Jeffrey's student loan of $28,500 at 3.62% compounded quarterly was amortized over 3 years with payments made at the end of every month. What was the principal balance on the loan after 1 year? Round to the nearest cent

To find the principal balance on Jeffrey's loan after 1 year, we need to follow these steps:

### Step 1: Identify the relevant parameters
- **Principal (P):** $28,500
- **Annual interest rate (r):** 3.62% or 0.0362
- **Compounding frequency:** Quarterly (4 times a year)
- **Loan term (n):** 3 years
- **Payment frequency:** Monthly (12 times a year)

### Step 2: Convert the annual interest rate to the quarterly interest rate
The nominal quarterly interest rate $ i $ can be calculated as:
\[
i = \frac{r}{4} = \frac{0.0362}{4} = 0.00905
\]

### Step 3: Calculate the number of quarterly periods in 3 years
The number of quarters in 3 years is:
\[
n_{\text{quarters}} = 3 \times 4 = 12 \text{ quarters}
\]

### Step 4: Calculate the monthly payment
To find the monthly payment, we use the amortization formula:
\[
PMT = \frac{P \times i}{1 - (1 + i)^{-n_{\text{months}}}}
\]
where:
- $ P $ is the principal amount,
- $ i $ is the monthly interest rate, and
- $ n_{\text{months}} $ is the total number of payments (36 months).

However, since the interest is compounded quarterly, the effective monthly interest rate $ j $ is:
\[
j = \left(1 + \frac{r}{4}\right)^{\frac{1}{3}} - 1 = \left(1 + 0.00905\right)^{\frac{1}{3}} - 1
\]

Calculating $ j $:
\[
j \approx 0.003015
\]

Now calculate the monthly payment $ PMT $:
\[
PMT = \frac{28500 \times 0.003015}{1 - (1 + 0.003015)^{-36}}
\]
\[
PMT \approx 828.25
\]

### Step 5: Calculate the principal balance after 1 year
After 1 year (12 payments), the remaining balance can be found using the amortization formula:
\[
B = P \times (1 + j)^{12} - PMT \times \frac{(1 + j)^{12} - 1}{j}
\]
where $ B $ is the remaining balance.

Substitute the known values:
\[
B = 28500 \times (1 + 0.003015)^{12} - 828.25 \times \frac{(1 + 0.003015)^{12} - 1}{0.003015}
\]
\[
B \approx 28500 \times 1.0368 - 828.25 \times \frac{0.0368}{0.003015}
\]
\[
B \approx 29549.6 - 9,900
\]
\[
B \approx 19,649.60
\]

So, the principal balance on the loan after 1 year is approximately **$19,649.60**.

Would you like more details or have any questions?

Here are some related questions:

1. How would the remaining balance change if the interest rate were 4%?
2. What would be the remaining balance after 2 years?
3. How much total interest would Jeffrey pay over the life of the loan?
4. How would a higher payment frequency affect the remaining balance?
5. What would the monthly payment be if the loan term were extended to 5 years?

**Tip:** To quickly approximate the balance after a certain period, you can use online amortization calculators, which automate the complex formulas.

Find out the principal balance on Jeffrey's $28,500 student loan after 1 year of amortization with a 3.62% annual interest rate, compounded quarterly and paid monthly. Get a detailed solution with step-by-step calculations. Suitable for advanced high school and college students.

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