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12 Jan 2026, 10:22
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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
A. 48
B. 88
C. 120
D. 132
E. 136
12 Jan 2026, 10:22
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Official Solution:
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
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 \(y = \frac{x}{3}\). Thus:
\(308 = 220 + 6x + \frac{x}{3} + x\).
\(x = 12\)
Therefore:
Only NT \(= x = 12\)
Only DA \(= 6x = 72\)
Only PM \(= 4x = 48\)
Participants attending exactly one workshop:
\(72 + 48 + 12 = 132\).
Answer: D
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
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 \(y = \frac{x}{3}\). Thus:
\(308 = 220 + 6x + \frac{x}{3} + x\).
\(x = 12\)
Therefore:
Only NT \(= x = 12\)
Only DA \(= 6x = 72\)
Only PM \(= 4x = 48\)
Participants attending exactly one workshop:
\(72 + 48 + 12 = 132\).
Answer: D
