Problem 99 - Beginning and Intermediate Algebra

Question

The expression \(\frac{I}{P T}\) represents the rate of interest being charged if a loan of \(P\) dollars for \(T\) years required \(I\) dollars in interest to be paid. Find the interest rate if a \(\$ 650\) loan for 3 years to buy a used IBM personal computer requires \(\$ 126.75\) in interest to be paid.

Step-by-Step Solution

Answer

The interest rate is 6.5\%.

Step 1: Identify the Variables

First, recognize and identify the values given in the problem. Here, \( I = 126.75 \), \( P = 650 \), and \( T = 3 \).

Step 2: Write the Formula for Interest Rate

Use the formula for the interest rate given by \( R = \frac{I}{P T} \) where \( R \) is the interest rate, \( I \) is the interest, \( P \) is the principal, and \( T \) is the time in years.

Step 3: Substitute Known Values

Substitute the known values into the formula: \( R = \frac{126.75}{650 \, \times\, 3} \).

Step 4: Calculate the Denominator

Calculate the denominator: \( 650 \, \times \, 3 = 1950 \).

Step 5: Perform the Division

Now, divide the numerator by the calculated denominator: \( R = \frac{126.75}{1950} \). This equals \( R = 0.065 \) or 6.5\%.

Key Concepts

Interest Rate Formula

Interest is the extra money paid for borrowing money, often expressed as a percentage of the principal. To find the actual percentage, you can use the interest rate formula. This formula is a fundamental tool in financial mathematics. The formula for calculating the interest rate, denoted as \( R \), is:\[R = \frac{I}{P \times T}\]Here:

• \( I \) is the total interest paid during the entire period.
• \( P \) stands for the principal amount, or the initial sum of money lent or borrowed.
• \( T \) denotes the time period for which the money is borrowed, expressed in years.

By using this formula, you can determine how much interest you will pay each year as a percentage of the principal amount. This is useful whether you are taking out a loan or depositing money in a savings account. The computed rate can help you compare different financial products and find the best deal.

Principal Amount

The principal amount is the initial sum of money either lent or invested. It's the base amount on which interest calculations are performed. Knowing the principal is vital for both lenders and borrowers, as it serves as the foundation for all subsequent interest-based calculations. For example, if you borrow \( \$650 \) to buy a used IBM personal computer, as in the original exercise, this amount is your principal. It does not change over the life of the loan or investment unless you make partial repayments. Nevertheless, the principal is crucial as it determines how much interest will accrue over time.

Why is Principal Important?

Consider these factors since they highlight the principal's significance:

• Bigger principal amounts mean more interest accrues, as interest is a percentage of the principal.
• Knowing the principal helps in understanding your or your lender's risk exposure.
• Principal values are used in myriad financial calculations, from loan schedules to profitability assessments.

Understanding the principal amount lets you grasp your overall financial responsibility or potential return on investment.

Time Period in Years

Time plays a crucial role in determining the amount of interest you will pay or earn. With simple interest, the time period \( T \) is usually expressed in years, indicating how long the money is borrowed or invested. In our earlier exercise with the used computer, the loan lasts for 3 years. This means interest is calculated for each year, with the value of time always in the denominator of our interest rate formula.

The Importance of Time in Interest Calculations

The length of time directly influences the total amount of interest you will ultimately pay or earn:

• Longer time periods increase the total interest due as more time allows more interest to accumulate.
• Shorter time periods lessen the financial burden of interest, a crucial point when aiming to minimize cost.
• Financial planning benefits from knowing the time period, letting you manage cash flows and expenses wisely.

Understanding how time affects simple interest is critical for making informed and financially sound decisions. Always consider how many years you're engaging in a loan or investment, as it drastically impacts your financial outcomes.