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Understanding Venn Diagrams for Problem Solving

Apr 21, 2025

Venn Diagrams and Problem Solving

Introduction

  • Focus on solving problems associated with Venn diagrams.
  • Example: Survey of students regarding pet ownership.

Example 1: Pet Ownership

  • Survey Details:
    • 20 students own cats.
    • 25 students own dogs.
    • 8 students own both cats and dogs.
    • 12 students own neither.
  • Venn Diagram:
    • Two overlapping circles (Cats and Dogs).
    • Overlapping region represents students owning both (8 students).
  • Part B: Cats but no dogs
    • Total cat owners: 20
    • Cats only: 20 - 8 = 12 students.
  • Part C: Dogs but no cats
    • Total dog owners: 25
    • Dogs only: 25 - 8 = 17 students.
  • Part D: Total students surveyed
    • Sum: 12 (Cats only) + 17 (Dogs only) + 8 (Both) + 12 (Neither) = 49 students.
    • Alternative Calculation:
      • 20 (Cats) + 25 (Dogs) - 8 (Both) + 12 (Neither) = 49 students.

Example 2: Language Study

  • Survey Details:
    • 40 students in a math class.
    • 16 students studying Spanish.
    • 19 students studying French.
    • 5 students studying both.
  • Venn Diagram:
    • Overlapping region for both Spanish and French (5 students).
  • Part A: Spanish but not French
    • Total Spanish students: 16
    • Spanish only: 16 - 5 = 11 students.
  • Part B: French but not Spanish
    • Total French students: 19
    • French only: 19 - 5 = 14 students.
  • Part C: Neither Spanish nor French
    • Total: 40 students.
    • Equation: 11 (Spanish only) + 14 (French only) + 5 (Both) + X (Neither) = 40.
    • X = 10 students.

Example 3: Car and SUV Ownership

  • Survey Details:
    • 100 college students surveyed.
    • 40 own a car.
    • 25 own an SUV.
    • 47 own neither.
  • Goal: Calculate how many own both.
  • Formula:
    • Method 1: Total = C + S - B + N
    • Method 2: Total = (C - B) + (S - B) + B + N
    • Both methods simplify to: C + S - B + N
  • Calculation:
    • Total: 100, C = 40, S = 25, N = 47
    • Equation: 40 + 25 - B + 47 = 100
    • B = 12 (Both own car and SUV).
  • Part B: Car but not SUV
    • 40 (Car) - 12 (Both) = 28 students.
  • Part C: SUV but not car
    • 25 (SUV) - 12 (Both) = 13 students.
  • Verification:
    • Sum: 28 (Car only) + 12 (Both) + 13 (SUV only) + 47 (Neither) = 100 students.

Conclusion

  • Solved three types of Venn diagram word problems involving two categories.
  • Key variations involve calculating totals, neither, or both categories.

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