Jeremy takes out a 30-year mortgage of 230000 dollars at an annual interest rate of 8.25 percent compounded monthly, with the first payment due in one month. How much does he owe on the loan immediately after the 87th payment?

To determine how much Jeremy owes on his mortgage after the 87th payment, we will use the following approach:

### Step 1: Monthly mortgage payment calculation
The formula for calculating the monthly mortgage payment (also called the amortized payment) is:

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

Where:
- $ M $ is the monthly mortgage payment
- $ P = 230,000 $ is the principal (loan amount)
- $ r = \frac{0.0825}{12} = 0.006875 $ is the monthly interest rate (8.25% annually compounded monthly)
- $ n = 30 \times 12 = 360 $ is the total number of payments (for a 30-year mortgage)

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

### Step 2: Remaining balance after 87 payments
To calculate the remaining balance after 87 payments, we use the following formula for the balance of a loan after $ k $ payments:

\[
B_k = P (1 + r)^k - M \left( \frac{(1 + r)^k - 1}{r} \right)
\]

Where:
- $ B_k $ is the remaining balance after $ k = 87 $ payments
- $ P $, $ r $, and $ M $ are as defined above
- $ k = 87 $

Now let's compute both $ M $ and $ B_k $.Jeremy's monthly mortgage payment is approximately **$1,727.91**. After making 87 payments, the remaining balance on the mortgage is approximately **$212,614.02**.

Would you like further clarification on any part of the solution?

Here are 5 related questions to consider:
1. How would the balance change if the interest rate were lower?
2. What would be the impact of making additional payments on the principal?
3. How does a mortgage's amortization schedule affect the balance over time?
4. What is the total amount Jeremy will pay over the life of the loan?
5. How would the monthly payment change with a different loan term?

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

Discover how to calculate the remaining balance on a 30-year mortgage of $230,000 with an 8.25% interest rate after 87 payments. Learn key formulas for amortized payments and the remaining balance calculation. Suitable for high school students and above.

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