Problem 13 - Algebra for College Students

Question

Solve \(A=P+P r t\) for \(P\), given that \(A=\$ 326, r=7 \%\), and \(t=9\) years.

Step-by-Step Solution

Answer

The value of \(P\) is approximately \(200\).

Step 1: Write Down the Given Formula

The given formula is \(A = P + P r t\). We need to solve this formula for \(P\).

Step 2: Substitute Given Values

We know that \(A = 326\), \(r = 0.07\) (since the percentage needs to be in decimal form), and \(t = 9\). Substitute these into the equation: \(326 = P + P \times 0.07 \times 9\).

Step 3: Simplify the Equation

Simplify the right side of the equation by combining the terms involving \(P\): \(P + P \times 0.07 \times 9 = P(1 + 0.07 \times 9) = P(1 + 0.63) = P \times 1.63\).

Step 4: Solve for P

Now that the equation is simplified to \(326 = P \times 1.63\), divide both sides by \(1.63\) to solve for \(P\): \(P = \frac{326}{1.63}\).

Step 5: Calculate P

Perform the division to find \(P\): \(P \approx 200\).

Key Concepts

Solving Equations

Solving equations is a fundamental concept in algebra. It involves finding the value of an unknown variable that satisfies a given equation. In this exercise, we need to solve for the variable \(P\) in the equation \(A = P + Prt\). The key steps involve isolating the variable \(P\) by manipulating the equation through algebraic operations like addition, subtraction, multiplication, and division.

• Start by identifying the equation you need to solve, writing down any known values.
• Substitute the known values into the equation. This converts it into a simpler expression involving the unknown variable.
• Perform algebraic operations to isolate the unknown variable on one side of the equation. This often involves simplification and rearranging the terms.
• Once the variable is isolated, solve the equation by performing the required arithmetic operations.

In our example, by substituting the numerical values into the original equation and simplifying, we can solve for \(P\) by dividing both sides by the simplified coefficient of \(P\). This process ensures we find the correct value of \(P\) that satisfies the original equation.

Interest Rate Problems

Interest rate problems involve calculations based on the concept of interest added to a principal amount over a period of time. In these calculations, you typically encounter terms like rate (\(r\)), time (\(t\)), principal (\(P\)), and the total amount (\(A\)). The formula \(A = P + Prt\) represents the future value of the principal after interest accrues over a specific timeframe.

• The principal \(P\) is the initial amount of money invested or borrowed.
• The rate \(r\) is usually expressed as a percentage but must be converted to a decimal (e.g., 7% becomes 0.07) for calculation purposes.
• The time \(t\) is often measured in years.
• The amount \(A\) is the total money accumulated after interest is applied over the period.

To solve interest-related questions, it's crucial to understand how these variables interact. Replace each variable with its known value to simplify the equation. This approach reveals not only how much interest will be earned or paid but also helps in evaluating investment opportunities or loan costs accurately.

Step-by-Step Solutions

Breaking down a problem into a step-by-step solution offers a clear path to understanding and solving equations. This methodical approach allows you to focus on each part of the equation independently, providing a deep understanding of the mathematical operations involved. When tackling a problem, especially in algebra, working systematically through the steps ensures accuracy and builds confidence.

• Organize the Information: Start by noting down the equation and identifying the knowns and unknowns.
• Substitution: Fill in the known values into the equation, simplifying the process of solving for the unknown.
• Simplification: Focus on reducing complex expressions by combining like terms and simplifying factors.
• Solution: Apply arithmetic operations to isolate the variable, verifying each step to ensure accuracy.
• Verification: Double-check your result by substituting it back into the original equation to verify its correctness.

In our example, this methodical breakdown not only revealed \(P\)'s value as 200 but also demonstrated how these steps build towards finding solutions in complex expressions, applying logical reasoning and precision in each step.