Question

For Exercises \(15-30,\) find the final balance and interest. \(\$ 650\) at \(5 \%\) compounded annually for 3 years

Short answer

Answer: The final balance is approximately $749.43, and the interest earned is approximately $99.43.

Step-by-step solution

Understand the compound interest formula

The formula for compound interest is: \(A = P(1 + \frac{r}{n})^{nt}\) Where: - \(A\) is the final balance - \(P\) is the principal (initial deposit) - \(r\) is the annual interest rate (as a decimal) - \(n\) is the number of times the interest is compounded per year - \(t\) is the number of years In this problem, we have \(P = 650\), \(r = 0.05\) (5% as a decimal), \(n = 1\) (compounded annually), and \(t = 3\) years.

Find the final balance

Plug the given values into the compound interest formula and calculate the final balance (A): \(A = 650(1 + \frac{0.05}{1})^{1 \cdot 3}\) \(A = 650(1 + 0.05)^3\) \(A = 650(1.05)^3\) \(A \approx 749.43\) The final balance after 3 years is approximately $749.43.

Calculate the interest earned

To find the interest earned, subtract the initial deposit from the final balance: \(Interest = A - P\) \(Interest = 749.43 - 650\) \(Interest \approx 99.43\) After 3 years, the interest earned is approximately $99.43. Therefore, the final balance is about \(749.43, and the interest earned is approximately \)99.43.

Key concepts

Final Balance Calculation

Understanding the final balance calculation in the realm of compound interest is essential for managing personal finances and making informed investment decisions. Simply put, it's how you find out how much money you'll have in the future based on your current savings and the interest it accumulates over time.

To calculate the final balance using the compound interest formula, you'll need to know your initial deposit (the principal), the annual interest rate, how often the interest is compounded (once per year in this scenario), and for how many years you plan to save or invest your money. In essence, you're looking at the growth of your initial deposit due to the interest that is added to that deposit over and over again over a period.

In the given exercise, for a principal amount of \(650, an annual interest rate of 5%, compounded annually for 3 years, the formula morphs into \(A = 650(1 + 0.05)^3\), resulting in a final balance of approximately \)749.43. This result highlights the power of compound interest in growing your initial investment.

Annual Interest Rate

The annual interest rate is the percentage that indicates how much interest you earn on an investment or pay on a loan over the span of a year. It's a pivotal factor in financial matters as it influences how much you can expect your savings to grow or how much your debt may increase over time.

In the world of compound interest, the annual interest rate should be represented as a decimal when plugged into the formula. For instance, a 5% interest rate is expressed as 0.05. This allows for accurate calculations without having to convert percentages in the midst of your math. The higher your annual interest rate, and the more frequently that interest is compounded, the faster your money grows.

It's important to note that while the annual rate gives an idea of the earnings over a year, the actual growth of your investment depends on the number of compounding periods—the focus of our next concept.

Interest Compounding

Interest compounding is akin to a snowball rolling down a hill, where it grows in size as it accumulates more snow—in this case, 'snow' being the interest. It's the process by which you earn interest on both your initial deposit (the principal) and the interest that has been added to this deposit over time.

The frequency of compounding can greatly bolster the effect of compound interest. Commonly, interest could be compounded annually, semi-annually, quarterly, monthly, or even daily. As the frequency of compounding increases, so does the total amount of interest accrued: the principal is updated more frequently, and each interest calculation includes the previously earned interest.

Understanding the concept of compounding is crucial because it helps investors and savers make smarter decisions. Realizing that money can earn interest upon interest, you are empowered to gauge how your investments or savings will grow over time and the significance of starting to save early.