Problem 46 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve for the indicated variable. Investment at Simple Interest Solve for \(r: A=P+P r t\)

Step-by-Step Solution

Answer

The rate of interest \(r\) is given by \(r = \frac{A - P}{P t}\).

Step 1: Isolate terms containing \(r\)

Subtract \(P\) from both sides of the equation to isolate terms that contain \(r\). The equation becomes: \(A - P = P r t\).

Step 2: Isolate \(r\)

Divide both sides of the equation by \(P t\) to isolate \(r\). This will give: \(r = \frac{A - P}{P t}\).

Key Concepts

Investment Problems

Investment problems often involve dealing with various financial calculations to determine interest, time, or rate of return. When dealing with investments, especially those related to simple interest, it's crucial to understand the basic formula:

• Simple Interest Formula: The formula to calculate simple interest is given by: \[ A = P + Prt \]where:
  • \( A \) = the total amount after interest,
  • \( P \) = the principal amount or initial investment,
  • \( r \) = the rate of interest,
  • \( t \) = the time period for which the interest is calculated.

To solve investment problems, you are usually asked to find one of the variables in that formula given the others. The key is to properly rearrange the formula to solve for the unknown, depending on what information you have.

Solving Equations

Solving equations involves finding the value of the variable that makes the equation true. In this context, we're solving for the variable \( r \), the rate of interest in an equation derived from the simple interest formula. The goal is to isolate \( r \) on one side of the equation so you can determine its value straightforwardly. To tackle such a problem, you need to:

• Identify the expressions or terms involving \( r \) and aim to have them on one side.
• Reorganize the equation to simplify and solve it.
• Never forget to perform the same operation on both sides of the equation to maintain equality.

In our specific example, we moved all terms containing \( r \) to one side to make solving feasible.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying equations to find solutions for unknown variables. This involves strategic operations such as adding, subtracting, multiplying, or dividing both sides of an equation. For our exercise, we did:

• Step 1: Subtract \( P \) from both sides. This helps in isolating parts of the equation that contain \( r \): \[ A - P = Prt \]
• Step 2: Divide both sides by \( Pt \). This isolates \( r \) and solves the equation: \[ r = \frac{A - P}{Pt} \]

Working systematically through each step ensures clarity and precision, minimizing errors. By mastering algebraic manipulation, you can solve various kinds of equations with confidence.