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605-655 (Medium)|   Overlapping Sets|         
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12 Days of Christmas GMAT Competition 2025 - Day 6: At a professional 22 Dec 2025, 09:00
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84% (02:28) correct 16% (03:13) wrong based on 158 sessions

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At a professional conference, 308 participants signed up for at least one of three evening workshops: Data Analysis, Project Management, and Negotiation Training. Exactly six times as many participants attended only Data Analysis as attended only Negotiation Training. The number of participants who attended only Project Management was twelve times the number who attended both Data Analysis and Negotiation Training but not Project Management. The number who attended only Negotiation Training was one-quarter of the number who attended only Project Management. If 220 participants attended the Project Management workshop, how many participants attended exactly one of the three workshops?

A. 48
B. 88
C. 120
D. 132
E. 136

M46-18

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D
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12 Days of Christmas GMAT Competition 2025 - Day 6: At a professional 22 Dec 2025, 10:00
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Bunuel
At a professional conference, 308 participants signed up for at least one of three evening workshops: Data Analysis, Project Management, and Negotiation Training. Exactly six times as many participants attended only Data Analysis as attended only Negotiation Training. The number of participants who attended only Project Management was twelve times the number who attended both Data Analysis and Negotiation Training but not Project Management. The number who attended only Negotiation Training was one-quarter of the number who attended only Project Management. If 220 participants attended the Project Management workshop, how many participants attended exactly one of the three workshops?

A. 48
B. 88
C. 120
D. 132
E. 136
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GMAT Club Official Explanation:



Construct a Venn diagram based on the relationships in the problem:




Only NT = x
Only DA = 6x
Only DA and NT = y
Only PM = 12y

Given that only NT (x) is one-quarter of only PM:
Only PM = 12y = 4x

Total participants = Total PM + 6x + y + x
308 = 220 + 6x + y + x.

Since from 12y = 4x, we get y = x/3

Thus:
308 = 220 + 6x + x/3 + x.
x = 12

Therefore:
Only NT = x = 12
Only DA = 6x = 72
Only PM = 4x = 48.

Participants attending exactly one workshop:

72 + 48 + 12 = 132.

Answer: D.

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12 Days of Christmas GMAT Competition 2025 - Day 6: At a professional 22 Dec 2025, 09:14
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Let b be both NT and DA

PM only = 12b
NT only = 1/4 * 12b = 3b
DA only = 6 * 3b = 18b

PM = 220
Total = 308

Total = 18b + b + 3b +PM
22b + 220 = 308
22b = 88
b = 4

Exactly one = 18b + 12b +3b = 33b = 33*4 = 132

Answer D
Bunuel
At a professional conference, 308 participants signed up for at least one of three evening workshops: Data Analysis, Project Management, and Negotiation Training. Exactly six times as many participants attended only Data Analysis as attended only Negotiation Training. The number of participants who attended only Project Management was twelve times the number who attended both Data Analysis and Negotiation Training but not Project Management. The number who attended only Negotiation Training was one-quarter of the number who attended only Project Management. If 220 participants attended the Project Management workshop, how many participants attended exactly one of the three workshops?

A. 48
B. 88
C. 120
D. 132
E. 136

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12 Days of Christmas GMAT Competition 2025 - Day 6: At a professional 22 Dec 2025, 09:23
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Let number of those who attended both DA and NT but not PM = A
Then number of those who attended only PM = 12A
Only NT = 1/4 of only PM = 3A
Only DA = 6 times of only NT = 18A

Those who attended PM = 220
i.e., remaining only DA + only NT + both DA & NT = 308-220 = 88
18A + 3A + A = 88
22A = 88
A = 4

Only one workshop = only DA + only PM + only NT = 18A + 12A + 3A
=> 33A
=> 33 x 4
=> 132

Hence, answer Option D
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12 Days of Christmas GMAT Competition 2025 - Day 6: At a professional 22 Dec 2025, 09:24
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Bunuel
At a professional conference, 308 participants signed up for at least one of three evening workshops: Data Analysis, Project Management, and Negotiation Training. Exactly six times as many participants attended only Data Analysis as attended only Negotiation Training. The number of participants who attended only Project Management was twelve times the number who attended both Data Analysis and Negotiation Training but not Project Management. The number who attended only Negotiation Training was one-quarter of the number who attended only Project Management. If 220 participants attended the Project Management workshop, how many participants attended exactly one of the three workshops?

