The Venn Diagrams from the logical reasoning section would enhance your rational thinking skills to prepare you for various entrances such as CAT, CMAT, MAT, XAT, IIFT, SNAP etc.
Directions (1 -4): Read the following information and answer the questions below.
In a coaching institute there are total 170 students and they studied different subject’s viz. Economics, Maths and English.
The ratio of students studying all 3 subjects to students studying at least 2 subjects is 2:9. The ratio of students studying only one subject to students studying at least 2 subjects is 8:9.
Number of students taking Maths only exceeds number of students of Economics only by 14.
Number of students studying English only exceeds number of students of Economics only by 12. Number of students studying English, Economics, and Maths is 90, 93, 97 respectively.

1. Number of students studying all three subjects is:
a) 18
b) 12
c) 20
d) None of these
Explanation:

Let’s solve this step by step:

Total students = 170

Let’s denote:

• E = Economics
• M = Maths
• En = English

x = students studying all 3 subjects

y = students studying at least 2 subjects

z = students studying only one subject

From the problem, we have the following information:

• The ratio of students studying all 3 subjects to students studying at least 2 subjects is 2:9. So, we can write this as x/y = 2/9.
• The ratio of students studying only one subject to students studying at least 2 subjects is 8:9. So, we can write this as z/y = 8/9.
• Number of students taking Maths only exceeds number of students of Economics only by 14.
• Number of students studying English only exceeds number of students of Economics only by 12.
• Number of students studying English, Economics, and Maths is 90, 93, 97 respectively.

From these equations, we can solve for x, y, and z. After solving these equations, we find that the number of students studying all three subjects is 20.

So, the answer is C 20.

2. Number of students studying no more than one subject is:
a) 76
b) 80
c) 60
d) can’t be determined
Explanation:

Let’s solve this step by step:

Total students = 170

Let’s denote:

• E = Economics
• M = Maths
• En = English
• x = students studying all 3 subjects
• y = students studying at least 2 subjects
• z = students studying only one subject

From the problem, we have the following information:

• The ratio of students studying all 3 subjects to students studying at least 2 subjects is 2:9. So, we can write this as x/y = 2/9.
• The ratio of students studying only one subject to students studying at least 2 subjects is 8:9. So, we can write this as z/y = 8/9.
• Number of students taking Maths only exceeds number of students of Economics only by 14.
• Number of students studying English only exceeds number of students of Economics only by 12.
• Number of students studying English, Economics, and Maths is 90, 93, 97 respectively.

From these equations, we can solve for x, y, and z. After solving these equations, we find that the number of students studying no more than one subject (the same as the number of students studying only one subject) is 80.

So, the answer is (b) 80.

3. Number of students using exactly two subjects is:
a) 38
b) 55
c) 70
d) none of these
Explanation:

Let’s solve this step by step:

1. Total students = 170
2. Let’s denote:
• E = Economics
• M = Maths
• En = English
• x = students studying all 3 subjects
• y = students studying at least 2 subjects
• z = students studying only one subject

From the problem, we have the following information:

• The ratio of students studying all 3 subjects to students studying at least 2 subjects is 2:9. So, we can write this as x/y = 2/9.
• The ratio of students studying only one subject to students studying at least 2 subjects is 8:9. So, we can write this as z/y = 8/9.
• Number of students taking Maths only exceeds number of students of Economics only by 14.
• Number of students studying English only exceeds number of students of Economics only by 12.
• Number of students studying English, Economics, and Maths is 90, 93, 97 respectively.

From these equations, we can solve for x, y, and z. After solving these equations, we find that the number of students studying exactly two subjects is 70.

So, the answer is C 70.

4. The number of students who are studying both Economics and Maths but not English is:
a) 23
b) 40
c) 36
d) data insufficient
Explanation:

a+b+c = α, x+y+z = β, k = γ
α+β+γ= 170
α + 2β + 3γ = 90 + 93 + 97 = 280
γ:(β+γ)= 2:9
⇒ γ : β = 2 : 7
and α:(β+γ) = 8 : 9
⇒ α:β:γ = 8 : 7 : 2
∴ α = 80, β = 70 and γ = 20
⇒ a+b+c = 80, x+y+ z = 70 ...(1)
and k = 20
Again c - b =14 and a - b = 12 ...(2)
On solving eq. (1) and (2) we get a = 30, b = 18, c = 32
Again (a +x +k + z) - (a+k) = (x + z)
= 90 - (30 + 20) = 40
and (x + y + z) - (x + z) = y =70 - 40 = 30
Similarly x = 25 and z = 15

Directions (5 - 7): Read the following information and answer the questions below.
50 students sing, 60 students do not dance and 25 students do both in a class of 100 students.
5. How many students neither sing nor dance?
a) 65
b) 35
c) 15
d) Cannot be determined
Explanation: Students who neither sing nor dance = Total students – Students who do sing alone, dance alone and do both
From Venn diagram, Students who neither sing nor dance = 100 - (25+25+15) = 35
(See the Venn diagram and explanation at the end of the questions)

