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GMAT Club Olympics 2025 (Day 10): At a company with 36 employees, 23

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Rahilgaur

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12 Jul 2025, 04:42

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Bunuel

Total employees = 36

Project Alpha + Project Beta – Both + Neither = Total
23+18-x+y=36
x-y = 5 .... x=? , y=?

1. Five employees are not assigned to either project = y=5
X=10. -Sufficient

2. Exactly 21 employees are assigned to only one of the two projects
21= Project Alpha + Project Beta – 2*Both
21= 23+18-2*x
2x=20
X=10 Sufficient.

Both sufficient alone – D is the answer.

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12 Jul 2025, 05:21

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Company with 36 employees, 23 are assigned to Project Alpha, and 18 are assigned to Project Beta

Let x be the number of employees assigned to both Alpha and Beta
=> Total number of employees assigned to atleast one of the projects = 23 + 18 - x = 41 -x

Given statements,
(1) Five employees are not assigned to either project.

We know that total number of employees = 36
=> Number of employees assigned to atleast one of the project = 36 - 5 = 31

We also know that, employees assigned to atleast one of the project = 41 -x
=> 41 - x = 31
=> x = 10

Statement (1) is sufficient

(2) Exactly 21 employees are assigned to only one of the two projects.

Given the above statement, lets say that,

Employees assigned to only Alpha = 23−x
Employees assigned to only Beta = 18−x

=> 23 - x + 18 - x = 21
=> x = 10

Statement(2) is sufficient

=> D. Each statement alone is sufficient.

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12 Jul 2025, 05:39

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Total= 36= 23+18-both+none
S1 Gives us the missing link meaning 36= 23+18-both+5= both 10 thus suffcient
S2 only gives tells us 23-21=2 members of project Alpha are also in Beta but no information about how many of those in Beta are also in Alpha thus insufficient
ANS A

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12 Jul 2025, 05:47

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Bunuel

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Option D is my answer

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12 Jul 2025, 05:57

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Total = A + B – Both + Neither
36 = 23 + 18 – Both + Neither
Both - Neither = 41-36 = 5

(1)
Neither=5
Both - Neither = Both - 5 = 5
Both = 10

Statement (1) alone is sufficient.

(2)
(A - Both) + (B - Both) = 21
A + B - 2*Both = 21
2*Both = 23+18-21 = 20
Both = 10

Statement (2) alone is sufficient.

Answer is D

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12 Jul 2025, 05:59

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Given :
Total emp. = 36
No. of emp. in Alpha = 23
No. of emp in Beta = 18
We have to find no. of emp. assigned to both projects. Let that no. be x

(1) Five employees are not assigned to either project.
Emp. in at least one project. -> 36 − 5 = 31
23 + 18 counts the people in both twice, so
23 + 18 − x = 31
41 − x = 31
x = 10
Sufficient

(2) Exactly 21 employees are assigned to only one of the two projects.
23 + 18 counts the people in both twice, so
-> 23 + 18 − 2x = 21
-> 41 − 2x = 21
-> 2x = 20
-> x = 10
Sufficient

Ans. = D

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12 Jul 2025, 06:00

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Option D is the correct answer.

Lets understand the question before answer it.

So the question starts by telling us that "At a company with 36 employees, 23 are assigned to Project Alpha, and 18 are assigned to Project Beta". Then it asks us to find many employees are assigned to both projects.

Lets assume the number of employees assigned only to Project Alpha to be 'x', employees assigned only to Project Beta to be 'y' and employees assigned to both Project Alpha & Beta to be 'z'. Be can also see this in the Venn Diagram below.

Now from the question be get that "23 are assigned to Project Alpha" i.e x+z = 23, it also tells us that "18 are assigned to Project Beta" i.e. y+z = 18.

Lets go through the given statements to see whether we can find our answer with their help or not.

Statement 1: "Five employees are not assigned to either project". Now this statement tells us that exactly 5 employees are not part of any of the project. Now lets use overlapping set formula here.
=> Total = A + B - Both + Neither
=> 36 = 23 + 18 - z + 5
=> 36 = 46 - z
=> z = 10
This statement gives us the unique value to the answer that's Sufficient to answer the question.

Statement 2: "Exactly 21 employees are assigned to only one of the two projects". This statement tells us that their are a total of 21 employees who are assigned to only one of the projects.
So, Employees assigned to exactly one project = (Alpha - Both) + (Beta - Both)
=> 21 = (23 - z) + (18 - z)
=> 21 = 41 - 2z
=> 2z = 20
=> z = 10
This statement also gives us the unique value to the answer that's Sufficient to answer the question.

So, as both the statements alone are Sufficient to answer the question that's why Option D is our answer.

Bunuel

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12 Jul 2025, 06:01

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At a company with 36 employees, 23 are assigned to Project Alpha, and 18 are assigned to Project Beta. How many employees are assigned to both projects?

