Venn Diagram Questions For
CAT PDF Set-2
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Question 1: In a school, 150 students have to opt for at least one subject out of English, Physics
and Mathematics. If 71 students opted for English, 98 students opted for Maths, 68 students opted
for Physics and 14 students opted for all the three subjects, find the number of students who opted
for exactly two subjects.
a) 49
b) 59
c) 39
d) 69
Question 2: In an exam conducted for 38 students of class XII 15 passed in Biology and 25
passed in Chemistry. If atleast 1 student failed in both the subjects,find the maximum number of
students who passed in only Biology ?
a) 12
b) 15
c) 10
d) 13
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Question 3: In a locality, residents read at least one of the three newspapers - Hindu, Times and
Express. 60% read Hindu, 80% read Times and 55% read Express. What can be the minimum
and maximum percentage of residents who read exactly two newspapers?
a) 2.5%, 92.5%
b) 5%, 95%
c) 10%, 90%
d) 0%, 95%
Question 4: Out of 60 families living in a building, all those families which own a car own a
scooter as well. No family has just a scooter and a bike. 16 families have both a car and a bike.
Every family owns at least one type of vehicle and the number of families that own exactly one
type of vehicle is more than the number of families that own more than one type of vehicle. What
is the sum of the maximum and minimum number of families that own only a bike?
a) 24
b) 34
c) 44
d) 54
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Question 5: 207 people who attend “Bold Gym” in Kondapur take four types juices Apple,
Orange, Pomegranate and Mango. There are a few people who do not take any of the juices. It
is known that for every person in the Gym who takes atleast ‘N’ types of juices there are 2
persons who take atleast ‘N-1’ juices for N = 2, 3 and 4. If the number of people who take all four
types of juices is equal to the number of people who do not take any juice at all, what is the
number of people who take exactly 2 types of juices?
a) 23
b) 46
c) 69
d) 92
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Answers & Solutions:
1) Answer (B)
2) Answer (A)
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From the Venn diagram , a + b = 15 - (1)
b + c = 25 - (2)
a + b + c + n = 38 - (3)
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(3) - (2) - (1) = b - n = 2
as n\geq≥1 the minimum value of b is 3.
From (1) the maximum value of a is 12.
3) Answer (D)
Let ‘a’ be the percentage of residents who read only one newspaper. Let ‘b’ be the percentage of
residents who read exactly two newspapers. Let ‘c’ be the percentage of residents who read all
three newspapers.
a + b + c = 100
a + 2b + 3c = 60 + 80 + 55 = 195
=> b + 2c = 95
To find the maximum percentage of residents who read exactly two newspapers, ‘b’ has to be
maximum and ‘c’ has to be minimum. If c = 0, then b = 95, in which case a = 5.
To find the minimum percentage of residents who read exactly two newspapers, ‘b’ has to be
minimum. Since b + 2c = 95, if b = 0, c = 47.5, in which case a = 52.5
So, the minimum percentage is 0% and the maximum percentage is 95%
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4) Answer (C)
From the information given in the question, the following Venn Diagram can be constructed:
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So, in order to maximize the number of families that own only a bike, we can put the remaining 44
families in ‘only bike’ region. Similarly, in order to minimize the number of families that own only a
bike, we can put the remaining 44 families in ‘only scooter’ region.
So, the maximum number of families that own only a bike is 44 and the minimum number of
families that own only a bike is 0.
So, sum = 44 + 0 = 44
5) Answer (B)
Let the number of persons who do not take any juice and the number of persons who take all 4
types of juices be ‘x’.
As there are ‘x’ people who take atleast 4 types of juices there should be ‘2x’ people who take
atleast 3 types of juices.
This means the number of people who take exactly 3 types of juices = 2x - x = x.
As there are ‘2x’ people who take atleast 3 types of juices there should be ‘4x’ people who take
atleast 2 types of juices.
This means the number of people who take exactly 2 types of juices = 4x - x - x = 2x.
As there are ‘4x’ people who take atleast 2 types of juices there should be ‘8x’ people who take
atleast 1 type of juices.
This means the number of people who take exactly 1 type of juices = 8x - 4x = 4x.
Thus the total number of people in the Gym = x + 4x + 2x + x + x = 9x.
9x = 207 => x = 23.
The number of people who take exactly 2 types of juices = 2x = 46.
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