A. 48
B. 88
C. 120
D. 132
E. 136

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Let:
x = number who attended only Negotiation Training
Then only Data Analysis = 6x
Only Project Management = 4x (since Only Negotiation is one quarter of Only PM)
y = number who attended DA & Neg. only
Then only PM = 12y
So, 4x=12y; x=3y

People attending PM include: only PM (4x); Data+PM (a); PM+Neg. (b); All three (c)
4x+a+b+c=220 - (1)

Total Participants = 308
Sum of all disjoint regions: 6x+4x+x+y+a+b+c = 308 - (2)

Subtract (1) from (2)
(11x+y+a+b+c)-(4x+a+b+c)=308-220
7x+3y=88
Since, x=3y
7(3y)+y=88
y=4
x=12; Only PM: 6x=72; Only D: 4x=48; Only NT=12
Exactly One = 72+48+12 = 132 (OPTION D)

Note: This can more effectively be done using Venn Dig.
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12 Days of Christmas GMAT Competition 2025 - Day 6: At a professional 22 Dec 2025, 09:26
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A data analysis
B Project management
C Negotiation
A=6C
B=12DN
C(1/4)B
B=4C
4C=12DN
DN=(1/3)C
Total
attendees 308=A+B+C+DP+DN+PN+T
For project management 220=B+DP+PN+T
A+B+C=6C+4C=C=11C
letX=DP+PN+T
X=220-B=220-4C
308=(A+B+C)+DN+X=11C+1/3C+(220-4C)=220+22/3C
So C, 308-220=22/3C
22/3C=88, C=12
A+B+11C=132
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12 Days of Christmas GMAT Competition 2025 - Day 6: At a professional 22 Dec 2025, 09:36
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At a professional conference, 308 participants signed up for at least one of three evening workshops: Data Analysis, Project Management, and Negotiation Training. Exactly six times as many participants attended only Data Analysis as attended only Negotiation Training. The number of participants who attended only Project Management was twelve times the number who attended both Data Analysis and Negotiation Training but not Project Management. The number who attended only Negotiation Training was one-quarter of the number who attended only Project Management. If 220 participants attended the Project Management workshop, how many participants attended exactly one of the three workshops?

A. 48
B. 88
C. 120
D. 132
E. 136

Work shops
Data = a
PM= b
N= c
attended Data & N , but not PM = x
" Data & PM, but not N = y
" N & PM , but not data = z
all three =v
a+b+c+x+y+z+v= 308

given from info
a = 6c
b= 12 x
c=b/4
b+y+z+v=220
solve for values we get
a= 18x , c = 3x , b=12x
18x+3x+12x+(x+y+z)+v=308
34x+(y+z+v)= 308
PM = b+y+z+v=220
y+z+v=220-b

34x+220-b=308
34x-12x= 88
x=4
a= 72, b= 12, c= 48
a+b+c = 132

OPTION D ; 132
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12 Days of Christmas GMAT Competition 2025 - Day 6: At a professional 22 Dec 2025, 09:39
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Bunuel
At a professional conference, 308 participants signed up for at least one of three evening workshops: Data Analysis, Project Management, and Negotiation Training. Exactly six times as many participants attended only Data Analysis as attended only Negotiation Training. The number of participants who attended only Project Management was twelve times the number who attended both Data Analysis and Negotiation Training but not Project Management. The number who attended only Negotiation Training was one-quarter of the number who attended only Project Management. If 220 participants attended the Project Management workshop, how many participants attended exactly one of the three workshops?