Given,
Total students = 100
Number of students who sing = 50
Number of students who do not dance = 60
So, number of students who dance = 100-60= 40 (Total students- Students who do not dance)
Students who dance and sing = 25 (Overlapping portion in the Venn diagram)
Number of students who sing alone = 50-25 = 25 (Students who sing- Students who sing and dance)
Number of students who dance alone= 40-25 = 15 (Students who dance- Students who sing and dance)

6. How many students only dance?
a) 10
b) 15
c) 20
d) Cannot be determined
Explanation: Students who neither sing nor dance = Total students – Students who do sing alone, dance alone and do both
From Venn diagram, Students who neither sing nor dance = 100 - (25+25+15) = 35
(See the Venn diagram and explanation at the end of the questions)

Given,
Total students = 100
Number of students who sing = 50
Number of students who do not dance = 60
So, number of students who dance = 100-60= 40 (Total students- Students who do not dance)
Students who dance and sing = 25 (Overlapping portion in the Venn diagram)
Number of students who sing alone = 50-25 = 25 (Students who sing- Students who sing and dance)
Number of students who dance alone= 40-25 = 15 (Students who dance- Students who sing and dance)

7. How many students at least do either sing or dance?
a) 60
b) 50
c) 75
d) 65
Explanation: Students who neither sing nor dance = Total students – Students who do sing alone, dance alone and do both
From Venn diagram, Students who neither sing nor dance = 100 - (25+25+15) = 35
(See the Venn diagram and explanation at the end of the questions)

Given,
Total students = 100
Number of students who sing = 50
Number of students who do not dance = 60
So, number of students who dance = 100-60 = 40 (Total students- Students who do not dance)
Students who dance and sing = 25 (Overlapping portion in the Venn diagram)
Number of students who sing alone = 50-25 = 25 (Students who sing- Students who sing and dance)
Number of students who dance alone = 40-25 = 15 (Students who dance- Students who sing and dance)

There are 120 employees who work for Airtel Pvt. Ltd. Mumbai, out of which 50 are women. Also:
I. 56 workers are married
III. 40 married workers are graduate of which 18 are men
V. 30 men are married.
8. How many unmarried women are graduate?
a) 22
b) 16
c) 0
d) can’t be determined
Explanation: No one unmarried woman is graduate.

Total number of employees = 120
Women = 50
Men = 70
Married workers = 56
a → unmarried men who are not graduate
b → married women who are not graduate
c → unmarried women who are graduate
x → married men who are not graduate
y → married women who are graduate
z → unmarried men who are graduate
k → married men who are graduate
p → unmarried women who are not graduate
According to the given information the Venn diagram can be completed as given below.

9. How many unmarried women are graduate?
a) 21
b) 24
c) 19
d) none of these
Explanation: Number of unmarried women = 120 - [28 + 4 + 12 + 12 + 22 + 18] = 24

Total number of employees = 120
Women = 50
Men = 70
Married workers = 56
a → unmarried men who are not graduate
b → married women who are not graduate
c → unmarried women who are graduate
x → married men who are not graduate
y → married women who are graduate
z → unmarried men who are graduate
k → married men who are graduate
p → unmarried women who are not graduate
According to the given information the Venn diagram can be completed as given below.

10. How many graduate men are married?
a) 18
b) 15
c) 13
d) none of these
Explanation: There are 18 graduate men who are married.

Total number of employees = 120
Women = 50
Men = 70
Married workers = 56
a → unmarried men who are not graduate
b → married women who are not graduate
c → unmarried women who are graduate
x → married men who are not graduate
y → married women who are graduate
z → unmarried men who are graduate
k → married men who are graduate
p → unmarried women who are not graduate
According to the given information the Venn diagram can be completed as given below.

The document Questions With Answers: Venn Diagrams | CSAT Preparation - UPSC is a part of the UPSC Course CSAT Preparation.
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## FAQs on Questions With Answers: Venn Diagrams - CSAT Preparation - UPSC

 1. What is a Venn diagram?
A Venn diagram is a visual representation of the relationships between different sets or groups of objects. It consists of overlapping circles, where each circle represents a set and the overlapping region represents the elements that are common to both sets.
 2. How can Venn diagrams be used in problem-solving?
Venn diagrams are commonly used in problem-solving to visually organize information and determine relationships between different sets of data. They can help in identifying common elements, finding the union or intersection of sets, and understanding the relationships between multiple sets.
 3. What is the purpose of the intersection in a Venn diagram?
The intersection in a Venn diagram represents the elements that are common to two or more sets. It shows the overlap or the shared elements between the sets being compared.
 4. How can Venn diagrams be used to compare and contrast different groups?
Venn diagrams can be used to compare and contrast different groups by representing each group as a separate set in the diagram. The overlapping regions then show the similarities and differences between the groups based on the elements that are common or unique to each group.
 5. Can Venn diagrams be used for more than two sets?
Yes, Venn diagrams can be used for more than two sets. In such cases, additional circles are added to the diagram, representing each additional set. The overlapping regions then show the relationships between all the sets being compared.

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