Question

For Exercises \(15-30,\) find the final balance and interest. \(\Rightarrow 500\) at \(11 \%\) compounded semiannually for 5 years

Short answer

Answer: To solve this, we first find the period interest rate: \(R = \frac{11}{2} \% = 5.5\% \). Then, we calculate the final balance using the compound interest formula: \(A = 500(1 + \frac{0.11}{2})^{(2 \times 5)}\). Lastly, we find the total interest by subtracting the principal amount from the final balance: \(TotalInterest = A - 500\). Upon completing the calculations, we find the final balance and total interest.

Step-by-step solution

Calculate the interest rate for each period

Since the given interest rate is compounded semiannually, we need to find the interest rate for each period. To do this, simply divide the annual interest rate (11%) by the number of compounding periods in a year (2): \[PeriodInterestRate = R = \frac{11}{2} \% \] #calculate final balance using compound interest formula#

Calculate final balance using compound interest formula

We will use the compound interest formula to determine the final balance (A): \[A = P(1 + \frac{r}{n})^{(nt)}\] Where: A = final balance P = principal amount ($500) r = annual interest rate (as a decimal, hence 0.11) n = number of times compounded per year (2, semiannually) t = time in years (5 years) Plug in the given values and solve for A: \[A = 500(1 + \frac{0.11}{2})^{(2 \times 5)}\] #calculate total interest#

Calculate total interest

Once we have the final balance, we can calculate the total interest by subtracting the principal amount from the final balance: \[TotalInterest = A - P\] Now, complete the calculations and find the final balance and total interest.

Key concepts

Calculating Compound Interest

Calculating compound interest involves understanding how an investment or loan grows over time when the interest earned is reinvested to earn more interest. This concept differs from simple interest, where you earn interest only on the principal amount.

In our exercise, we're faced with an initial investment, or principal, of $500, which grows with an 11% annual interest rate. However, because the interest is compounded semiannually, the calculation must account for the interest being added to the principal twice a year, not just once at the end of the year.

To reveal the magic of compound interest, you should visualize it as a snowball rolling down a hill, gathering more snow along the way. With each semiannual period, our financial snowball picks up additional interest, causing it to grow at an increasing rate.

Semiannual Compounding

Semiannual compounding means that the interest is calculated and added to the principal twice a year. It's important to grasp that although the annual interest rate might be fixed, the actual compounding frequency can heavily impact the growth of your investment or loan.

For example, with semiannual compounding, an 11% annual rate is effectively divided into two 5.5% periods within the year. As a result, interest is computed and added to the balance at the 6-month mark, which then earns interest itself in the following 6 months. This cycle continues, with each half-year's interest earning its own interest in the subsequent periods.

The effect of semiannual compounding is more significant over time, as each compounding period adds to the foundation that further interest is calculated upon, leading to exponential growth of the investment.

Compound Interest Formula

To crunch the numbers for compound interest, we use a specific formula:
\[A = P(1 + \frac{r}{n})^{(nt)}\]
Where:

• \(A\) represents the final amount of money after t years, including interest,
• \(P\) is the principal amount (the initial amount of money),
• \(r\) stands for the annual interest rate (expressed as a decimal),
• \(n\) indicates the number of times the interest is compounded per year,
• and \(t\) is the time in years.

By plugging the numbers from our exercise into this formula, you can see how your initial investment evolves over the span of 5 years with interest compounding semiannually. Understanding and applying this formula is crucial for making informed financial decisions and being able to predict the future value of an investment.