$560,000 with $60,000 down , 25 year mortgage at a fixed rate of 6 3/8%. How much is the monthly payment? After the first payment, what would be the balance of the principal?

To solve this, we'll use the formula for calculating the monthly mortgage payment (known as the **loan amortization formula**):

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

Where:
- $ M $ = Monthly payment
- $ P $ = Loan amount (principal) = $ 560,000 - 60,000 = 500,000 $
- $ r $ = Monthly interest rate = Annual interest rate / 12 = $ \frac{6 \frac{3}{8}\%}{12} = \frac{0.06375}{12} \approx 0.0053125 $
- $ n $ = Total number of payments (25 years $\times$ 12 months/year) = 300

### Step 1: Calculate the Monthly Payment

Now we substitute into the formula:

\[
M = 500,000 \cdot \frac{0.0053125(1 + 0.0053125)^{300}}{(1 + 0.0053125)^{300} - 1}
\]

I'll compute this for you.The monthly mortgage payment is approximately **$3,337.09**.

### Step 2: Calculate the Balance of the Principal After the First Payment

To find the balance of the principal after the first payment, we need to determine how much of the first payment goes toward interest and how much goes toward the principal. The formula for the interest portion of the first payment is:

\[
\text{Interest} = P \times r
\]

The principal portion of the payment is:

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

The remaining balance after the first payment is:

\[
\text{Remaining Balance} = P - \text{Principal Payment}
\]

I'll compute this for you now.After the first payment:
- The interest portion is **$2,656.25**.
- The principal portion is approximately **$680.84**.
- The remaining balance on the principal is approximately **$499,319.16**.

Would you like further details or have any questions?

Here are 5 related questions to consider:
1. How much total interest will be paid over the life of the loan?
2. How would the monthly payment change with a different interest rate?
3. What is the impact of paying extra toward the principal each month?
4. How does the loan balance decrease over the first year?
5. What is the difference in total payments if the mortgage term is reduced to 20 years?

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

$560,000 with $60,000 down, 25-year mortgage at a fixed rate of 6 3/8%. How much is the monthly payment? After the first payment, what would be the balance of the principal?

Find out how to calculate the monthly payment and principal balance for a $560,000 mortgage with a $60,000 down payment, 6 3/8% interest rate, and 25-year term. This solution involves loan amortization formulas and compound interest calculations, suitable for students in Grades 10-12.

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