Problem 70 - A Graphical Approach to College Algebra

Question

Use the appropriate compound interest formula to find the amount that will be in each account, given the stated conditions. \(\$ 35,000\) invested at \(4.2 \%\) annual interest for 3 years compounded (a) annually; (b) quarterly

Step-by-Step Solution

Answer

(a) $39,585; (b) $39,662.

Step 1: Understand the Formula for Compound Interest

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

Step 2: Set Parameters for Annual Compounding

For part (a), the interest is compounded annually. Thus, \( P = 35000 \), \( r = 0.042 \), \( n = 1 \), and \( t = 3 \).

Step 3: Calculate Amount for Annual Compounding

Plug the values into the formula: \( A = 35000 \left(1 + \frac{0.042}{1}\right)^{1 \times 3} = 35000 \times (1.042)^3 \). Calculate the result: \( A \approx 35000 \times 1.131 \approx 39585 \).

Step 4: Set Parameters for Quarterly Compounding

For part (b), the interest is compounded quarterly. Thus, \( P = 35000 \), \( r = 0.042 \), \( n = 4 \), and \( t = 3 \).

Step 5: Calculate Amount for Quarterly Compounding

Plug the values into the formula: \( A = 35000 \left(1 + \frac{0.042}{4}\right)^{4 \times 3} = 35000 \times (1.0105)^{12} \). Calculate the result: \( A \approx 35000 \times 1.1332 \approx 39662 \).

Key Concepts

Annual Compounding

When we talk about annual compounding, it means that the interest is calculated and added to the principal once per year. In other words, the investment grows at the end of each year based on that year's interest.

The formula used for calculating compound interest is:

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

Where

• \(A\) is the amount on the account after interest is applied,
• \(P\) is the principal or initial amount invested,
• \(r\) is the annual interest rate (expressed as a decimal),
• \(n\) is the number of times interest is compounded per year,
• \(t\) is the number of years the money is invested.

For annual compounding, \(n = 1\), because the interest is added once a year. This might seem straightforward, but understanding compounding can greatly affect how rapidly an investment grows over time. In our example, investing \(35,000 at an annual rate of 4.2% compounded annually results in an amount of approximately \)39,585 after 3 years. This shows how even a modest interest rate can enhance savings significantly over time.

Quarterly Compounding

Quarterly compounding involves interest being calculated and added to the principal four times a year. This means more interest is applied throughout the year, leading to a different amount compared to annual compounding, assuming the same rate.

For quarterly compounding, the same compound interest formula applies, but with \(n = 4\) for quarterly calculations:

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

This approach shares a key point:

• Each quarter, one-fourth of the annual rate is applied to the principal.

When you apply this over 3 years for our initial amount of \(35,000 at a 4.2% annual rate, the result after calculating is approximately \)39,662. This higher amount compared to annual compounding underscores the benefits of more frequent compounding, as it yields greater returns by applying interest more often.

Investment Calculation

Understanding how to calculate the value of an investment is essential for making informed financial decisions.

Investment growth calculations incorporate the use of already earned interest to earn more interest, a concept known as compound interest. This approach accelerates investment growth compared to simple interest.

For the formula:

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

Each parameter plays a vital role:

• \(P\): The principal is the starting point from which all growth stems.
• \(r\): The annual rate must be represented as a decimal to properly scale interest.
• \(n\) and \(t\): Together they determine the frequency of compounding and time, crucial in assessing the total growth.

The efficient computation resulting in either scenario (annual or quarterly) helps estimate future worth of investments, paving the way towards meeting financial goals.