Problem 72 - Algebra for College Students

Question

Solve \(A=P+\) Prt for \(P\), given that \(A=\$ 1423.50\), \(r=9 \frac{1}{2} \%\), and \(t=1\) year.

Step-by-Step Solution

Answer

The principal amount \( P \) is \( 1300 \).

Step 1: Identify the Formula

The original formula given is \( A = P + Prt \). This formula represents the future value \( A \) which includes the principal amount \( P \) and the simple interest over time \( Prt \).

Step 2: Substitute the Given Values

We are given that \( A = 1423.50 \), \( r = \frac{9.5}{100} = 0.095 \), and \( t = 1 \) year. Substitute these values into the formula: \( 1423.50 = P + P \times 0.095 \times 1 \).

Step 3: Simplify the Expression

Now, simplify the expression by replacing \( r \times t \) with its value: \( 1423.50 = P + 0.095P \). This can be simplified to \( 1423.50 = P(1 + 0.095) \).

Step 4: Solve for P

Solve for \( P \) by dividing both sides by \( 1.095 \): \( P = \frac{1423.50}{1.095} \).

Step 5: Calculate the Value of P

Perform the division to find the value of \( P \): \( P = 1300 \).

Key Concepts

Future Value Calculation

Future value calculation involves determining how much a current sum of money will be worth after a specified period, when interest is applied. In the context of simple interest, the formula to find the future value, denoted by \(A\), is \(A = P + Prt\). Here:

• \(P\) is the principal or initial amount invested or borrowed.
• \(r\) is the rate of interest per year (expressed as a decimal).
• \(t\) is the time period, generally in years, for which the money is invested or borrowed.

By substituting these values into the formula, you can calculate the total amount \(A\) you would accumulate. It's essential to understand this concept as it forms the foundation for many financial calculations, from loans to savings plans. The formula simplifies to \(A = P(1 + rt)\), emphasizing how it's tied directly to the principal and interest factors.

Solving Equations

When faced with an equation, the goal is to isolate the variable you need to find. In the given problem, the equation is \(1423.50 = P(1 + 0.095)\). This is a simple equation involving multiplication. Here's how to solve it:

• First, simplify the bracket: \(1 + 0.095 = 1.095\).
• This leads to the equation \(1423.50 = P \times 1.095\).
• To isolate \(P\), divide both sides by \(1.095\): \(P = \frac{1423.50}{1.095}\).
• Finally, perform the division to find \(P\).

Each step reduces the complexity of the formula, gradually honing in on the value of \(P\). Practicing this approach builds confidence, enabling you to tackle similar problems efficiently, whether in algebra or finance.

Interest Rate Conversion

Often, interest rates are given as percentages, but mathematical calculations need them in decimal form. Such conversion is straightforward and crucial for accurate calculations. For instance, in the given problem:

• The interest rate \(r\) is given as \(9\frac{1}{2}\%\).
• Convert it to a decimal by dividing by 100: \(r = \frac{9.5}{100} = 0.095\).

This decimal form is then used in calculations to determine the interest accrued. Understanding how to convert percentages to decimals ensures that any percentage-based calculation, such as interest on savings or loans, remains precise and yield correct results. It's a fundamental skill that prevents errors in more complex calculations.