Problem 80 - A Graphical Approach to Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

The solution for \( p \) is \( \frac{24f - AB}{AB} \).

Step 1: Isolate the Denominator

Start by isolating the denominator that contains the variable \( p \). Multiply both sides of the equation by \( B(p+1) \):\[ A imes B(p+1) = 24f \]

Step 2: Expand the Equation

Now, expand the left side of the equation:\[ AB(p+1) = 24f \]This becomes:\[ ABp + AB = 24f \]

Step 3: Isolate Terms Containing p

We need to separate the term containing \( p \) from the rest of the equation. Subtract \( AB \) from both sides:\[ ABp = 24f - AB \]

Step 4: Solve for p

Finally, divide both sides of the equation by \( AB \) to solve for \( p \):\[ p = \frac{24f - AB}{AB} \]

Key Concepts

Solving Equations

Solving equations involves finding the value of a variable that makes the equation true. It’s like a quest where you look for the answer which completes the puzzle. To solve an equation, you often need to perform a series of steps or operations to find the desired variable.
In our original exercise, we're trying to find the value of the variable \( p \). This requires understanding the equation and then systematically breaking it down to isolate \( p \). The trick with solving equations is recognizing which mathematical operations to apply, such as adding, subtracting, multiplying, or dividing, to simplify the equation step by step.
By the end of the process, we aim to have \( p \) on one side of the equation and all other terms on the opposite side.

Isolating Variables

Isolating variables means rearranging an equation so that one specific variable appears alone on one side. This process is crucial when you want to determine the specific value of a variable.
In the exercise, the variable \( p \) is entangled in several terms, including the fraction \( \frac{24f}{B(p+1)} \). To pinpoint \( p \), you follow a series of logical steps. Starting by eliminating any terms or factors blocking the path to \( p \) is key. This might involve multiplying or dividing entire terms to simplify the equation.
Our journey includes specific actions such as multiplying both sides by a term to get rid of the denominator and subsequently subtracting or adding terms. Each move brings us closer to having \( p \) isolated and alone, which is our goal.

Formula Rearrangement

Formula rearrangement involves reorganizing an equation into a form that's more useful for the problem at hand. It allows for easier evaluation or computation, especially when different variables need to be solved for different scenarios.
In the example of rearranging the given formula for \( p \), it's a bit like reordering a sentence for clarity. You want all elements in the right place, ensuring the formula reflects what you aim to solve without changing its inherent meaning.

• First, eliminate fractions by clearing denominators or numerators as necessary.
• Next, rearrange terms methodically, keeping variable terms on one side and constant terms on the other.

This approach ensures that, eventually, the specific formula for \( p \) comes out neatly, ready for use in calculations or further evaluations.

Annual Interest Rate

The annual interest rate is a percentage that represents the cost of borrowing money or the gain from lending money over a year. It plays a crucial role in finance, affecting loans, savings, and investments.
When dealing with equations, like in this exercise, the variable you isolate (\( p \) in this context) might represent the annual interest rate periodically used in formulas. Understanding its role within the formula helps frame your calculation in a practical financial context.
By correctly isolating \( p \), you uncover a calculation reliable for determining this key financial metric. This understanding enriches the practical application of algebraic manipulations, showing the real-world significance behind mathematical rearrangements.