You borrow $180,000 to be repaid in monthly installments over the next 25 years. The first payment occurs one month from today. If the annualized interest rate for the loan is5.1%, how much principal is amortized in payment #51? Enter your answer as a positive number, and round to the nearest dollar.

To calculate how much principal is amortized in payment #51, we'll follow these steps:

### Step 1: Calculate the monthly payment amount using the loan amortization formula:
The formula for the monthly payment $ M $ is:

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

where:
- $ P $ is the loan amount = $180,000
- $ r $ is the monthly interest rate = $\frac{5.1\%}{12} = \frac{0.051}{12}$
- $ n $ is the total number of payments = 25 years $\times 12 = 300$ months

### Step 2: Find the outstanding balance after 50 payments
We will use the remaining balance formula for an amortizing loan after $ t $ payments:

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

where $ t = 50 $ in this case.

### Step 3: Calculate the interest portion of payment #51
The interest portion of any payment is calculated as:

\[
\text{Interest at Payment } t = r \times B(t-1)
\]

where $ B(t-1) $ is the outstanding balance after 50 payments.

### Step 4: Calculate the principal amortized in payment #51
Finally, the principal amortized in any payment is simply the total payment minus the interest portion for that payment:

\[
\text{Principal Amortized} = M - \text{Interest at Payment } t
\]

Let's calculate this step by step.The amount of principal amortized in payment #51 is \$368, rounded to the nearest dollar.

Do you want more details on any specific part of the calculation, or do you have any other questions?

Here are five related questions to deepen your understanding:
1. How do changes in the interest rate affect the monthly payment?
2. What is the impact of a shorter loan term (e.g., 15 years instead of 25)?
3. How would the principal amortized in earlier payments compare to later ones?
4. How does extra payment towards the principal affect the loan amortization schedule?
5. What happens if the interest is compounded daily instead of monthly?

**Tip:** The interest portion of a loan payment decreases over time, while the principal portion increases as more payments are made.

You borrow $180,000 to be repaid in monthly installments over the next 25 years. The first payment occurs one month from today. If the annualized interest rate for the loan is 5.1%, how much principal is amortized in payment #51? Enter your answer as a positive number, and round to the nearest dollar.

Discover how to calculate the principal amortized in payment #51 for a $180,000 loan over 25 years with a 5.1% annual interest rate. Using key loan amortization formulas, this example breaks down the principal and interest calculation for advanced financial mathematics students.

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