity = b
Attended cloud computing = c
Attended neither = d
We’re asked: What fraction of participants attended both sessions?
Let’s call the number of people who attended both sessions = x

Total number of participants who attended at least one session = b + c – x
We're subtracting x because participants who attended both got counted twice (once in b and once in c).
So:
Attended at least one session = b + c – x
We also know:
Attended neither session = d
So total participants = (Attended at least one) + (Attended neither)
a=(b+c–x)+da = (b + c – x) + da=(b+c–x)+d
Rearranging:
a=b+c–x+d⇒x=b+c+d–aa = b + c – x + d \Rightarrow x = b + c + d – aa=b+c–x+d⇒x=b+c+d–a
That’s the number of people who attended both.
We’re asked for the fraction of participants who attended both = x / a
From above:
x = b + c + d – a
So:
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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Correct answer is C, below attached is the solution for the question
Attachments

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total are A and b,c are part of individual pool if x is the no present in both then neither would be = a - (b+C+x)=d
d-a+(b+c)/a
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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 x = people who attended both sessions
Number who attended at least one = b+c-x
Number who attended neither = d
a = (b+c-x) + d
x = b+c+d-a
x/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, 08:13
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a= b+c -(Both)+d
(Both)=b+c+d-a
Both/a = (b+c+d-a)/a
Hence Ans C
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let both cs and cc be x.
so, total = only cs + only cc + (cs and cc) + neither( cs and cc)

a = (b-x) + (c-x) + x + d

x = b+c+d-a
so fraction = x/a = (b+c+d-a)/a

ans is C
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Total participant = a
Participant attended cybersecurity = b
Participant attended cloud computing = c
Participant attended neither = d
Participant attended both = x

a-d = b+c-x
Hence C
x = b+c+d-a
Fraction = b+c+d-a/a
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Given information:
- Total number of participants = a
- Number who attended cybersecurity session = b
- Number who attended cloud computing session = c
- Number who attended neither session = d
Let's define x as the number who attended both sessions (the overlap region).
Using the formula for the total number of elements:
Total = (Cybersecurity only) + (Cloud only) + (Both) + (Neither)
We can express this as:
a = (b -x) + (c -x) +x+ d
a = b+c-x+d
Solving for x (the number who attended both sessions): b+c-x+d= a
-x= a - b-c-d
x=b+c+d-a
The question asks for the fraction of participants who attended both sessions:
Fraction = x/a = (b+ c+ d - a)/a
this matches with choice C (b+c+d-a)/a
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