Test: Venn Diagrams- 1 - UPSC MCQ

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

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.

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.

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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

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).

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).

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).

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).

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).