Problem 22 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

CoMMUTER TRENDS Of 50 employees of a store located in downtown Boston, 18 people take the subway to work, 12 take the bus, and 7 take both the subway and the bus. How many employees a. Take the subway or the bus to work? b. Take only the bus to work? c. Take either the bus or the subway to work? d. Get to work by some other means?

Step-by-Step Solution

Answer

a. \(23\) employees take the subway or the bus to work. b. \(5\) employees take only the bus to work. c. \(23\) employees take either the bus or the subway to work. d. \(27\) employees get to work by some other means.

Step 1: Use the Principle of Inclusion-Exclusion

According to the Principle of Inclusion-Exclusion: Total taking subway or bus = Total taking subway + Total taking bus - Total taking both subway and bus. For our problem, this would translate to: Total taking subway or bus = 18 (subway) + 12 (bus) - 7 (both)

Step 2: Calculate the number of employees taking subway or bus

Using the formula from Step 1, we can find out how many employees take the subway or the bus to work: Total taking subway or bus = 18 + 12 - 7 = 23 So, 23 employees take the subway or the bus to work.

Step 3: Calculate the number of employees taking only the bus

To find how many employees take only the bus to work, subtract the employees taking both subway and bus from the total taking the bus: Only bus = Total taking bus - Total taking both subway and bus Only bus = 12 - 7 = 5 So, 5 employees take only the bus to work.

Step 4: Calculate the number of employees taking either the bus or the subway

Since the number of employees taking the subway or bus has been already calculated in Step 2, we don't need to do extra calculations for this. The answer is: 23 employees take either the bus or the subway to work.

Step 5: Calculate the number of employees using other means of transportation

To find how many employees get to work by some other means, subtract the employees taking subway or bus from the total number of employees: Other means = Total employees - Total taking subway or bus Other means = 50 - 23 = 27 So, 27 employees get to work by some other means. #Final Answers# a. 23 employees take the subway or the bus to work. b. 5 employees take only the bus to work. c. 23 employees take either the bus or the subway to work. d. 27 employees get to work by some other means.

Key Concepts

Principle of Inclusion-Exclusion

When solving problems that involve overlapping groups, like commuters using different modes of transport, the Principle of Inclusion-Exclusion (PIE) helps determine the total number of individual items across those groups. Imagine you have two overlapping circles—a Venn diagram—with each circle representing a different set of people. The overlap in the middle counts individuals categorized in both sets, such as those taking both the bus and subway.
By applying PIE, we ensure we don't count this overlap twice. The formula is as follows:

• Total in at least one of the groups = (Total in first group) + (Total in second group) - (Total in both groups).

For example, if 18 people take the subway and 12 take the bus, and 7 take both, PIE tells us to add the number of subway and bus commuters, then subtract those counted twice. This results in 18 + 12 - 7 = 23. Thus, 23 employees either take the subway or bus to work.

Commuter Trends

Commuter habits are fascinating as they provide insights into people's daily routines and logistical preferences. Understanding these trends in a workplace can impact decisions on infrastructure and benefits.
In this exercise, we explore how employees at a store in downtown Boston commute to work. Knowing that 18 employees take the subway and 12 take the bus, with 7 using both methods, helps us explore different parts of this trend like:

• Who takes only one mode of transport?
• Who prefers a combination of options?

By recognizing these patterns, businesses might enhance employee experiences by considering incentives for public transportation or adding facilities like bike racks for those who commute by other means.

Problem Solving

Problem solving in mathematics often involves breaking down complex scenarios into manageable steps. This helps tackle the problem in a logical sequence, making it easier to solve.
Here, the problem is to determine how employees commute to a store in relation to several categories. By using a systematic approach:

• Firstly, identify the main components—subway, bus, both, and other methods.
• Calculate how many use each method, subtracting overlaps with the Principle of Inclusion-Exclusion.

Each step aligns with the final goal of understanding the commuting preferences of the employees.
This logical progression from understanding the components and their overlaps ensures that our answers reflect the true commuter trends at the workplace.

Set Theory

Set Theory is a fundamental principle of mathematics focused on the grouping of objects. It's particularly useful for analyzing relationships between different sets, like commuters using distinct modes of transport. Here, each type of transportation to work—subway, bus, both—acts as a different set.
Visualizing these like a Venn diagram:

• The subway set overlaps the bus set, creating a subset for both modes of transport.

This conceptual framework helps us use the Principle of Inclusion-Exclusion to ensure we accurately account for every individual without double-counting.
Set Theory also supports the clarity of results, offering an organized method. It's a cornerstone of resolving real-world problems, making abstractly challenging ideas concrete and manageable.