Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

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

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

Step by step solution

01

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.
02

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: \[ |A \bigcup B| = 150 - 20 = 130 \]
03

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: \[ 130 = 80 + 100 - |A \bigcap B| \]
04

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.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with Vaia!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Train A left Centerville Station, heading toward Dale City Station, at 3: 00 p.m. Train B left Dale City Station, heading toward Centerville Station, at \(3: 20 \mathrm{p} . \mathrm{m}\). on the same day. The trains rode on straight tracks that were parallel to each other. If Train A traveled at a constant speed of 30 miles per hour and Train B traveled at a constant speed of 10 miles per hour, and the distance between the Centerville Station and Dale City Station is 90 miles, when did the trains pass each other? A \(4: 45 \mathrm{p.m}\) B \(5: 00 \mathrm{p.m}\) C \(5: 20 \mathrm{p.m}\) D \(5: 35 \mathrm{p.m}\) E \(6: 00 \mathrm{p.m}\)

Did Jon complete a journey of 40 kilometers in less time than it took Ann to complete the same journey? (A) Jon traveled at an average speed of 30 kilometers per hour for the first 10 kilometers and then at an average speed of 15 kilometers per hour for the rest of the journey. (B) Ann traveled at an average speed of 20 kilometers per hour for the entire journey.

The average (arithmetic mean) of all scores on a certain algebra test was \(90 .\) If the average of the 8 male students' grades was \(87,\) and the average of the female students' grades was \(92,\) how many female students took the test? A. $$8$$ B. $$9$$ C. $$10$$ D. $$11$$ E. $$12$$

A canoeist paddled upstream at 10 meters per minute, turned around, and drifted downstream at 15 meters per minute. If the distance traveled in each direction was the same, and the time spent turning the canoe around was negligible, what was the canoeist's average speed over the course of the journey, in meters per minute? A 11.5 B 12 C 12.5 D 13 E 13.5

A truck owner will refuel his vehicle at a rate of 2 gallons of diesel fuel every 5 seconds. If diesel fuel costs \(\$ 1.25\) per gallon, how long will it take for the truck owner to refuel his vehicle with \(\$ 40\) worth of diesel fuel? A 20 seconds B 1 minute C 1 minute 20 seconds D 20 minutes E 1 hour 20 minutes

See all solutions

Recommended explanations on English Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free