Q.1. A survey was conducted, it was found that the number of people who prefer only Burger, only Pizza, both Burger and Pizza and neither of them are 40, 45, 18 & 22 respectively. Find the number of people surveyed.
Sol : Option D
Let A be the number of persons who like Burger & B be the number of persons who like Pizza.
A∪;B = A + B - A ∩ B
People who like at least one of them = 40 + 45 – 18 = 67
67 people like at least one of the two things & 22 people like none. Therefore, a total of 89 persons were surveyed.
Q.2. In a college of 350 students, every student needs to choose among the three subjects (i.e. Economics, Accounts & Taxation) offered along with the main course. The students who chose each of these subjects are 120, 80 & 95. The number of students who chose more than one of the three is 28 more than the number of students who chose all the three subjects. If there are no students who chose none of the three subjects, how many students study all the three subjects?
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Sol : Option A
There are 350 students in this question. The number of students who choose more than one ( i.e. either of two or three subjects ) is 28 more than those who chose three subjects. This means that number of students who chose exactly two subjects is 28. Now,
(A∪B∪C) = A + B + C - A ∩ B - B ∩ C- C ∩ A + A ∩ B ∩ C
350 = 120 + 80 + 95 – 28 + A ∩ B ∩ C
A ∩ B ∩ C = 83
Q.3. In a group of 120 people, 54 like Coca Cola and 84 like Pepsi and each person likes at least one of the two beverages. How many like both Coca Cola & Pepsi?
A. 24
B. 18
C. 32
D. 12
Sol : Option B
Let A = Set of people who like Coca Cola.
B = Set of people who like Pepsi.
Given
(A ∪ B) = 120
n(A) = 54
n(B) = 84 then;
n(A ∩ B) = n(A) + n(B) - n(A ∪ B)
= 54 + 84 - 120
= 138 - 120 = 18
= 18
Therefore, 18 people like both beverages.
Q.4. After sports day in a college, college awarded medals in different categories. 72 medals in Athletics, 24 medals in weightlifting and 36 medals in swimming. If these medals went to a total of 90 persons and only 8 persons got medals in all the three sports, how many received medals in exactly two of these sports?
A. 6
B. 8
C. 10
D. 12
Sol : Option A
Let A = set of all the persons who got medals in athletics.
B = set of all the persons who got medals in weightlifting.
C = set of all the persons who got medals in swimming.
Given,
n(A) = 72
n(B) = 24
n(C) = 36
n(A ∪ B ∪ C) = 90
n(A ∩ B ∩ C) = 8
We know that number of elements belonging to exactly two of these three sets A, B, C can be written as
= n(A ∩ B) + n(B ∩ C) + n(A ∩ C) - 3n(A ∩ B ∩ C)
= n(A ∩ B) + n(B ∩ C) + n(A ∩ C) - 3 × 8 ........(i)
From (i) required number of students = n(A) + n(B) + n(C) + n(A ∩ B ∩ C) - n(A ∪ B ∪ C) - 12
= 72 + 24 + 36 + 8 - 90 - 24
= 140 - 134
= 6
Q.5. In a group of 300 students, 216 students study Hindi and 129 study Punjabi. How many study Hindi only? How many study Punjabi only and how many study both Hindi & Punjabi?
A. Rs. 75
B. Rs. 100
C. Rs. 72
D. Rs. 77.5
Sol : Option A
Let A be the set of students who study Hindi.
B be the set of students who study Punjabi.
A - B be the set of students who study Hindi but not Punjabi.
B - A be the set of students who study Punjabi but not Hindi.
A ∩ B be the set of students who study both Hindi and Punjabi.
Given,
n(A) = 216
n(B) = 129
n(A ∪ B) = 300
Now, n(A ∩ B) = n(A) + n(B) - n(A ∪ B)
= 216 + 129 - 300
= 345 - 300
= 45
Therefore, Number of students who study both Hindi & Punjabi = 45
n(A) = n(A - B) + n(A ∩ B) ⇒
n(A - B) = n(A) - n(A ∩ B)
= 216 - 45
= 171
and n(B - A) = n(B) - n(A ∩ B)
= 129 - 45
= 84
Therefore, Number of students studying Hindi only = 57
Number of students studying Punjabi only = 28
Q.6. In a class of 212 students, each student studies at least one of the three subjects Mandarin, French and Spanish. 96 of them study Mandarin, 102 studies French and 106 Spanish. 32 studies Mandarin and French, 34 study Mandarin and Spanish and 36 study French and Spanish. How many students study all the three subjects?
A. 12
B. 10
C. 8
D. 6
Sol : Option B
Let A be the set of students who study Mandarin
Let B be the set of students who study French
Let C be the set of students who study Spanish
A∪B∪C = A + B + C - (A n B + B n C + C n A) + (A n B n C)
212 = 96 + 102 + 106 -32- 34 – 36 + x
212 = 202 + x
X = 10
Q.7. A survey was conducted among certain people, it was found that the number of people who like only Chocolate, only Vanilla, both Chocolate and Vanilla and neither of them are 120, 135, 54 & 66 respectively. Find the number of people surveyed.
A. Rs. 207
B. Rs. 234
C. Rs. 246
D. Rs. 267
Sol : Option D
Let A be the number of persons who like Chocolate & B be the number of persons who like Vanilla.
A∪B = A + B - A ∩ B
People who like at least one of them = 120 + 135 – 54 = 201
201 people like at least one of the two beverages & 66 people like none. Therefore, a total of 267 persons were surveyed.
Q8. In a class of 120 students, 36 enrolled for both Quant and Verbal. 66 enrolled for Verbal. If the students of the class enrolled for at least one of the two subjects, then how many students enrolled for only Quant and not Verbal?
A. 90
B. 30
C. 54
D. 84
Sol : Option C
Let A be the number of studnets who enrolled for Quant & B be the number of students who enrolled for Verbal
A∪B = A + B - A ∩ B
120 = x + 66 – 36
X = 90
Now, x is the number of students who have enrolled for quant. If we subtract 36 from 90, we get 54, that is these students have enrolled only for Quant but not Verbal.
Q9. In a party with 600 guests, 180 guests ate Vanilla as well as Chocolate Ice cream. 330 ate Chocolate Ice Cream. If all the guests at the party ate at least one of the two ice creams, then how many guests ate only Vanilla and not Chocolate?
A. 450
B. 150
C. 270
D. 420
Sol : Option C
Let A be the number of guests who ate Chocolate & B be the number of guests who ate Vanilla
A∪B = A + B - A ∩ B
600 = x + 330 – 180
X = 450
Now, x is the number of guests who have ate Chocolate. If we subtract 180 from 450, we get 270, that is the number of guests who have eaten only Chocolate & not Vanilla.
Q10. In a sports club of 530 members, each member plays at least one of the three sports Cricket, Football and Squash. 240 of them play Cricket, 255 play Football and 265 Squash. 80 play Cricket and Football, 85 play Cricket and Squash and 90 play Football and Squash. How many members play all the three sports?
A. 30
B. 25
C. 20
D. 15
Sol : Option B
Let A be the set of members who play Cricket
Let B be the set of members who play Football
Let C be the set of members who play Squash
A∪B∪C = A + B + C - (A n B + B n C + C n A) + (A n B n C)
530 = 240 + 255 + 265 -80- 85 – 90 + x
530 = 505 + x
X = 25