Problem 40 - Elementary Algebra

Question

Use the formula \(i=P n\) to reach a solution. (Objective A) How much interest will be charged on a student loan if \(\$ 8000\) is borrowed for 9 months at a \(19.2 \%\) annual interest rate?

Step-by-Step Solution

Answer

The interest charged is $1152.

Step 1: Identify the parameters

First, we need to identify the values for each component of the formula \(i = Pn\). Here, \(P\) is the principal amount (the initial amount of money borrowed), \(n\) represents the rate and time period in a suitable format.

Step 2: Determine the principal \(P\)

From the question, it is given that the principal \(P\) is the amount borrowed, which is \(P = 8000\) dollars.

Step 3: Convert annual interest rate to monthly rate

The annual interest rate is given as \(19.2\%\). To find the monthly interest rate, divide the annual rate by 12 months: \(\frac{19.2}{12}\%\), which simplifies to \(1.6\%\) per month.

Step 4: Calculate \(n\) using the monthly interest

Since the loan is for 9 months with a monthly interest rate of \(1.6\%\), we convert \(1.6\%\) to a decimal for \(n\). This gives us:\[n = 1.6\% \times 9 = 0.016 \times 9 = 0.144\].

Step 5: Calculate the interest \(i\)

Now, using the formula \(i = Pn\):\[i = 8000 \times 0.144 = 1152\].Thus, the interest charged on the student loan for 9 months is \(\$1152\).

Key Concepts

Principal Amount

The principal amount is the initial sum of money borrowed or invested before any interest is applied. In the context of loans, it represents the amount of money the borrower requests from a lender. It's essential to determine the principal amount accurately because it forms the basis on which interest will be calculated.

When it comes to calculating simple interest, the principal amount is denoted by the symbol \( P \). In our exercise, the principal amount is \( \$8000 \). This value affects how much interest you end up paying or earning, so it’s crucial to understand it means the initial amount without any additional charges or interest.

• Always confirm the principal amount as the starting point for any simple interest calculation.
• The interest paid is directly proportional to the principal amount. More principal often means more total interest.

Annual Interest Rate

Interest is the cost of borrowing money or the gain from investing it, expressed as a percentage of the principal over a specific period, usually annually. The annual interest rate is the percentage of the principal that is charged as interest each year.

For this exercise, the annual interest rate is \( 19.2\% \). This means that if the loan was taken for one full year, the interest charged would be \( 19.2\% \) of the principal amount, \( \$8000 \) in this case. However, since the loan term here is 9 months, we'll need to adjust this rate to reflect the shorter period.

• Convert annual interest rates to other time frames (e.g., monthly) to match the borrowing period.
• An understanding of annual rates helps to foresee the potential cost of long-term borrowing.

Monthly Interest Rate

A monthly interest rate is a portion of the annual interest rate that applies to the portion of the year represented by each month. To find the monthly rate, divide the annual rate by 12, as there are 12 months in a year.

In this exercise, the annual rate of \( 19.2\% \) is divided by 12 months to find a monthly rate of \( 1.6\% \). This conversion allows for an accurate calculation of interest accumulated over a specific number of months.

• Monthly rates provide precision in short-term interest calculations like loans lasting a few months.
• Always convert percentages to decimals in formulas: here, \( 1.6\% \) becomes \( 0.016 \) when using in calculations.