Problem 21 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

In a poll conducted among 200 active investors, it was found that 120 use discount brokers, 126 use fullservice brokers, and 64 use both discount and full-service brokers. How many investors a. Use at least one kind of broker? b. Use exactly one kind of hroker? c. Use only discount brokers? d. Don't use a broker?

Step-by-Step Solution

Answer

a. \(182\) investors use at least one kind of broker. b. \(118\) investors use exactly one kind of broker. c. \(56\) investors use only discount brokers. d. \(18\) investors don't use a broker.

Step 1: Draw and label a Venn diagram

Draw two overlapping circles within a large rectangle. Label the left circle as "Discount Brokers" representing the number of investors who use discount brokers only or both (120). Label the right circle as "Full-service Brokers," representing the number of investors who use full-service brokers only or both (126). Name the rectangle as "Total Investors" representing all investors (200).

Step 2: Find the intersection

Since we know that 64 investors use both discount and full-service brokers, this number represents the intersection of the two circles. Label the overlapping region in the Venn diagram with 64.

Step 3: Determine the remaining values for each circle

To find the remaining values in each circle, subtract the number of investors in the intersection from the total number of investors for each kind of broker: - For discount brokers: 120 - 64 = 56; label the remaining area in the Discount Brokers circle with 56. This represents the investors who use only discount brokers. - For full-service brokers: 126 - 64 = 62; label the remaining area in the Full-service Brokers circle with 62. This represents the investors who use only full-service brokers.

Step 4: Calculate the investors who don't use a broker

To find the number of investors who don't use a broker, subtract the sum of investors who use only discount brokers, only full-service brokers, and both brokers from the total number of investors: 200 - (56 + 62 + 64) = 200 - 182 = 18 Now, we can answer the questions: a. Use at least one kind of broker?

Step 5: To find the investors who use at least one kind of broker, add the investors who use only discount brokers, only full-service brokers, and both brokers: 56 + 62 + 64 = 182. b. Use exactly one kind of broker?

To find the investors who use exactly one kind of broker, add the investors who use only discount brokers and those who use only full-service brokers: 56 + 62 = 118. c. Use only discount brokers?

Step 6: We have already found this value in the Discount Brokers circle: 56. d. Don't use a broker?

We have already found this value when calculating the investors who don't use a broker: 18.

Key Concepts

Set Theory

Understanding the principles of set theory is vital when approaching problems that involve grouping different items. A set is a collection of distinct objects, considered as an object in its own right. In the case of the Venn diagram exercise, 'Discount Brokers', 'Full-service Brokers', and 'Both' are considered sets, each containing the number of investors that fall into the respective categories.

When determining the number of investors in each set, it's crucial to consider that some investors may belong to multiple sets. This overlap is where the concept of the intersection becomes important. The intersection represents elements that are common to two or more sets—in this problem, investors who use both types of brokers. By calculating the intersection, we find the number of investors that belong exclusively to a single set or to none of the sets at all.

Working with Subsets

Each circle within our Venn diagram represents a subset of the total set of investors. This allows us to sort the investors into mutually exclusive and comprehensive categories. For instance, sorting investors who use exactly one kind of broker helps to create a clearer picture of the market preferences.

Probability

Probability is a measure of the likelihood that an event will occur. It is often used in conjunction with set theory when analyzing Venn diagrams. In our exercise, we can think of each category of broker usage as an event. If we were to randomly select an investor, probability theory would allow us to calculate the chances of different outcomes, such as the investor using a discount broker, a full-service broker, both, or none.

For example, the probability that a randomly selected investor uses only a full-service broker could be calculated by taking the number of investors in that subset (62) and dividing it by the total number of investors (200). This integration of probability into set theory demonstrates the interconnectedness of mathematical concepts and how they can provide meaningful insights into real-world situations.

Mathematical Reasoning

Mathematical reasoning is the logic behind solving mathematical problems, such as the reasoning used to solve our Venn diagram exercise. It involves identifying relationships, making assumptions, and drawing conclusions based on the information and rules given.

To answer questions about the usage of brokers among investors accurately, we had to understand the steps to take and why to take them. We completed steps such as subtracting the intersection from the totals to find the number of investors who use exactly one kind of broker. Each part of the solution builds on the previous steps, and mathematical reasoning ensures that each conclusion is logically sound.

Critical Thinking in Math

This exercise required critical thinking to decide how to represent the data in the Venn diagram and how to interpret this visual representation. It's a skill that's fundamental to all branches of mathematics and science, as it helps students approach problems methodically and logically.