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Difficulty: MediumLogical Venn Diagrams

At an annual fitness convention, 220220 individuals registered for various sessions. The convention offered three main workshop categories: Yoga (YY), Pilates (PP), and Aerobics (AA).

According to the attendance records:
- 100100 individuals participated in Yoga.
- 110110 individuals participated in Pilates.
- 9090 individuals participated in Aerobics.
- 5050 individuals participated in exactly two of these workshop categories.
- 2020 individuals did not participate in any of the three workshops.

Based on this information, how many individuals participated in all three workshop categories?

  1. A
    15
  2. 25Answer
  3. C
    50
  4. D
    75

Answer

25
To find the number of people in all three categories, we analyze the structure of the sets. The sum of people in the individual categories is 100+110+90=300100 + 110 + 90 = 300. The actual number of active participants (the union) is the total population minus those who did not participate: 220−20=200220 - 20 = 200. When we subtract the union from the sum of the individual sets (300−200=100300 - 200 = 100), this difference represents the overcounted overlaps. Specifically, people in exactly two categories (E2E_2) were counted twice, leaving one extra count. People in all three categories (E3E_3) were counted three times, leaving two extra counts. Therefore, 100=E2+2E3100 = E_2 + 2E_3. Since we are given E2=50E_2 = 50, we substitute to get 100=50+2E3100 = 50 + 2E_3, which simplifies to 2E3=502E_3 = 50, meaning E3=25E_3 = 25.

Step-by-Step Solution

1
Determine the number of individuals who participated in at least one workshop (the union of all three sets).
220−20=200220 - 20 = 200 individuals.
The universal set contains 220220 people, but 2020 participated in none. The union (N(Y∪P∪A)N(Y \cup P \cup A)) is the total minus the 'none' group.
2
Calculate the total sum of individuals across the three individual sets.
100+110+90=300100 + 110 + 90 = 300.
This sum counts individuals in exactly one category once, individuals in exactly two categories twice, and individuals in all three categories three times.
3
Set up an equation relating the sum of individual sets, the union, the 'exactly two' group (E2E_2), and the 'all three' group (E3E_3).
Sum of Sets−Union=E2+2E3\text{Sum of Sets} - \text{Union} = E_2 + 2E_3. Substituting values: 300−200=50+2E3300 - 200 = 50 + 2E_3.
Subtracting the union (which counts every participant exactly once) from the sum of sets leaves behind the extra counts: one extra count for everyone in exactly two, and two extra counts for everyone in all three.
4
Solve the algebraic equation for E3E_3.
100=50+2E3  ⟹  50=2E3  ⟹  E3=25100 = 50 + 2E_3 \implies 50 = 2E_3 \implies E_3 = 25.
Isolating E3E_3 provides the exact number of individuals who participated in all three workshops.

Key Concept

Logical Venn Diagrams and the application of categorical region sums (Inclusion-Exclusion Principle).
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Topics

Logical Venn Diagrams

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