Problem 11 - Algebra for College Students

Question

11\. Solve \(A=P+\) 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: Understand the Formula

The formula provided is the interest formula: \( A = P + Prt \), where \( A \) is the total amount, \( P \) is the principal amount, \( r \) is the rate of interest, and \( t \) is the time in years.

Step 2: Rearrange the Formula to Solve for r

To solve for \( r \), we need to isolate \( r \) on one side of the equation. Start by subtracting \( P \) from both sides, giving us: \( A - P = Prt \). Then divide both sides by \( Pt \) to get \( r = \frac{A - P}{Pt} \).

Step 3: Substitute Given Values

We are given \( A = 1372 \), \( P = 700 \), and \( t = 12 \). Substitute these values into the equation: \( r = \frac{1372 - 700}{700 \times 12} \).

Step 4: Calculate the Value Inside the Fraction

First compute \( 1372 - 700 \) to get \( 672 \). Next, calculate \( 700 \times 12 \) to get \( 8400 \). So, the equation becomes \( r = \frac{672}{8400} \).

Step 5: Calculate r

Now, perform the division \( \frac{672}{8400} = 0.08 \).

Step 6: Convert r to a Percentage

Multiply \( r \) by 100 to convert it into a percentage: \( 0.08 \times 100 = 8\% \). This means the interest rate is \( 8\% \).

Key Concepts

Solving for Variables

Understanding how to solve for variables is crucial in algebra and any mathematical calculations. When we are given an equation, we often need to isolate a specific variable within that equation to find its value. In the context of the interest formula, the variable we are solving for is the interest rate, denoted as \( r \).

To solve for \( r \) in the equation \( A = P + Prt \), the first step is to make \( r \) the subject of the equation. This involves rearranging the equation to isolate \( r \) on one side. It typically involves these steps:

• Subtract \( P \) from both sides to eliminate the added principal: \( A - P = Prt \).


• Divide both sides by \( Pt \) to solve for \( r \): \( r = \frac{A - P}{Pt} \).

These are fundamental steps to ensure you systematically isolate the variable and can accurately solve the equation.

Interest Calculation

Interest calculation is a significant aspect of finance, commonly used to determine how much extra money one gains or owes due to interest. The interest formula \( A = P + Prt \), where \( A \) is the total amount, comprises both principal and interest earned over time, helps us compute these values. In this problem, our goal was to identify the interest rate \( r \).

Steps for calculating interest usually involve:

• Identifying the principal amount \( P \), which represents the initial sum of money.


• Determining the time \( t \) for which the interest accumulates, measured in years.


• Substituting the known values into the rearranged formula to compute the unknown, in this case, \( r \).

By computing \( 1372 - 700 \) and \( 700 \times 12 \), and subsequently dividing \( 672 \) by \( 8400 \), we found the rate \( r \), which helps in understanding the proportion of interest accrued over the period.

Percentage Conversion

Converting a decimal to a percentage is a vital mathematical skill that is especially practical in financial contexts. Oftentimes, interest rates are expressed as percentages to make them easier to understand and compare. After calculating \( r \) in its decimal form, \( 0.08 \) in this instance, the final step is converting it to a percentage.

The process is straightforward:

• Multiply the decimal result by 100. In this example, \( 0.08 \times 100 = 8 \% \).

This conversion helps clarify the rate, which, in finance, indicates the annual return or cost represented per unit of 100. Percentages are generally more indicative to everyday monetary understanding, aiding in decision-making related to loans, savings, and investments.