Problem 74 - A Graphical Approach to College Algebra

Question

Solve each formula for the specified variable.} \(A=\frac{24 f}{B(p+1)}\) for \(p\) (Approximate annual interest rate)

Step-by-Step Solution

Answer

\( p = \frac{24f - AB}{AB} \)

Step 1: Multiply Both Sides by the Denominator

We need to get rid of the fraction. Since \( A = \frac{24f}{B(p+1)} \), multiply both sides by \( B(p+1) \) to eliminate the denominator. This gives us: \( A \cdot B(p+1) = 24f \).

Step 2: Distribute A into the Parenthesis

Expand the left side of the equation by distributing \( A \cdot B \) across \( (p + 1) \). This results in: \( ABp + AB = 24f \).

Step 3: Isolate the Term Containing p

Move \( AB \) to the right side of the equation by subtracting it from both sides: \( ABp = 24f - AB \).

Step 4: Solve for p

Divide both sides by \( AB \) to solve for \( p \): \( p = \frac{24f - AB}{AB} \).

Key Concepts

Isolate Variable

When solving equations, one of the primary goals is to isolate the variable you want to find. In this context, isolating a variable means rewriting the equation so that the desired variable is by itself on one side of the equation.
For instance, in the given exercise where we need to solve for \( p \), isolating \( p \) involves rearranging the equation to have \( p \) on one side. Initially, the fraction \( \frac{24f}{B(p+1)} \) makes direct isolation challenging. Therefore, each step systematically moves us closer to having \( p \) alone. This progresses from clearing fractions, distributing values, and ultimately dividing to solve for \( p \).
The goal of isolating variables is crucial in diverse fields like finance, physics, and engineering, enabling the calculation of unknown quantities in various formulas and equations.

Academic Formula

An academic formula is a mathematical representation used to express relationships between different variables. These formulas are essential tools in all fields of study, enabling scholars and professionals to model real-world situations mathematically.
In the context of the given problem, the formula \( A = \frac{24f}{B(p+1)} \) relates several variables in a financial setting, potentially involving interest rates. This particular formula may come into play when considering how amounts change over time or under varying conditions depending on principal or interest rates.
Understanding how to manipulate academic formulas—such as changing the subject of the formula to solve for a specific variable—requires a grasp of algebraic operations like reversing multiplication with division, or clearing fractions, as we've seen in the provided exercise.

Step-by-Step Solution

A step-by-step solution is an instructional method used to solve equations or other mathematical problems by breaking down the process into clear, manageable stages. This approach provides transparency and helps students follow the logic behind each action taken.
In the original problem's solution, the process is divided into distinct steps:

• Step 1: Multiply both sides by the denominator to eliminate the fraction.
• Step 2: Distribute terms to simplify the equation and make isolation of the variable easier.
• Step 3: Move terms strategically to isolate \( p \) on one side.
• Step 4: Solve for \( p \) by dividing to fully isolate it.

By following each step in sequence, students learn how to handle more complex equations with confidence, applying the same method to other similar problems they might encounter.