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

1 2 3 4

BongBideshini

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11 Jul 2025, 10:01

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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?

Expressing data in tabular form

	A	$\bar{A}$	Total
B			18
$\bar{B}$			18
	23	13	36

Statement 1: Five employees are not assigned to either project

	A	$\bar{A} $	Total
B	10		18
$\bar{B}$	13	5	18
	23	13	36

AB=10
So, Statement 1 is sufficient.

Statement 2: Exactly 21 employees are assigned to only one of the two projects.

That means we know that ${B}\bar{A}$ + ${A}\bar{B}$ = 21
Or, AB+ $\bar{A}\bar{B}$ = 36-21 = 15,
But, we need AB . So, statement-2 is not sufficient

IMO, Ans A

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Statement 1 provides us with the number of employees who are not assigned to either project, using this information we can determine P(Total) = P(Project Alpha) + P(Project Beta) - P(Both) + P(Neither) --> 36 = 23 + 18 - x + 5 --> x = 10. Thus Statement 1 is sufficient. Statement two provides us with the not enough information to answer the question. Answer A.

Regards,
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Total 36
Project Alpha = 23
Project Beta = 18
Question- How many present in both projects

Total = Alpha + Beta - Both + Neither
Statement 1:
(1) Five employees are not assigned to either project.
Neither =5
=> 36 = 23+18-Both+5 => Both = 10
Sufficient

Statement 2:
(2) Exactly 21 employees are assigned to only one of the two projects.
Alpha only + Beta only = 21
Alpha + Beta - 2*Both= Alpha only + Beta only
=> 41 - 2*Both = 21=>Both = 10
Sufficient

Hence answer is D

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11 Jul 2025, 10:35

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Let,
A = N of employees in Project Alpha
B = N of employees in Project Beta
b = N of employees in both
n = N of employees in neither
$A_0$ = N of employees in Project Alpha ONLY
$B_0$ = N of employees in Project Beta ONLY

We know in sets, Total = A + B - b +n
36 = 23+18-b+n
b-n=5

We need to find the value of b

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

This means, n=5

using our equation, b-n=5, we get, b=10

Sufficient.

Statement 2:
Exactly 21 employees are assigned to only one of the two projects.

This says that, $A_0 + B_0 = 21$

we know that, A + B = 23 + 18 = 41
Further, $A_0 + b = A$ and $B_0 + b = B$

Plugging them in the first equation, we get,

$A_0 + B_0 + 2b = 41$
using our result from statement 2, we know, $A_0 + B_0 = 21$

Thus, b = 10

Sufficient.

Answer D.

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	A	Not A	Total
B	?		18
Not B			18
Total	23	13	36

(1) Five employees are not assigned to either project - sufficient

	A	Not A	Total
B	10	8	18
Not B	13	5	18
Total	23	13	36

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

	A	Not A	Total
B	10	b	18
Not B	a=13	13-b	18
Total	23	13	36

a + b = 21 (eq 1)
a + 13 - b = 18
a - b = 5 (eq 2)

from eq 1 and 2

a=13. b=5
Hence, sufficient.

Answer is (D)

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11 Jul 2025, 11:19

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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.
We can find the number of people assigned to both projects

Suff

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

We can find the number of people assigned to both projects
Suff

Ans D

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Bunuel

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A company has a total of 36 employees.

23 are assigned to Project Alpha.

18 are assigned to Project Beta.

Let x be the number of employees who are assigned to both Alpha and Beta. X = ?

Statement 1:

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

Let’s assume n as the number of employees who are not assigned either of the project.

(23-x) + x + (18-x) + n = 36

41 - x + 5 = 36

x = 10

Hence, Sufficient

Statement 2:

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

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

41 - 2x = 21

20 = 2x

x = 10

Hence, Sufficient

Option D

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Project A= 23 Employees
Project B= 18 employees
x = number of employees assigned to both projects
n = number of employees assigned to neither project
Total employees=36

(A+B-x)+n=36
23+18-x+n=36
x-5=n

Statement (1): Five employees are not assigned to either project.
Given n= 5
x-5=5
x=10
Sufficient.

Statement (2): Exactly 21 employees are assigned to only one of the two projects.
Employees assigned to exactly one project
Given :
(A-x)+(B-x)=21
A+B-2x=21
23+18-2x=21
2x=20
x=10
Sufficient.

Hence, D) Each statement alone is sufficient.

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11 Jul 2025, 12:42

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Given data that,

total employees (36) = Some employees work on Project Alpha + Some employees work on Project Beta + Some employees work on both Project Alpha and Project Beta + Some employees work on neither project.

we have to fond how many employees work on both Project Alpha and Project Beta = ?

Total employees = 36
Employees on Project Alpha = 23
Employees on Project Beta = 18

So, the number of people on at least one project is = (Employees on Alpha) + (Employees on Beta) - (Employees on Both)
=> (23 + 18) - (Employees on Both) = 41 - (Employees on Both)

Now, let's check the given statements:

(1) Five employees are not assigned to either project.
This means 5 people are doing neither Project Alpha nor Project Beta.
If 5 people are doing neither, then the rest of the 36 people must be doing at least one project.
So, the number of employees on at least one project = Total employees - Employees on neither
= 36 - 5 = 31 people.

Now we can use our previous idea:
31 (people on at least one project) = 41 - (Employees on Both)
To find "Employees on Both":
Employees on Both = 41 - 31
Employees on Both = 10.
Since we found a clear number for "Employees on Both," Statement (1) is enough to answer the question.

