Question

the manager of a campus gym is planning the schedule of fitness classes for a new school year, and will decide how often to hold certain classes based on the interests of the students. she polls 64 students at various times of day, asking what type of classes theyd be interested in attending. the results are summarized in the table below.
type of class | students interested
--- | ---
yoga | 17
pilates | 23
spinning | 15
yoga and pilates | 9
pilates and spinning | 8
yoga and spinning | 7
all three | 3

(a) construct a venn diagram illustrating these results.

Explanation:

Step1: Calculate only Yoga

Subtract overlaps from total Yoga:
$17 - 9 - 7 + 3 = 4$

Step2: Calculate only Pilates

Subtract overlaps from total Pilates:
$23 - 9 - 8 + 3 = 9$

Step3: Calculate only Spinning

Subtract overlaps from total Spinning:
$15 - 7 - 8 + 3 = 3$

Step4: Calculate outside all classes

Subtract all groups from total students:
$64 - (4 + 9 + 3 + 9 + 7 + 8 + 3) = 21$

Step5: Map to Venn diagram

• 3-circle Venn (Yoga, Pilates, Spinning):
• Center (all 3): 3
• Yoga-Pilates only: $9 - 3 = 6$
• Yoga-Spinning only: $7 - 3 = 4$
• Pilates-Spinning only: $8 - 3 = 5$
• Only Yoga: 4; Only Pilates:9; Only Spinning:3
• Outside all circles:21

Answer:

The Venn diagram will have three overlapping circles labeled Yoga, Pilates, and Spinning within a universal set (total 64 students) with the following values:

• Only Yoga: 4
• Only Pilates: 9
• Only Spinning: 3
• Yoga and Pilates only: 6
• Yoga and Spinning only: 4
• Pilates and Spinning only: 5
• All three classes: 3
• No interest in any class: 21