Your home loan has an initial principal balance of $290,000, monthly payments over the next 30 years, and an annualized interest rate of 3.4%. What is the interest component of payment #32? Enter your answer as a positive number (in dollars), and round to the nearest dollar.

To calculate the interest component of payment #32, we use the formula for the interest portion of a loan payment, which changes each month as the principal balance decreases.

Here's the step-by-step solution:

### Step 1: Calculate the Monthly Payment
The monthly payment $ M $ for a fixed-rate mortgage is calculated using the formula:
\[
M = P \cdot \frac{r(1 + r)^n}{(1 + r)^n - 1}
\]
where:
- $ P = 290,000 $ (initial loan principal),
- $ r = \frac{0.034}{12} \approx 0.0028333 $ (monthly interest rate),
- $ n = 30 \times 12 = 360 $ (total number of payments).

### Step 2: Determine the Outstanding Balance Before Payment #32
The loan balance before a specific payment $ k $ can be calculated using the formula:
\[
B_k = P \cdot \frac{(1 + r)^n - (1 + r)^k}{(1 + r)^n - 1}
\]
where $ k = 31 $ (since we want the balance after 31 payments, just before the 32nd payment).

### Step 3: Calculate the Interest Portion of Payment #32
The interest portion of any payment $ k $ is:
\[
\text{Interest}_{32} = B_{31} \cdot r
\]
where $ B_{31} $ is the remaining balance before the 32nd payment.

Let’s calculate this systematically:The interest component of payment #32 is **$779**.

Would you like more details or have questions about this calculation?

Here are five related questions to further explore this topic:
1. How does the interest component change over the course of the loan?
2. What would the interest component be for the first payment?
3. How much principal would be paid off by the 32nd payment?
4. How would a higher interest rate, such as 5%, affect the monthly payment?
5. What would the total interest paid over the life of the loan be?

**Tip:** In amortized loans, the interest portion decreases with each payment, while the principal portion increases, keeping the monthly payment constant.

Learn how to find the interest portion of payment #32 on a $290,000 home loan with a 3.4% annual interest rate and a 30-year repayment period. This guide covers key amortization concepts, interest calculations, and formulas for loan repayment schedules, making it suitable for college-level finance or advanced high school math students.

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