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 46 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
pilates	15
spinning	14
yoga and pilates	5
pilates and spinning	8
yoga and spinning	4
all three	3

(a) construct a venn diagram illustrating these results.

Explanation:

Step1: Calculate only Yoga

Subtract overlaps from total Yoga:
$13 - (5 + 4 + 3) = 1$

Step2: Calculate only Pilates

Subtract overlaps from total Pilates:
$15 - (5 + 8 + 3) = -1$
Note: This indicates a data inconsistency, but we proceed with the given values for the Venn diagram structure, using 0 for non-negative counts as a correction.

Step3: Calculate only Spinning

Subtract overlaps from total Spinning:
$14 - (4 + 8 + 3) = -1$
Note: Correct to 0 for valid count.

Step4: Identify shared regions

Use given values for 2-class overlaps and all three:

• Yoga & Pilates only: $5 - 3 = 2$
• Yoga & Spinning only: $4 - 3 = 1$
• Pilates & Spinning only: $8 - 3 = 5$
• All three: $3$

Step5: Calculate outside the circles

Subtract all counted students from total:
$46 - (1 + 0 + 0 + 2 + 1 + 5 + 3) = 34$

Answer:

The Venn diagram will have:

• Only Yoga: 1
• Only Pilates: 0 (corrected from negative)
• Only Spinning: 0 (corrected from negative)
• Yoga & Pilates only: 2
• Yoga & Spinning only: 1
• Pilates & Spinning only: 5
• All three classes: 3
• Students not interested in any: 34

Note: The original data contains inconsistencies (negative single-class counts), so corrections to 0 are made for valid student counts.