Problem 33 - College Algebra with Corequisite Support

Question

For the following exercises, use the compound interest formula, \(A(t)=P\left(1+\frac{r}{n}\right)^{n t}\). Solve the compound interest formula for the principal, \(P\).

Step-by-Step Solution

Answer

P = \(\frac{A(t)}{\left(1+\frac{r}{n}\right)^{nt}}\).

Step 1: Identify the given formula

We are given the compound interest formula: \(A(t)=P\left(1+\frac{r}{n}\right)^{n t}\). Our goal is to solve this formula for the principal \(P\).

Step 2: Isolate the principal P

To solve for \(P\), we need to rearrange the formula to get \(P\) by itself. The current formula is \(A(t) = P \left(1 + \frac{r}{n}\right)^{nt}\). We can isolate \(P\) by dividing both sides by \(\left(1+\frac{r}{n}\right)^{nt}\).

Step 3: Formula after isolation

After dividing both sides by \(\left(1+\frac{r}{n}\right)^{nt}\), we get: \[P = \frac{A(t)}{\left(1+\frac{r}{n}\right)^{nt}}\]. This is the formula for the principal \(P\).

Key Concepts

Principal

In financial mathematics, the term "principal" refers to the original sum of money that is deposited, invested, or borrowed before any interest is applied. It is the amount at the starting point of a financial calculation and serves as the base amount upon which interest is calculated.
Understanding the principal is crucial because it influences the total amount of interest earned or paid. The larger the principal, the more potential for increased interest earnings or payments over time.

• Initial Investment or Loan: The principal represents this initial sum.
• Interest Calculation: Interest is always calculated on the principal amount.

The compound interest formula, which is widespread in financial applications, specifically pinpoints the principal to determine how much the original amount will grow or how much needs to be repaid over time. Recognizing the principal in financial equations is the starting step for solving most financial problems.

Isolate Variable

Isolating a variable is a fundamental algebraic skill used to solve equations. This involves manipulating the equation so that the desired variable stands alone on one side, making it easier to understand its value or role in the equation. In our context, isolating the principal, \(P\), from the compound interest formula is critical.
To isolate \(P\), you begin with the original equation: \(A(t) = P \left(1 + \frac{r}{n}\right)^{nt}\). The goal is to have \(P\) by itself on one side of the equation.
Here's how you can do it:

• Identify the operations affecting \(P\).
• Reverse these operations by applying inverse mathematical operations.
• Perform the reverse operation on both sides of the equation to maintain equality.

In this situation, dividing both sides by \(\left(1+\frac{r}{n}\right)^{nt}\) isolates \(P\), resulting in the equation: \[P = \frac{A(t)}{\left(1+\frac{r}{n}\right)^{nt}}\]. Isolating variables helps in solving a range of problems and understanding relationships between different components in a formula.

Algebraic Rearrangement

Algebraic rearrangement is the process of rearranging an equation to solve for a specific variable or simplify it. This important mathematical process ensures that equations are more accessible for analysis and interpretation.
In the context of the compound interest formula, algebraic rearrangement involves rearranging the formula to determine the principal \(P\). The steps involve understanding how different parts of the equation interact and applying arithmetic operations to both sides of the equation until \(P\) is isolated.

• Understand the structure: Recognize that the formula is structured with multiplication, exponents, and fractions.
• Plan the steps: Decide what operations will bring the desired variable to one side noting inverse operations.
• Execute the steps: Methodically perform each operation equally on both sides of the equation until the desired form is achieved.

Effective algebraic rearrangement makes solving complex equations manageable and clear, especially in financial mathematics, where compound interest must be addressed precisely and efficiently.

Financial Mathematics

Financial mathematics is a field that applies mathematical methods to financial problems. It includes calculations that help individuals and businesses make informed financial decisions.
One key concept within financial mathematics is the compound interest formula. The formula \(A(t) = P \left(1 + \frac{r}{n}\right)^{nt}\) is crucial for understanding how investments grow over time or how debts accumulate.

• Investment Growth: Compound interest shows how the principal increases when interest is calculated on both the initial sum and accumulated interest.
• Debt Management: It helps in evaluating how much one will pay over time when interests are aggregated.
• Decision Making: Understanding these calculations assists in financial planning and decision-making.

Financial mathematics is not just about number crunching. It’s a vital tool for making smart financial decisions, recognizing potential growth opportunities, and navigating complex financial landscapes with confidence.