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Practice Question - 12 (Venn Diagram) | 100 DILR Questions for CAT Preparation PDF Download

There are 15 girls and some boys among the graduating students in a class. They are planning a get-together, which can be either a 1-day event, or a 2-day event, or a 3-day event. There are 6 singers in the class, 4 of them are boys. There are 10 dancers in the class, 4 of them are girls. No dancer in the class is a singer.
Some students are not interested in attending the get-together. Those students who are interested in attending a 3-day event are also interested in attending a 2-day event; those who are interested in attending a 2-day event are also interested in attending a 1-day event.
The following facts are also known:
1. All the girls and 80% of the boys are interested in attending a 1-day event. 60% of the boys are interested in attending a 2-day event.
2. Some of the girls are interested in attending a 1-day event, but not a 2-day event; some of the other girls are interested in attending both.
3. 70% of the boys who are interested in attending a 2-day event are neither singers nor dancers. 60% of the girls who are interested in attending a 2-day event are neither singers nor dancers.
4. No girl is interested in attending a 3-day event. All male singers and 2 of the dancers are interested in attending a 3-day event.
5. The number of singers interested in attending a 2-day event is one more than the number of dancers interested in attending a 2-day event.

Q1: How many boys are there in the class?

Practice Question - 12 (Venn Diagram) | 100 DILR Questions for CAT PreparationView Answer  Practice Question - 12 (Venn Diagram) | 100 DILR Questions for CAT Preparation

Ans: 50
No. of girls = 15
Let the no. of boys be x
No. of singers = 6
No of boys who are singers = 4
Therefore, no of girls who are singers = 2
No of dancers = 10
No of boys who are dancers = 6
Therefore, no. of girls who are dancers = 4
No. of boys who are neither singers nor dancers = x-10
No. of girls who are neither singers nor dancers = 9

Practice Question - 12 (Venn Diagram) | 100 DILR Questions for CAT Preparation

Now we fill the above table, using statements 1 and 2, we get the following table

Practice Question - 12 (Venn Diagram) | 100 DILR Questions for CAT Preparation

Let the number of girls who are interested in attending a 2-day event be a and the number of girls who are dancers and are interested in 2-day event be b.
Now using statements 3 and 4, we get

Practice Question - 12 (Venn Diagram) | 100 DILR Questions for CAT Preparation

20.18x462 ≤ 0.18x − 4 ≤ 6
6 ≤ 0.18x ≤ 10
0.18x should be integer for which x should be a multiple of 50, and 0.18x lies between 6 and 10; therefore, the only possible value of x is 50.

Practice Question - 12 (Venn Diagram) | 100 DILR Questions for CAT Preparation

From statement 5, we can say that,
4+ 0.4a - b = 5+b +1
or, 0.4a = 2+2b
or, a = 5(1+b)
a should be a multiple of 5 as b is a whole number. So possible values of a can be 5, 10 or 15. Now, as the maximum value of b can be 4 and the maximum value of 0.4a-b can be 2, so the only possible value of a satisfying the conditions above is 5. If a= 5 then b=1.

Practice Question - 12 (Venn Diagram) | 100 DILR Questions for CAT Preparation

Q2: Which of the following can be determined from the given information?
I. The number of boys who are interested in attending a 1-day event and are neither dancers nor singers.
II. The number of female dancers who are interested in attending a 1-day event.
(a) Only I
(b) Neither I nor II
(c) Only II
(d) Both I and II

Practice Question - 12 (Venn Diagram) | 100 DILR Questions for CAT PreparationView Answer  Practice Question - 12 (Venn Diagram) | 100 DILR Questions for CAT Preparation

Ans: (c) 
No. of girls = 15
Let the no. of boys be x
No. of singers = 6
No of boys who are singers = 4
Therefore, no of girls who are singers = 2
No of dancers = 10
No of boys who are dancers = 6
Therefore, no. of girls who are dancers = 4
No. of boys who are neither singers nor dancers = x-10
No. of girls who are neither singers nor dancers = 9

Practice Question - 12 (Venn Diagram) | 100 DILR Questions for CAT Preparation

Now we fill the above table, using statements 1 and 2, we get the following table

Practice Question - 12 (Venn Diagram) | 100 DILR Questions for CAT Preparation

Let the number of girls who are interested in attending a 2-day event be a and the number of girls who are dancers and are interested in 2-day event be b.
Now using statements 3 and 4, we get

Practice Question - 12 (Venn Diagram) | 100 DILR Questions for CAT Preparation

20.18x462 ≤ 0.18x − 4 ≤ 6
6 ≤ 0.18x ≤ 10
0.18x should be integer for which x should be a multiple of 50, and 0.18x lies between 6 and 10; therefore, the only possible value of x is 50.

