- A tech conference tracks attendee participation in three types of sessions: Technical Workshops (T), Business Panels (B), and Networking Events (N). With 500 total attendees, the data shows: 300 attended technical workshops, 250 attended business panels, 200 attended networking events, 140 attended both workshops and panels, 120 attended both workshops and networking events, 100 attended both panels and networking events, and 70 attended all three types of sessions. Conference organizers want to send a follow-up survey to attendees who participated in networking events but did not attend technical workshops. How many attendees should receive this survey?
Answer: 80 attendees should receive the follow-up survey about networking experiences
We need |N - T| = |N| - |N∩T|. We know |N| = 200 and |N∩T| = 120. Therefore, |N - T| = 200 - 120 = 80. This represents attendees who went to networking events but not technical workshops. We can verify this by calculating directly: attendees in N only = 200 - 120 - 100 + 70 = 50, and attendees in N∩B but not T = (100 - 70) = 30. So total attendees in N but not T = 50 + 30 = 80.
- A fitness app tracks user activities across three categories: Running (R), Cycling (C), and Swimming (S). In January, among 400 active users, the data shows: 240 users logged running activities, 180 logged cycling, 120 logged swimming, 100 logged both running and cycling, 60 logged both running and swimming, 40 logged both cycling and swimming, and 20 logged all three activities. The app wants to create a specialized training program for users who participate in swimming but not in both of the other two activities simultaneously. How many users would qualify for this program?
Answer: 100 users qualify for the specialized swimming-focused training program
We need users who swim but are NOT in the set (R∩C). Users who swim = 120. Among these, we need to exclude those who do running AND cycling simultaneously. Users who do all three activities (R∩C∩S) = 20. So users who swim but don't do both running and cycling = 120 - 20 = 100. This includes: users who swim only, users who swim and run only, and users who swim and cycle only. Let me verify: S only = 120 - 60 - 40 + 20 = 40, (S∩R) only = 60 - 20 = 40, (S∩C) only = 40 - 20 = 20. Total = 40 + 40 + 20 = 100. This confirms our answer.
- A health club tracks member participation in three main activity areas: Cardio equipment (C), Weight training (W), and Group fitness classes (G). In a given week, among 350 active members, the usage data shows: 200 used cardio equipment, 180 participated in weight training, 140 attended group fitness classes, 95 did both cardio and weight training, 75 did both cardio and group fitness, 60 did both weight training and group fitness, and 35 participated in all three activity areas. The club management wants to offer personalized training consultations to members who participate in weight training OR group fitness classes, but they want to exclude members who also use cardio equipment heavily (since those members already receive cardiovascular assessments). How many members would be eligible for these personalized consultations?
Answer: 125 members are eligible for personalized training consultations without cardio focus
We need |(W∪G) - C|. First, |W∪G| = 180 + 140 - 60 = 260. Next, |(W∪G)∩C| = |W∩C| + |G∩C| - |W∩G∩C| = 95 + 75 - 35 = 135. Therefore, members who do weight training OR group fitness but NOT cardio = 260 - 135 = 125. We can verify: W only = 180 - 95 - 60 + 35 = 60, G only = 140 - 75 - 60 + 35 = 40, (W∩G) only = 60 - 35 = 25. Total = 60 + 40 + 25 = 125. This confirms our answer.
