Questions: 33,000 is deposited for 2 years in an account earning 3% interest. (Round your answers to two decimal places.) (a) Calculate the interest earned if interest is compounded semiannually, (b) Calculate the interest earned if interest is compounded quarterly. (c) How much more interest is earned on the account when the interest is compounded quarterly?

Solution Steps

To solve these problems, we will use the formula for compound interest:

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

where:

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

(a) For semiannual compounding, \( n = 2 \).

(b) For quarterly compounding, \( n = 4 \).

(c) To find how much more interest is earned with quarterly compounding, subtract the interest earned with semiannual compounding from the interest earned with quarterly compounding.

Using the formula for compound interest:

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

For semiannual compounding, where \( n = 2 \):

\[ A_{\text{semiannual}} = 33000 \left(1 + \frac{0.03}{2}\right)^{2 \cdot 2} = 35024.9972 \]

The interest earned is:

\[ \text{Interest}_{\text{semiannual}} = A_{\text{semiannual}} - P = 35024.9972 - 33000 = 2024.9972 \]

For quarterly compounding, where \( n = 4 \):

\[ A_{\text{quarterly}} = 33000 \left(1 + \frac{0.03}{4}\right)^{4 \cdot 2} = 35032.7620 \]

The interest earned is:

\[ \text{Interest}_{\text{quarterly}} = A_{\text{quarterly}} - P = 35032.7620 - 33000 = 2032.7620 \]

To find how much more interest is earned with quarterly compounding compared to semiannual compounding:

\[ \text{Interest Difference} = \text{Interest}_{\text{quarterly}} - \text{Interest}_{\text{semiannual}} = 2032.7620 - 2024.9972 = 7.7648 \]

Final Answer