(1) Five employees are not assigned to either project.
(2) Exactly 21 employees are assigned to only one of the two projects.

Let x= employees in both project
b= neither
Project Alpha only = 23-x
Project Beta= 18-x

36 = 23-x+18-x+x+b

STatement 1
b= 5

36= 23-x+18-x+x+5
36=41+5-x
x= 10

Statement 1 is sufficient

Statement 2

23-x+18-x= 21

41-2x=21
x=10

statement 2 is sufficient

Both statements alone are sufficient. The answer is D

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12 Jul 2025, 06:07

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Total Employees (T) = 36

Employees assigned to Project Alpha (A) = 23
Emp. assigned to Project Beta (B) = 18

Q: Employees assigned to both the projects (A and B)?

Let N = Employees assigned to neither projects

T - N = A + B - AB

36 - N = 23+18 - AB

AB - N = 5

ST1: N = 5

AB = 10. Sufficient.

ST2: A or B (AuB) = 21

A or B = A + B - 2AB
21 = 23+18 - 2AB
2AB = 20
AB = 10. Sufficient

Option D

Bunuel

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12 Jul 2025, 07:12

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Group 1 + Group 2 – Both + Neither = Total
23 + 18 – Both + Neither = 36
Both - Neither = 5

(1)
Neither = 5
Both = 5 + 5 = 10

Sufficient

(2)
only Group 1 + only Group 2 + Both + Neither = Total
21 + Both + Neither = 36
Both + Neither = 15

Using also:
Both - Neither = 5

Solving:
Both = 10

Sufficient

Correct answer is D

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12 Jul 2025, 07:13

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Let Both Projects = "b" people

Project Alpha exclusive = "a" people
Project Alpha= a + b = 23 -----------(1)

Project beta exclusive = "c" people
Project Beta = b + c = 18 -----------(2)

So, Total (36) = Alpha exclusive + Beta exclusive + Both + neither
Or, Total = a + c + b + neither

Statement 1 : neither = 5
36 = a + c + b + 5
We also know a + b = 23 from (1)
So, 36 = 23 + c + 5
c = 8 people

Substituting the value of c on (2) : b + 8 = 18
b = 10 --------------(Statement 1 : Sufficient)

Statement 2 : Exactly 21 people are assigned to only one of the two projects i.e., a + c = 21 but we have no information on people who are on neither of the projects.
So, (Statement 1 : Not Sufficient)

Option A

Bunuel

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12 Jul 2025, 07:27

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Using set theory (AB is both):
T = A + B - AB + N
36 = 23 + 18 - AB + N
AB - N = 5

AB?

(1)
N=5
AB = 5+N = 10

Statement is sufficient

(2)
(23 - AB) + (18 - AB) = 21
41 - 2*AB = 21
2*AB = 20
AB = 10

Statement is sufficient

The right answer is D

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12 Jul 2025, 07:32

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E=36, A=23,B=18 , both=?
(1) Five employees are not assigned to either project.
23+18+5-both=36
46-both=36
both=10
Sufficient
(2) Exactly 21 employees are assigned to only one of the two projects.
21=23+18-2(both)
21=41-2(both)
-20=-2(both)
both=10
Sufficient
IMO:D

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12 Jul 2025, 07:34

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Total = Alpha + Beta - Both + Neither
36 = 23 + 18 - Both + Neither
Both - Neither = 5

Both?

(1)
Neither = 5
Both - 5 = 5
Both = 10

SUFFICIENT

(2)
Alpha + Beta - Both = onlyAlpha + onlyBeta + Both
23 + 18 - Both = 21 + Both
2*Both = 20
Both = 10

SUFFICIENT

IMO D

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12 Jul 2025, 07:44

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Given -> 36 employees, 23 are assigned to Project Alpha, and 18 are assigned to Project Beta.
To determine -> Employees assigned to both?

Statement 1 -> Five employees are not assigned to either project.
Alpha+Beta-Both+Neither = 36
23+18-Both+5=36
Solvable for Both. Sufficient.

Statement 2 -> Exactly 21 employees are assigned to only one of the two projects.
Only Alpha + Only Beta = 21
(23-Both)+(18-Both) = 21
Solvable for Both. Sufficient

Answer - D

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12 Jul 2025, 07:59

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Given,
Total = 36 employees,
Project Alpha = 23
Project Beta = 18

Assume both = x

Statement (1): Five employees are not assigned to either project.

Total = Group1 + Group2 - Both + Neither
36 = 23 + 18 - x +5
x = 10

Statement 1 is sufficient.

(2) Exactly 21 employees are assigned to only one of the two projects.

Only Alpha = 23 - x

Only Beta = 18−x

Only one 21 = (23 - x) + (18−x) = 41−2x
21 = 41−2x
x = 10
Statement 2 is also sufficient.
Each statement alone is enough to find the answer.