Problem 10 - Algebra for College Students

Question

10\. Solve \(A=P+P r t\) for \(A\), given that \(P=\$ 850\), \(r=9 \frac{1}{2} \%\), and \(t=10\) years.

Step-by-Step Solution

Answer

\(A = 1657.5\) dollars.

Step 1: Identify Given Variables

We need to identify the given values for the variables in the equation. From the problem statement: \(P = \$850\), \(r=9.5\%\) (which can be written as \(0.095\)), and \(t=10\) years.

Step 2: Write the Equation with Values

Replace the variables in the equation \(A = P + Prt\) with the given values: \(A = 850 + 850 \times 0.095 \times 10\).

Step 3: Calculate the Interest

Calculate \(Prt\) by multiplying the principal \(P\), rate \(r\), and time \(t\). So \(850 \times 0.095 \times 10 = 807.5\).

Step 4: Calculate Total Amount \(A\)

Add the principal \(P\) to the calculated interest \(Prt\) to find \(A\). Thus, \(A = 850 + 807.5 = 1657.5\).

Key Concepts

Simple Interest Formula

The Simple Interest Formula is a helpful tool, especially when calculating the interest earned on a fixed deposit over a period of time. The formula is expressed as: \( A = P + P \, r \, t \), where:

• \( A \) is the total amount after time \( t \).
• \( P \) is the principal or original sum of money.
• \( r \) represents the interest rate per period as a decimal.
• \( t \) is the time duration that the money is invested or borrowed for, in years.

To use this formula effectively, it's essential to convert the interest rate from a percentage to a decimal by dividing by 100. This ensures consistency in the calculation.
Using this formula, you can determine both the final amount \( A \) and the interest amount \( P \, r \, t \) separately. It's a straightforward method of calculating interest that only requires simple multiplication and addition.

Equation Solving

Solving an equation in algebra can involve several steps. In our case, the equation was \( A = P + P \, r \, t \). Here, we solved for \( A \), which means finding the total amount that includes both the initial investment and the interest earned.
Consider these steps:

• Substitute known values: Replace variables with the given numerical values, maintaining the integrity of the equation.
• Perform arithmetic operations: Carry out multiplication and addition as required by the equation.

Through these steps, the process becomes a matter of straightforward calculation, where logic and order of operations paved the way to the solution.

Step-by-Step Calculation

Breaking down problems into smaller steps makes them more manageable. Let’s see how each part of the equation is handled:

• Identify Components: Start by noting all given parts of the problem, such as \( P = \$850 \), the interest rate \( r = 9.5\% = 0.095 \), and the time \( t = 10 \) years.
• Substitute: Place these values into the simple interest formula: \( A = 850 + 850 \times 0.095 \times 10 \).
• Interest Calculation: Calculate the interest \( P \, r \, t \). Here, it will be \( 850 \times 0.095 \times 10 = 807.5 \).
• Final Amount: Finally, add the initial principal to the interest to find \( A \). So, \( A = 850 + 807.5 = 1657.5 \).

Applying these steps ensures a clear path to solving the problem, demonstrating a practical approach to algebraic problems.