Logical Venn Diagram Test-1

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Logical Venn Diagram Test-1

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Question 1
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Shyam visited Ram during his brief vacation. In the mornings they both would go for yoga. In the evenings they would play tennis. To have more fun, they indulge only in one activity per day, i.e. either they went for yoga or played tennis each day. There were days when they were lazy and stayed home all day long. There were 24 mornings when they did nothing, 14 evenings when they stayed at home, and a total of 22 days when they did yoga or played tennis. For how many days Shyam stayed with Ram?

A
32

B
24

C
30

D
None of these

Solution

Let the number of days in the vacation be x
They played tennis for x - 14 days
They did yoga for x - 24 days
So, they did yoga or played tennis for x - 14 + x - 24 = 2x - 38 days
2x –38 = 22
=>x = 30
Question 2
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How many even integers n, where 100 ≤n ≤200, are divisible neither by seven nor by nine?

A
40

B
37

C
39

D
38

Solution

Between 100 and 200 both included there are 51 even nos.
There are 7 even nos which are divisible by 7 and 6 nos which are divisible by 9 and 1 no divisible by both ie 42.
Hence in total 51 –(7+6-1) = 39
There is one more method through which we can find the answer.
Since we have to find even numbers, consider the numbers which are divisible by 14, 18 and 126 between 100 and 200.
These are 7, 6 and 1 respectively.
Question 3
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There are 3 clubs A, B &C in a town with 40, 50 &60 members respectively. While 10 people are members of all 3 clubs, 70 are members in only one club. How many belong to exactly two clubs?

A
20

B
25

C
50

D
70

Solution

We know that x + y + z = T and x + 2y + 3z = R, where
x = number of members belonging to exactly 1 set = 70
y = number of members belonging to exactly 2 sets
z = number of members belonging to exactly 3 sets = 10
T = Total number of members
R = Repeated total of all the members = (40 + 50 + 60) = 150
Thus we have two equations and two unknowns. Solving this we get y = 25
So, 25 people belong to exactly 2 clubs.
Question 4
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In a class of 60, along with English as a common subject, students can opt to major in Mathematics, Physics, Biology or a combination of any two. 6 students major in both Mathematics and Physics, 15 major in both Physics and Biology, but no one majors in both Mathematics and Biology. In an English test, the average mark scored by students majoring in Mathematics is 45 and that of students majoring in Biology is 60. However, the combined average mark in English, of students of these two majors, is 50. What is the maximum possible number of students who major ONLY in Physics?

A
15

B
25

C
20

D
30

Solution
Question 5
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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
54

D
44

Solution

From the information given in the question, the following Venn Diagram can be constructed:

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
Question 6
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400 students were admitted to the 2018-19 MBA batch. 200 of them did not choose “Business Statistics ”. 100 of them did not choose “International Management ’. There were 80 students who did not choose any of the two subjects. Find the number of students who chose both Business Statistics and International Management.

A
220

B
180

C
280

D
300

Solution

Number of students who chose Business statistics = 400 −200 = 200
Number of students who chose International Management = 400 −100 = 300
Number of students who chose at least one of the two subjects = 400 −80 = 320
∴Number of students who chose both the subjects = 200 + 300 −320 = 500 −320 = 180
Hence, option (b).
Question 7
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A premier B-school, which is in process of getting an AACSB accreditation, has 360 second year students. To incorporate sustainability into their curriculum, it has offered 3 new elective subjects in the second year namely Green Supply Chain, Global Climate Change &Business and Corporate Governance. Twelve students have taken all the three electives, and 120 students study Green Supply Chain. There are twice as many students who study Green Supply Chain and Corporate Governance but not Global Climate Change and Business, as those who study both Green Supply Chain and Global Climate Change &Business but not Corporate Governance, and 4 times as many who study all the three. 124 students study Corporate Governance. There are 72 students who could not muster up the courage to take up any of these subjects. The group of students who study both Green Supply Chain and Corporate Governance but not global Climate Change &Business is exactly the same as the group made up to the students who study both Global Climate Change &Business and Corporate Governance. How many students study Global Climate Change &Business only?

A
176

B
104

C
152

D
188

Solution

The number of students who study each combination of subjects (based on the direct data) given is as shown below:

It is given that: (GSC and CG but not GCCB) = 4 times (all three electives)
∴2x = 4(12) i.e. x = 24
Also: (GSC and CG but not GCCB) = (all three electives) + (GCCB and CG but not GSC)
∴(GCCB and CG but not GSC) = 2x −12 = 2(24) −12 = 36
So, the figure becomes:

Now, CG only = 124 −(48 + 12 + 36) = 28
∴GCCB alone = 360 −120 −36 −28 −72 = 104
Hence, option (b).
Question 8
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In an amusement park along with the entry pass a visitor gets two of the three available rides (A, B and C) free. On a particular day 77 opted for ride A, 55 opted for B and 50 opted for C; 25 visitors opted for both A and C, 22 opted for both A and B, while no visitor opted for both B and C. 40 visitors did not opt for ride A and B, or both. How many visited with the entry pass on that day?

A
102

B
115

C
130

D
150

Solution

Let the Venn diagram be as shown in the figure,

No one can take all three rides, hence g = 0.
22 people take rides A and B,
∴d = 22
25 people take rides A and C,
∴f = 25
50 people take ride C,
∴c = 50 –25 = 25.
40 people don ’t take A or B or both,
∴40 = c + h
⇒h = 40 –25 = 15
∴Total number of people visiting the park = (77 + 55 + 50 –25 –22) + 15 = 150.
Hence, option (d).
Question 9
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290 students of MBA (International Business) in a reputed Business School have to study foreign language in Trimesters IV and V. Suppose the following information are given
(i) 120 students study Spanish
(ii) 100 students study Mandarin
(iii) At least 80 students, who study a foreign language, study neither Spanish nor Mandarin
Then the number of students who study Spanish but not Mandarin could be any number from

A
20 to 110

B
80 to 100

C
50 to 80

D
80 to 170

Solution

Atleast 80 students study neither Spanish nor Mandarin.
Hence, maximum number of students who study atleast one language = 290 –80 = 210
Minimum number of students who study both languages = 100 + 120 –210 = 10
∴Maximum number of students who study Spanish but not Mandarin = 120 –10 = 110
Maximum number of students who study both languages = smaller value of 100 and 120 = 100
∴Minimum number of students who study Spanish but not Mandarin = 120 –100 = 20
Hence, the range could be any number from 20 to 110.
Hence, option (a).
Question 10
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In a certain village, 22% of the families own agricultural land, 18% own a mobile phone and 1600 families own both agricultural land and a mobile phone. If 68% of the families neither own agricultural land nor a mobile phone, then the total number of families living in the village is:

A
20000

B
10000

C
8000

D
5000

Solution

Let total number of families in the village be T
Number of families own agricultural land, n(A) = 0.22T
Number of families own mobile phone, n(M) = 0.18T
Number of families own both agricultural land and mobile phone, n(A ⋂M) = 1600
Number of families own agricultural land or mobile phone, n(A ⋃M) = T –0.68T = 0.32T
∴n(A ⋃M) = n(A) + n(M) –n(A ⋂M)
∴n(A ⋂M) = 0.08T
0.08T = 1600 ⇒T = 20000
Hence, option (a).