Problem 102 - Introductory and Intermediate Algebra for College Students 4th

Question

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. Because there are four quarters in a dollar, I can use the formula \(q=4 d\) to determine the number of quarters, \(q,\) in \(d\) dollars.

Step-by-Step Solution

Answer

The statement 'Because there are four quarters in a dollar, I can use the formula \(q=4 d\) to determine the number of quarters, \(q,\) in \(d\) dollars' makes sense. This is because the formula accurately represents the relationship between the number of quarters and the total in dollars.

Step 1: Understand Real-world Values

In reality, one dollar is equivalent to four quarters. Therefore, 'd' dollars would be equivalent to '4d' quarters.

Step 2: Compare with Given Formula

Looking at the provided formula 'q = 4d', it suggests that the number of quarters, 'q', in 'd' dollars is '4d'. This is consistent with the fact that each dollar is made up of four quarters.

Step 3: Reasoning

Since the provided formula matches the reality of the number of quarters in a dollar, it can be concluded that the statement makes sense.

Key Concepts

Mathematical formulas

Mathematical formulas are essential tools in algebra and many other areas of mathematics. A formula is a concise way of expressing information symbolically. In the context of the exercise, the formula \( q = 4d \) is used to determine the number of quarters (\( q \)) in a given amount of dollars (\( d \)).
Formulas often use variables to represent numbers in equations. Here, \( q \) and \( d \) are variables. The number "4" is a constant, representing there are four quarters in a dollar. As a formula, \( q = 4d \) indicates that for every dollar, you have four quarters. When learning about formulas, it's important to:

• Identify the variables and constants involved.
• Understand how each relates to real-world quantities.
• Apply the formula to calculate or verify values.

Learning how to interpret and use formulas is crucial for solving problems efficiently.

Real-world applications

Real-world applications of algebraic formulas allow us to solve everyday problems efficiently. The exercise's example, where quarters are used to count dollars, is a direct application of a real-world context through algebra.
Using formulas such as \( q = 4d \) shows how mathematical concepts can be applied to count money quickly and logically. This concept can be applied in various scenarios:

• Budgeting: We can use formulas to manage and calculate budgets, determining quantities like savings or expenses.
• Conversions: Formulas can convert between different units of measure, like dollars to quarters or miles to kilometers.
• Logical reasoning: Understanding formulas through real-world examples sharpens math and logical reasoning skills.

When you apply algebra in real life, it helps bridge the gap between abstract mathematical concepts and practical usage.

Statement validation

Statement validation is the process of determining if a given statement aligns with logical or factual correctness. When validating algebraic statements, it's important to compare them against known facts or logical reasoning.
In the exercise, the statement "Because there are four quarters in a dollar, I can use the formula \( q=4d \) to determine the number of quarters, \( q, \) in \( d \) dollars" is analyzed. The validation process involves:

• Comparing the statement with real-world facts (each dollar indeed equals four quarters).
• Checking the correctness of the corresponding algebraic formula.
• Ensuring logical reasoning is sound.

By concluding that the statement "makes sense," we have validated it based on consistent logic and factual alignment. Mastering statement validation helps improve critical thinking skills and ensures high precision in mathematical applications.