Question

Of the 150 employees at company \(X, 80\) are full-time and 100 have worked at company \(X\) for at least a year. There are 20 employees at company \(X\) who aren't full-time and haven't worked at company \(X\) for at least a year. How many fulltime employees of company \(X\) have worked at the company for at least a year? A 20 B 30 C 50 D 80 E 100

Short answer

The number of full-time employees who have worked for at least a year is 50 (option C).

Step-by-step solution

Define the sets and their intersections

Let set A represent the full-time employees and set B represent the employees who have worked at the company for at least a year. We are given that: 1) \( |A| = 80 \) (number of full-time employees) 2) \( |B| = 100 \) (number of employees who have worked for at least a year) 3) |total employees| = 150 4) There are 20 employees who are neither full-time nor have worked for at least a year.

Calculate the number of employees who are either full-time or have worked at least a year

We know that the total number of employees is 150, and 20 of these are not in sets A or B. Therefore, the number of employees in set A union B is: \[ |A \bigcup B| = 150 - 20 = 130 \]

Apply the principle of inclusion-exclusion

According to the principle of inclusion-exclusion for two sets, we have: \[ |A \bigcup B| = |A| + |B| - |A \bigcap B| \] Plugging in the values, we get: \[ 130 = 80 + 100 - |A \bigcap B| \]

Solve for the intersection set

Rearrange the equation to solve for \( |A \bigcap B| \): \[ |A \bigcap B| = 80 + 100 - 130 = 50 \] Thus, the number of full-time employees who have worked at the company for at least a year is 50.

Key concepts

Venn diagrams

Venn diagrams are a powerful visual tool to represent sets and their relationships with each other.
They help us see how different groups overlap and interact.
In this exercise, we can use a Venn diagram to understand the relationship between full-time employees and those who have worked at the company for at least a year.
The circles represent the sets of full-time employees and employees who have been with the company for a year.
Where the circles overlap, we find employees who belong to both sets.
Venn diagrams make it easier to manage complex information and simplify problem-solving by visualizing what we are trying to calculate.

sets and intersections

Sets are collections of distinct items, and in our exercise, we have two sets:
full-time employees and those who have worked at the company for at least a year.
Set A (full-time employees) has 80 members, and Set B (employees who have worked for a year) has 100 members.
Intersections represent elements common to both sets, which we need to find.
The intersection between two sets (A and B) can be found using the principle of inclusion-exclusion.
This principle helps calculate the number of items that belong to both sets, providing valuable insights into overlapping categories.
Understanding intersections is crucial for answering many common questions related to groups and categories.

principle of inclusion-exclusion

The principle of inclusion-exclusion is a fundamental concept in set theory.
It is used to calculate the size of the union of two or more sets, taking into account the overlap between them.
For two sets A and B, the principle is expressed as:
ewline ewline ewline \[ |A \bigcup B| = |A| + |B| - |A \bigcap B| \]
ewline ewlineewlineewlineewline This formula helps us accurately determine the number of items in the union of two sets by subtracting the double-counted items in the intersection.
In our exercise, we first calculated the union of full-time employees and employees who worked for a year:ewline \[ |A \bigcup B| = 150 - 20 = 130 \] ewlineewline Then, we applied the principle of inclusion-exclusion to find the intersection:
\[130 = 80 + 100 - |A \bigcap B|\] ewlineewline Solving this, we found that: \[|A \bigcap B| = 50 \]
This showed us there are 50 full-time employees who have worked at the company for at least a year.

quantitative reasoning

Quantitative reasoning involves applying mathematical concepts and techniques to solve problems.
In this exercise, we used quantitative reasoning to solve for the number of employees who meet both criteria:
being full-time and having worked at the company for at least a year.
We did this by defining our sets, performing calculations, and applying the principle of inclusion-exclusion.
Quantitative reasoning also includes understanding relationships and dependencies between different variables.
By breaking down the problem into smaller, manageable parts and using logical steps, we can achieve accurate solutions.
Skills in quantitative reasoning are essential for making informed decisions based on numerical data and relationships.