Problem 9 - Algebra for College Students

Question

Solve \(A=P+\) Prt for \(A\), given that \(P=\$ 1000\), \(r=12 \%\), and \(t=5\) years.

Step-by-Step Solution

Answer

The final amount \(A\) is $1600.

Step 1: Identify the Formula

The exercise involves using the formula for simple interest: \[ A = P + Prt \] This can also be expressed as: \[ A = P (1 + rt) \] where: - \(A\) is the final amount- \(P\) is the principal amount- \(r\) is the interest rate (expressed as a decimal)- \(t\) is the time in years.

Step 2: Convert Percentage Rate to Decimal

The interest rate \(r\) is given as 12%, which needs to be converted to a decimal for calculations.To do this, divide the percentage by 100:\[ r = \frac{12}{100} = 0.12 \]

Step 3: Substitute Known Values

Substitute the known values into the formula.Given: \(P = 1000\), \(r = 0.12\), \(t = 5\).The formula becomes: \[ A = 1000 + 1000 \times 0.12 \times 5 \]

Step 4: Calculate the Interest

Calculate the interest by multiplying the values of \(P\), \(r\), and \(t\):\[ Prt = 1000 \times 0.12 \times 5 \]\[ Prt = 1000 \times 0.6 \]\[ Prt = 600 \]

Step 5: Compute the Final Amount

Add the calculated interest back to the principal:\[ A = 1000 + 600 \]\[ A = 1600 \]

Key Concepts

Mathematical Formulas

Formulas provide a structure to solve mathematical problems. In the context of simple interest calculations, the formula you need is straightforward:

• Complete formula: \( A = P + Prt \)
• Simplified formula: \( A = P(1 + rt) \)

Both forms will accomplish the task of calculating how much money you will have after earning interest over a given period. \( A \) represents the total amount, \( P \) is the principal (starting) amount, \( r \) stands for the interest rate converted to a decimal, and \( t \) is the time in years. By using these formulas, you're able to clearly see how each parameter contributes to calculating the final amount. These are critical tools in finance and basic algebra.

Percentage Conversion

Understanding percentages and how to convert them to decimals is crucial. This is especially important in financial calculations like interest. A percentage is simply a fraction of 100. For example, converting 12% into a decimal requires dividing by 100. This process helps to simplify calculations.

• Original percentage: 12%
• Conversion formula: \( r = \frac{12}{100} \)
• Result as a decimal: \( r = 0.12 \)

By converting a percentage to a decimal, you can easily use it within mathematical equations like the simple interest formula, ensuring accuracy and simplifying the multiplication process.

Interest Calculation

Calculating interest is a step-by-step process that combines different values like the principal amount, interest rate, and time. Here's how you break it down:First, substitute the known values into the formula: \( P = 1000 \), \( r = 0.12 \), \( t = 5 \).The calculation steps are:

• Calculate \( Prt \): \( 1000 \times 0.12 \times 5 \)
• Intermediate steps yield \( Prt = 1000 \times 0.6 \)
• Final result of \( Prt = 600 \)

By multiplying these components, you find the interest amount accumulated over the specified period.

Algebraic Substitution

Algebraic substitution is the step where you place known values into a formula to find an unknown quantity. In the case of simple interest, you've already outlined the formula. The next step is using this formula:Substitution requires plugging the values of \( P \), \( r \), and \( t \) into the equation:

• Start by positioning values \( P = 1000 \), \( r = 0.12 \), \( t = 5 \) into \( A = P + Prt \)
• The equation then reads \( A = 1000 + 1000 \times 0.12 \times 5 \)
• Solve to find \( A = 1600 \)

Through substitution, the answer is revealed, showing both the intermediate and final calculations to ensure steps are clear and mistakes are minimized.