Determining Airplane Boarding Capacity Using Inequalities A Math Problem

Introduction

In this article, we'll break down a common mathematical problem involving passenger capacity and inequalities. We'll focus on a scenario where Li's family is flying on a plane with a maximum seating capacity, and we need to determine how many more passengers can board the aircraft. This type of problem often arises in real-world situations, making it crucial to understand how to approach it effectively. Understanding passenger capacity and how it relates to inequalities is essential for solving problems related to resource management and limitations. This particular problem uses a straightforward scenario to illustrate the application of inequalities in determining the available space on an airplane. We'll explore how to translate the given information into a mathematical inequality that accurately represents the situation. Moreover, we'll delve into the thought process behind choosing the correct inequality symbol, ensuring that we capture the constraint imposed by the plane's maximum seating capacity. This skill is not just valuable for solving math problems but also for making informed decisions in various real-life contexts, such as planning events or managing inventory. By the end of this article, you'll have a solid understanding of how to set up and interpret inequalities related to capacity problems, equipping you with a practical tool for problem-solving in mathematics and beyond. Through a clear explanation and step-by-step approach, we aim to demystify the process of translating word problems into mathematical expressions, making it easier for you to tackle similar challenges in the future. Whether you're a student learning about inequalities or simply someone interested in sharpening your problem-solving skills, this article provides a valuable resource for understanding and applying mathematical concepts in a practical context.

Problem Statement

The core of our discussion revolves around a specific problem: The airplane Li's family will be flying on can seat up to 149 passengers. If 96 passengers are currently on the plane, which inequality can be used to determine how many more people can board? This problem requires us to translate a real-world situation into a mathematical expression. To effectively solve it, we need to identify the key information provided and understand how it relates to the concept of inequalities. The phrase "can seat up to 149 passengers" is crucial because it sets the upper limit for the number of passengers the plane can accommodate. This limit is the foundation for constructing our inequality. We also know that there are already 96 passengers on board, and we want to find out how many more people can board the plane. This unknown quantity can be represented by a variable, such as 'x', which denotes the additional number of passengers that can be accommodated. The challenge is to combine these pieces of information—the existing passengers, the additional passengers, and the maximum capacity—into an inequality that accurately reflects the situation. We need to carefully consider the relationship between these quantities and choose the appropriate inequality symbol to ensure that our mathematical expression correctly captures the constraint imposed by the plane's seating capacity. This involves understanding the difference between symbols like less than (<), greater than (>), less than or equal to (≤), and greater than or equal to (≥), and selecting the one that best fits the context of the problem. By breaking down the problem statement and carefully analyzing the given information, we can construct an inequality that not only solves the problem but also provides valuable insights into the real-world scenario it represents.

Understanding Inequalities

Before diving into the solution, let's clarify what inequalities are and how they work. Inequalities are mathematical expressions that compare two values, indicating that they are not necessarily equal. They use symbols like < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to) to show the relationship between the values. In our context, understanding inequalities is paramount because the problem involves a limit – the maximum number of passengers the plane can hold. This limit creates a constraint that needs to be expressed using an inequality. The "less than or equal to" symbol (≤) is particularly relevant here because it allows us to represent situations where the total number of passengers can be either less than the maximum capacity or equal to it, but not greater. To grasp this concept more intuitively, consider a simple example: If a room can hold a maximum of 50 people, the number of people inside the room must be less than or equal to 50. We cannot exceed this limit. Similarly, in our airplane problem, the total number of passengers, including those already on board and those who will board, must be less than or equal to the plane's seating capacity. Understanding the nuances of these inequality symbols is crucial for correctly translating real-world scenarios into mathematical expressions. The choice of symbol directly affects the solution and the interpretation of the problem. For instance, using "less than" (<) instead of "less than or equal to" (≤) would exclude the possibility of the plane being fully occupied, which might not accurately represent the situation. Therefore, a solid understanding of inequalities is not just a mathematical skill but also a practical tool for representing and solving problems involving constraints and limitations.

Identifying the Correct Inequality

Now, let's analyze the given options and determine the correct inequality. We have the following choices:

• $96 + x ext{ ≤ } 149$
• $96 + x ext{ ≥ } 149$
• $96 + x > 149$
• $96 + x < 149$

Remember, 96 represents the number of passengers already on the plane, and 'x' represents the additional number of passengers that can board. The sum of these two values ($96 + x$) must be less than or equal to the plane's capacity, which is 149. Therefore, we need an inequality that reflects this constraint. To pinpoint the right inequality, let's break down each option and assess its suitability. The inequality $96 + x ext{ ≥ } 149$ implies that the total number of passengers must be greater than or equal to 149, which contradicts the fact that the plane can seat up to 149 passengers. Similarly, $96 + x > 149$ suggests that the total number of passengers must exceed the plane's capacity, which is not possible. The inequality $96 + x < 149$ indicates that the total number of passengers must be strictly less than 149, but it doesn't account for the possibility of the plane being fully occupied. This leaves us with the inequality $96 + x ext{ ≤ } 149$, which accurately represents the situation. This inequality states that the sum of the existing passengers and the additional passengers must be less than or equal to 149. This aligns perfectly with the problem statement, which specifies that the plane can seat up to 149 passengers. By systematically evaluating each option and relating it back to the problem's constraints, we can confidently identify the correct inequality that models the scenario.

Step-by-Step Solution

Let's walk through the solution step-by-step to reinforce our understanding. The core of our problem lies in translating the word problem into a mathematical inequality. Translating real-world scenarios into mathematical expressions is a critical skill in problem-solving. Here’s how we approach it in this case:

1. Identify the knowns: We know that the plane can seat up to 149 passengers, and there are already 96 passengers on board.
2. Define the unknown: Let 'x' represent the number of additional passengers that can board.
3. Formulate the inequality: The total number of passengers (96 + x) must be less than or equal to the plane's capacity (149). This gives us the inequality $96 + x ext{ ≤ } 149$.

To further illustrate this process, consider a similar scenario where a bus can hold a maximum of 60 people, and 25 people are already on the bus. To determine how many more people can board, we follow the same steps. We identify the knowns (bus capacity of 60, 25 people already on board), define the unknown (let 'y' be the additional number of passengers), and formulate the inequality: $25 + y ext{ ≤ } 60$. This methodical approach ensures that we capture the essence of the problem in a mathematical expression, making it easier to solve. By breaking down the problem into smaller, manageable steps, we can avoid confusion and ensure accuracy in our solution. This step-by-step process is not just applicable to this specific problem but can be generalized to a wide range of mathematical challenges, making it a valuable tool in your problem-solving arsenal.

Conclusion

The correct inequality to determine how many more people can board the plane is $96 + x ext{ ≤ } 149$. This inequality accurately represents the constraint that the total number of passengers cannot exceed the plane's capacity. In summary, we've explored a problem that combines real-world context with mathematical concepts, specifically inequalities. Understanding how to apply inequalities in practical scenarios is a valuable skill that extends beyond the classroom. We started by dissecting the problem statement, identifying the key information, and defining the unknown variable. Then, we delved into the concept of inequalities, emphasizing the importance of choosing the correct symbol to accurately represent the relationship between the quantities involved. We methodically evaluated each of the given options, eliminating those that didn't align with the problem's constraints and ultimately arriving at the correct inequality: $96 + x ext{ ≤ } 149$. This inequality succinctly captures the idea that the sum of the existing passengers and the additional passengers must be less than or equal to the plane's maximum capacity. Moreover, we highlighted the broader applicability of this problem-solving approach. The ability to translate word problems into mathematical expressions is a fundamental skill that can be applied to a wide range of situations, from managing resources to making informed decisions in everyday life. By mastering this skill, you'll be better equipped to tackle challenges that require quantitative reasoning and logical thinking. Whether you're a student preparing for an exam or simply someone looking to enhance your problem-solving abilities, the principles and techniques discussed in this article will serve as a valuable resource.

This problem serves as a great example of how math can be used to solve real-world scenarios. By understanding the concepts and applying them methodically, we can tackle similar problems with confidence.