Problem 45 - Intermediate Algebra

Question

Translate each verbal model into a mathematical model. Answers may vary depending on the variables chosen. The product of the number of boxes of crayons in a case and 12 gives the number of crayons in a case.

Step-by-Step Solution

Answer

The mathematical model is \( 12b = c \).

Step 1: Identify Key Quantities

First, we need to identify the quantities involved in the verbal model. Here, we have 'the number of boxes of crayons in a case' and 'the number of crayons in a case'. Let's denote the number of boxes of crayons by the variable \( b \) and the total number of crayons in the case by \( c \).

Step 2: Understand the Relationship

The problem states that the 'product of the number of boxes of crayons in a case and 12 gives the number of crayons in a case'. This means that if you multiply the number of boxes \( b \) by 12, you will get the total number of crayons \( c \).

Step 3: Formulate the Mathematical Model

Translate the verbal relationship into a mathematical equation. According to the problem, the equation is \( 12b = c \). This equation models the relationship described in the verbal statement.

Key Concepts

Verbal Models

In mathematics, verbal models are useful tools for translating everyday language into mathematical language. They help bridge the gap between spoken or written problems and their mathematical counterparts. For instance, the sentence "The product of the number of boxes of crayons in a case and 12 gives the number of crayons in a case" is a verbal model. Here, the key phrases describe a mathematical relationship.

• Product: This word signals multiplication. In the example, it indicates that two quantities should be multiplied.
• Number of boxes: This is a quantity which can vary, and hence needs a variable.
• Gives the number: This suggests equality or the result of a calculation, leading to the equal sign in a mathematical model.

Translating such verbal models into mathematical expressions requires recognizing important terms and their corresponding mathematical operations. This process simplifies complex relationships and makes them solvable using algebra.

Algebraic Equations

When translating a verbal model into an algebraic equation, we create a mathematical statement that shows equivalence. An algebraic equation uses variables to denote unknown quantities, which can then be solved. In our case, an equation was formed: \[ 12b = c \]This equation uses the following components:

• Variables: In the example, \( b \) represents the number of boxes of crayons, and \( c \) represents the total number of crayons in a case.
• Constant: The number 12 is a constant multiplier, representing the number of crayons in each box.
• Expression: "12b" indicates that the number of boxes is multiplied by 12 to obtain the total count of crayons.

By setting these elements equal, we have formed an equation that quantitatively describes the relationship stated verbally. By substituting known values into such equations, we can solve for unknowns and understand various scenarios.

Quantitative Reasoning

Quantitative reasoning involves using mathematical understanding to solve real-world problems. It combines the translation of verbal information into mathematical models with analytical thinking. Let's explore how to apply it using our exercise as an example.

• Identify Relationships: Understand how different quantities relate. Recognize multiplication keywords, like "product," to set up relationships.
• Use Approaches: Translate the verbal description into an equation that accurately represents the situation. Refine the model as necessary if more information is known.
• Interpret Results: Once the solution is found, interpret it in context. For example, if \( 12b = c \), knowing \( b \) tells us exactly how many crayons are in a case.

Quantitative reasoning enhances problem-solving skills by encouraging one to think critically about numerical information. It contributes significantly to effectively tackling mathematical challenges.