Problem 18 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Of 100 clock radios with digital tuners and/or CD players sold recently in a department store, 70 had digital tuners and 90 had CD players. How many radios had both digital tuners and CD players?

Step-by-Step Solution

Answer

There are \(60\) radios that have both digital tuners and CD players.

Step 1: Define the sets

Let's define set A as the set of radios with digital tuners, and set B as the set of radios with CD players. We are looking for the intersection of these two sets, which we'll call set C: the set of radios with both digital tuners and CD players.

Step 2: Use the principle of inclusion-exclusion

According to the principle of inclusion-exclusion, the total number of elements in the union of two sets is the sum of the elements in each set, minus the elements in their intersection. In our case, this corresponds to the total number of radios: Total radios = Radios with digital tuners + Radios with CD players - Radios with both digital tuners and CD players Given the information, we have: 100 = 70 + 90 - C

Step 3: Solve for the intersection

Now let's solve for C, which represents the number of radios with both digital tuners and CD players: 100 - 90 - 70 + C = 0 C = 60 So there are 60 radios that have both digital tuners and CD players.

Key Concepts

Principle of Inclusion-Exclusion

The Principle of Inclusion-Exclusion is a fundamental concept in set theory. It provides a way to calculate the size of the union of multiple sets. The main idea is that we can add up the sizes of single sets, then subtract the sizes of all pairwise intersections to avoid double-counting, and so on for intersections of more sets.

In the context of our problem with clock radios, we used this principle to figure out how many radios had both digital tuners and CD players. The formula goes as follows:

• Let the number of radios with digital tuners be 70.
• Let the number of radios with CD players be 90.
• To find out how many had both features, we use the formula for two sets: \[ |A \cup B| = |A| + |B| - |A \cap B| \]
• Inserting our numbers: \[ 100 = 70 + 90 - |A \cap B| \]
• Solving for the intersection: \[ |A \cap B| = 70 + 90 - 100 = 60 \]

Therefore, 60 radios had both a digital tuner and a CD player.

Intersection of Sets

The intersection of sets focuses on the common elements shared between sets. If two sets, say A and B, have any elements that are present in both sets, those elements are part of their intersection, denoted as \(A \cap B\).

In solving the radio problem, we needed to identify the intersection between two specific sets:

• Set A: Radios with digital tuners.
• Set B: Radios with CD players.

The intersection \(A \cap B\) included radios with both digital tuners and CD players. Our task was to find the size of this intersection, which was 60 radios.

Understanding intersections is vital in set theory as it helps identify commonalities between different groups, making it a powerful tool in various mathematical applications.

Applied Mathematics

Applied Mathematics refers to the use of mathematical methods and concepts to solve real-world problems. In this context, the problem of calculating the number of clock radios with both digital tuners and CD players is a practical example.

By employing set theory and the Principle of Inclusion-Exclusion, we move from theoretical concepts to solving real-life challenges. Here's how the mathematical approach was applied:

• We defined relevant sets representing our groups (radios with digital tuners, radios with CD players).
• We applied a formula that allowed us to find the intersection, giving us the number of radios with both features.
• Ultimately, we provided a solution that not only offers an answer (60 radios) but also demonstrates how mathematical principles can break down complex queries into manageable parts.

This process encourages critical thinking and problem-solving, illustrating the seamless connection between mathematics and its applications in various fields.