(2) Exactly 21 employees are assigned to only one of the two projects.
"Only one project" means they are either in Project Alpha only OR Project Beta only.
So, (Employees on Alpha only) + (Employees on Beta only) = 21.

We also know that:
Employees on Alpha only = (Total Alpha employees) - (Employees on Both)
Employees on Beta only = (Total Beta employees) - (Employees on Both)

Let's put this together:
21 = (23 - Employees on Both) + (18 - Employees on Both)
21 = 23 + 18 - 2 * (Employees on Both)
21 = 41 - 2 * (Employees on Both)

Now, let's solve for "Employees on Both":
2 * (Employees on Both) = 41 - 21
2 * (Employees on Both) = 20
(Employees on Both) = 20 / 2
Employees on Both = 10.
Since we found a clear number for "Employees on Both," Statement (2) is also enough to answer the question.

Conclusion:
Since both Statement (1) alone and Statement (2) alone are enough to figure out how many employees are assigned to both projects, the answer is D.

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11 Jul 2025, 12:44

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Let us first organise the given information in a 2 by 2 grid.

	A	~A
B			18
~B			18
	23	13	36

We need to find AB.

Statement 1:
~A~B = 5

	A	~A
B	10	8	18
~B		5	18
	23	13	36

Hence SUFFICIENT

Statement 2:

	A	~A
B	18-x	x	18
~B	y		18
	23	13	36

x+y = 21
18-x + y = 23
y = 13
18 -x = 10
SUFFICIENT

Hence, the answer is D.

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11 Jul 2025, 12:57

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Solution:

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

	Project Alfa	No Project Alfa	Total
Project Beta	10	8	18
No Project Beta	13	5	18
	23	13	36

Statement 2: Exactly 21 employees are assigned to only one of the two projects. Insufficient

	Project Alfa	No Project Alfa	Total
Project Beta		Not possible	18
No Project Beta	Not possible		18
	23	13	36

Hence, Option A is the answer.

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11 Jul 2025, 13:36

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Option A is correct answer.

Statement I:

Total = alpha + beta - both + neither
36 = 23 + 18 - both + 5
both = 10

sufficient.

Statement II:
We do not know anything about , employees are not assigned to either project. Insufficient.

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11 Jul 2025, 13:48

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

(1) Five employees are not assigned to either project.
This gives Neither = 5, putting in the above equation will give us Both (Employees in both projects). Sufficient

(2) Exactly 21 employees are assigned to only one of the two projects.
Remaining employees = 36 - 21 = 15, these 15 include employees in both projects & employees in neither project. Hence Not Sufficient

Ans A

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A. (1) is sufficient by itself.

a=Project Alpha
b=Project Beta
c=Both
d=None

(1)
36-d=a+b-c
36-5=23+18-c
c=10
ENOUGH

(2)
a+b=21
36=a+b-c+d
36=21-c+d
15=d-c
NOT ENOUGH

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The correct answer choice is (D) both statements are sufficient on their own

Statement 1 - if 5 employees of the 36 are not assigned, then we know that the other 31 employees are assigned and using the numbers on each project = 23 + 18 = 41 and subtract the assigned employees 41 - 31 = 10 we can get a definitive number for those on 2 projects at once.

Statement 2 - if 21 employees are assigned to only one project of the 36, we know that the remaining 15 are on both projects.

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Bunuel

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1) Alone sufficient

We are looking at a basic overlapping question.
Therefore, we look at two overlapping quantities and one underlying set, which contains all the employees.
We have:
- Non workers
- Only Alpha
- Only Beta
- Alpha and Beta

Absolute A and B are given, as well as the total.

Knowing, that there are 5 employees not assigned to any project means, that we have to distribute the 31 Workers in a way, that we have 23 on A and 18 on B. That is only possible in one way. Therefore -> Sufficient.

2) Alone sufficient

With the information of 21 Employees at only one (A OR B), we can derive to the same result as at 1), as the same logic applies here.

Therefore, the answer is D)

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given n(a)=23 n(b)=18 and total no of employees=36
n(a int. b)=?
1. n(none)=5
n(a U b)=36-5=31
n(a int. b)=n(a)+n(b) - n(a U b)=10 SUFF
2. n(E1)=21
since n(a int. b)={n(a)+n(b)-n(E1)}/2=(41-21)/2=10 SUFF
hence D

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If B employees are assigned to both projects and N employees are assigned to neither of the projects, we have:
$36=23+18-B+N$,
$B=N+5$
The question asks for the value of B.

Statement 1:
This statement says that $N=5$, so we have enough information to calculate B, which will be 10.
Statement 1 is Sufficient.

Statement 2:
This statement tells us that $(23-B)+(18-B)=21$, which will give us $B=10$.
Statement 2 is Sufficient.

The answer is 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?

Statement 1
Five employees are not assigned to either project.

Employees who have been project =36-5=31
so 23+16-X=31, X=10
Statement 1 is sufficient

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

Suppose A be the employees doing 1 project and B doing 2 project
A intersection B

A+B-2*(A intersection B) =21
18+23-2X=21
2X=41-21=20
X=10

So both statements alone are sufficient. Answer is D

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Statement 1: 5 employees are not working on any of the project

Total employees = 36

Employees working on a project = 36-5 = 31

Project Alpha = 23
Project Beta = 18

Working on Alpha + Working on Beta - Working on both = 31

23 + 18 - x = 31

x = 10

sufficient

Statement 2: Exactly 21 are assigned to any one of the two projects

Working on Alpha = 23
Working on Beta = 18

23 + 18 = 41 (includes employees working on only one project plus double counted employees working on both)

41 - 21 = 20 = 2*employees working on both

employees working on both = 10

sufficient

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