Practice Question - 12 (Venn Diagram) | 100 DILR Questions for CAT Preparation

From statement 5, we can say that,
4+ 0.4a - b = 5+b +1
or, 0.4a = 2+2b
or, a = 5(1+b)
a should be a multiple of 5 as b is a whole number. So possible values of a can be 5, 10 or 15. Now, as the maximum value of b can be 4 and the maximum value of 0.4a-b can be 2, so the only possible value of a satisfying the conditions above is 5. If a = 5 then b =1.

Practice Question - 12 (Venn Diagram) | 100 DILR Questions for CAT Preparation

Q3: What fraction of the class are interested in attending a 2-day event?
(a) 7/10
(b) 7/13
(c) 9/13
(d) 2/3

Practice Question - 12 (Venn Diagram) | 100 DILR Questions for CAT PreparationView Answer  Practice Question - 12 (Venn Diagram) | 100 DILR Questions for CAT Preparation

Ans: (b)
No. of girls = 15
Let the no. of boys be x
No. of singers = 6
No of boys who are singers = 4
Therefore, no of girls who are singers = 2
No of dancers = 10
No of boys who are dancers = 6
Therefore, no. of girls who are dancers = 4
No. of boys who are neither singers nor dancers = x-10
No. of girls who are neither singers nor dancers = 9

Practice Question - 12 (Venn Diagram) | 100 DILR Questions for CAT Preparation

Now we fill the above table, using statements 1 and 2, we get the following table

Practice Question - 12 (Venn Diagram) | 100 DILR Questions for CAT Preparation

Let the number of girls who are interested in attending a 2-day event be a and the number of girls who are dancers and are interested in 2-day event be b.
Now using statements 3 and 4, we get

Practice Question - 12 (Venn Diagram) | 100 DILR Questions for CAT Preparation

2 ≤ 0.18x − 4 ≤ 6
6 ≤ 0.18x ≤ 10
0.18x should be integer for which x should be a multiple of 50, and 0.18x lies between 6 and 10; therefore, the only possible value of x is 50.

Practice Question - 12 (Venn Diagram) | 100 DILR Questions for CAT Preparation

From statement 5, we can say that,
4+ 0.4a - b = 5+b +1
or, 0.4a = 2+2b
or, a = 5(1+b)
a should be a multiple of 5 as b is a whole number. So possible values of a can be 5, 10 or 15. Now, as the maximum value of b can be 4 and the maximum value of 0.4a-b can be 2, so the only possible value of a satisfying the conditions above is 5. If a= 5 then b=1.

Practice Question - 12 (Venn Diagram) | 100 DILR Questions for CAT Preparation

Total students interested in 2-day event = 30+5 = 35
Total students = 15 + 50 = 65
Fraction = 35/65 = 7/13

Q4:  What BEST can be concluded about the number of male dancers who are interested in attending a 1-day event?
(a) 5 or 6
(b) 6
(c) 5
(d) 4 or 6

Practice Question - 12 (Venn Diagram) | 100 DILR Questions for CAT PreparationView Answer  Practice Question - 12 (Venn Diagram) | 100 DILR Questions for CAT Preparation

Ans: (a)
No. of girls = 15
Let the no. of boys be x
No. of singers = 6
No of boys who are singers = 4
Therefore, no of girls who are singers = 2
No of dancers = 10
No of boys who are dancers = 6
Therefore, no. of girls who are dancers = 4
No. of boys who are neither singers nor dancers = x-10
No. of girls who are neither singers nor dancers = 9

Practice Question - 12 (Venn Diagram) | 100 DILR Questions for CAT Preparation

Now we fill the above table, using statements 1 and 2, we get the following table

Practice Question - 12 (Venn Diagram) | 100 DILR Questions for CAT Preparation

Let the number of girls who are interested in attending a 2-day event be a and the number of girls who are dancers and are interested in 2-day event be b.
Now using statements 3 and 4, we get

Practice Question - 12 (Venn Diagram) | 100 DILR Questions for CAT Preparation

20.18x462 ≤ 0.18x − 4 ≤ 6
6 ≤ 0.18x ≤ 10
0.18x should be integer for which x should be a multiple of 50, and 0.18x lies between 6 and 10; therefore, the only possible value of x is 50.

