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This is a problem on sets.

Total Participants = a
Cybersecurity = b
Cloud Computing = c
Neither sessions = d
Both sessions = x

cybersecurity alone = (b - x)
Cloud Computing alone = (c - x)

Now a = (b - x) + (c - x) + d + x
∴ a = b + c + d - x
∴ fraction of the participants attended both sessions = x/a = (b + c + d - a)/a

Correct option is Option C

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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 = Cyber security + Cloud Computing – Both + None
a = b + c – Both + d
Both = b+c+d-a

Answer = Both/Total Participants
= (b+c+d-a)/a (C)

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Let total participants = a
Let x = number who attended both sessions

a = b + c - x + d

x = b + c + d - a

So, the fraction who attended both=
x/ a = b + c + d - a/ a

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

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The total = attended + not attended
a = (b+c-(Attended Both))+d
Attend both = a-b-c-d
Fraction = (a-b-c-d)/a

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Total no. of people attending or not attending the sessions = a ----- (1)
Total no. of people not attending either of the sessions = d ----- (2)

Let no. of people who attended both sessions = z ----- (3)
No. of people attended cybersecurity session exclusively = b - z ----- (4)
No. of people attended cloud computing session exclusively = c - z ----- (5)

No. of people attended cybersecurity session exclusively + No. of people attended cloud computing session exclusively + no. of people who attended both sessions + no. of people not attending either of the sessions = Total no. of people attending or not attending the sessions --------- (6)

Putting respective values of (1), (2), (3), (4), (5) in (6)

b - z + c - z + z + d = a
b + c + d - z = a
b + c + d - a = z

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

Bunuel

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C.

The number of participants who attended at least 1 session is A-D. The number of participants who attended either sessions B or C, or both, is B+C-X. We will set A-D=B+C-X and solve for X to get the number of participants who attended both sessions, which is B+C+D-A. Since the question is looking for a fraction, the denominator will be the total number of participants, which is A.

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02 Jul 2025, 01:56

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Cyber security - b
Cloud Computing - c
neither - d
Total - a
Both - x ( need to find this to find the fraction)

Only cyber security - b-x
Only cloud computing - c- x

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

fraction - (b + c + d - a)
a

Answer C

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Bunuel

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For overlapping sets like these, the formula I use is "Total = A + B - Both + Neither", where "Both" is subtracted because it has been counted twice in both A and B.
It is easier to see it on a Venn diagram.

For this question, if we input the given information, we get "a = b + c - Both + d".
We can put "Both" on one side of the equation by:
1. subtracting b + c + d from both sides => "a - b - c - d = (- Both)"
2. multiplying both sides by (-1) => "(- a) + b + c + d = Both" or "b + c + d - a = Both"

The question is asking for the fraction of participants that attended both sections, which means we need to find "$\frac{Both}{a}$".
By substituting our result for "Both" from step 2, we can get "$\frac{Both}{a}=\frac{b+c+d-a}{a}$", which matches answer choice C.

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C. Total = Group1 + Group2 - Both + Neither

Total: a (total participants)
Group1: b (attended cybersecurity session)
Group2: c (attended cloud computing session)
Neither: d (attended neither session)
Both: 'x'.

Substitute these into the formula:
a = b + c - x + d

Solving for 'x'
x = b + c + d - a

Finding the Fraction

This is the number of people who attended both (x) divided by the total number of participants (a):
Fraction = x / a = (b + c +d - a) / a

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

Bunuel

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Total = a
Cyber = b
Cloud = c
Neither = d

Question: "what fraction of the participants attended both sessions?" -> Both/Total?

Cyber + Cloud - Both + Neither = Total
-> b + c - Both + d = a
-> Both : b + c + d - a
-> Both/Total = Both/a = (b + c + d - a) / a

Answer choice C.

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Bunuel

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The right answer is c we can also solve it by taking numbers

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Can solve the sets using the table:

	Cyber	n/Cyber
Cloud		a-b-d	c
n/Cloud		d	a-c
	b	a-b	a

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

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Using a Venn Diagram the total participants is b+c-x+d = a where x is the number of those that attended both
Rearrange the equation and x becomes b+c+d-a
To get the fraction you divide x by the total a which gives us choice C

Bunuel

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I drew a table to calculate the fraction of those attending both events:

	Cybersecurity	No cybersecurity	Total
Cloud Computing	x		c
not Cloud Computing		d
Total	b		a

b + c - x + d = a
or A + B - both + neither = Total

=> x = b + c + d - a
Fraction = (b + c + d - a)/a, where a is the total

Choose C

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Total (a) = At least (b) + At least (c) - Both + None (d)
--> Both = b + c + d - a
--> Fraction = Both/Total = (b + c + d - a)/a

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so we have

a = total people
b = cyberscurity session
c = cloud computing session
d = people who went to neither

i used that inclusion-exclusion thing. basically, people who went to atleast one session = total - people who went to neither = a - d
but also, people who went to atleast one = cybersecurity + cloud - both sessions
so: a - d = b + c - both

solving for both
both = b + c - (a - d)
both = b + c - a + d
so the frction is (b + c - a + d)/a
rearranging: (b + c + d - a)/a

thats option C.

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Answer - C

Total participants = a
Participants who attended Cybersecurity = b
Participants who attended cloud computing = c
Attended Neither = d

Total = attended cybersecurity + attended cloud computing - both + neither
a= b+c-both+d

Attended both = b+c+d-a
Attended both ratio = (b+c+d-a)/a

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To determine the fraction of participants who attended both the sessions, fill up the table as shown below (follow the numbers to determine the expression step wise)

Note that in either case , the numerator of the required fraction will simplify to the same expression i.e.
c - [(a-b) - d] or
b - [(a-c) - d]

And the denominator will be total participants -> a
Thus, we can quickly eliminate Option (E)

We can now observe in options (A), (B) and (D), the numerator expressions all have '+a' terms.
But from the expressions that we came up with, all the terms simplify to '-a' terms.

So we can eliminate options (A), (B) and (D) and select
Option (C) as the correct choice.

(Just to confirm, the expressions simplify to)
c - [(a-b) - d] or
b - [(a-c) - d]
----------------
= b - [ a-c - d]
= b - a + c + d
= b + c + d - a (same as option C numerator)

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To answer this question we can create a set matrix

a-ca-b

	Cyber	Not Cyber	Total
Cloud	b-a+c+d	a-b-d	c
Not Cloud	a-c-d	d
Total	b	a

The marked areas (yellow) are the areas we can determine based on the information provided which are the cells with no colour.
Now we can use the set matrix to determine the fraction of the participants attending both sessions to be [b-a+c+d][/a]. Thus, answer C.

Regards,
Lucas

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in a venn diagram,
total participant =a
participant for cybersecurity=b
participant cloud computing =c
neither=d
let who attended both be x
a=b-x+c-x+x+d
so,
x=b+c+d-a
x/a=b+c+d-a/a ---- ans (C)

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