Lois received a 9-year subsidized student loan of $39,000 at an annual interest rate of 3.625%. Determine her monthly payment on the loan after she graduates in 3 years.

To determine Lois's monthly payment after she graduates in 3 years, we'll assume that the loan is structured such that she starts making payments only after graduation. Let's break down the process in the following steps:

### Step 1: Calculate the loan balance at the time of repayment.
Since Lois will not start repaying the loan until after 3 years, we need to find the loan balance after this period. Interest will accumulate during these 3 years, assuming she is not making any payments during this time.

The formula for the future value $ A $ of a loan with compound interest is:

\[
A = P \times (1 + r)^t
\]

Where:
- $ P = 39,000 $ is the principal loan amount.
- $ r = \frac{3.625\%}{12} = \frac{0.03625}{12} \approx 0.00302083 $ is the monthly interest rate.
- $ t = 3 \times 12 = 36 $ is the total number of months over which the loan accrues interest before she starts repayment.

So:

\[
A = 39,000 \times (1 + 0.00302083)^{36}
\]

### Step 2: Calculate the monthly payment
Once we have the total loan balance after 3 years, Lois will begin repaying it over the remaining 6 years. The formula for calculating the monthly payment $ M $ on an amortized loan is:

\[
M = \frac{A \times r}{1 - (1 + r)^{-n}}
\]

Where:
- $ A $ is the loan balance after 3 years (from Step 1).
- $ r = 0.00302083 $ is the monthly interest rate.
- $ n = 6 \times 12 = 72 $ is the number of months over which the loan is repaid.

Let me compute these steps for you.After 3 years of accrued interest, the loan balance will be approximately **$43,473.33**.

Lois will need to make a monthly payment of approximately **$672.75** over the remaining 6 years to pay off the loan.

Would you like more details on any part of this process, or do you have further questions?

Here are 5 related questions you might consider:
1. How does the loan balance change with a higher interest rate?
2. What would the monthly payment be if the repayment period was extended to 10 years?
3. How is the loan affected if she makes partial payments during the first 3 years?
4. What is the total amount paid over the life of the loan?
5. How does compound interest differ from simple interest in this scenario?

**Tip:** When calculating interest on a loan, remember that small changes in the interest rate or loan term can lead to significant differences in the total amount paid.

Learn how to calculate the monthly payment for a $39,000 student loan with a 3.625% annual interest rate after a 3-year grace period. This problem involves using compound interest and loan amortization formulas. Suitable for high school students in Grades 11-12.

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