Problem 10 - Algebra and Trigonometry

Question

Write an algebraic formula for the given quantity.. The number \(N\) of cents in \(q\) quarters

Step-by-Step Solution

Answer

The formula is \( N = 25q \).

Step 1: Understand the Problem

You need to create an algebraic formula that expresses the total number of cents, \( N \), when you have \( q \) quarters.

Step 2: Determine the Value of a Single Quarter

Each quarter is worth 25 cents. This is crucial for forming an expression for \( N \).

Step 3: Formulate the Algebraic Expression

To find the total number of cents from \( q \) quarters, multiply the number of quarters by the value of each quarter. So, the expression becomes \( N = 25q \).

Step 4: Verify the Expression

Double-check the formula by plugging in a simple example. If \( q = 1 \), then \( N = 25 \times 1 = 25 \) cents, which is correct.

Key Concepts

Problem-Solving in Algebra

Algebra is all about expressing relationships between different quantities using symbols and formulas. When dealing with any algebraic problem, the first step is always to clearly understand what needs to be solved.
For instance, consider the problem where you need to find the total number of cents in a certain number of quarters. Here, you're asked to create a formula that shows the relationship between the number of quarters and the total cents you've got.
To solve these kinds of problems, you should:

• Comprehend the question completely before jumping into calculations.
• Recognize the quantities involved, in this case, ‘quarters’ and ‘cents’.
• Identify the constant values; each quarter is always 25 cents.

Once you have a clear idea of these aspects, forming an algebraic expression becomes much simpler. It's all about finding a clear path from the given information to the unknown using logical and mathematical steps.

Understanding Quantities

When working with algebra, understanding the quantities involved is essential. Quantities can be anything you measure, calculate, or describe. They can be numbers, like the number of quarters, or total amounts, like cents.
The core of the original problem is about these quantities: quarters and their total value in cents. Understanding that one quarter equals 25 cents allows you to translate the problem into a mathematical form.
Here’s a breakdown of how to approach this:

• Identify the known quantity: In this problem, it's the value of each quarter, which is 25 cents.
• Identify the variable quantity: The number of quarters, represented by \( q \).
• Determine the relationship between them: \( N = 25q \), where \( N \) represents the total number of cents.

Recognizing these relationships enables you to convert real-world problems into an algebraic expression, making the math straightforward.

Verifying Mathematical Expressions

Once you've formulated an equation, the next step is verification. Verification in algebra means making sure that your expression correctly represents the relationship between the quantities.
To check if the formula \( N = 25q \) is accurate, you can plug in a simple value for \( q \).
Here’s how you can do it:

• Choose a small number to test; for example, \( q = 1 \).
• Calculate using the formula: \( N = 25 \times 1 = 25 \).
• Check if this result makes sense: indeed, one quarter yields 25 cents.

By verifying like this, you ensure your formula is correct and you can confidently use it with other numbers. It instills reliability in your mathematical work and is an essential practice in problem-solving.