Problem 9 - Elementary Algebra

Question

Calculate the simple interest earned on a 3-year investment of \(\$ 750\) at an annual interest rate of \(8 \% .\)

Step-by-Step Solution

Answer

The simple interest earned is \(\$180\).

Step 1: Understanding the Formula

To calculate the simple interest earned, we use the formula: \[ I = P \times r \times t \] where \(I\) is the interest, \(P\) is the principal amount, \(r\) is the annual interest rate (in decimal), and \(t\) is the time in years.

Step 2: Substitute Known Values

From the problem, we know the principal \(P = 750\), the rate \(r = 8\% = 0.08\), and the time \(t = 3\) years. Substitute these values into the formula so that it becomes: \[ I = 750 \times 0.08 \times 3 \]

Step 3: Calculate the Interest

Perform the multiplication: \[ I = 750 \times 0.08 \times 3 = 60 \times 3 = 180 \] So, the simple interest earned is \(\$180\).

Key Concepts

Interest Calculation

To calculate interest, especially simple interest, we rely on a straightforward mathematical formula. The primary formula for simple interest is \( I = P \times r \times t \), where:

• \( I \) represents the interest you earn over time.
• \( P \) stands for the principal amount, which is your initial investment or the initial sum of money on which the interest is calculated.
• \( r \) denotes the annual interest rate, expressed in decimal form.
• \( t \) is the number of years the money is invested or borrowed.

Using this formula, you can easily determine the amount of interest that will accumulate over a given period. To convert an interest rate from a percentage to a decimal, divide by 100. For example, an interest rate of 8% becomes 0.08. Simply plug in the known values into the formula to get your answer. This calculation helps in understanding how your money grows over time or how much extra you will owe on a loan.

Investment

Investment refers to the allocation of resources, such as money, with the expectation of generating future returns or profit. When you invest, you essentially put your money to work.

This can be accomplished through various vehicles like stocks, bonds, real estate, or savings accounts. In the context of our example, the investment involves depositing a sum of money, called the principal, into an account that yields a specified annual interest rate.

• This principal amount is your starting point.
• The investment period is determined by how long you plan to keep your money invested, which in the given problem is 3 years.
• After this period, the interest earned becomes an additional benefit to the initial amount.

Choosing the right type of investment and understanding its implications are crucial steps. By learning to calculate the simple interest, you can better evaluate the potential returns on different investments and make more informed financial decisions.

Annual Interest Rate

The annual interest rate is a critical component in the calculation of interest as it indicates the yearly percentage increase on the principal amount. In simple terms, it tells you how much extra money you will earn or owe each year.

• Expressed as a percentage, this rate is often found in the terms and conditions of the investment or loan.
• To use it in calculations, convert it to a decimal by dividing by 100.
• In our example, an 8% annual interest rate becomes 0.08 when expressed as a decimal.

This annual rate directly influences the amount of interest accrued over time. A higher interest rate typically means more interest earned or paid over the same period. Keeping track of the annual interest rate helps investors or borrowers in planning their finances and anticipating future financial status.