GMAT Focus EditionQuantitative ReasoningMedium
A market researcher is conducting a survey. She finds that 60% of respondents like Product X, 50% like Product Y, and 30% like both Product X and Product Y. What percentage of respondents like neither Product X nor Product Y?
- A10%
- B40%
- C30%
- D20%
Show answer & explanationAnswer & explanation
Correct answer: D. 20%
Using the Principle of Inclusion-Exclusion for two sets: P(X or Y) = P(X) + P(Y) - P(X and Y). So, P(X or Y) = 60% + 50% - 30% = 110% - 30% = 80%. The percentage of respondents who like neither is 100% - P(X or Y) = 100% - 80% = 20%.
Why the other options are wrong
- A. This is 50% - 30% (only Y).
- B. This is 60% - 30% (only X) + 50% - 30% (only Y) = 30% + 20% = 50% who like exactly one product.
- C. This is the percentage that likes both products.
Inclusion-Exclusion Principle (Two Sets)
For two sets A and B, the principle of inclusion-exclusion states that the number of elements in their union is |A ∪ B| = |A| + |B| - |A ∩ B|. This prevents double-counting elements common to both sets.
- Used to find the size of the union of sets.
- Subtracts the intersection to correct for overlap.
- Can be extended to three or more sets.
Memory trick: Union is Sum Minus Overlap.