Sarah secured a bank loan of $190,000 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 in 5 years, we need to calculate the remaining balance on the mortgage. This can be done using the amortization formula.

The formula for the remaining balance $ B(t) $ after $ t $ months on an amortized loan is:

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

Where:
- $ P $ is the principal (loan amount),
- $ r $ is the monthly interest rate,
- $ N $ is the total number of payments (loan term in months),
- $ t $ is the number of payments already made.

### Step 1: Define the values
- Principal $ P = 190,000 $ dollars,
- Annual interest rate = 3% or 0.03,
- Monthly interest rate $ r = \frac{0.03}{12} = 0.0025 $,
- Term $ N = 15 \times 12 = 180 $ months,
- Time after 5 years $ t = 5 \times 12 = 60 $ months.

### Step 2: Apply these values into the formula

Now, I will calculate the remaining balance using the amortization formula.After 5 years, Sarah will still owe approximately **$135,883.91** on her house.

Would you like further details on how this calculation was made, or do you have any questions?

Here are 5 related questions to explore:
1. How can Sarah reduce the amount owed after 5 years?
2. What would her monthly payment be over the 15-year term?
3. How does the amortization schedule break down between interest and principal over time?
4. What would the remaining balance be if the interest rate were 4% instead of 3%?
5. How would early payments or extra payments affect her remaining balance?

**Tip:** Making extra payments directly toward the principal can significantly reduce the total interest paid over the loan term.

Sarah secured a bank loan of $190,000 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.)

Discover how to calculate the remaining balance on a $190,000 mortgage after 5 years, with a 3% annual interest rate compounded monthly. This problem involves key concepts of amortization, loan repayment schedules, and compound interest. Perfect for advanced high school or college-level students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation