Modeling Mileage: Equations For Jen & Amelie's Cars

by Alex Johnson 52 views

Let's dive into a common type of problem you might encounter in mathematics: modeling real-world scenarios with equations. This particular problem involves comparing the mileage of two cars, Jen's and Amelie's. The goal is to identify which equations accurately represent the given information. Understanding how to translate word problems into mathematical expressions is a crucial skill, not just for exams, but also for everyday problem-solving. So, grab your thinking cap, and let's break this down step by step.

Understanding the Problem

Before we jump into the equations, let's make sure we fully understand the scenario. The core information is:

  • Jen has a newer car than Amelie.
  • Jen's car has 1/3 the miles of Amelie's car.
  • Amelie's car has 45,000 miles.

The question asks us to select the equations that model this situation. This means we need to identify equations where the mathematical relationships accurately reflect the relationships described in the word problem. The key here is the phrase "1/3 the miles." This indicates a multiplicative relationship – Jen's mileage is a fraction of Amelie's mileage. To properly model this, we need to understand how fractions translate into equations, and how to represent an unknown quantity (Jen's mileage) with a variable.

Key Concepts: Variables and Equations

In algebra, we use variables to represent unknown quantities. In this case, let's use the variable 'm' to represent the number of miles on Jen's car. An equation is a mathematical statement that shows the equality between two expressions. It typically involves variables, constants, and mathematical operations. The equations provided as options aim to express the relationship between Jen's mileage (m) and Amelie's mileage (45,000 miles), considering the given fraction (1/3). Understanding these basics is crucial as we proceed to evaluate each equation and determine its validity in representing the given scenario. It's the foundation for translating real-world problems into mathematical models. Remember, each part of the sentence gives us a clue on how to build the equation, and careful reading can make the task much easier.

Evaluating the Equations

Now, let's look at the equations provided and see which ones fit the scenario. We'll analyze each option individually:

A. 3(m) = 45,000

This equation states that three times Jen's mileage (3 * m) is equal to Amelie's mileage (45,000). Let's think about this in the context of the problem. If Jen's car has 1/3 the miles of Amelie's car, then Amelie's car has three times the miles of Jen's car. This equation accurately reflects that relationship. To further clarify, if we were to solve this equation for 'm', we would be dividing Amelie's mileage by 3, which corresponds to finding 1/3 of her mileage. This is a critical step in verifying the equation's accuracy. Thus, the equation 3(m) = 45,000 correctly models the situation.

B. 1/3 + m = 45,000

This equation states that 1/3 plus Jen's mileage (m) equals Amelie's mileage (45,000). This doesn't align with the problem description. The problem states that Jen's mileage is 1/3 of Amelie's mileage, not that it's a sum involving 1/3. This equation implies an additive relationship, where a fraction is being added to Jen's mileage to reach Amelie's mileage, which is not what the problem describes. The correct interpretation should involve multiplying Amelie's mileage by 1/3 to get Jen's mileage. This equation incorrectly represents the proportional relationship and is therefore not a valid model for the given scenario. Hence, option B can be ruled out.

Conclusion: Selecting the Correct Equations

After carefully analyzing the equations, we've determined that only one of the provided options accurately models the given situation:

  • A. 3(m) = 45,000

This equation correctly represents the relationship between Jen's and Amelie's mileage. It highlights that Amelie's mileage is three times Jen's mileage, which is the inverse of the statement that Jen's mileage is 1/3 of Amelie's. The key takeaway here is the importance of careful interpretation of word problems and translating those relationships into mathematical equations. Understanding the fundamental operations, like multiplication for fractions of a quantity, is crucial. This exercise demonstrates how to break down a word problem, identify key information, and translate it into a mathematical model, a skill that is invaluable in both academic and real-world scenarios.

Remember, mathematics is not just about numbers; it's about understanding relationships and patterns. By practicing these types of problems, you'll become more confident in your ability to model complex situations and solve them effectively. For additional resources on mathematical modeling, you might find helpful information on websites like Khan Academy's Algebra I course.