Question

For Exercises \(15-30,\) find the final balance and interest. \(\$ 290\) at \(9 \%\) compounded semiannually for 4 years

Short answer

Answer: The final balance is approximately $436.08, and the interest earned is approximately $146.08.

Step-by-step solution

Identify the variables in the problem

We have the following variables given in the problem: Initial investment (Principal): P = $290 Annual interest rate: r = \(9\%\) = \(0.09\) Number of compounds per year (Semiannually): n = 2 Time in years: t = 4

Compound interest formula

To find the final balance, we'll use the compound interest formula: \(A = P(1 + \frac{r}{n})^{nt}\) Where: A is the final balance P is the principal (initial investment) r is the annual interest rate (as a decimal) n is the number of compounds per year t is the time in years

Calculate the final balance

Plug the given values into the compound interest formula: \(A = 290(1 + \frac{0.09}{2})^{2(4)}\) \(A = 290(1 + 0.045)^{8}\) \(A = 290(1.045)^{8}\) \(A \approx \$436.08\) So, the final balance is approximately $436.08.

Calculate the interest

To find the interest earned, subtract the initial investment from the final balance: Interest = Final balance - Initial investment Interest = \(436.08 - 290\) Interest = \(\$146.08\) The interest earned is approximately $146.08.

Key concepts

Annual Interest Rate

The annual interest rate is a key component in calculating compound interest. It is expressed as a percentage and represents the rate at which your investment will grow over a year. In this particular problem, we have an annual interest rate of 9%.

To effectively use this in your calculations, convert the percentage to a decimal by dividing by 100. Hence, 9% becomes 0.09, which will be used in the compound interest formula.

Understanding this rate is crucial because it tells how much extra money will be made on top of the initial amount over time. The higher the rate, the more potential growth for your investment.

Semiannual Compounding

Semiannual compounding means the interest is calculated and added to the principal twice a year. This results in interest being paid more frequently than annual compounding. Here, the number of times interest is compounded per year is 2.

When compounding semiannually, you divide the annual interest rate by the number of compounding periods per year. In our case, that's 9% divided by 2, which equals 4.5% per six months, or 0.045 when converted to decimal form.

This frequent compounding can have a significant impact on the growth of your investment. More frequent compounding generally means more accumulated interest over the same period compared to less frequent compounding.

Compound Interest Formula

The compound interest formula is a mathematical tool used to calculate the final balance of an investment over time. The formula is \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] where:

• \( A \) is the amount (final balance) after time \( t \).
• \( P \) is the principal amount (initial investment).
• \( r \) is the annual interest rate (as a decimal).
• \( n \) is the number of times that interest is compounded per year.
• \( t \) is the time the money is invested for in years.

Plugging in values: - For an initial investment of $290, - An interest rate of 9% compounded semiannually, - Over 4 years, We calculate: \[ A = 290 \left(1 + \frac{0.09}{2}\right)^{2 \times 4} \] Which simplifies to: \[ A = 290 \times 1.045^8 \] And gives a final balance.

Final Balance

The final balance represents the total amount of money in the account after a certain period, including both the initial principal and the interest earned. Calculating this is the end goal of applying the compound interest formula.

From our example, using an initial amount of $290 compounded at a 9% annual rate semiannually for 4 years, we found the final balance to be approximately $436.08.

This balance includes both the principal and the interest of $146.08, which was accumulated over the period due to compounding. Understanding the final balance helps in evaluating the growth of your investment and making informed financial decisions.