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Difficulty: EasyOverlapping Sets and Venn Diagrams

In a market research study of 100100 consumers, 6565 consumers stated that they regularly purchase Brand A, and 4545 consumers stated that they regularly purchase Brand B. If 2020 consumers purchase neither Brand A nor Brand B, how many consumers regularly purchase both Brand A and Brand B?

Answer: 30 consumers

Answer

The number of consumers who regularly purchase both Brand A and Brand B is 3030.
According to the principle of inclusion-exclusion for two sets, Total=Brand A+Brand B−Both+Neither\text{Total} = \text{Brand A} + \text{Brand B} - \text{Both} + \text{Neither}. Substituting the given numbers into this relationship gives 100=65+45−Both+20100 = 65 + 45 - \text{Both} + 20, which simplifies to 100=130−Both100 = 130 - \text{Both}. Subtracting 100100 from 130130 yields Both=30\text{Both} = 30.

Step-by-Step Solution

1
Identify the given values from the problem statement.
Total=100\text{Total} = 100, Brand A=65\text{Brand A} = 65, Brand B=45\text{Brand B} = 45, Neither=20\text{Neither} = 20.
Extracting all known set quantities to set up the overlapping sets equation.
2
Apply the inclusion-exclusion formula for two overlapping sets.
Total=Brand A+Brand B−Both+Neither\text{Total} = \text{Brand A} + \text{Brand B} - \text{Both} + \text{Neither}.
Subtracting the overlap prevents double-counting elements belonging to both groups.
3
Substitute the values and solve for the unknown overlap.
100=65+45−Both+20  ⟹  100=130−Both  ⟹  Both=30100 = 65 + 45 - \text{Both} + 20 \implies 100 = 130 - \text{Both} \implies \text{Both} = 30.
Algebraically determining the number of consumers in the intersection.

Key Concept

Two-Set Inclusion-Exclusion Principle
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Topics

Overlapping Sets and Venn Diagrams

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