Megan took out a loan of $85,600 at an interest rate of 11.5%, compounded monthly. She paid off the loan in full over the course of exactly twelve years. Assuming that Megan made the same payment every month for twelve years, what was her monthly payment?

To find Megan's monthly payment for the loan, we can use the formula for the monthly payment of a loan with compound interest, which is derived from the **loan amortization formula**:

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

Where:
- $ M $ is the monthly payment,
- $ P $ is the principal loan amount (in this case, $85,600),
- $ r $ is the monthly interest rate (annual rate divided by 12),
- $ n $ is the total number of payments (12 years * 12 months/year).

### Step 1: Calculate the monthly interest rate
The annual interest rate is 11.5%, so the monthly interest rate is:

\[
r = \frac{11.5\%}{12} = \frac{0.115}{12} = 0.0095833
\]

### Step 2: Calculate the total number of payments
Megan made payments for 12 years, and since payments are monthly, the total number of payments is:

\[
n = 12 \times 12 = 144 \text{ months}
\]

### Step 3: Plug values into the formula
Now we can substitute the values into the loan amortization formula:

\[
M = \frac{85,600 \times 0.0095833 \times (1 + 0.0095833)^{144}}{(1 + 0.0095833)^{144} - 1}
\]

### Step 4: Calculate the monthly payment
Let's compute the result.

\[
M = \frac{85,600 \times 0.0095833 \times (1.0095833)^{144}}{(1.0095833)^{144} - 1}
\]

\[
M \approx \frac{85,600 \times 0.0095833 \times 3.434503}{3.434503 - 1}
\]

\[
M \approx \frac{85,600 \times 0.032876}{2.434503}
\]

\[
M \approx \frac{2,815.46}{2.434503}
\]

\[
M \approx 1,158.50
\]

So, Megan's monthly payment is approximately **$1,158.50**.

Would you like a more detailed breakdown of the calculations, or do you have any questions?

Here are 5 follow-up questions you could explore:
1. What would the total amount paid over the 12 years be, including interest?
2. How would the monthly payment change if the loan term were shorter or longer?
3. How would a higher or lower interest rate affect the monthly payment?
4. How can we calculate the total interest paid over the course of the loan?
5. What happens if Megan pays extra toward the principal each month?

**Tip:** If you're unsure about making monthly payments, it's helpful to try using a loan calculator to verify your results or experiment with different terms and interest rates.

Learn how to calculate the monthly payment for a loan of $85,600 with an interest rate of 11.5%, compounded monthly. This solution uses the loan amortization formula to determine the monthly payment over a 12-year period. Suitable for students in Grades 10-12 studying finance or algebra.

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