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

The Financial News Daily has 25 reporters covering Asia, 20 covering Europe, and 20 covering North America. Four reporters cover Asia and Europe but not North America, 6 reporters cover Asia and North America but not Europe, and 7 reporters cover Europe and North America but not Asia. How many reporters cover all of the 3 continents (Asia, Europe, and North America)? (1) The Financial News Daily has 38 reporters in total covering at least 1 of the following continents: Asia, Europe, and North America. (2) There are more Financial News Daily reporters covering only Asia than there are Financial News Daily reporters covering only North America.

Short Answer

Expert verified
There are 4 reporters covering all three continents.

Step by step solution

01

- Define Variables

Let A be the number of reporters covering Asia, E be the number covering Europe, and N be the number covering North America. Let x be the number of reporters covering all three continents.
02

- Given Numbers

The problem states: A = 25, E = 20, N = 20. Additionally, there are 4 reporters covering Asia and Europe but not North America, 6 covering Asia and North America but not Europe, and 7 covering Europe and North America but not Asia.
03

- Set Up the Inclusion-Exclusion Principle

According to the principle: \[ |A \cup E \cup N| = |A| + |E| + |N| - |A \cap E| - |A \cap N| - |E \cap N| + |A \cap E \cap N| \]
04

- Account for Intersections

From the problem: \[ |A \cap E| = 4 + x \]\[ |A \cap N| = 6 + x \]\[ |E \cap N| = 7 + x \]
05

- Insert Values into the Inclusion-Exclusion Formula

Substitute all values including intersections into the inclusion-exclusion formula: \[ 38 = 25 + 20 + 20 - (4 + x) - (6 + x) - (7 + x) + x \]
06

- Solve the Equation

Simplify and solve: \[ 38 = 65 - 17 \] which simplifies to: \[ 38 - x = 48 - x \] resulting in: \[ x = 4 \]

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.

Inclusion-Exclusion Principle
The Inclusion-Exclusion Principle is a powerful tool in set theory and probability. It's especially useful when dealing with overlapping sets.
In essence, this principle helps you find the union of multiple sets by adjusting for overlap. If you just add the sizes of the sets, you overcount the elements that appear in more than one set.
The formula looks like this:
\[ |A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C| \] The terms subtracted represent the overlaps of the sets taken two at a time, and the term added back represents the overlap of all three sets.
This tool is very powerful for solving problems involving unions and intersections of multiple sets.
Set Intersections
Intersections help us understand the commonality between sets.
When we talk about the intersection of two sets like A and E, we denote it as \(|A \cap E|\). This represents elements that are in both sets A and E.
Here's how it works in our problem:
  • \(||A \cap E|| = 4 + x\), where 4 reporters cover both Asia and Europe but not North America, and x represents those who cover all three continents.
  • Similarly, \(||A \cap N|| = 6 + x\) and \(||E \cap N|| = 7 + x\).
This intersection method provides a granular view of the overlaps and helps in applying the Inclusion-Exclusion Principle effectively.
Logical Reasoning
Logical reasoning is crucial for problem-solving. It helps you break down complex problems into manageable parts.
In our example, understand the logical flow of the problem:
  • We define variables representing the number of reporters covering each continent and intersections.
  • Next, we account for given and hidden overlaps.
  • We then set up the problem using the Inclusion-Exclusion Principle and solve it step-by-step.
By following this logical path, you ensure all parts of the problem are addressed.
This logical flow enables you to navigate through the intricacies of the problem methodically.
Problem-Solving in Mathematics
Effective problem-solving involves several steps:
1. Clearly understand the problem. Read it multiple times if needed.
2. Identify all given information and what is being asked.
3. Define variables for unknown quantities.
4. Break down the problem using fundamental principles like set theory or the Inclusion-Exclusion Principle.
5. Perform step-by-step calculations.
6. Double-check your work to ensure accuracy.
In our example, by following these steps, we derived the solution for the number of reporters covering all three continents:
We first organized our data and then used the Inclusion-Exclusion Principle to set up an equation
Solving it step by step gave us the correct count for those covering all continents.
This structured approach not only makes the problem solvable but also enables you to apply similar steps to different mathematical problems.

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

The amount of an investment will double in approximately \(70 / p\) years, where \(p\) is the percent interest, compounded annually. If Thelma invests \(\$ 40,000\) in a long-term \(\mathrm{CD}\) that pays 5 percent interest, compounded annually, what will be the approximate total value of the investment when Thelma is ready to retire 42 years later? A \(\$ 280,000\) B \(\$ 320,000\) C \(\$ 360,000\) D \(\$ 450,000\) E \(\$ 540,000\)

Truck \(\mathrm{X}\) is 13 miles ahead of Truck \(\mathrm{Y}\), which is traveling the same direction along the same route as Truck X. If Truck \(\mathrm{X}\) is traveling at an average speed of 47 miles per hour and Truck \(Y\) is traveling at an average speed of 53 miles per hour, how long will it take Truck Y to overtake and drive 5 miles ahead of Truck X? A 2 hours B 2 hours 20 minutes C 2 hours 30 minutes D 2 hours 45 minutes E 3 hours

If John invested \(\$ 1\) at 5 percent interest compounded annually, the total value of the investment, in dollars, at the end of 4 years would be A $$(1.5)^{4}$$ B $$4(1.5)$$ C $$(1.05)^{4}$$ D $$1+(0.05)^{4}$$ E $$1+4(0.05)$$

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

Three hundred students at College \(Q\) study a foreign language. Of these, 110 of those students study French and 170 study Spanish. If at least 90 students who study a foreign language at College \(Q\) study neither French nor Spanish, then the number of students who study Spanish but not French could be any number from A 10 to 40 B 40 to 100 C 60 to 100 D 60 to 110 E 70 to 110

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