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

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Bunuel

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GMAT Club Official Explanation:

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?

Total = A + B - Both + Neither
36 = 23 + 18 - Both + Neither

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

This statement implies Neither = 5, so:

36 = 23 + 18 - Both + 5
Both = 10

Sufficient.

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

This statement implies:

(A - Both) + (B - Both) = 21

(23 - Both) + (18 - Both) = 21

Both = 10.

Sufficient.

Answer: D.

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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?
solve using 2x2 matirx

------A----NA-----total
B------ ---- ------18
NB---- ---- ---------18
total--23---- 13----36

#1
Five employees are not assigned to either project.

------A----NA-----total
B------ 10 ---- 8 ------18
NB---- 13 ---- 5---------18
total--23---- 13----36

both projects 10 ; sufficient

#2

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

------A----NA-----total
B------ 10 ---- 8 ------18
NB---- 13 ---- 5---------18
total--23---- 13----36
both projects 10 ; sufficient

IMO option D is correct

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There are 36 employees in total, with 23 assigned to Project Alpha and 18 to Project Beta.
Total= A + B - Both + None >>>>this relationship means that the number of employees assigned to neither project is always five less than the number assigned to both projects. Statement (1) tells us that 5 employees are assigned to neither project, which means the number assigned to both projects must be 10. Statement (2) tells us that 21 employees are assigned to only one project, which when combined with the totals for Alpha and Beta also leads to the conclusion that 10 employees are assigned to both projects. Because both statements independently allow us to determine the number of employees in both projects, each statement alone is sufficient to answer the question.

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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?

	Alpha	~Alpha	Total
Beta	x = ?		18
~Beta			36-18=18
Total	23	36-23 = 13	36

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

	Alpha	~Alpha	Total
Beta	x = 23-13=18-8 = 10	13-5=8	18
~Beta	18-5=13	5	18
Total	23	13	36

10 employees are assigned to both projects

SUFFICIENT

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

	Alpha	~Alpha	Total
Beta	x = ?	y	18
~Beta	21-y	y-3	18
Total	23	13=2y-3	36

2y-3 = 13; y = 8

	Alpha	~Alpha	Total
Beta	x = 10	8	18
~Beta	13	5	18
Total	23	13	36

10 employees are assigned to both projects

SUFFICIENT

IMO D

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11 Jul 2025, 08:46

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Using our variables:
- A = employees assigned to Alpha only
-B = employees assigned to Beta only
- AB = employees assigned to both projects
- N= employees assigned to neither project
We know:
A + B+ AB+ N=36
A + AB = 23
B+ AB = 18
Statement (1): Five employees are not assigned to either project.
This tells us N= 5.
So: A + B+ AB =36-5=31
Now we have:
A + B+ AB=31
A + AB = 23
B+ AB = 18
Subtracting the second equation from the first:
B = 31-23 = 8
And now we can find AB:
B+ AB = 18
8+ AB= 18
AB = 10
Statement (1) is sufficient.
Statement (2): Exactly 21 employees are assigned to only one of the two projects.
This means A + B = 21.
From our original equations:
A+ AB = 23
B+ AB = 18
A+ B=21
Adding the first two equations:
A + AB+B+ AB= 23 + 18
A+B+ 2AB =41
Substituting A + B= 21:
21 + 2AB= 41
2AB = 20
AB = 10
Statement (2) is also sufficient.
Therefore, the answer is D: Each statement alone is sufficient.

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11 Jul 2025, 08:49

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Bunuel

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we have 36 employees 5 are not assigned to both ,therefore 23/31 are assigned to alpha, 18/31 are assigned beta therefore 23+18=41 are assigned to either but 21 are assigned to exactly one, implies 13 not assigned to beta for both or 8 not assigned to alpha ie 13+8=21. this proves that those not assigned are assigned to only one .therefore we subtract 21 from 41 implies 20 are assigned to both

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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?
..........B............NB............T
A....... X .........—-.......... 23
NA....——........—............13
T.......18...........18...............36
X=?

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

..........B............NB............T
A....... X .........—-.......... 23
NA....——........(5)—............13
T.......18...........18...............36
X=23-13=10
Sufficient

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

..........B............NB............T
A....... X .........(23-x).......... 23
NA....(18-x)........—............13
T.......18...........18...............36

(23-x)+(18-x)=21
23+18-21=2x
2x=23+18-21=20
x=10
Sufficient

D

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Total = 36
A = 23
B = 18
Both = ?

Statement 1 :
None = 5
Total = A+B - Both + None
Since we know all other values , we can calculate for Both.
Thus, SUFFICIENT

Statement 2 :
A+B-Both = 21
Since we know all other values , we can calculate for Both.
Thus, SUFFICIENT

Answer is D.

