Problem 66 - College Algebra with Modeling and Visualization

Question

\(\$ 3300\) at \(8 \%\) compounded quarterly for 2 years

Step-by-Step Solution

Answer

The amount after 2 years is approximately $3866.47.

Step 1: Identify the Given Variables

We have the principal amount \( P = 3300 \), the annual interest rate \( r = 0.08 \), and the time duration \( t = 2 \) years. The compound frequency is quarterly, so the number of times that interest is applied per year \( n = 4 \).

Step 2: Use the Compound Interest Formula

The compound interest formula is \( A = P \left(1 + \frac{r}{n}\right)^{nt} \). We will substitute the known values into this formula: \( A = 3300 \left(1 + \frac{0.08}{4}\right)^{4 \times 2} \).

Step 3: Calculate the Quarterly Interest Rate

First, determine the quarterly interest rate as \( \frac{r}{n} = \frac{0.08}{4} = 0.02 \). The interest rate each quarter is 2%.

Step 4: Compute the Total Number of Compoundings

Since the interest is compounded quarterly, calculate the total number of compounding periods, \( nt = 4 \times 2 = 8 \).

Step 5: Substitute Values into the Formula

Now, substitute all these calculated values back into the compound interest formula: \[ A = 3300 \left(1 + 0.02 \right)^8 \].

Step 6: Calculate the Future Value

Compute \( A\) by calculating \( 1.02^8 \), which approximately equals \( 1.171659 \). Therefore, \( A = 3300 \times 1.171659 \approx 3866.47 \).

Step 7: Interpret the Result

The future value, \( A \), represents the total amount of money accumulated after 2 years, including interest.

Key Concepts

Interest Rate Calculation

When discussing compound interest, understanding how interest rates work is fundamental. Interest rate calculation determines how much extra money will be earned on a principal sum over a time period. In the context of compound interest like in our example, the initial principal (P) is used repeatedly to calculate interest.

• Annual Interest Rate: Represents the percentage of the principal earned as interest each year. Here, it is 8%.
• To find the rate for shorter intervals like quarters, this rate needs to be divided. So we take the annual rate (8%) and divide it by 4, since interest is compounded quarterly.
• This gives us a quarterly interest rate of: \[ \frac{0.08}{4} = 0.02 \text{, or 2% per quarter} \]

Understanding this breakdown ensures accuracy in calculating how money grows over time when compounded.

Quarterly Compounding

Quarterly compounding is a critical concept in compound interest scenarios where interest is calculated four times per year. This means that interest is not only earned on the initial principal, but also on the interest accumulated in previous quarters.

• This involves dividing the year into quarters, with four compounding periods annually.
• Each compounding period affects the principal, enabling the amount to grow exponentially.
• In our case, since interest is compounded four times a year, over two years, \( 4 \times 2 = 8 \) compounding periods occur.

Understanding quarterly compounding helps us see how calculations lead to a larger future value more quickly than annual compounding.

Future Value Calculation

The future value calculation tells us how much an investment will be worth after a set period, taking interest compounding into account. It is an essential part of financial planning and decision-making.To calculate this:

• Use the future value formula: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
• Substitute the known values. For this example:\( P = 3300 \),\( r = 0.08 \),\( n = 4 \) (quarterly compounding), and\( t = 2 \) (years).
• This gives us:\[ A = 3300 \left(1 + 0.02 \right)^8 \]
• Calculate \( 1.02^8 \approx 1.171659 \). Multiply this by the principal: \( 3300 \times 1.171659 \approx 3866.47 \).

The resulting future value of approximately $3866.47 shows both principal and the interest earned. This calculation is crucial for understanding the benefits of re-investing earnings over time.