A. 48
B. 88
C. 120
D. 132
E. 136

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Let
D P N - are worshops
according to given info let P be the paticipants who took only P worshop, DN be the particpants who took both D and N

DN = y
N = 3y
D = 18y
P = 12y

let NP = z
and DP = w

accordign to eqn
total P will be P + NP + DP
= 12y + z + w = 220

and total all peopl with aleast 1 worskop
18y + 12y + 3y + y + z + w = 34y + z +w = 308

using 1 eqn we get
22y = 88
y=4

now to calculate only 1 worshop - 33y = 132
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12 Days of Christmas GMAT Competition 2025 - Day 6: At a professional 22 Dec 2025, 09:51
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Participants who attended only D = x ; Only P = y ; Only N = z
Only D&P= u ; Only D&N=v ; Only P&N=w ; All D&P&N=t
Total participants = x+y+z+u+v+w+t=308
Also, it is given that,
x=6z ..............................(1)
y=12v or 4z=12v or v=(1/3)z ........(2)
z=(1/4)y or y=4z ..............................(3)
220 attended P = y+u+w+t=220 .........(4)
Total participants = 308 = x+y+z+u+v+w+t
6z+4z+z+u+(z/3)+w+t=308
(34/3)z +u+w+t =308 ............(5)

From (4) ...y+u+w+t=220 (y=4z)
u+w+t=220-4z

In (5) (34/3)z +220-4z =308
((34-12)/3)z =308-220
22z=88*3
z=12

x=6z=6*12=72
y=4z=4*12=48

Participants who attended exactly one of three workshop= x+y+z = 12+72+48=132

D
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12 Days of Christmas GMAT Competition 2025 - Day 6: At a professional 22 Dec 2025, 10:01
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Given total participants = 308
Taking number of people attended only Negotiation training to be 'x', no of people attending only data analysis will be '6x'
Taking number of people who attended both negotiation and data analysis training without project mangaement (PM) as 'a'
Then no. of people attending only PM will be 12a
Taking total people attending all three and group of data analysis and PM, Negotiation and PM as one group 'b'
Now total PM = 12a + b = 220
Total people = 34a + b = 308; Solving we get a = 4
Hence people attending exactly one of training = 33a = 132 Option D
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12 Days of Christmas GMAT Competition 2025 - Day 6: At a professional 22 Dec 2025, 10:03
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say, only data - a
only project - b
only negotiation - c
now there are zero people in neither of this activity.

a= 6* c = 6 * 2g= 18g
b = 12*g
c= 12g/4 = 3g

e= both in data and project
f= both in project and negotiation
g= both in negotiation and data
d= all three

we also know people in project- 220
b= p-d-e-f
12g= 220-d-e-f
d+e+f= 220-12g

a+b+c+d+e+f+g = 308
18g+12g+3g+220-12g+g = 308
g=4

a+b+c= 132

Bunuel
At a professional conference, 308 participants signed up for at least one of three evening workshops: Data Analysis, Project Management, and Negotiation Training. Exactly six times as many participants attended only Data Analysis as attended only Negotiation Training. The number of participants who attended only Project Management was twelve times the number who attended both Data Analysis and Negotiation Training but not Project Management. The number who attended only Negotiation Training was one-quarter of the number who attended only Project Management. If 220 participants attended the Project Management workshop, how many participants attended exactly one of the three workshops?

A. 48
B. 88
C. 120
D. 132
E. 136

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Total=308;
Let Only DA=6x, NT=x, PM=4x, DAandPM=a, DAandNT=x/3, PMandNT=b, DAandNTandPM=c
We have, 4x+a+b+c=220
6x+x/3+x+4x+a+b+c=308
34x/3+220-4x=308 [a+b+c=220-4x]
x=12
So, DA+NT+PM= (6*12) + 12 + (4*12)= 134
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12 Days of Christmas GMAT Competition 2025 - Day 6: At a professional 22 Dec 2025, 10:43
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Given that there were total 308 participants. Let us call the workshop names in short form.

Let x be only NT participants and y be only NT and DA participants but not PM.

