Problem 69 - Introductory and Intermediate Algebra for College Students 4th

Question

You have \(\$ 10,000\) to invest. One bank pays \(5 \%\) interest compounded quarterly and the other pays \(4.5 \%\) interest compounded monthly. a. Use the formula for compound interest to write a function for the balance in each account at any time \(t\) in years. b. Use a graphing utility to graph both functions in an appropriate viewing rectangle. Based on the graphs, which bank offers the better return on your money?

Step-by-Step Solution

Answer

The functions for the balance at any time \(t\) are \(A_1(t) = 10000(1 + 0.05/4)^(4t)\) and \(A_2(t) = 10000(1 + 0.045/12)^(12t)\) for the first and second bank respectively. The comparison depends on the values of \(t\) but mostly, the graph of the second function will be above that of the first function because it compounds more often despite having a lower interest rate.

Step 1: Formulate the Function for the First Bank

Using the formula for compound interest, the function for the first bank that pays \(5 \%\) interest compounded quarterly is \(A_1(t) = 10000(1 + 0.05/4)^(4t)\)

Step 2: Formulate the Function for the Second Bank

For the second bank that pays \(4.5 \%\) interest compounded monthly, the function is \(A_2(t) = 10000(1 + 0.045/12)^(12t)\)

Step 3: Graph Both Functions

Next, graph both functions using a graphing utility, where \(x\) axis represents time in years \(t\) and \(y\) axis represents the balance in the accounts. This can help to compare visually.

Step 4: Comparison and Final Decision

After graphing, compare the two banks. The bank with the higher balance at a given time \(t\) offers the better return. Note that despite the lower interest rate, the second bank compounds more often and may provide a higher return in the long run.

Key Concepts

Interest Rates

Interest rates are crucial when it comes to understanding investments and savings. In our exercise, we have two different interest rates offered by banks. The first bank provides a 5% interest rate, while the second offers 4.5%. This percentage represents the annual rate at which your investment grows. However, what's particularly interesting is how often this interest is applied or compounded. Banks often compound interest more frequently than once a year, which affects the total amount you earn over time.
Understanding the nature of interest rates and how they work can significantly impact your financial decisions. It's not just about which rate is higher, but also about how the compounding frequency can influence the total returns, as seen in the second bank's offer.

Investment Comparison

When comparing investments, such as deciding between bank offers, the key is to analyze the total returns after a period. Here, two investment options are presented:

• Bank 1: 5% interest compounded quarterly.
• Bank 2: 4.5% interest compounded monthly.

Initially, a higher percentage like 5% may seem more attractive. However, the compounding intervals play a significant role. The compounding frequency, which we'll explore further, is a critical factor in determining the better investment. Over different periods, one account might yield higher returns than the other despite seemingly lower interest. Investing in either bank requires using the compound interest formula to predict potential earnings over time. This allows for informed investment decisions.

Graphing Functions

Graphing functions can be an excellent way to visually explore and compare investments. By plotting the two functions given by our exercise, we can easily see how the balance in each account grows over time. In our case:

• For Bank 1: The function is \(A_1(t) = 10000(1 + 0.05/4)^{4t}\).
• For Bank 2: The function is \(A_2(t) = 10000(1 + 0.045/12)^{12t}\).

On a graph, the x-axis represents time (in years), and the y-axis represents the account balance. By using a graphing utility, students can input these equations and observe their behavior over different periods. This approach helps in picturing which investment grows faster over time, visually supporting the decision-making process.

Compounding Frequency

Compounding frequency refers to how often the interest is calculated and added to the account's balance. The more frequently interest is compounded, the quicker the investment grows. In our exercise, the first bank compounds quarterly, while the second compounds monthly. Here's what that means:

• Quarterly compounding: Interest is applied four times a year.
• Monthly compounding: Interest is applied twelve times a year.

Though Bank 2 offers a lower interest rate annually, the more frequent compounding can lead to a larger total return over time than Bank 1's option. Understanding compounding frequency is essential for evaluating investments accurately as it affects the exponential growth of your money over time.