Consider a home mortgage of ​$200 comma 000 at a fixed APR of 4.5​% for 15 years.
a. Calculate the monthly payment. (Do not round until the final answer. Then round to the nearest cent as​ needed.)
b. Determine the total amount paid over the term of the loan.
c. Of the total amount​ paid, what percentage is paid toward the principal and what percentage is paid for interest.

We will solve this problem step by step using the formula for calculating the monthly payment for a fixed-rate mortgage, and we'll also determine the total amount paid and the interest-to-principal ratio.

### Part a: Calculate the Monthly Payment
The formula for calculating the monthly mortgage payment is given by the amortization formula:

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

Where:
- $ M $ is the monthly payment.
- $ P $ is the loan amount (principal), which is $ 200,000 $ USD.
- $ r $ is the monthly interest rate, which is the annual interest rate divided by 12. Here, the annual rate is $ 4.5\% $, so $ r = \frac{4.5\%}{12} = 0.00375 $.
- $ n $ is the total number of monthly payments, which is $ 15 \times 12 = 180 $ payments for 15 years.

Now, we'll calculate $ M $, the monthly payment.

### Part b: Determine the Total Amount Paid Over the Term
Once we have the monthly payment $ M $, the total amount paid over the term is simply:

\[
\text{Total amount paid} = M \times n
\]

### Part c: Calculate the Percentage Paid Toward Principal and Interest
- The total amount paid for the principal is the original loan amount, $ 200,000 $.
- The total amount paid for interest is the difference between the total amount paid and the principal.

The percentage paid toward the principal is:

\[
\text{Percentage toward principal} = \left(\frac{200,000}{\text{Total amount paid}}\right) \times 100
\]

The percentage paid toward interest is:

\[
\text{Percentage toward interest} = 100 - \text{Percentage toward principal}
\]

Let's calculate all these values.### Part a: Monthly Payment
The monthly mortgage payment is approximately **$1,529.99**.

### Part b: Total Amount Paid Over the Term
The total amount paid over the 15 years is approximately **$275,397.58**.

### Part c: Percentage Paid Toward Principal and Interest
- The percentage of the total amount paid toward the principal is approximately **72.62%**.
- The percentage of the total amount paid toward interest is approximately **27.38%**.

Would you like any further details or explanations?

Here are 5 related questions for further exploration:
1. How would the monthly payment change if the interest rate increased to 5%?
2. What would the total interest paid be if the loan term was extended to 30 years?
3. How does making extra payments each month affect the total interest paid?
4. What is the effect of making a larger down payment on the loan?
5. How does refinancing at a lower interest rate impact the loan term?

**Tip:** Making additional principal payments early in the mortgage can significantly reduce the total interest paid over time.

Consider a home mortgage of $200,000 at a fixed APR of 4.5% for 15 years.
a. Calculate the monthly payment. (Do not round until the final answer. Then round to the nearest cent as needed.)
b. Determine the total amount paid over the term of the loan.
c. Of the total amount paid, what percentage is paid toward the principal and what percentage is paid for interest.

Calculate the monthly payment, total amount paid, and the percentage of payment towards principal and interest for a $200,000 mortgage at a 4.5% APR over 15 years. This detailed step-by-step solution covers key financial mathematics concepts such as loan amortization and interest rate calculations, suitable for high school students in Grades 10-12.

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