We have only DA as 6x and only PM as 12y

Since x=1/4*12y =3y

We can also say that 220+6x+x+y= 308

220+7x+x/3 = 308. Solving for x gives 12

We are supposed to find 6x+x+12y or 7x+4x =11x = 132

Therefore, Option D
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12 Days of Christmas GMAT Competition 2025 - Day 6: At a professional 22 Dec 2025, 10:45
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Bunuel
At a professional conference, 308 participants signed up for at least one of three evening workshops: Data Analysis, Project Management, and Negotiation Training. Exactly six times as many participants attended only Data Analysis as attended only Negotiation Training. The number of participants who attended only Project Management was twelve times the number who attended both Data Analysis and Negotiation Training but not Project Management. The number who attended only Negotiation Training was one-quarter of the number who attended only Project Management. If 220 participants attended the Project Management workshop, how many participants attended exactly one of the three workshops?

A. 48
B. 88
C. 120
D. 132
E. 136

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Total = 308
All who attended Project Mgmt = 220
EQN -1 : 308-220 = Only Data Ana + only negotiation trg + (Both Data & Negotiation but not Project)....use venn diagram for better understanding.

Now Only Data = 6*Only Negotiation
Only negotiation = Only Project mgmt/4
Only Project = 12*(Data and negotiation but not project)
Substitute in these 3 in 1st equation:
88 = 6*Project only/4 +Project only/4 + Project only/12
Gives , Project only = 48

Using above relations :
Participants attending exactly 1 workshop => Data only + Negotiation only + Project only = (6/4 + 1/4 + 1)Project only
Substitue and we get = 132.
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12 Days of Christmas GMAT Competition 2025 - Day 6: At a professional 22 Dec 2025, 10:47
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Let us assume a Venn Diagram with Data Analysis (D), Project Management (P) and Negotiations Training (N).
x: Only N
y: Only D and N (but not P)
z: Only P
a: Only D
b: D and P (but not N)
c: N and P (but not D)
d: All three (D, P, and N)

Based on the problem description, we can establish the following relationships:
Exactly six times as many attended only D as attended only N: a = 6x
Only P was twelve times the number who attended both D and N but not P: z = 12y
Only N was one-quarter of the number who attended only P:x = (1/4)z (Substituting z = 12y into this: x = (1/4)(12y) = 3y)

From these, we can express a, x, and z all in terms of y: x = 3y
z = 12y
a = 6x = 6(3y) = 18y

Using the Group Totals We know two main totals:
Total Participants: a + x + z + y + b + c + d = 308
Total for Project Management (P): z + b + c + d = 220
We can subtract the P total from the overall total to find the participants who are not in the Project Management circle (a + x + y): (a + x + z + y + b + c + d) - (z + b + c + d) = 308 - 220
a + x + y = 88

Solving for y and replacing a, b and c we found above:
18y + 12 y + y = 88
y = 4.

Calculating values of a, x and z from above:
a = 18y = 18(4) = 72
x= 3 y = 3 (4) = 12
z = 12 y = 12 (4)

a + x + z = 132
Ans D.

Can someone help me with a shorter way? I feel mine is really long.
Bunuel
At a professional conference, 308 participants signed up for at least one of three evening workshops: Data Analysis, Project Management, and Negotiation Training. Exactly six times as many participants attended only Data Analysis as attended only Negotiation Training. The number of participants who attended only Project Management was twelve times the number who attended both Data Analysis and Negotiation Training but not Project Management. The number who attended only Negotiation Training was one-quarter of the number who attended only Project Management. If 220 participants attended the Project Management workshop, how many participants attended exactly one of the three workshops?

A. 48
B. 88
C. 120
D. 132
E. 136

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12 Days of Christmas GMAT Competition 2025 - Day 6: At a professional 22 Dec 2025, 10:49
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In this types of problems, I find it helpful to visualise the data with a Venn diagram.
onlyD = 6 * onlyN
onlyP = 12 * DandN -> DandN = onlyN / 3
onlyN = onlyP / 4
P = 220
Total = 308

Drawing the Venn diagram we can see that:
onlyD + onlyN + DandN + P = Total
6*onlyN + onlyN + onlyN/3 + 220 = 308
solve this to find onlyN = 12

onlyD = 6*12 = 72
DandN = 12/3 = 4
onlyP = 12 * 4 = 48

People who attended EXACTLY one of the workshops:
onlyD + onlyP + onlyN = 72+48+12 = 132

Answer D
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12 Days of Christmas GMAT Competition 2025 - Day 6: At a professional 22 Dec 2025, 10:50
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The number of participants who attended only data analysis (DA) = 6 * The number of participants who attended only negotiation training (NT) (x) = 6x

