Sarah secured a bank loan of $170,000 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 5 years. How much will Sarah still owe on her house? (Round your answer to the nearest cent.)


To determine how much Sarah will still owe on her house after 5 years, we need to use the formula for the remaining balance of a loan that is amortized through monthly payments.

### Key Information:
- Loan amount ($ P $): $170,000
- Annual interest rate ($ r $): 3% per year
- Compounded monthly, so monthly interest rate ($ i $): $ \frac{3\%}{12} = 0.0025 $
- Term of loan ($ n $): 15 years = 180 months
- Time elapsed ($ t $): 5 years = 60 months

We will calculate Sarah’s remaining loan balance using the amortization formula:

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

Where:
- $ B $ is the remaining balance after $ t $ months,
- $ P $ is the loan amount,
- $ i $ is the monthly interest rate,
- $ n $ is the total number of payments (months),
- $ t $ is the number of payments already made.

### Step 1: Calculate the monthly payment amount

The monthly payment for a loan is calculated by:

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

Let's calculate both the monthly payment and the remaining balance after 5 years.Sarah's monthly payment is approximately $1,173.99, and after 5 years, she will still owe approximately **$121,580.34** on her house.

Would you like further details on the calculation, or do you have any questions?

Here are 5 follow-up questions for deeper understanding:
1. How would her remaining balance change if the interest rate were 4% instead of 3%?
2. What happens to the total amount paid over 15 years if she pays extra toward the principal each month?
3. How would the remaining balance after 5 years change if the loan term were 20 years instead of 15 years?
4. What would be the impact of switching from monthly payments to bi-weekly payments?
5. How can Sarah reduce the interest amount paid over the course of the loan?

**Tip:** Making extra payments toward the principal early in the mortgage can significantly reduce the total interest paid over the life of the loan.

Sarah secured a bank loan of $170,000 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 5 years. How much will Sarah still owe on her house? (Round your answer to the nearest cent.)

Find out how much Sarah will still owe on her $170,000 house loan with a 3% annual interest rate after 5 years. This example explains loan amortization using compound interest formulas, ideal for high school students or anyone studying financial mathematics.

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