Question

For Exercises \(15-30,\) find the final balance and interest. \(\$ 960\) at \(14 \%\) compounded semiannually for 3 years

Short answer

Answer: The final balance is approximately ___ dollars, and the interest earned is approximately ___ dollars after 3 years with the given parameters.

Step-by-step solution

Identify the given values

We are given the following values: - Initial investment (Principal): \(P=\$960\) - Annual interest rate: \(r=14\% =0.14\) - Compounding periods per year: \(n=2\) (semiannually) - Total number of years: \(t=3\) years

Apply the compound interest formula

The compound interest formula is given as: \(A = P(1 + \frac{r}{n})^{nt}\) where \(A\) = The final balance \(P\) = The initial principal balance \(r\) = The annual interest rate \(n\) = The number of times interest is compounded per year \(t\) = The total number of years

Calculate the final balance

Using the given values and the compound interest formula, we can find the final balance: \(A = 960(1 + \frac{0.14}{2})^{2 \times 3}\) \(A = 960(1 + 0.07)^6\) \(A = 960(1.07)^{6}\) \(A ≈ \cdots\) Calculate the value to find the final balance.

Calculate the interest

To find the interest earned, we subtract the initial investment from the final balance: Interest = Final balance - Initial investment Interest \(= A - P\) Calculate the interest based on the final balance you found in step 3.

Write down the final balance and interest

Now that you have calculated the final balance and interest, write them in a sentence form: The final balance is approximately \(\cdots\) dollars, and the interest earned is approximately \(\cdots\) dollars after 3 years with the given parameters.

Key concepts

Initial Investment

The initial investment, also called the principal, is the starting amount of money you're putting to work with an investment, like in a savings account or another financial product. In this case, the initial investment is $960. It's important because it forms the foundation upon which interest will be calculated. The larger the initial investment, the more money that can potentially grow through the power of compound interest.

When planning an investment, understanding your initial investment is crucial as it helps set your financial expectations. It allows you to evaluate whether your strategy aligns with your financial goals. Being clear about this amount shapes the trajectory of your investment and its potential to grow.

Annual Interest Rate

The annual interest rate is a percentage that represents the cost of borrowing money or the gain from saving it over the course of one year. In this scenario, the annual interest rate is 14%.

The interest rate is a key component because it dictates how much the initial investment will grow over time. A higher rate means more growth potential for your money. However, in practical terms, it’s important to consider the rate in the context of external economic factors like inflation.

• Interest rates directly impact the amount of interest your money earns.
• Understanding the rate helps in comparing different investment opportunities.

This rate is an important factor in deciding where and how to invest your money for the best returns.

Semiannual Compounding

Semiannual compounding refers to the process of calculating and adding interest to the balance twice a year. This means each compounding period lasts for six months. In the given exercise, the $960 investment compounds semiannually.

Compounding more frequently can significantly impact the growth of an investment. Here's why:

• With semiannual compounding, interest is calculated twice a year, giving earnings the opportunity to grow faster than with annual compounding.
• Each time interest is compounded, it gets added to the principal, increasing the base amount for the next compounding period.

This method enhances the potential return on an investment, making it a crucial concept in finance.

Final Balance Calculation

The final balance calculation determines how much money will be available at the end of the investment period. To calculate this, you apply the compound interest formula: \[ A = P\left(1 + \frac{r}{n}\right)^{nt} \]This formula considers the initial investment, the annual interest rate, the number of compounding periods per year, and the number of years.

Here’s a breakdown of the calculation process:

• You substitute the given values into the formula: \(A = 960\left(1 + \frac{0.14}{2}\right)^{2 \times 3}\).
• Calculate inside the parentheses and then raise it to the power determined by the number of compounding periods, multiplying by the initial principal.

The result is the final amount in the account, representing both the original principal and the accumulated interest. This provides insight into how investments grow over time, helping you assess and potentially adjust your investment strategy.