The number who attended only Project management (PM) training = 12* The number who attended DA & NT but not PM (y) = 12

The number who attended only Negotiation Training = x = 12y/4 = 3y

Let y + z = The number who attended 2 or 3 trainings

12y + z = 220

18y + y + 3y + 12 + z = 308

22y = 308 - 220 = 88

y = 4

The number who attended exactly one training = 18y + 3y + 12y = 33y = 132

IMO D
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12 Days of Christmas GMAT Competition 2025 - Day 6: At a professional 22 Dec 2025, 10:57
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This is 3 circle ven diagram problem:

Assume a, b and c as exactly one out of the DE, PM and NT respectively,
d being both DA and PM and not NT
e being both DA and NT and not PM
f being both PM and NT and not DA
and g being all three workshops:

Now given:
a = 6c
b = 12e
c = b/4
b + d + f + g = 220
a + b +c + d +e + f + g = 308
we need a + b + c = ?

Since:
a + b + c + d + e + f + g = 308

So, a + e +c = 88

putting given info in above equation will give:
c = 12
Hence a = 72
b = 48

So total is 132
Hence D
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12 Days of Christmas GMAT Competition 2025 - Day 6: At a professional 22 Dec 2025, 10:58
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Bunuel
At a professional conference, 308 participants signed up for at least one of three evening workshops: Data Analysis, Project Management, and Negotiation Training. Exactly six times as many participants attended only Data Analysis as attended only Negotiation Training. The number of participants who attended only Project Management was twelve times the number who attended both Data Analysis and Negotiation Training but not Project Management. The number who attended only Negotiation Training was one-quarter of the number who attended only Project Management. If 220 participants attended the Project Management workshop, how many participants attended exactly one of the three workshops?

A. 48
B. 88
C. 120
D. 132
E. 136

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308 participants attended one of the three events. So, neither group is ZERO.

This is a three set Venn diagram, Data Analysis , Project Management and Negotiation Training.

Let a denote Data Analysis only,

Let b denote Project Management only,

Let c denote Negotiation training only.

Let d denote Data Analysis and Project Management.

Let e denote Data Analysis and Negotiation Training.

Let f denote Project Management and Negotiation Training.

Let g denote all the three groups.

Given that: Exactly six times as many participants attended only Data Analysis as attended only negotiation training.

a = 6* c

The number of participants who attended project management only is 12 times the number of participants who attended data analysis and negotiation training, but not project management.

b = 12*e

The number of participants who attended Negotiations Training only is one quarter the number of participants who attended Project Management only.

c = (1/4)*b

the ratio of :

a:c = 6:1

c:b = 1:4

Therefore, a: b: c = 6 : 4 : 1

b : e = 12 : 1

Then, the above equation becomes : a:b:c:e = 18:12:3:1

We need to find EXACTLY ONE = a+b+c = 18x + 12x + 3x = 33x

So, answer should be a multiple of 33.

And, only option which would fit is Option D = 132
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12 Days of Christmas GMAT Competition 2025 - Day 6: At a professional 22 Dec 2025, 10:59
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Only DA = 6x if only NT = x
Also, only PM = 12a if DA intersection NT = a

And only NT = 1/4 (only PM) = 1/4 (12a) = 3a
Thus, only DA = 6 (3a) = 18a

Now, 220+18a+a+3a = 308
22a = 88
a= 4

Only DA + Only NT + Only PM = 18a + 3a + 12a = 33a = 33*4 = 132 ...Option D
Bunuel
At a professional conference, 308 participants signed up for at least one of three evening workshops: Data Analysis, Project Management, and Negotiation Training. Exactly six times as many participants attended only Data Analysis as attended only Negotiation Training. The number of participants who attended only Project Management was twelve times the number who attended both Data Analysis and Negotiation Training but not Project Management. The number who attended only Negotiation Training was one-quarter of the number who attended only Project Management. If 220 participants attended the Project Management workshop, how many participants attended exactly one of the three workshops?

A. 48
B. 88
C. 120
D. 132
E. 136

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