You take out a 30-year $100,000 mortgage loan with an APR of 6% and monthly payments. In 12 years, you decide to sell your house and pay off the mortgage. What is the principal balance on the loan?


To find the principal balance of the mortgage loan after 12 years, we can use the formula for the remaining balance on a loan. The steps involve calculating the monthly payment, and then determining the remaining balance after 12 years (or 144 months) of payments.

### Step 1: Calculate the Monthly Payment
The formula for the monthly payment on a mortgage loan is:

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

Where:
- $ M $ is the monthly payment,
- $ P $ is the loan principal (initial amount borrowed),
- $ r $ is the monthly interest rate (APR divided by 12),
- $ n $ is the total number of payments (loan term in months).

Given:
- Principal $ P = 100,000 $,
- APR $ = 6\% $ (so monthly interest rate $ r = 0.06/12 = 0.005 $),
- Loan term $ n = 30 \times 12 = 360 $ months.

Now, let's calculate the monthly payment $ M $.

### Step 2: Calculate the Remaining Balance after 12 Years
The remaining balance after making $ k $ payments (in this case, $ k = 12 \times 12 = 144 $ months) is given by the formula:

\[
B_k = P \times \frac{(1 + r)^n - (1 + r)^k}{(1 + r)^n - 1}
\]

Where $ B_k $ is the remaining balance after $ k $ payments.

I'll calculate the remaining balance after 12 years (144 months).The monthly payment on the mortgage is approximately **$599.55**. After 12 years (or 144 months) of making these payments, the remaining principal balance on the loan is approximately **$79,079.44**.

Would you like further details or explanations on any of the steps? Here are 5 related questions you might find helpful:

1. How is the monthly payment affected by changes in the APR?
2. How would the remaining balance change if the loan term was shorter (e.g., 15 years)?
3. What happens to the remaining balance if extra payments are made each month?
4. How can the total interest paid over the life of the loan be calculated?
5. How does refinancing affect the mortgage balance?

**Tip:** Always consider the impact of extra payments on your mortgage—they can significantly reduce the total interest paid.

You take out a 30-year $100,000 mortgage loan with an APR of 6% and monthly payments. In 12 years, you decide to sell your house and pay off the mortgage. What is the principal balance on the loan?

Learn how to calculate the remaining principal balance on a 30-year $100,000 mortgage loan with a 6% APR after 12 years of payments. This problem covers key concepts in loan amortization and time value of money, including detailed formulas and step-by-step calculations. Suitable for advanced high school or college students.

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