Question

For Exercises \(15-30,\) find the final balance and interest. \(\$ 840\) at \(6 \%\) compounded annually for 2 years

Short answer

Answer: The final balance after 2 years is approximately $943.82, and the interest earned is approximately $103.82.

Step-by-step solution

Convert the annual interest rate to a decimal

In order to use the compound interest formula, we need to convert the annual interest rate to a decimal. To do this, divide the percentage by 100: \(6 \% = \frac{6}{100} = 0.06\) The annual interest rate as a decimal is 0.06.

Plug the values into the compound interest formula

Now that we have the interest rate as a decimal, we can plug the values into the compound interest formula: \(A = P(1 + \frac{r}{n})^{nt}\) Where P = \(840, r = 0.06, n = 1\) (compounded annually), and t = 2. Insert these values into the formula: \(A = 840(1 + \frac{0.06}{1})^{1 \cdot 2}\)

Compute the final balance

Next, we will compute the value of A, which is the final balance. First, calculate the expression inside the parentheses: \(1 + \frac{0.06}{1} = 1 + 0.06 = 1.06\) Then, raise this result to the power of \(1 \cdot 2 = 2\): \(1.06^{2} = 1.1236\) Finally, multiply this result by the principal of $840: \( A = 840 \cdot 1.1236 = 943.824\) The final balance after 2 years is approximately $943.82 (rounded to two decimal places).

Calculate the interest earned

To find the interest earned, subtract the principal (initial amount) from the final balance: \(Interest = A - P = 943.82 - 840 = 103.82\) The interest earned after 2 years is approximately $103.82.

Summary of final results

The final balance after 2 years is approximately \(943.82, and the interest earned is approximately \)103.82.

Key concepts

Compound Interest Formula

Understanding the compound interest formula is crucial for grasping how investments grow over time. The formula is given by:

\[ A = P(1 + \frac{r}{n})^{nt} \]

where:

• \(A\) represents the future value of the investment/loan, including interest.
• \(P\) stands for the principal amount (the initial sum of money).
• \(r\) is the annual interest rate (as a decimal).
• \(n\) indicates the number of times interest is compounded per year.
• \(t\) reflects the number of years the money is invested or borrowed for.

This formula embodies the principle of exponential growth, where returns are calculated on the accumulated returns of previous periods as well as the principal.

Converting Percentages to Decimals

To work with percentages in mathematical formulas, it is often essential to convert them into decimal form. This is a straightforward process:

To convert a percentage to a decimal, simply divide it by 100. For instance, an interest rate of \(6\%\) can be converted as follows:

\[6\% = \frac{6}{100} = 0.06\]

This step is fundamental to ensure accuracy when substituting values into the compound interest formula and performing calculations.

Exponential Growth

Exponential growth occurs when a quantity increases by a constant rate per unit time relative to its size. This is best described in financial contexts by the growth of investments through compound interest. Each period, the interest is calculated on the accumulated total, including past interest, not just the original principal. This leads to growth that accelerates over time, represented mathematically by an exponential function. The compound interest formula showcases this by raising the total growth factor, \(1 + \frac{r}{n}\), to the power of \(nt\), exemplifying how the returns expand multiplicatively.

Interest Calculation

Interest calculation involves finding out how much extra money will be earned (or paid) on top of the principal. For simple interest, it's a direct multiplication. In compound interest, as seen in our example, it requires a few more steps. After calculating the final balance with the formula, the interest earned is the difference between this final amount and the initial principal:

\[ Interest = A - P \]

In the example above, the initial investment was \(840, and after compounding interest, the balance grew to \)943.82 over 2 years. Therefore, the interest earned was approximately $103.82. By learning to calculate interest, students become equipped to make informed financial decisions and understand the potential growth of their savings or the cost of loans.