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01 Jul 2025, 08:52

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a total participants
b attended a session on cybersecurity,
c attended a session on cloud computing.
If exactly d participants attended neither session

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

answer is c

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01 Jul 2025, 08:56

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Use table method.

Total-= a
Total who participated in Cyber one= b
those who didnt= a-b

total who participated in cloud one- c
those who didnt- a-c

THose who did in neither = d

Thus those who did in cloud one but not cyber = a-b-d

Those who did in both = C - (a-b-d)
= c-a+b+d

fraction of total participants
= (b+c+d-a)/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?
Total participants= a
People who attended Cybersecurity Session= b
People who attended session on cloud computing= c
People who attended neither session = d
People who attended Cloud computing session but not Cybersecurity = a-b-d
People who attended both session= c-(a-b-d)= c-a+b+d
Fraction= (c-a+b+d)/a

C

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01 Jul 2025, 09:05

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Clearly,

a = b + c - Both + Neither
So, Both = b + c + d - a

Both as a fraction of total = \frac{ (b + c +d - a)}{a }

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If total number of participants are equal to a & we have b & c members attending sessions on cybersecurity & cloud computing respectively & if we assume there are x members attending both & there are d members attending neither, we can break down the problem as:

b-x (Only attend cyber sec)+x (Attend both) + c-x (Only attend cloud computing)+d(Attend neither)=a (Total members)

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

The fraction of the total participants will hence be equal to x/a = (b+c+d-a)/a. Hence the answer to this question is option (C) (b+c+d-a)/a

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

(b-x)= Attended Cybersecurity only
(c-x)= Attended cloud computing only

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

(b+c+d-a)/a is the fraction of people that attended both divided by the total.

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Point C states the summation of other candidates attending and non attending from the total on both sessions The remaining students attended both the sessions. Hence, Point C is the answer.

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

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

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We are given:
- Total participants: a
- Attended cybersecurity: b
- Attended cloud computing: c
- Attended neither: d
We need to find the fraction that attended both sessions.
Let x be the number who attended both.
Using the principle of inclusion-exclusion:
Total who attended at least one session = b + c - x
But the total who attended at least one session is also a - d.
So:
b + c - x = a - d
Solving for x:
x = b + c - (a - d)
x = b + c - a + d
The fraction is x / a = (b + c - a + d) / a
Hence answer is (b + c - a + d) / a

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01 Jul 2025, 09:12

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I ultimately had to resolve to option solving as I could not find a clear pathway through basic algebra. Realizing that in the end, the intersection U should be the only component left, the numerator (b+c+d−a) can be broken down as:
[b'+U+c'+U+d-(b'+U+c'+U+d)] where:
b' is only b, and c' is only c. The fraction is then simply to be divided by participants attended both sessions? so denominator becomes a-d.

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01 Jul 2025, 09:13

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We are given:
Total participants: a
Attended cybersecurity: b
Attended cloud computing: c
Attended neither session: d

Those who attended at least one session = total participants − those who attended neither = a-d

By set theory,
∣B∪C∣=∣B∣+∣C∣−∣B∩C∣ wherein we need to find ∣B∩C∣ = x

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

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

Answer 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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As a total we have a.
This total equals b+c+d+ All people, that do both.
Therefore, we are looking for a-b-c-d.

As we are looking for the fraction, we have to divide by the total number, that is a.

The result is therefore: (a-b-c-d)/a

Answer: B

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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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Plotting the information over a grid as shown below -

Overlap = c - a + b + d

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

IMO Option C

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01 Jul 2025, 09:18

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We are told:

Total participants = a

b attended cybersecurity

c attended cloud computing

d attended neither session

We are to find the fraction of participants who attended both sessions.

Step 1: Let’s use the principle of set inclusion:
Let:
Total number of people = a
Number who attended cybersecurity = b
Number who attended cloud computing = c
Number who attended neither = d

Let x be the number who attended both sessions

Then:

People who attended at least one session =
= Cyber only + Cloud only + Both
= b+c−x (since both are double-counted in b and c)

So:
People who attended at least one session=a−d

So set up the equation:
b+c−x=a−d

Solve for x:
x=b+c−(a−d)=b+c+d−a
So the number who attended both = x=b+c+d−a

We’re asked for the fraction of total participants who attended both:

x/a = (b+c+d−a)/a

Final Answer:C.
(b+c+d−a)/a

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my 2 cents-
The solution is really straightforward when solved using the matrix method

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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01 Jul 2025, 09:22

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x be the number of participants who attended both sessions.

Total who attended at least one session = a-d

Number who attended at least one session = b+c-x

Equating both: a-d = b+c-x

x= b+c+d-a

Fraction of total audience = x/a = b+c+d-a/a

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01 Jul 2025, 09:24

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

If x is the intersection of b and c, you can write:

a = b + c - x + d

The reason for this, is that b + c would count x two times.

Solve for x:

x = b + c + d - a

Divide by a to get the fraction:

x/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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01 Jul 2025, 09:26

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Total participants = a
Adding the participants in both seminars = b+c
Non participating =d
Now in (b+c), the participants who participated in b and c have overlapped
Hence, participants who participated in both = b+c+d-a
Fraction = (b+c+d-a)/a

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IMO should be C

I just took a to be 20, b and c each to be 6 and d to be 10. Only C satisfies the solution for the intersection between b and 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

Total = a
Not participating = d
Participating = a-d

Cyber Security = b
Cloud Computing = c
Both = x (Let's say)

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

Fraction : (b+c+d-a) / a

1 2 3 4 5 6 7 8 9 10 11

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