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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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This is a set theory style question :
Let the number of ppl who participate in both is x.
Out of total, the ones who participate at least on of the 2 sessions is a-d.
We also know that the above count is same as b+c - x

b+c - x = a-d, So x = d+b+c-a.

We need to find the ratio : x/a => $\frac{b + c + d - a}{a}$

IMO C

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Let e be number of students who attend both
The total number of participants a(all) = b(cybersecurity) + c(cloud computing) + d(neither) - e(both)
hence e = b + c + d - e
Question asks fraction of participants(a) who attended both session(e) = e/a = (b+c+d-a)/a

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Solution:

Total = b + c +d - Both

The correct answer seems to be C. $\frac{b + c + d - a}{a}$

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Total # of participants = # attended cybersecurity + # attended cloud computing - # attended both + # attended neither

Let's let the # attended both = x

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

As a fraction of all participants (a):

(b + c + d - a)/a

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

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I solved it through set theory,

Total Participants= a
Total cybersecurity Participants = b
Total Cloud computing Participants = c
None = d

Let the people who attended both cybersecurity and cloud computing be x

Then

Total = only cybersecurity+ only cloud computing + attended both + none

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

On solving, it would be x=b+c+d-a. Since it's asking for a fraction, divide it by a, and that's the answer.

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01 Jul 2025, 08:31

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Total participants (a) = b(Cybersecurity) + c(Cloud Computing) - x (who attended both) + d ( who attended neither)

SO fraction who attended both x/a = (b+c+d-a)/a

Option C.

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01 Jul 2025, 08:32

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Ans: C (b+c+d-a)/a
$\frac{b + c + d - a}{a}$

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?

from this lets say y = both in cyberc and computing. So,

a = (b-y) + y + (c-y) + d
from this we get
y = b + c + d - a

fraction = $\frac{b + c + d - a}{a}$

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I automatically set up a box for each of these questions now. Takes an extra 20 or so seconds but it has really helped me reduce stupid errors.

The table is below. Let x be the number of participants who attended both sessions, y be the number of participants who attended a session on cloud computing but not cybersecurity, and z be the total number of participants who did not attend a cybersecurity session. We want to find x as a fraction of all the participants (a):

	Cyber	Not Cyber	Total:
Cloud Comp	x	y	c
Not CC		d
Total:	b	z	a

z = a - b
y = z - d
x = c - y

substitute values:
y = (a - b) - d
x = c - ((a - b) - d) which can be rewritten as c - a + b + d or x = b + c + d - a

the number of participants who attended both sessions as a fraction of all the participants is x/a or $\frac{b + c + d - a}{a}$

Note: you get the same answer if assign variables to the other unknown values as well. For example, let x still be the number of participants who attended both sessions, but let p be the number of participants who attended a cyber session but not a cloud computing session, and let q be the total number of participants who did not attend a cloud computing session:

	Cyber	Not Cyber	Total:
Cloud Comp.	x		c
Not CC	p	d	q
Total:	b		a

then,
q = a - c
p = (a - c) - d
x = b - [(a - c) - d] => b - a + c + d

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First, the number of participants who attended at least one session is:
a−d

Using the formula for the union of two sets:
(Attended at least one)= b+c−(Attended both)

So we set the two expressions equal:
b+c−both=a−d
Solving for the number who attended both:
both=b+c−(a−d)=b+c+d−a

To express this as a fraction of total participants:
b+c+d−a/a

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Let [math]X[/math] be the number of participants that attented both the sessions.

Then, [math]a = b+c-X+d[/math]
[math]X = b+c+d-a[/math]

Now [math]X[/math] as a fraction of [math]a[/math] will be [math]\frac{X}{a}= \frac{b+c+d-a}{a}[/math]

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Total participants = a
Participants who attended Cybersecurity = b
Participants who attended Cloud computing = c
Participants who attended neither = d

From the formula : T-N = A + B - k => k = T - A - B - N , where k is the participants who attended both

Therefore, regarding the problem at hand, the participants who attended both, k = a - b - c - d

The fraction of those who attended both sessions to the total participants = k/a = (a - b - c - d) / a

Answer is B

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Let:

a = total participants
b = those in cybersecurity
c = those in cloud computing
d = those in neither

the number who attended at least one session = (total − neither) = a – d.

Also attended at least one=b+c−(both)

So:
a−d=b+c−(both)
both=b+c−(a−d)
both=b+c−a+d

To get a fraction from total, we divide by a:

(both/a)=( b+c−a+d)/a

That matches choice E.

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We are given:
a - Total Participants
b - Cyber Security
c - Cloud Computing
d - Neither

We know,
a = b + c - (b n c) + d
Therefore, b n c = b + c + d - a
The fraction would be (b + c + d - a) / a

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C. (b+c+d−a)/a

We have to find out b ∩ c

We know b ∪ c = b + c - (b ∩ c) -- 1
and we know reading the question i.e. b ∪ c = a - d

Substituting in 1 we get
a - d = b + c - (b ∩ c)
(b ∩ c) = b + c + d - a

Final answer = participants who attended both sessions/total participants = (b + c + d - a)/a

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Total participants=a
participants attended session on cybersecurity=b, so not attended session on cybersecurity=a-b
participants attended session on cloud computing=c, so not attended session on cloud=a-c
participants attended neither of session=d
also, we can compute the participants attended cloud computing but not cybersecurity=a-b-d
Therefore, we have c-a+b+d participants who attended both the sessions.
thus, (b+c+d-a)/a fraction of participants attended both sessions.
Option C

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let x be participants attended both sessions
total participants = who attended cyber security + cloud computing - (attended both) +neither
a= b+c-x+d
x= b+c+d-a

so the fraction will be b+c+d-a/a
Ans:C

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To find the fraction of participants who attended both sessions, we use the formula for two overlapping sets:

Total = Cybersecurity + Cloud Computing − Both + Neither or a = b + c − x + d

where a is the total number of participants, b attended the cybersecurity session, c attended the cloud computing session, d attended neither, and x is the number who attended both. Rearranging the equation to solve for x, we get x = b + c + d − a. To find the required fraction, we divide the number who attended both by the total: x / a = (b + c + d − a) / a. Thus, the final answer is option C.

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General formula is : n(A) + n(B) - N(A intersection B) + n( neither A or B) = Total
n(A)= B , n(B)= C, N(A intersection B)=X, n( neither A or B) = D , Total = A
A=B+C+D-X
X= B+C+D-A
Fraction of the participants attended both sessions =(B+C+D-A)/A

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Total participants = a

CS = b
CC = c

CS and CC = x

b+c-x+d=a

x = b+c+d-a

fraction of participants who attended both = b+c+d-a/a

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Total= A+B-both+neither
a=b+c-both+d
both=b+c+d-a
fraction= (b+c+d-a)/a

Ans C

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