Problem 53 - Precalculus: Functions and Graphs

Question

The formula occurs in the indicated application. Solve for the specified variable. \(A=P+P r t\) for \(r \quad\) (principal plus interest)

Step-by-Step Solution

Answer

\( r = \frac{A - P}{P t} \)

Step 1: Understand the Equation

The formula given is for the total amount accumulated after interest, where \( A \) is the total amount, \( P \) is the principal, \( r \) is the rate of interest, and \( t \) is the time. We need to solve for the variable \( r \).

Step 2: Isolate the Interest Term

First, rewrite the equation by isolating the interest term on one side. The equation \( A = P + P r t \) can be rearranged to \( A - P = P r t \). This allows us to focus on the terms involving \( r \).

Step 3: Solve for the Interest Rate \( r \)

To solve for \( r \), we need to divide both sides of the equation by \( P t \). This gives us \( \frac{A - P}{P t} = r \). Now, \( r \) is isolated and the solution is complete.

Key Concepts

Interest Rate

When dealing with financial calculations, understanding the concept of interest rate is crucial. The interest rate is essentially the cost of borrowing money, often expressed as a percentage of the principal over a time period. In the formula given, the interest rate \( r \) determines how much additional money you will have to pay back or earn on your original investment \( P \). Varying the interest rate can have a significant impact on the total amount \( A \) you accumulate over time.

Interest rates come in different forms, such as simple or compound. In our scenario, the formula reflects simple interest, where the interest is only calculated on the principal. Understanding how different interest rates work is key to making informed decisions on savings and borrowings.

• Concept of Simple Interest: Interest earned only on the original principal, without compounding.
• Affect on Total Amount: Directly proportional to the principal and time.

Algebraic Manipulation

Algebraic manipulation involves rearranging equations to isolate desired variables. This is a powerful tool that allows you to solve for unknowns in mathematical equations. The key here is to maintain the balance of the equation while performing various operations like adding, subtracting, multiplying, or dividing.

In our exercise, we began with the equation \( A = P + P r t \), where the goal is to solve for the interest rate \( r \). Through algebraic manipulation, we rearrange the terms to focus on \( r \). We subtracted \( P \) from both sides, which is a fundamental technique to isolate a variable. This step is crucial as it makes the equation simpler and focused.

• Maintaining Balance: Any operation done to one side must be done to the other to keep the equation valid.
• Simplification: Reducing equations to their simplest form by combining like terms and reducing fractions where possible.

Isolating Variables

Isolating variables is a fundamental aspect of solving equations. It essentially means rearranging the equation so that the variable of interest stands alone on one side of the equation. This technique is crucial in solving real-world problems where understanding of one specific quantity is needed.

To solve for \( r \) in our example, we isolated it by first moving other terms to the opposite side of the equation. Once arranged as \( A - P = P r t \), we divided both sides by \( P t \). This left \( r \) by itself, giving us the equation \( r = \frac{A - P}{P t} \).

• Rearranging Steps: Takes each side of the equation closer to isolating the desired variable.
• Division as an Isolating Tool: Used when multiplication involves the variable in question.

Understanding how to clearly isolate variables can significantly simplify complex problem solving, allowing for direct calculation of financial figures like interest rate, taxes, and more.