Problem 2 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Five different types of monthly commuter passes are offered by a city's local transit authority for each of three different groups of passengers: youths, adults, and senior citizens. How many different kinds of passes must be printed each month?

Step-by-Step Solution

Answer

The transit authority must print 15 different kinds of passes each month, as calculated using the counting principle: \( Total\ number\ of\ passes = 5\ types\ of\ passes * 3\ groups\ of\ passengers = 15\ passes \).

Step 1: Identify the given information

We have been given that there are five different types of monthly commuter passes and three different groups of passengers (youths, adults, and senior citizens).

Step 2: Apply the Counting Principle

According to the counting principle, to find the total number of different kinds of passes, we need to multiply the number of types of passes (5) with the number of groups of passengers (3).

Step 3: Calculate the total number of passes

Multiplying the number of types of passes and the number of groups of passengers, we get: Total number of passes = 5 types of passes * 3 groups of passengers = 15 passes Hence, the transit authority must print 15 different kinds of passes each month.

Key Concepts

Combinatorics

Combinatorics is an essential branch of mathematics that focuses on counting, arrangement, and combination. It answers the question of "how many" ways something can occur. In this exercise, combinatorics allows us to determine the total number of distinct commuter passes needed by calculating possible combinations. Each combination represents a unique pass that caters to different passenger needs and types. To apply combinatorics effectively:

• Identify all possible choices available.
• Understand the relationships between these choices.
• Use mathematical principles, like the Counting Principle, to find total combinations.

Recognizing these steps in combinatorics helps in a wide variety of problem-solving scenarios beyond just transit systems.

Commuter Passes Calculation

The exercise revolves around figuring out the number of various commuter passes a city's transit authority needs to print each month. The constituencies include three groups of commuters: youths, adults, and senior citizens. For each group, there are five distinct types of passes, possibly distinguished by routes, zones, or times they'll be valid. To calculate the total number of different types of passes:

• Identify the number of groups (here, 3).
• Identify the variances per group (here, 5 types).
• Multiply the group number by the type number (3 groups * 5 types).

This results in 15 unique commuter passes. Each type ensures that every group can select a pass tailored to their specific needs, optimizing usage and cost efficiency.

Problem-Solving in Applied Mathematics

Problem-solving in applied mathematics refers to tackling real-world scenarios using mathematical concepts. It involves:

• Understanding and interpreting the problem accurately.
• Applying relevant mathematical principles or formulas.
• Calculating and reaching a solution that addresses the problem's requirements.

In our case study, the transit authority needs to meet commuter demands efficiently by leveraging combinatorics. One key aspect is the Counting Principle, which helps systematically break down complex scenarios into manageable calculations. By applying it to determine the needed passes, the process becomes efficient and straightforward, demonstrating how math aids in organizing and strategizing in practical applications across numerous fields.