Question

For Exercises \(15-30,\) find the final balance and interest. \(\$ 1600\) at \(6 \%\) compounded semiannually for 3 years

Short answer

Answer: The final balance is approximately $1910.48, and the interest accumulated over 3 years is approximately $310.48.

Step-by-step solution

Gather the given information

We are given the following information: - Initial balance: \(P = \$1600\) - Annual interest rate: \(6\%\), so we convert it to decimal form: \(r = 0.06\) - Number of compounding periods per year: \(n = 2\), since it's compounded semiannually (twice per year) - Number of years: \(t = 3\)

Apply the compound interest formula

Now we will use the formula \(A=P(1+\frac{r}{n})^{nt}\) to find the final balance. A = \(1600\left(1+\frac{0.06}{2}\right)^{2\cdot3}\)

Simplify the expression

Before doing any calculations, let's first simplify the expression. A = \(1600\left(1+\frac{0.06}{2}\right)^{6}\) A = \(1600\left(1+0.03\right)^{6}\) A = \(1600\left(1.03\right)^{6}\)

Calculate the final balance

Now let's calculate the final balance: A ≈ \(1600\cdot 1.194052\) A ≈ \(1910.48\) The final balance is approximately \(1910.48\).

Calculate the interest

To calculate the interest, subtract the initial balance from the final balance: Interest = \(A - P\) Interest = \(1910.48 - 1600\) Interest ≈ \(310.48\) The interest accumulated over 3 years is approximately \(310.48\).

Key concepts

semiannual compounding

Semiannual compounding refers to the process of calculating interest on an initial balance in such a way that interest is added to the principal twice a year. In contrast to annual compounding, where interest is calculated once per year, semiannual compounding allows an investment to grow faster because the interest is calculated more frequently.

To understand it better, consider that:

• The number of compounding periods in a year for semiannual compounding is 2.
• Each year is divided into 2 equal periods (of 6 months each) during which the interest is calculated and added to the balance.
• The formula used for compound interest in this case takes the increased frequency of compounding into account, as seen in the exercise.

By understanding how often your interest is calculated and added back into the principal, you can see the difference it makes to the overall growth of your investment.

final balance calculation

Calculating the final balance of an investment using the compound interest formula gives you an idea of how much your investment has grown over a certain period, considering the interest rate and compounding frequency. The formula used for this calculation is:

• \[A = P\left(1+\frac{r}{n}\right)^{nt}\]
• Where:
• \(A\) is the final amount after interest is applied.
• \(P\) is the initial balance (the starting amount before interest).
• \(r\) is the annual nominal interest rate (as a decimal).
• \(n\) is the number of times interest is compounded per year.
• \(t\) is the time the money is invested for (in years).

In our example, the calculation of the final balance showed how the initial sum grows after 3 years with semiannual compounding.

interest calculation

Interest calculation is an important aspect of understanding how much extra money your investment or principal amount will earn over time. Once the final balance has been determined using the compound interest formula, the interest earned can be calculated by subtracting the initial balance from the final balance:

• The formula is: Interest = \(A - P\)
• Where:
• \(A\) represents the final balance.
• \(P\) is the initial balance.

By subtracting the initial investment from the final amount, you can see exactly how much money the interest has earned for you over the period.

initial balance

The initial balance, often referred to as the principal, is the starting amount of money you invest or deposit before any interest is accrued. This figure is crucial as it forms the base upon which interest calculations are made. In our given exercise example:

• The initial balance is \(\$1600\).
• This amount is used to begin calculating interest through the compound interest formula.
• Higher initial balances will result in higher total interest earned, assuming the interest rate and time period remain constant.

Understanding your initial balance helps in planning how much to invest and projecting future earnings.

interest rate

Interest rate is the percentage at which your investment grows or the cost of borrowing, usually expressed annually. It is one of the main factors that determine the amount of interest accrued on an initial balance. In the context of our exercise:

• The annual interest rate provided is 6%.
• It must be converted into decimal form for calculations (hence, 0.06).
• When compounding more frequently than annually, such as semiannually, the interest rate per period is adjusted in the compound interest formula, dividing it by the number of periods (2 in this case).

Comprehending how the interest rate affects your investment is fundamental. It affects not just the future value of your investment, but also influences important decisions on investment type and duration.