Problem 68 - Algebra and Trigonometry Real Mathematics, Real People

Question

Complete the table to determine the balance \(A\) for \(\$ 2500\) invested at rate \(r\) for \(t\) years and compounded \(n\) times per year. $$\begin{array}{|c|c|c|c|c|c|c|} \hline n & 1 & 2 & 4 & 12 & 365 & \text { Continuous } \\ \hline A & & & & & & \\ \hline \end{array}$$ $$r=3 \%, t=40 \text { years }$$

Step-by-Step Solution

Answer

The balance \(A\) after 40 years for differing compounding frequency \(n\) will be approximately the following based on the compound interest formula and continuous compound interest: \(n=1\): \(\$10,784.49\), \(n=2\): \(\$10,946.28\), \(n=4\): \(\$11,023.43\), \(n=12\): \(\$11,105.13\), \(n=365\): \(\$11,174.22\), Continuous: \(\$11,186.28\).

Step 1: Identifying the formula

The formula for compound interest is given by \(A = P(1 + r/n)^{nt}\) where \(A\) is the amount of money accumulated after n years, including interest, \(P\) is the principal amount (the initial amount of money), \(r\) is the annual interest rate (in decimal form), \(n\) is the number of times that interest is compounded per year and \(t\) is the time the money is invested for in years.

Step 2: Calculate the interest for different compound frequencies

Now, plug in the principal value \(P=\$2500\), interest rate \(r=0.03\) (remember to convert the percent to a decimal), and \(t=40\). We will vary \(n\) to be 1, 2, 4, 12, and 365 to find the balance \(A\) for different compounding frequencies. For example, when \(n=1\), the balance \(A\) will be calculated as \(A = $2500(1 + 0.03/1)^{1*40}\). Repeat these calculations for varying values of \(n\).

Step 3: Calculate the interest for continuous compounding

Finally, we'll calculate the balance for continuous compounding using the formula \(A = Pe^{rt}\), where \(e\) is the base of the natural logarithm, approximately equal to 2.71828. Here, \(P = $2500\) is the principal amount, \(r = 0.03\) is the annual interest rate, and \(t = 40\) is the number of years. This gives \(A = $2500*e^{0.03*40}\).

Key Concepts

Continuous Compounding

Continuous compounding is a method in finance where the interest earned on an initial principal amount is reinvested continuously. This means that the interest is calculated and added to the account balance at every possible instant. Like a clock that never stops ticking, the money in your account grows without interruption. To work out how much you'll have with continuous compounding, you use the formula: \[ A = Pe^{rt} \] Where:

• \(A\) is the amount of money accumulated after time \(t\), including interest.
• \(P\) is the principal amount or the initial amount of money.
• \(e\) is a mathematical constant approximately equal to 2.71828.
• \(r\) is the annual interest rate in decimal form.
• \(t\) is the time in years.

This method usually results in more interest being earned compared to other types of compounding, particularly when money is invested over a long period of time, like 40 years in our example.

Compound Interest Formula

The compound interest formula is crucial for calculating the future value of an investment or loan. It considers that accumulated interest is reinvested, earning more interest, unlike simple interest, which is calculated just on the initial principal. The formula used is: \[ A = P(1 + \frac{r}{n})^{nt} \] Here's what the variables stand for:

• \(A\) is the amount of money accumulated after \(n\) years, including interest.
• \(P\) is the principal amount (the initial amount of money).
• \(r\) is the annual interest rate (in decimal form, so 3% becomes 0.03).
• \(n\) is the number of times that interest is compounded per year.
• \(t\) is the time in years that the money is invested or borrowed.

This formula shows how frequently compounding occurs affects the total amount. With more frequent compounding periods per year, the compound interest earned increases.

Interest Rate

The interest rate is the percentage of the principal charged as interest, often specified annually, which affects how much future value an investment will grow to. In our formula, you usually see the rate in its decimal form. Here's why it's important: - It affects how quickly the principal grows. Higher rates mean faster growth. - It directly impacts how much interest is accumulated over time. - It tells you the return on investment for savings, or the cost of borrowing for loans. Conversion tip: To convert a percentage to a decimal, divide by 100. So a 3% rate becomes 0.03. This conversion is necessary for using most formulas correctly.

Principal Amount

The principal amount is the initial sum of money put into an investment or loan. It serves as the baseline for calculating interest. Whether you start with $2500 or any other amount, knowing your principal helps you understand the base upon which interest is calculated. Key points about the principal:

• It is what you start with before any interest is earned or added.
• The larger the principal, the more interest you can potentially earn.
• It is a deciding factor for the outcome of any investment equation.

The principal affects everything from the amount of interest earned to the eventual total balance of an investment. It's the foundation of your financial journey.