Problem 12 - Algebra for College Students

Question

12\. Solve \(A=P+P r t\) for \(r\), given that \(A=\$ 516, P=\$ 300\), and \(t=8\) years. Express \(r\) as a percent.

Step-by-Step Solution

Answer

The rate \(r\) is 9\%.

Step 1: Identify the Known Values

We are given the following values: \(A = 516\), \(P = 300\), and \(t = 8\) years. These are the values we will use to solve the equation for \(r\).

Step 2: Set Up the Equation

Substitute the known values into the formula \(A = P + Prt\). The equation becomes: \(516 = 300 + 300r \times 8\).

Step 3: Simplify the Equation

First, calculate \(300 \times 8\) to simplify the equation. This gives us \(2400r\), so the equation becomes \(516 = 300 + 2400r\).

Step 4: Isolate the Term with \(r\)

Subtract 300 from both sides to isolate the term containing \(r\): \(516 - 300 = 2400r\). This simplifies to \(216 = 2400r\).

Step 5: Solve for \(r\)

Divide both sides by 2400 to solve for \(r\): \(r = \frac{216}{2400}\).

Step 6: Calculate \(r\)

Perform the division to find \(r\): \(r = 0.09\).

Step 7: Convert \(r\) to a Percentage

Multiply \(r\) by 100 to express it as a percentage: \(r = 0.09 \times 100 = 9\%\).

Key Concepts

Solving Equations

The ability to solve equations is fundamental in algebra. It involves finding the values of variables that make the equation true. In this exercise, we are given an equation \(A=P+Prt\), and our task is to solve for \(r\), the interest rate. The equation includes values for total amount \(A\), principal \(P\), and time \(t\). Each of these values is plugged into the equation.

To isolate \(r\), we first substitute the given values into the equation, transforming it into a numerical expression. The goal is to manipulate this expression to isolate \(r\) on one side of the equation. This step-by-step isolation is what constitutes solving the equation. Understanding how to systematically rearrange the terms is crucial in this process.

Algebraic Manipulation

Algebraic manipulation involves performing operations to simplify and solve equations. In the exercise, we perform several key manipulations, such as multiplying and dividing, to solve for \(r\).

Here’s what happens:
- **Substitute the given values:** We plug \(A = 516\), \(P = 300\), and \(t = 8\) years into the formula.
- **Simplification:** Multiply \(300r\) by \(8\) to simplify it to \(2400r\).
- **Isolate \(r\):** By subtracting the principal \(300\) from both sides, we keep \(r\) on one side of the equation.
- **Solve for \(r\):** Divide each side by \(2400\) to fully isolate \(r\).

Through these steps, algebraic manipulation transforms the equation into an easily solvable form. Each operation is based on the properties of equality, ensuring that what you do to one side of the equation, you do to the other.

Percentage Calculation

Converting a decimal into a percentage is often necessary in solving mathematical problems. This conversion is especially important in finance, where rates are typically stated as percentages.

In our case, after solving for \(r\), the result is 0.09. This is a decimal representation of the interest rate. To convert it to a percentage, we multiply by 100, following the simple rule:

- **Decimal to Percentage:** \(r = 0.09 \times 100 = 9\%\)

This conversion ensures the rate is in a standard form that's easy to interpret. It's a quick calculation but vital for clarity in communication and understanding within any financial context.

Interest Rate Formula

The formula \(A = P + Prt\) represents the final amount \(A\), where \(P\) is the principal (initial amount), \(r\) is the interest rate (expressed as a decimal), and \(t\) is the time period.

This formula is a linear equation in terms of \(r\), assuming \(t\) is a fixed period. It is widely used to calculate simple interest in finance, providing a straightforward way to understand how money grows over time.

Key components include:

• **Principal \(P\):** The original sum of money.
• **Interest Rate \(r\):** Represents the cost of borrowing money, percentage form upon conversion.
• **Time \(t\):** The duration for which the money is borrowed or invested.

By mastering this formula, one gains a foundational tool for financial calculations, necessary for evaluating investment returns and loan costs.