Questions With Answers: Venn Diagrams | General Aptitude for GATE - Mechanical Engineering PDF Download

Venn diagrams help visualize the relationships between different sets, making it easier to solve problems involving logical connections, intersections, and exclusions. Through the questions and answers provided, students will strengthen their ability to analyze and interpret data, enhancing their logical thinking skills.

Directions (1 -4): Read the following information and answer the questions below.
In a coaching institute, there are a total of 170 students and they study 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.
The number of students taking Maths only exceeds the number of students of Economics only by 14.
The number of students studying English only exceeds the number of students of Economics only by 12. The 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
Answer: c

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

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
Answer: c

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
Answer: d

Explanation:
Let 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
Answer: b

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

6. How many students only dance?
a) 10
b) 15
c) 20
d) Cannot be determined
Answer: b

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

7. How many students at least do either sing or dance?
a) 60
b) 50
c) 75
d) 65
Answer: d

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 =

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

Students who sing- Students who sing and dance
50-25 = 25 
Number of students who dance alone = 

Students who dance- Students who sing and dance

= 40-25 = 15

Direction (8-9): Read the following information and answer the questions below.
There are 120 employees who work for Airtel Pvt. Ltd. Mumbai, out of which 50 are women. Also:
I. 56 workers are married
II. 52 workers are graduate
III. 40 married workers are graduate of which 18 are men
IV. 30 men are graduate
V. 30 men are married.
8. How many unmarried women are graduate?
a) 22
b) 16
c) 0
d) can’t be determined
Answer: c

Explanation: No one unmarried woman is graduate.

Number of men in total: Since there are 50 women, the total number of men is:
120 - 50 = 70
Number of married women: There are 56 married workers in total. Out of these, 18 are men (married graduates), so the number of married women is:
56 - 30 = 26 (since there are 30 married men)
Number of unmarried women: The total number of women is 50, and the number of married women is 26, so the number of unmarried women is:
50 - 26 = 24
Graduate women: Since there are 52 graduate workers in total, and 30 men are graduates, the number of graduate women is:
52 - 30 = 22
Graduate women who are married: We know that 40 married workers are graduates, and 18 of them are men. So, the number of married women who are graduates is:
40 - 18 = 22
Unmarried graduate women: The number of unmarried women who are graduates is:
22 - 22 = 0

9. How many graduate men are married?
a) 18
b) 15
c) 13
d) none of these
Answer: a

Explanation: 
From the information provided:
Total number of married workers = 56
Total number of graduate workers = 52
Graduate men = 30
Married graduate workers = 40
Married men = 30
Out of 40 married graduate workers, 18 are men.
Thus, the number of married graduate men is 18.

So, the correct answer is:

Option A) 18.