Stephen received a loan of $35,250, 6 years ago. The interest rate charged on the loan was 4.68% compounded quarterly for the first 6 months, 5.46% compounded semi-annually for the next 2 years, and 5.79% compounded monthly thereafter.

**a.** Calculate the accumulated value of the loan at the end of the first 6 months.

Round to the nearest cent

**b.** Calculate the accumulated value of the loan at the end of the next 2 year period.

Round to the nearest cent

**c.** Calculate the accumulated value of the loan today.

Round to the nearest cent

**d.** Calculate the amount of interest charged on this loan over the past 6 years.

Round to the nearest cent

To solve this problem, we'll break it down into three separate time periods and calculate the accumulated value at the end of each period using the compound interest formula.

### Given Data:
- **Initial Loan Amount (Principal)** $ P = 35,250 $
- **Period 1: First 6 months**
  - Interest rate: 4.68% compounded quarterly
  - Time: 6 months = 0.5 years
  - Compounding frequency $ n = 4 $ (quarterly)
  
- **Period 2: Next 2 years**
  - Interest rate: 5.46% compounded semi-annually
  - Time: 2 years
  - Compounding frequency $ n = 2 $ (semi-annually)
  
- **Period 3: Remaining 3.5 years**
  - Interest rate: 5.79% compounded monthly
  - Time: 3.5 years
  - Compounding frequency $ n = 12 $ (monthly)

### Step-by-Step Calculations:

1. **Calculate the accumulated value at the end of the first 6 months:**
   \[
   A_1 = P \left(1 + \frac{r_1}{n_1}\right)^{n_1 \times t_1}
   \]
   where:
   - $ r_1 = 4.68\% = 0.0468 $
   - $ n_1 = 4 $
   - $ t_1 = 0.5 $ years

2. **Calculate the accumulated value at the end of the next 2 years:**
   \[
   A_2 = A_1 \left(1 + \frac{r_2}{n_2}\right)^{n_2 \times t_2}
   \]
   where:
   - $ r_2 = 5.46\% = 0.0546 $
   - $ n_2 = 2 $
   - $ t_2 = 2 $ years

3. **Calculate the accumulated value at the end of the remaining 3.5 years:**
   \[
   A_3 = A_2 \left(1 + \frac{r_3}{n_3}\right)^{n_3 \times t_3}
   \]
   where:
   - $ r_3 = 5.79\% = 0.0579 $
   - $ n_3 = 12 $
   - $ t_3 = 3.5 $ years

4. **Calculate the amount of interest charged on the loan:**
   \[
   \text{Interest} = A_3 - P
   \]

Let's proceed with these calculations.### Results:

**a.** The accumulated value of the loan at the end of the first 6 months is **$36,079.68** (rounded to the nearest cent).

**b.** The accumulated value of the loan at the end of the next 2-year period is **$40,183.87** (rounded to the nearest cent).

**c.** The accumulated value of the loan today (after the full 6 years) is **$49,186.95** (rounded to the nearest cent).

**d.** The total amount of interest charged on this loan over the past 6 years is **$13,936.95** (rounded to the nearest cent).

Do you want more details or have any questions about these calculations?

Here are 8 related questions to consider:
1. What if the loan was for 5 years instead of 6 years with the same interest rates and compounding frequencies?
2. How would the accumulated value change if the interest rate for the final period was 6% instead of 5.79%?
3. What would the interest charged be if the principal amount was $40,000 instead of $35,250?
4. How would the accumulated value change if the loan was compounded annually for all periods?
5. What if the interest rate in the first period was 5% instead of 4.68%?
6. How would the total interest differ if the loan was compounded monthly for all periods?
7. What would be the impact on the accumulated value if the loan period was extended by another year?
8. How much would Stephen owe if he made a partial payment of $10,000 after the first 6 months?

**Tip:** When dealing with loans and interest rates, it's important to pay attention to the compounding frequency, as it significantly affects the total amount owed.

Learn how to calculate compound interest for a loan of $35,250 over 6 years with varying interest rates and compounding frequencies. Includes step-by-step solutions for each time period and the total interest charged. Suitable for college-level students and financial professionals.

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