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From first option,five employees are not assigned to any project
36=23+18-(people in both projects)+people not assigned any project
36=41-(people in both projects)+5
36=46-(people in both projects)
(people in both projects)=0

Hence first option is sufficient

from second option, 21 employees with only one of the two projects where x is people in both projects
41−2x=21
x=10

Hence second option is sufficent

Hence answer is D.

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Ans: D

At a company with 36 employees (T), 23 are assigned to Project Alpha (A) , and 18 are assigned to Project Beta (B). How many employees are assigned to both projects (C)?

(1) Five employees are not assigned to either project.
N = 5
We know from Venn diagram rules
Total = A + B - C + N
36 = 23 + 18 - C + 5
C = 0

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

A + B - C = 21
23 + 18 - C = 21
C = 20

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Given :
Total Employee = 36
Employees in Project Alpha = 23
Employees in Project Beta = 18

Not Given : Employees not involved in any of the projects

Asked : Number of employees assigned to both the projects.

Statement 1

Five employees are not assigned to either project.

=> Number of employees who are not part of any project =5
=> Number of employees distributed to both the projects is 36 - 5 = 31
=> No of employees who are part of both the projects = 23 + 18 - 31 = 10.

Hence, Statement 1 is sufficient.
Eliminate option B, C & E.

Statement 2

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

=> 21 employees are involved in one of the two projects only.
Therefore, number of employees in both the projects
=> Employees (Project Alpha + Project Beta) = Employees (in any one project) + 2 * Employees (in both projects)
=> 23 + 18 = 21 + 2 * Employees (in both projects)
=> 41 = 21 + 2 * Employees (in both projects)
=> Employees (in both projects) = (41 - 21) / 2 =10

Hence, Statement 2 is sufficient.

Therefore, since statements 1 & 2 are both sufficient,
Option D is the answer

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Bunuel

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Given: Number of employees assigned to Project Alpha = 23 and project Beta = 18 and total = 36

asked: Value of Both(employees in both the projects)?

Statement 1: Using Total = A + B - Both + neither, we can figure out the value of Both since we are given neither = 5. So sufficient

Statement 2: Using A + B - 2*Both = A only + B only, we can figure out the value of Both since we are given A only + B only = 21. So sufficient
Hence the answer is Option D.

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Statement 1: Sufficient.
Five employees are not assigned to either project,
Employees assigned to atleast one project= 36-5=31.
Number of employees on both projects: 31=23+18-x = 10.
So sufficient.

Statement 2: Sufficient.
Employees in alpha only: 23-x
Employees in beta only:18-x
So (23-x)+(18-x)=21
x= 10
So sufficient.

Answer: Option (D) . Each statement alone is sufficient,

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11 Jul 2025, 09:21

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36=23+18-both+neither

1. neither=5..SUFFICIENT
2. 23-Both+18-both=21...SUFFICIENT

Ans D

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If 5 people aren’t on either project, then the remaining 31 must belong to Alpha, Beta, or both. Since Alpha and Beta together account for 41 project‐slots (23+18), the only way to fill exactly 31 slots among employees is for ten of those slots to be double‐counted. meaning 10 people must be on both projects. That gives a unique answer from statement (1) alone.

Similarly, if exactly 21 employees work on just one of the two projects, then the remaining slots (forty‐one total minus those twenty‐one single‐project assignments) must come from people counted twice. That again forces ten people to hold two assignments. So statement (2) by itself also fixes the overlap.

Because each statement independently yields a single, consistent count of employees on both projects, the correct choice is that each one alone is sufficient, hence D!

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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. =>
So We know XUY = 36
X= 23
Y= 18
SO XUY = X + Y - Both
36 = 23+18 - Both
so Both = 10, hence Sufficient

(2) Exactly 21 employees are assigned to only one of the two projects. =>
X only + Y only = 21

X + Y - 2* Both = 21
23 + 18 -2*Both = 21
20 = 2 * Both
So Both = 10
Hence Sufficient

Hence Ans D

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Out of 36 employees, 5 have not been assigned to any project, whereas 31 has been assigned. Out of 31, 21 is there for one project, we can conclude of employees with both assignments.

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T= A+B-b+n, where b = both and n=none/no assignment
36=23+18-b+n
b-n=5

b=?

S1
n=5
b-5=5
b=10
Sufficient

S2
23-b+18-b=21
20=2b
b=10
Sufficient

Answer D

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11 Jul 2025, 09:40

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A = Number of people in Project Alpha = 23
B = Number of people in Project Beta = 18
x = Number of people in both projects
N = Number of people in neither project

Total company size is 36. Using the "addition principle":
Number in at least one project + N = 36
And "in at least one" = |A ∪ B| = |A| + |B| – |A∩B| = 23 + 18 – x = 41 – x

Translation:

All 36 = 41 - X + N

Condition One:
By providing N,
we can calculate
36 = 41 - X + 5
X = 10
The condition is sufficient.

Condition Two:
A - X + B - X = 21
41 - 2X = 21
X = 10
This condition is also sufficient.