Question

In our coaching there were 200 students enrolled for DI, 150 for English and 150 for Maths. Of these 80 students enrolled for both DI and English. 60 students enrolled for Maths and English, while 70 students enrolled for DI and Maths. Some of these students enrolled for all the three subjects. Diwakar teaches those students who are enrolled for DI classes only. Priyanka teaches those students who are enrolled for English only and Varun teaches those students who are enrolled for Maths only. Sarvesh is a senior most faculty therefore, he can teach all the three subjects. Students always prefer a . specialist for their respective subjects. If Diwakar teaches 80 : students then the other three faculty can be arranged in terms . of the number of students taught as: (a) Sarvesh > Varun > Priyanka (b) Sarvesh > Priyanka > Varun (c) Varun > Sarvesh \(>\) Priyanka (d) none of the aboveDirections : Answer the question no. 27 to 29 on the basis of the information given below: Sanskaram Karoti (SK) is a spiritual organisation involved in performing spiritual rites. Currently it has 37 volunteers. They are involved in three jobs : Body Massage (BM), Yoga and Pooja. Each volunteer working with Sanskaram Karoti (SK) has to be involved in atleast one of the three jobs mentioned above. A maximum number of volunteers are involved in Yoga. Among them, the number of volunteers involved in Yoga alone is equal to the volunteers having additional involvement in the Pooja. The number of volunteers involved in Pooja alone is double the number of volunteers involved in all the three jobs. 17 volunteers are involved in Body Massage (BM). The number of volunteers involved in Body Massage alone is one less than the number of volunteers involved in Pooja alone. Ten volunteers involved in the Body Massage are also involved in atleast one more job.

Short answer

Explain your answer. Answer: No, it is not possible to determine the arrangement of the other three faculty members in terms of the number of students they teach. The given information leads to an unrealistic solution with negative values for the number of students. This indicates that the problem may be unsolvable as presented or that there is incorrect or insufficient information provided.

Step-by-step solution

Identifying the given facts from the exercise

From the given data, we have the following information for students enrolled in different classes: 1. Total DI (D) students = 200 2. Total English (E) students = 150 3. Total Maths (M) students = 150 4. DI and English students = 80 5. Maths and English students = 60 6. DI and Maths students = 70 7. Diwakar teaches DI students only (80 students). We need to find the arrangement of the other three faculty members in terms of students they teach.

Drawing a Venn diagram to represent the scenario

Draw a Venn diagram consisting of three overlapping circles representing DI, English, and Maths students respectively. Label the intersections and individual sections as follows: (a) DI only = x (b) English only = y (c) Maths only = z (d) DI and English only = 80 (given) (e) Maths and English only = 60 (given) (f) DI and Maths only = 70 (given) (g) All three = t (unknown)

Writing equations based on Venn diagram and given data

From the Venn diagram, we can write the following equations based on given data: 1. x + 80 + 70 + t = 200 (total DI students) 2. 80 + y + 60 + t = 150 (total English students) 3. 70 + 60 + z + t = 150 (total Maths students) Also, given that Diwakar teaches DI students only, we can write: 4. x = 80

Solving the equations

From equation 4, we know x = 80. Substitute this into the equation 1: 80 + 80 + 70 + t = 200 t = 200 - 230 = -30 However, this solution is not reasonable because the number of students (t) cannot be negative. We have made an error in interpreting the given information. We were informed that there were a total of 80 students who were enrolled in both DI and English, and Diwakar taught 80 students. This means Diwakar taught only DI students who were not enrolled in English classes. Therefore, the correct equation would be: x - 80 = 80 x = 160 Now, substituting the new value of x in equation 1: 160 + 80 + 70 + t = 200 t = 200 - 310 = -110 Again, the value of t cannot be negative. Recheck the interpretation of the information and confirm that equation 1 is correctly set up: x + 80 + 70 + t = 200 Since x = 160, then 160 + 80 + 70 + t = 200 t = -110 This result again implies there could be incorrect information given or the problem may be unsolvable as it is presented. Although we continued to get negative values for 't', from this solution approach, we could conclude that the given information might be either incorrect or insufficient to uniquely find the arrangement of the other three faculty members in terms of students they teach.

Key concepts

Venn Diagram

A Venn diagram is an essential tool for visualizing relationships between different sets or groups. These diagrams consist of overlapping circles, each representing a set with shared and unique characteristics. In the context of quantitative aptitude problems, Venn diagrams are tremendously helpful in understanding complex scenarios, such as multiple groups of students enrolled in different classes.

When constructing a Venn diagram for our given example, each circle would represent the students enrolled in DI, English, and Maths. By overlapping these circles, we can visually divide the students into seven distinct groups: those enrolled in just one subject, those enrolled in any two subjects, and those enrolled in all three. Each section is labeled with variables or known values, making it simple to write equations representing the relationships between the different groups.

For example, if we are told that Diwakar teaches 80 students from the DI-only group, we place that number in the respective section of the diagram. Creating a Venn diagram is the first step in unraveling the complexities of the question and lays the foundation for the formation of equations used in problem-solving.

Problem-Solving

In problem-solving, particularly in quantitative aptitude, we strategically follow steps to find a solution to a complex question. Each step is like a building block, clarifying components of the problem and getting us closer to the answer.

Initially, we identify the known quantities and information from the given scenario. Next, we represent this information visually with a Venn diagram, as seen in the exercise involving faculty and student enrollment. After that, we establish equations based on this visual representation. However, problem-solving isn't just about setting up equations; it also involves analyzing our solutions. If an equation yields an illogical outcome, such as a negative number of students, we must scrutinize our assumptions and the structure of our equations. Sometimes, errors or discrepancies in the given information become apparent only as we delve deeper into solving the problem, prompting a reevaluation of our approach.

Furthermore, it's crucial to realize that not all problems might have a conventional solution; some could be designed to test the ability to identify inconsistencies within the provided data. The act of problem-solving is as much about solving the problems as it is about detecting and adapting to potential flaws in the initial data or approach.

Set Theory

Set theory is a mathematical concept that deals with the collection of objects or elements, known as 'sets.' It's fundamental in understanding Venn diagrams and problem-solving, as it provides the language and notation to describe groups and their interactions. In set theory, groups of students in a class can be thought of as sets, with students as elements of these sets.

When we talk about students enrolled in DI, English, and Maths, we're referring to three sets with certain overlaps—these are intersections in set theory. For example, the students enrolled in both DI and Maths form the intersection of these two sets.

Moreover, set theory is about understanding not only the intersection but also the union (all students in any of the classes) and the relative complement (students in one set but not in another). Understanding these concepts is critical in interpreting the given data correctly. For instance, if Diwakar only teaches students enrolled exclusively in DI, those students are an element of the DI set but not of the intersections involving DI and any other subject.

Understanding set theory enables a clearer translation of word problems into mathematical representation—a pivotal step in problem-solving.