Problem 27 - Intermediate Algebra

Question

Suppose that Nora invested $$\$ 500$$ at \(8.25 \%\) compounded annually for 5 years, and Patti invested $$\$ 500$$ at \(8 \%\) compounded quarterly for 5 years. At the end of 5 years, who will have the most money and by how much?

Step-by-Step Solution

Answer

Patti will have 5 cents more than Nora after 5 years.

Step 1: Understanding Compound Interest

To solve this, we need to understand how compound interest works. The compound interest formula is given by \( A = P \left(1 + \frac{r}{n} \right)^{nt} \), where \( A \) is the amount after time \( t \), \( P \) is the principal amount, \( r \) is the annual interest rate as a decimal, \( n \) is the number of times interest is compounded per year, and \( t \) is the time in years.

Step 2: Calculating Nora's Investment

Nora's investment is compounded annually, so \( n = 1 \). The formula becomes \( A = 500 \left(1 + \frac{0.0825}{1} \right)^{1 \times 5} \). Calculating this gives: \[ A = 500 \times (1.0825)^5 \approx 500 \times 1.485947 \approx 742.97 \text{ dollars} \].

Step 3: Calculating Patti's Investment

Patti's investment is compounded quarterly, so \( n = 4 \). Using the same formula: \( A = 500 \left(1 + \frac{0.08}{4} \right)^{4 \times 5} \), we calculate: \[ A = 500 \times (1.02)^{20} \approx 500 \times 1.485947 \approx 743.02 \text{ dollars} \].

Step 4: Comparing the Investments

Now we compare the two amounts: Nora's \( 742.97 \text{ dollars} \) versus Patti's \( 743.02 \text{ dollars} \). Patti's investment is slightly larger.

Step 5: Conclusion

Patti will have more money at the end of 5 years. The difference is \( 743.02 - 742.97 = 0.05 \text{ dollars} \), meaning Patti ends up with 5 cents more than Nora.

Key Concepts

Investment Calculation

Investment calculation using compound interest is key to understanding how your money can grow over time. It's a way of calculating the total amount of money you will have after investing a certain principal amount at a specified interest rate.

The compound interest formula is:

• \( A = P \left(1 + \frac{r}{n} \right)^{nt} \)

In this formula:

• \( A \) is the amount of money accumulated after n years, including interest.
• \( P \) is the principal investment amount (the initial amount).
• \( r \) is the annual interest rate (in decimal form).
• \( n \) is the number of times that interest is compounded per year.
• \( t \) is the number of years the money is invested for.

Understanding this formula is crucial because it shows how different factors like the interest rate and the number of compounding periods influence the final amount.

Annual vs Quarterly Compounding

When comparing annual and quarterly compounding, it's important to understand that the difference lies in the frequency of interest application.

In annual compounding, interest is added to the initial investment once a year. This means:

• The principal plus any earned interest from the previous year is used as the new principal for the next year's calculations.
• For annual compounding, the formula becomes: \( A = P \left(1 + r \right)^{t} \).

Quarterly compounding, however, divides the year into four parts, applying interest every three months. This means:

• Each quarter, the principal grows by the interest gained during that period.
• For quarterly calculations, you modify the formula's interest rate and time to account for four compounding periods per year: \( A = P \left(1 + \frac{r}{4} \right)^{4t} \).

By compounding quarterly, you essentially get interest on interest more frequently, which can lead to slightly larger returns over the same period.

Interest Rate Comparison

Comparing interest rates across different compounding intervals can be insightful. This exercise demonstrates that, at first glance, a higher annual interest rate may not always lead to a better outcome if compounded less frequently.

• Nora’s investment with an 8.25% rate compounded annually resulted in \( 742.97 \) dollars.
• Patti’s investment with an 8% rate compounded quarterly came out to \( 743.02 \) dollars.

Despite the lower nominal interest rate, quarterly compounding gave Patti a slight edge. This is because the interest applies more frequently, allowing the investment to grow faster in smaller increments. It's a subtle yet powerful demonstration of how compounding frequency can impact the growth of an investment.

Thus, when comparing investments, it's crucial to consider not just the interest rate but also how often it compounds. This understanding will help in making more informed investment decisions.