Problem 44 - Intermediate Algebra

Question

Translate each verbal model into a mathematical model. Answers may vary depending on the variables chosen. The weight of a super-size order of French fries is twice that of a regular- size order.

Step-by-Step Solution

Answer

The mathematical model is \( s = 2r \).

Step 1: Understand the Given Problem

We need to translate the verbal statement into a mathematical model. The statement mentions two types of orders: super-size and regular-size, and their relationship involving weight.

Step 2: Declare Variables

Let's declare two variables to represent the weights. Let \( r \) be the weight of a regular-size order of French fries and \( s \) be the weight of a super-size order.

Step 3: Formulate a Mathematical Model

According to the problem, 'the weight of a super-size order is twice that of a regular-size order.' This can be written mathematically as: \( s = 2r \).

Key Concepts

Verbal to Mathematical Translation

Understanding how to convert verbal descriptions into mathematical expressions is essential for solving real-world problems using algebra. In the given exercise, we are presented with a scenario involving two different sizes of French fries orders, described verbally. Our goal is to translate this narrative into a mathematical statement.

To achieve this translation, there are some steps and techniques that you can use effectively:

• Identify key phrases: Look for terms in the verbal description that describe relationships, quantities, or operations. In our example, phrases such as "twice that of" or "the weight of" guide us in identifying mathematical operations and relationships.
• Select appropriate variables: Choose symbols to represent unknown quantities. Here, we used two variables, where \( r \) is for the regular fries and \( s \) for the super-size fries.
• Translate relationships: Convert relational words into mathematical symbols. Thus, "twice" translates into multiplication by 2, forming the equation \( s = 2r \).

By following these steps, you can successfully transform real-world verbal problems into precise mathematical language, enhancing problem-solving abilities.

Variables in Equations

Variables are symbols used to represent numbers in equations and expressions. They are powerful tools in algebra that allow us to generalize mathematical relationships and solve problems effectively. In the exercise, variables are employed to describe the weights of French fries orders.

Here's how the concept works:

• Representation: Variables, like \( r \) and \( s \) in our task, are placeholders for numbers that we might not initially know but want to solve for or describe relationships.
• Flexibility: They provide flexibility in forming equations that can describe multiple scenarios with a single expression.
• Solving Equations: By developing equations, such as \( s = 2r \), involving variables, we can perform algebraic operations to find variable values, explaining the weight relation between order sizes.

Variables navigate us through complex problems by simplifying descriptions into manageable parts, making abstract thinking concrete.

Linear Relationships

Linear relationships represent a direct proportionality between two quantities. In algebra, these relationships are often expressed in a form where one variable is a multiple of another, plus or minus a constant. Our exercise describes such a linear relationship between the two types of French-fry orders.

Here's how linear relationships are relevant:

• Proportionality: The equation \( s = 2r \) is a classic example where the weight of a super-sized order of French fries is proportional to the weight of a regular order.
• Simplicity: Linear equations, like this one, are simple to understand and solve, as they often feature straightforward arithmetic operations.
• Graphical Representation: When graphed, linear relationships create straight lines, which visually represent constant rates of change. In our case, doubling the weight at every point.

Understanding linear relationships is pivotal in algebra since they frequently occur in everyday problems, allowing for simple yet powerful models of real-world scenarios.