Problem 43 - Elementary Algebra

Question

Use the formula \(i=P n\) to reach a solution. (Objective A) What will be the interest earned on a \(\$ 5000\) certificate of deposit invested at \(3.8 \%\) annual interest for 10 years?

Step-by-Step Solution

Answer

The interest earned is \$1900.

Step 1: Identify the variables

We need to determine the values of the variables in our formula. In this exercise, the principal amount \(P\) is \$5000\, the annual interest rate \(r\) is 3.8%, which can be written as 0.038, and the time \(n\) is 10 years.

Step 2: Substitute values into the formula

The formula to calculate simple interest is \(i = P \times r \times n\). Substitute the given values into the formula: \(i = 5000 \times 0.038 \times 10\).

Step 3: Perform the multiplication

Calculate the product to find the interest: \(i = 5000 \times 0.038 = 190\). Then multiply this result by \(10\) to find the total interest: \(i = 190 \times 10 = 1900\).

Step 4: Verify your calculations

Double-check your calculations to ensure accuracy: Begin by multiplying the principal by the interest rate, and then by the number of years, confirming the result is the same: \(5000 \times 0.038 \times 10 = 1900\).

Key Concepts

Principal Amount

The principal amount refers to the initial sum of money that is invested or loaned without considering any interest. In simple interest calculations, the principal is the base number on which the interest will be calculated.
For example, if you invest $5000 into a savings account that offers interest, the $5000 is your principal amount.
This amount remains constant in simple interest calculations, meaning it does not change regardless of how long the money is invested.
In the exercise, the principal amount is $5000, which means that all interest earned will be calculated based on this figure. A few key points about the principal amount include:

• It is the starting point in any simple interest equation.
• It does not include interest or additional deposits added after the initial investment.
• Knowing the principal amount is essential because it directly impacts the total interest earned.

Interest Rate

The interest rate is the percentage charged on the principal amount for a particular period. In simple interest, it is expressed as an annual percentage rate (APR) and determines how much interest will be added to the principal.
An easy understanding of the interest rate is that it is like a fee you earn for investing or a base cost you pay for borrowing money.
In the provided example, the interest rate is 3.8%.
This means that for every $100 invested, the investor earns $3.80 in interest each year. It must be converted into a decimal form for calculation, i.e., 3.8% becomes 0.038.
The interest rate plays a significant role in determining the total interest earned over the investment period. Here's what to remember about interest rates:

• They must be converted from percentages into decimals for calculation purposes.
• The rate determines how quickly money grows—for higher rates, you earn more interest.
• In simple interest, the interest rate is constant throughout the entire period.

Time Period

The time period in simple interest calculations is the length of time over which the interest is calculated. It is usually expressed in years.
This time period is crucial because it tells us how long the principal amount will earn interest at the specified rate.
The formula uses the time period as a multiplier alongside the principal amount and interest rate to calculate total interest.
In the example, the time period is 10 years. This means the principal will earn interest at the specified rate every year for a decade. A few important aspects of the time period:

• It is usually measured in consistent units, such as years or months, to match the interest rate period.
• The longer the time period, the more interest accumulates, as the principal earns interest each time unit passes.
• Accurate measurement of the time period is key for precise interest calculation.