Question

in a group of 80 employees, the number of employees who are engineers is twice that of the employees who are MBAs. The number of employees who are not engineers is 32 and that of those who are not MiLAs is \(56 .\) The number of employees who are both engineers and MBAs is twice that of the employees who are only MBAs. How many employees are neither engineer (B. Tech) nor MBAs ? (a) 24 (b) 38 (c) 36 (d) can't be determined

Short answer

Based on our analysis and three step solution, we can determine that there are 24 employees who are neither engineers nor MBAs. Step 1: We determined the total of MBAs and engineers based on the information given. Step 2: We found out the number of employees who are solely MBAs as well as those who are both engineers and MBAs. Step 3: We found out the number of employees who are neither engineers nor MBAs by subtracting the number of employees who are either engineers, MBAs or both from the total number of employees. This turned out to be 24. So, the correct answer is: (a) 24.

Step-by-step solution

Find the total number of MBAs and engineers

According to the given information, the number of engineers is twice the number of MBAs. We know that the number of employees who are not engineers is 32, so the number of engineers must be the total employees minus those who are not engineers, which is 80 - 32 = 48. Since engineers are twice the number of MBAs, the total number of MBAs is 48 / 2 = |\frac{48}{2}|= 24.

Find the number of employees who are only MBAs and those who are both engineers and MBAs

We are given that the number of employees who are both engineers and MBAs is twice that of employees who are only MBAs. Using the total number of employees who are MBAs, we define x as the number of employees who are only MBAs. So the number of employees who are both engineers and MBAs is 2x. Adding these two quantities up gives us the total number of MBAs: x + 2x = 24. Solving for x, we have 3x = 24 => x = |\frac{24}{3}| = 8. So there are 8 employees who are only MBAs and 16 employees who are both engineers and MBAs.

Find the number of employees who are neither engineers nor MBAs

The number of employees who are only engineers is calculated as the total number of engineers minus the employees who are both engineers and MBAs: 48 - 16 = 32. Now, we can find the total number of employees who are engineers, MBAs or both: 32 (only engineers) + 8 (only MBAs) + 16 (both engineers and MBAs) = 56. Finally, we find the number of employees who are neither engineers nor MBAs by subtracting the number of employees who are engineers, MBAs or both from the total number of employees: 80 - 56 = 24 So there are 24 employees who are neither engineers nor MBAs. The correct answer is: (a) 24.

Key concepts

Set Theory

Set theory is a crucial concept in mathematics that helps us understand collections of objects, known as sets. It provides the tools necessary to organize and analyze data based on shared characteristics. Here are some important points to grasp:

• **Sets**: A set is a collection of distinct objects, typically represented within curly brackets, like \{A, B, C\}.
• **Operations**: We use operations such as unions and intersections to combine or compare sets. For example, the union of two sets A and B, denoted as \(A \cup B\), includes all elements from both A and B.
• **Venn Diagrams**: These allow us to visually represent sets and their relationships, making it easier to understand complex combinations of data.

In our exercise, the group of engineers and MBAs represents two separate sets with some overlap. Employees who are both engineers and MBAs belong to the intersection of these sets. By using set theory principles, we figure out how many are neither by subtracting the union of both sets from the total population.

Problem Solving

Problem solving involves applying logical and mathematical strategies to find a solution to a given question or problem. In tackling problems like the one in our exercise, we use a few defining approaches:

• **Understanding the Problem**: Break down the problem statement to capture essential data and requirements. Identify fixed numbers or relationships.
• **Developing a Plan**: Formulate a strategy using known relationships and constraints. For instance, in this problem, you have the total employees and relationships between engineers and MBAs to map out a solution path.
• **Executing the Plan**: Implement mathematical calculations step by step. Our plan includes determining the number of engineers and MBAs based on the provided relationships.
• **Review**: Double-check calculations and logic to ensure the answer makes sense.

Our problem-solving process in this context helps us find how many employees are neither engineers nor MBAs by leveraging known data and performing systematic deductions.

Logical Reasoning

Logical reasoning gives us the framework to draw sound conclusions from given premises or data points. It involves:

• **Analyzing Relationships**: Understanding the relationships between different data points, such as the link between the number of engineers and MBAs.
• **Making Inferences**: Derive conclusions based on established facts. In our exercise, facts like the total number of engineers and the specific relationships between these groups guide our logical deductions.
• **Deductive and Inductive Reasoning**: Deductive reasoning uses facts to derive one specific conclusion, while inductive reasoning uses patterns to suggest probable outcomes.

In the exercise, logical reasoning helps us incrementally connect the relationships between sets, resulting in the answer. We identify mutually exclusive and overlapping groups, determining employees who fit neither category through logical progression.