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Practice Question - 13 (Venn Diagram) | 100 DILR Questions for CAT Preparation PDF Download

A survey of 600 schools in India was conducted to gather information about their online teaching learning processes (OTLP). The following four facilities were studied. 
F1: Own software for OTLP
F2: Trained teachers for OTLP
F3: Training materials for OTLP
F4: All students having Laptops 
The following observations were summarized from the survey. 
1. 80 schools did not have any of the four facilities - F1, F2, F3, F4.
2. 40 schools had all four facilities.
3. The number of schools with only F1, only F2, only F3, and only F4 was 25, 30, 26 and 20 respectively.
4. The number of schools with exactly three of the facilities was the same irrespective of which three were considered.
5. 313 schools had F2.
6. 26 schools had only F2 and F3 (but neither F1 nor F4).
7. Among the schools having F4, 24 had only F3, and 45 had only F2.
8. 162 schools had both F1 and F2.
9. The number of schools having F1 was the same as the number of schools having F4.

Q1: What was the total number of schools having exactly three of the four facilities?
(a) 64
(b) 50
(c) 200
(d) 80

Practice Question - 13 (Venn Diagram) | 100 DILR Questions for CAT PreparationView Answer  Practice Question - 13 (Venn Diagram) | 100 DILR Questions for CAT Preparation

Ans: (c) 
Let the number of schools with exactly three of the facilities was the same irrespective of which three were considered be x.
Number of schools with none of the facilities be 'n' from 1, n=80. 
Number of schools with only F1 and F2 be 'b'
Number of schools with only F1 and F3 be 'c'
Number of schools with only F1 and F4 be 'd'
From the information given in the question we will get the following Venn diagram.
Practice Question - 13 (Venn Diagram) | 100 DILR Questions for CAT Preparation

From 5, b+141+3x=313 => b+3x=172....(i)
From 8, b+x+40+x=162 => b+2x=122....(ii)
(ii)-(i) gives x=50 => b=22
From 9, 237+3x+c+d=279+3x=d => c=42
Total number of schools =600 => 313+25+c+x+d+26+24+20+80=600 => d=20.
The final table looks like:
Practice Question - 13 (Venn Diagram) | 100 DILR Questions for CAT Preparation

The total number of schools with exactly three of the four facilities= 4x = 200.

Q2: What was the number of schools having facilities F2 and F4?
(a) 185
(b) 95
(c) 45
(d) 85

Practice Question - 13 (Venn Diagram) | 100 DILR Questions for CAT PreparationView Answer  Practice Question - 13 (Venn Diagram) | 100 DILR Questions for CAT Preparation

Ans: (a)
Let the number of schools with exactly three of the facilities was the same irrespective of which three were considered be x.
Number of schools with none of the facilities be 'n' from 1, n=80. 
Number of schools with only F1 and F2 be 'b'
Number of schools with only F1 and F3 be 'c'
Number of schools with only F1 and F4 be 'd'
From the information given in the question we will get the following Venn diagram.
Practice Question - 13 (Venn Diagram) | 100 DILR Questions for CAT Preparation

From 5, b+141+3x=313 => b+3x=172....(i)
From 8, b+x+40+x=162 => b+2x=122....(ii)
(ii)-(i) gives x=50 => b=22
From 9, 237+3x+c+d=279+3x=d => c=42
Total number of schools =600 => 313+25+c+x+d+26+24+20+80=600 => d=20.
The final table looks like:
Practice Question - 13 (Venn Diagram) | 100 DILR Questions for CAT Preparation

The total number of schools having facilities F4 and F2= 45+50+50+40=185.

Q3: What was the number of schools having only facilities F1 and F3?

Practice Question - 13 (Venn Diagram) | 100 DILR Questions for CAT PreparationView Answer  Practice Question - 13 (Venn Diagram) | 100 DILR Questions for CAT Preparation

Ans: 42
Let the number of schools with exactly three of the facilities was the same irrespective of which three were considered be x.
Number of schools with none of the facilities be 'n' from 1, n=80. 
Number of schools with only F1 and F2 be 'b'
Number of schools with only F1 and F3 be 'c'
Number of schools with only F1 and F4 be 'd'
From the information given in the question we will get the following Venn diagram.
Practice Question - 13 (Venn Diagram) | 100 DILR Questions for CAT Preparation

From 5, b+141+3x=313 => b+3x=172....(i)
From 8, b+x+40+x=162 => b+2x=122....(ii)
(ii)-(i) gives x=50 => b=22
From 9, 237+3x+c+d=279+3x=d => c=42
Total number of schools =600 => 313+25+c+x+d+26+24+20+80=600 => d=20.
The final table looks like:
Practice Question - 13 (Venn Diagram) | 100 DILR Questions for CAT Preparation

The total number of schools having only F1 and F3 = c = 42

Q4: What was the number of schools having only facilities F1 and F4?

Practice Question - 13 (Venn Diagram) | 100 DILR Questions for CAT PreparationView Answer  Practice Question - 13 (Venn Diagram) | 100 DILR Questions for CAT Preparation

Ans: 20
Let the number of schools with exactly three of the facilities was the same irrespective of which three were considered be x.
Number of schools with none of the facilities be 'n' from 1, n=80. 
Number of schools with only F1 and F2 be 'b'
Number of schools with only F1 and F3 be 'c'
Number of schools with only F1 and F4 be 'd'
From the information given in the question we will get the following Venn diagram.
Practice Question - 13 (Venn Diagram) | 100 DILR Questions for CAT Preparation

From 5, b+141+3x=313 => b+3x=172....(i)
From 8, b+x+40+x=162 => b+2x=122....(ii)
(ii)-(i) gives x=50 => b=22
From 9, 237+3x+c+d=279+3x=d => c=42
Total number of schools =600 => 313+25+c+x+d+26+24+20+80=600 => d=20.
The final table looks like:
Practice Question - 13 (Venn Diagram) | 100 DILR Questions for CAT Preparation

The total number of schools having only F1 and F4=20.

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FAQs on Practice Question - 13 (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 that uses overlapping circles to illustrate the relationships between different sets. In problem-solving, it helps to identify commonalities and differences among groups, making it easier to analyze and interpret data.
2. How can Venn diagrams be applied in competitive exams?
Ans. Venn diagrams are commonly used in competitive exams to assess analytical skills. They help candidates visualize problems involving sets, such as determining the number of elements in various categories, which is crucial for answering questions related to logical reasoning and data interpretation.
3. What types of questions can be solved using Venn diagrams in exams?
Ans. Questions that can be solved using Venn diagrams include those that ask for the number of elements in union, intersection, or exclusive sections of sets. For example, questions may involve calculating how many people belong to one group but not another or how many belong to both groups.
4. What are the common mistakes to avoid when using Venn diagrams?
Ans. Common mistakes include mislabeling the circles, failing to account for overlaps correctly, and misunderstanding the question's requirements regarding union and intersection. It's important to carefully read the problem and double-check the placement of elements in the diagram.
5. Can Venn diagrams be used for more than two sets?
Ans. Yes, Venn diagrams can be extended to represent three or more sets. While they become more complex with additional sets, they remain a useful tool for visualizing relationships and intersections among multiple groups, aiding in the analysis of larger data sets.
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