Problem 24 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

In a survey of 400 households regarding the ownership of VCRs and DVD players, the following data were obtained: 360 households own one or more VCRs. 170 households own one or more VCRs and one or more DVD players. 19 households do not own a VCR or a DVD player. How many households own only one or more DVD players?

Step-by-Step Solution

Answer

There are 191 households that own only one or more DVD players.

Step 1: Identify the given information

We have the following information from the survey: 1. Total households surveyed: 400 2. Households with one or more VCRs: 360 3. Households with one or more VCRs and one or more DVD players: 170 4. Households without a VCR or a DVD player: 19

Step 2: Create a Venn diagram

Let A be the set of households with VCRs and B be the set of households with DVD players. We represent this information in a Venn diagram with two circles, one for each set A and B.

Step 3: Calculate the households with VCRs only

Since there are 360 households with VCRs, and 170 households with VCRs and DVD players, there are 360 - 170 = 190 households with only VCRs.

Step 4: Calculate the households with DVD players only

We know that there are 19 households without a VCR or DVD player, so there must be 400 - 19 = 381 households with any of VCR or DVD player. Now, subtract the households with VCRs (360) and add back the households with both VCRs and DVD players (170) to get the total households with DVD players only: 381 - 360 + 170 = 191 households own only one or more DVD players. Hence, there are 191 households that own only one or more DVD players.

Key Concepts

Venn Diagram

Understanding how to use a Venn diagram is essential when analyzing survey data. In our exercise, Venn diagrams have been used to visualize the relationship between households owning VCRs and DVD players. Imagine two overlapping circles: one circle represents households with VCRs, and the other households with DVD players. The overlapping region shows households with both.

Venn diagrams are incredibly powerful tools in set theory, allowing us to see logical relationships and perform computations like unions, intersections, and complements, simply by looking at how the circles intersect and what they include or exclude. To enhance clarity for students, the Venn diagram could be supplemented with color-coding or labels that differentiate the sections and represent the exclusive and common elements of the sets.

Household Survey Analysis

Survey analysis in the context of household items like VCRs and DVD players involves summarizing and interpreting data collected from a sample of households. This kind of analysis is integral in generating insights about consumer behavior, market trends, and more.

For our exercise, we approached the survey analysis by working through the given numerical data. It is crucial for students to carefully examine each piece of information, relate them to real-world implications, and translate them through mathematical representations like the Venn diagram. When improving comprehension for this subject, practical examples and real data sets can make a significant difference by connecting theory with real-life applications.

Applied Mathematics

Applied mathematics can often feel abstract, but it comes to life when tied to real-world problems like our survey. The exercise utilizes basic concepts of set theory and operations – intersections, unions, and complements – to find a practical solution. By expressing real-world problems mathematically, applied mathematics not only provides answers to tangible questions but also develops critical thinking and analytical skills.

When aiming to help students understand applied mathematics, context is essential. Relating mathematical operations to everyday issues, as demonstrated with the household survey, empowers students to see the value in what they're learning and better grasp complex concepts. Emphasizing this connection can be done through interactive problem-solving and real-time scenario analysis.