SET Practice DS SET – 3
DS: Set Theory – Level 2
1.
How many people play exactly one sport (Tennis or Golf)?
(1) 40 people play Tennis and 30 play Golf.
(2) There are 60 people total, and everyone plays at least one.
(1) 40 people play Tennis and 30 play Golf.
(2) There are 60 people total, and everyone plays at least one.
Solution (C):
(1) Gives Total Tennis and Total Golf. Missing Intersection. Not sufficient.
(2) Gives Union. Missing group sizes. Not sufficient.
Combined: Union = 60. \( 60 = 40 + 30 – \text{Both} \implies \text{Both} = 10 \).
Exactly One = Union – Both = \( 60 – 10 = 50 \). Sufficient.
2.
In a group of 50 students, how many study ONLY French?
(1) 20 study French, and 5 study both French and Spanish.
(2) 30 study Spanish.
(1) 20 study French, and 5 study both French and Spanish.
(2) 30 study Spanish.
Solution (A):
(1) Total French = 20. Both = 5. Only French = \( 20 – 5 = 15 \). Sufficient.
(2) Gives Spanish. Tells nothing about French. Not sufficient.
3.
What is the value of \( n(A \cap B) \)?
(1) \( n(A) = 2n(B) \).
(2) \( n(A \cup B) = 30 \) and \( n(A) + n(B) = 35 \).
(1) \( n(A) = 2n(B) \).
(2) \( n(A \cup B) = 30 \) and \( n(A) + n(B) = 35 \).
Solution (B):
(1) Ratio of sets. No values. Not sufficient.
(2) \( n(A \cup B) = n(A) + n(B) – n(A \cap B) \).
\( 30 = 35 – n(A \cap B) \implies n(A \cap B) = 5 \). Sufficient.
4.
Is \( n(A) > n(B) \)?
(1) \( n(A \cup B) = 40 \).
(2) \( n(A \cap B) = 5 \).
(1) \( n(A \cup B) = 40 \).
(2) \( n(A \cap B) = 5 \).
Solution (E):
Combined: \( 40 = A + B – 5 \implies A + B = 45 \).
We have a sum, but no info on which is larger. A could be 40 and B could be 5, or vice versa. Not sufficient.
5.
What percentage of employees have a car but NOT a bike?
(1) 70% have a car and 40% have a bike.
(2) 20% have both.
(1) 70% have a car and 40% have a bike.
(2) 20% have both.
Solution (C):
We want “Only Car” = Total Car – Both.
(1) Gives Total Car (70%). Missing Both. Not sufficient.
(2) Gives Both (20%). Missing Total Car. Not sufficient.
Combined: \( 70 – 20 = 50\% \). Sufficient.
6.
Are there any students who study neither Math nor Physics?
(1) There are 100 students.
(2) 50 study Math, 40 study Physics, and 10 study both.
(1) There are 100 students.
(2) 50 study Math, 40 study Physics, and 10 study both.
Solution (C):
(1) Gives Total. No subset info. Not sufficient.
(2) Gives subset sizes. Union = \( 50 + 40 – 10 = 80 \). Without Total, we don’t know if there are leftovers. Not sufficient.
Combined: Total = 100. Union = 80. Neither = \( 100 – 80 = 20 \). Yes, there are 20. Sufficient.
7.
If \( A \cap B = \emptyset \) (disjoint), what is \( n(A) \)?
(1) \( n(A \cup B) = 30 \).
(2) \( n(B) = 10 \).
(1) \( n(A \cup B) = 30 \).
(2) \( n(B) = 10 \).
Solution (C):
If disjoint, \( n(A \cup B) = n(A) + n(B) \).
(1) \( n(A) + n(B) = 30 \). Two unknowns. Not sufficient.
(2) \( n(B) = 10 \). \( n(A) \) unknown. Not sufficient.
Combined: \( n(A) + 10 = 30 \implies n(A) = 20 \). Sufficient.
8.
Of the employees, what percent have a master’s degree?
(1) 60% of employees are over 40 years old, and 50% of them have a master’s degree.
(2) Of the employees 40 or younger, 20% have a master’s degree.
(1) 60% of employees are over 40 years old, and 50% of them have a master’s degree.
(2) Of the employees 40 or younger, 20% have a master’s degree.
Solution (C):
Total Master’s = (Masters in Over 40) + (Masters in Under 40).
(1) Over 40 group: \( 0.50 \times 0.60 = 0.30 \) (30% of total). Missing Under 40 info. Not sufficient.
(2) Under 40 group size is \( 100 – 60 = 40\% \). Masters = \( 0.20 \times 0.40 = 0.08 \) (8% of total). Missing Over 40 info. Not sufficient.
Combined: \( 30\% + 8\% = 38\% \). Sufficient.
9.
Is \( n(A) = n(B) \)?
(1) \( n(A \cup B) = 20 \).
(2) \( n(A – B) = n(B – A) \).
(1) \( n(A \cup B) = 20 \).
(2) \( n(A – B) = n(B – A) \).
Solution (B):
\( n(A) = n(A – B) + n(A \cap B) \).
\( n(B) = n(B – A) + n(A \cap B) \).
(1) Gives Union. Doesn’t compare sizes. Not sufficient.
(2) \( n(A – B) = n(B – A) \). Since they share the same intersection component, if the “only” parts are equal, the totals must be equal. Yes. Sufficient.
10.
In a survey, do more people like Cats than Dogs?
(1) 40% like Cats.
(2) 30% like Dogs.
(1) 40% like Cats.
(2) 30% like Dogs.
Solution (C):
(1) Gives Cats only.
(2) Gives Dogs only.
Combined: Cats = 40%, Dogs = 30%. Since 40 > 30, Yes, more people like Cats. Sufficient.
Score: 0 / 0