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To solve the problem, we need to find out how much Sarah still owes on her house after 10 years of payments, given the loan details.

### Step 1: Understanding the parameters
- **Loan amount (P)**: $165,000
- **Interest rate (annual)**: 3% (compounded monthly)
- **Loan term**: 15 years
- **Payments are made monthly**
- **Sarah plans to sell after 10 years**

We need to determine how much is still owed after 10 years.

### Step 2: Formula for calculating the monthly payment
The monthly payment for a loan can be calculated using the formula:

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

Where:
- $ M $ = Monthly payment
- $ P $ = Loan principal = $165,000
- $ r $ = Monthly interest rate = $\frac{3\%}{12} = 0.0025$
- $ n $ = Total number of payments = $15 \times 12 = 180$

### Step 3: Calculate monthly payment

Substituting the values into the formula:

\[
M = \frac{165,000 \cdot 0.0025 \cdot (1+0.0025)^{180}}{(1+0.0025)^{180} - 1}
\]

Let me calculate this for you.The monthly payment $ M $ for Sarah's loan is approximately **$1,139.46**.

### Step 4: Calculate the remaining balance after 10 years
After 10 years (or 120 months), Sarah will still owe part of the loan. The formula to calculate the remaining balance after $t$ months is:

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

Where:
- $ B $ = Remaining balance
- $ t $ = 10 years $ \times 12 = 120 $ months
- $ r $ = Monthly interest rate
- $ n $ = Total number of payments (180 months)

Now, let me calculate the remaining balance after 120 months.After 10 years (120 months), Sarah will still owe approximately **$63,413.62** on her house.

Would you like more details or have any questions on this?

Here are some related questions to further your understanding:

1. How would the remaining balance change if the interest rate were higher, say 5%?
2. How does the remaining balance change if Sarah plans to sell the house after 5 years instead of 10 years?
3. What would be the total interest paid over the entire 15-year period if Sarah never sold the house?
4. How does changing the loan term (e.g., to 30 years) affect the monthly payment and balance?
5. Can we calculate how much equity Sarah would have built up in her home after 10 years?

**Tip**: When calculating loans, it's important to consider how small changes in the interest rate or loan duration can significantly affect both monthly payments and the overall cost of the loan.

Sarah secured a bank loan of $165,000 for the purchase of a house. The mortgage is to be amortized through monthly payments for a term of 15 years, with an interest rate of 3%/year compounded monthly on the unpaid balance. She plans to sell her house in 10 years. How much will Sarah still owe on her house?

This problem calculates the remaining balance on a $165,000 loan with a 3% annual interest rate after 10 years. Learn how to apply loan amortization formulas to determine remaining debt.

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