Problem 11 - Algebra for College Students

Question

Solve \(A=P+\operatorname{Prt}\) for \(r\), given that \(A=\$ 1372\), \(P=\$ 700\), and \(t=12\) years. Express \(r\) as a percent.

Step-by-Step Solution

Answer

The interest rate \(r\) is 8\%.

Step 1: Identify the given values

We are provided with the equation \(A = P + P \, r \, t\). The values given are \(A = 1372\), \(P = 700\), and \(t = 12\) years. Our task is to solve for \(r\).

Step 2: Substitute values into the equation

Substitute the known values into the equation: \[ 1372 = 700 + 700 \, r \, \times 12 \]

Step 3: Isolate the term containing \(r\)

To isolate the term containing \(r\), first, subtract \(700\) from both sides of the equation:\[ 1372 - 700 = 700 \, r \, \times 12 \]This simplifies to:\[ 672 = 700 \, r \, \times 12 \]

Step 4: Solve for \(r\)

Now, solve for \(r\) by dividing both sides of the equation by \(700 \times 12\):\[ r = \frac{672}{700 \times 12} \]

Step 5: Calculate the value of \(r\)

Perform the calculation:\[ r = \frac{672}{8400} \approx 0.08 \]

Step 6: Express \(r\) as a percentage

Since we need \(r\) as a percentage, multiply by 100:\[ r = 0.08 \times 100 = 8\% \]

Key Concepts

Algebraic Manipulation

Algebraic manipulation is a skill that allows us to rearrange equations to find unknown variables. In our exercise, manipulating the given formula enabled us to solve for the interest rate, \( r \). When given an equation like \( A = P + P \, r \, t \), our goal was to isolate \( r \). This meant moving other terms to the opposite side of the equation and performing operations to simplify.
In general, these are some of the algebraic manipulation techniques commonly used:

• Moving Terms: This includes adding or subtracting terms on both sides to isolate the variable you're solving for.
• Factoring: If needed, factor out common terms to simplify the equation.
• Division or Multiplication: Use these operations to get the variable alone on one side of the equation.

In this problem, subtracting \( 700 \) from both sides helped isolate the term associated with \( r \). This step is key to simplifying the expression to our target unknown.

Equation Solving Steps

The process of solving equations involves systematic steps to find the unknown variable. Let's breakdown those steps as used in our example for clarity.

• Identify Known Values: Recognize all elements and what they represent. Here, \( A \), \( P \), and \( t \) were given.
• Substitute Values: Place these known values into the equation. This makes the equation numeric instead of algebraic initially.
• Isolate the Variable: Rearrange the equation so the desired variable is isolated on one side. This involves using inverse operations such as subtraction or division.

Following these steps allows simplification of even complex-looking equations to find the unknown. Each phase methodically brings us closer to the solution in a structured manner, ensuring clarity and accuracy.

Interest Rate Calculation

Interest rate calculation is a crucial part of many financial computations, allowing individuals to understand how interest affects loan amounts over time. Given the formula: \( A = P + P \, r \, t \) Here's the breakdown of understanding interest rate calculations better:

• Principal (\(P\)): This is the initial amount of money on which interest is calculated.
• Rate (\(r\)): The percentage of interest charged per period. Solving for this was the task in our exercise.
• Time (\(t\)): The period over which the interest is calculated, often in years.
• Amount (\(A\)): This is the total value after interest is added.

In our exercise, once \( r \) was calculated, expressing it as a percentage by multiplying by 100 provided a clearer financial perspective. This shows how the interest affects the principal over a given time.