Practice Question - 12 (Venn Diagram) | 100 DILR Questions for CAT Preparation

From statement 5, we can say that,
4+ 0.4a - b = 5+b +1
or, 0.4a = 2+2b
or, a = 5(1+b)
a should be a multiple of 5 as b is a whole number. So possible values of a can be 5, 10 or 15. Now, as the maximum value of b can be 4 and the maximum value of 0.4a-b can be 2, so the only possible value of a satisfying the conditions above is 5. If a = 5 then b = 1.

Practice Question - 12 (Venn Diagram) | 100 DILR Questions for CAT Preparation

Q5: How many female dancers are interested in attending a 2-day event?
(a) 2
(b) 1
(c) 0
(d) Cannot be determined

Practice Question - 12 (Venn Diagram) | 100 DILR Questions for CAT PreparationView Answer  Practice Question - 12 (Venn Diagram) | 100 DILR Questions for CAT Preparation

Ans: (c)
No. of girls = 15
Let the no. of boys be x
No. of singers = 6
No of boys who are singers = 4
Therefore, no of girls who are singers = 2
No of dancers = 10
No of boys who are dancers = 6
Therefore, no. of girls who are dancers = 4
No. of boys who are neither singers nor dancers = x-10
No. of girls who are neither singers nor dancers = 9

Practice Question - 12 (Venn Diagram) | 100 DILR Questions for CAT Preparation

Now we fill the above table, using statements 1 and 2, we get the following table

Practice Question - 12 (Venn Diagram) | 100 DILR Questions for CAT Preparation

Let the number of girls who are interested in attending a 2-day event be a and the number of girls who are dancers and are interested in 2-day event be b.
Now using statements 3 and 4, we get

Practice Question - 12 (Venn Diagram) | 100 DILR Questions for CAT Preparation

20.18x462 ≤ 0.18x − 4 ≤ 6
6 ≤ 0.18x ≤ 10
0.18x should be integer for which x should be a multiple of 50, and 0.18x lies between 6 and 10; therefore, the only possible value of x is 50.

Practice Question - 12 (Venn Diagram) | 100 DILR Questions for CAT Preparation

From statement 5, we can say that,
4+ 0.4a - b = 5+b +1
or, 0.4a = 2+2b
or, a = 5(1+b)
a should be a multiple of 5 as b is a whole number. So possible values of a can be 5, 10 or 15. Now, as the maximum value of b can be 4 and the maximum value of 0.4a-b can be 2, so the only possible value of a satisfying the conditions above is 5. If a= 5 then b=1.

Practice Question - 12 (Venn Diagram) | 100 DILR Questions for CAT Preparation

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FAQs on Practice Question - 12 (Venn Diagram) - 100 DILR Questions for CAT Preparation

1. What is a Venn diagram, and how is it used in problem-solving?
Ans. A Venn diagram is a visual representation used to illustrate the relationships between different sets. It consists of overlapping circles, where each circle represents a set, and the overlaps indicate common elements. In problem-solving, Venn diagrams help to organize information, identify relationships, and make comparisons between sets, making it easier to analyze data and draw conclusions.
2. How can Venn diagrams be applied to solve problems in competitive exams?
Ans. In competitive exams, Venn diagrams are often used to solve problems related to set theory, logic, and probability. They allow candidates to visually map out the relationships between different groups, making it easier to answer questions about the number of elements in each set, those that belong to multiple sets, or to find the union and intersection of sets. This visual aid simplifies complex problems, enabling quicker and more accurate responses.
3. What types of questions can be expected in exams that involve Venn diagrams?
Ans. Questions involving Venn diagrams can vary widely but typically include scenarios where candidates must determine the number of elements in specific categories, understand the relationships between different groups, or calculate probabilities based on set interactions. Common question types include finding the union (total elements in all sets), intersection (elements common to all sets), and exclusive elements (those in one set but not in others).
4. What are some common mistakes to avoid when interpreting Venn diagrams?
Ans. Common mistakes include miscounting elements in each section, confusing the relationships between sets (e.g., assuming all elements in one set are also in another), and neglecting to account for overlaps correctly. It's essential to carefully analyze each section of the diagram and ensure a clear understanding of how the sets interact before arriving at a conclusion.
5. Can Venn diagrams be used beyond mathematics, and if so, how?
Ans. Yes, Venn diagrams can be applied in various fields beyond mathematics, including logic, statistics, computer science, and even everyday decision-making. They are useful for visualizing relationships in data sets, comparing features of different products, analyzing survey results, and categorizing information in a clear, structured manner. This versatility makes them a valuable tool in both academic and professional settings.
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