a house cost $140000. the term of the sale are 20% down payment and the remainder to be paid in monthly payments over 10 years at an annual rate of 3.75% compounded monthly. what are the monthly payments?

To calculate the monthly payments for this loan, we need to follow the steps for solving a standard amortization problem. The formula to calculate monthly payments is:

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

Where:
- $ M $ is the monthly payment.
- $ P $ is the loan principal (the amount to be financed).
- $ r $ is the monthly interest rate (annual interest rate divided by 12).
- $ n $ is the total number of payments (number of months).

Let's break it down step by step.

### 1. **Find the loan principal, $ P $**:
Since the buyer makes a 20% down payment, the loan principal is 80% of the house's price.

\[
P = 140,000 \times (1 - 0.20) = 140,000 \times 0.80 = 112,000
\]

### 2. **Find the monthly interest rate, $ r $**:
The annual interest rate is 3.75%, so the monthly interest rate is:

\[
r = \frac{3.75\%}{12} = \frac{0.0375}{12} = 0.003125
\]

### 3. **Find the total number of payments, $ n $**:
Since the loan term is 10 years, and the payments are monthly, we calculate the total number of payments:

\[
n = 10 \times 12 = 120 \, \text{months}
\]

### 4. **Calculate the monthly payment, $ M $**:

Now, using the formula:

\[
M = \frac{112,000 \times 0.003125 \times (1 + 0.003125)^{120}}{(1 + 0.003125)^{120} - 1}
\]

First, calculate $ (1 + r)^{120} $:

\[
(1 + 0.003125)^{120} = 1.003125^{120} \approx 1.4690
\]

Now plug this into the formula:

\[
M = \frac{112,000 \times 0.003125 \times 1.4690}{1.4690 - 1} = \frac{112,000 \times 0.003125 \times 1.4690}{0.4690}
\]

\[
M \approx \frac{512.50 \times 1.4690}{0.4690} \approx \frac{752.79}{0.4690} \approx 1,605.89
\]

### Final Answer:
The monthly payments are approximately **$1,605.89**.

Would you like further details or have any questions about this calculation?

Here are 5 related questions you might find useful:
1. How do different interest rates affect the monthly payment amount?
2. What would the monthly payments be if the down payment were 25% instead of 20%?
3. How does compounding frequency impact the overall cost of the loan?
4. How can extra payments reduce the total loan cost and payoff time?
5. What is the total interest paid over the course of this 10-year loan?

**Tip**: Making even small additional payments toward the principal each month can significantly reduce the total interest paid over time.

A house costs $140,000. The terms of the sale are 20% down payment and the remainder to be paid in monthly payments over 10 years at an annual rate of 3.75% compounded monthly. What are the monthly payments?

Learn how to calculate monthly loan payments for a $140,000 house with a 20% down payment and a 3.75% annual interest rate, compounded monthly. Follow a detailed step-by-step solution using the loan amortization formula. Suitable for Grades 11-12 students and anyone learning about interest rates and loan payments.

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