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Difficulty: EasyOverlapping Sets, Statistics, and Data Distributions

A total of 3030 tourists visited a city, and each tourist visited Museum X, Museum Y, or both. How many of the tourists visited both Museum X and Museum Y?

(1) 2020 of the tourists visited Museum X.
(2) 1515 of the tourists visited Museum Y.

  1. A
    Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
  2. B
    Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
  3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.Answer
  4. D
    EACH statement ALONE is sufficient.
  5. E
    Statements (1) and (2) TOGETHER are NOT sufficient.

Answer

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
The correct option states that both statements together are sufficient, but neither statement alone is sufficient. By applying the formula Total=Set A+Set B−Both\text{Total} = \text{Set A} + \text{Set B} - \text{Both}, we see that knowing the total (3030) along with both individual sets (2020 and 1515) allows us to solve for the overlap uniquely (55). Neither statement alone provides both set totals.

Step-by-Step Solution

1
Rephrase the question stem using the overlapping set formula.
The total number of tourists is given by Total=N(X)+N(Y)−N(X∩Y)\text{Total} = N(X) + N(Y) - N(X \cap Y), where Total=30\text{Total} = 30. To find N(X∩Y)N(X \cap Y), we need the sum N(X)+N(Y)N(X) + N(Y).
Establishing the algebraic relation clarifies what specific information is required from the statements.
2
Evaluate Statement (1) independently.
Statement (1) gives N(X)=20N(X) = 20. Substituting into the equation gives 30=20+N(Y)−N(X∩Y)⇒N(X∩Y)=N(Y)−1030 = 20 + N(Y) - N(X \cap Y) \Rightarrow N(X \cap Y) = N(Y) - 10. Since N(Y)N(Y) is unknown, N(X∩Y)N(X \cap Y) cannot be determined.
Determines whether Statement (1) alone yields a unique answer.
3
Evaluate Statement (2) independently.
Statement (2) gives N(Y)=15N(Y) = 15. Substituting into the equation gives 30=N(X)+15−N(X∩Y)⇒N(X∩Y)=N(X)−1530 = N(X) + 15 - N(X \cap Y) \Rightarrow N(X \cap Y) = N(X) - 15. Since N(X)N(X) is unknown, N(X∩Y)N(X \cap Y) cannot be determined.
Determines whether Statement (2) alone yields a unique answer.
4
Evaluate both statements together.
Combining Statement (1) and Statement (2) provides N(X)=20N(X) = 20 and N(Y)=15N(Y) = 15. Plugging both into the formula: 30=20+15−N(X∩Y)⇒30=35−N(X∩Y)⇒N(X∩Y)=530 = 20 + 15 - N(X \cap Y) \Rightarrow 30 = 35 - N(X \cap Y) \Rightarrow N(X \cap Y) = 5. This gives a single, unique value.
Determines if combining both statements provides sufficient information.

Key Concept

Two-Group Overlapping Sets Principle
Estimated Time:1m 0s
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Topics

Overlapping Sets, Statistics, and Data Distributions

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