Consider the first payment against a $200,000 mortgage that last for 25 years. Fixed repayments are made on a monthly basis. The first row of the amortization schedule is shown below.

 

Payment #	Payment	Interest	Debt Payment	Balance
1	d	716.67	p1	b1
2	…	…	…	…
 

Calculate d, the fixed monthly payments that are made. 
Round your answer to the nearest cent.

To calculate the fixed monthly payment $ d $ for a mortgage, we use the formula for fixed monthly payments on a loan, which is derived from the amortization formula:

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

Where:
- $ P $ is the loan principal (the total loan amount),
- $ r $ is the monthly interest rate,
- $ n $ is the total number of payments.

### Given Data:
- Loan principal, $ P = 200,000 $ dollars.
- Annual interest rate: The first interest payment is 716.67, which is based on a monthly interest rate of $ \frac{r}{12} \times 200,000 = 716.67 $.
  
We first calculate the monthly interest rate $ r $:

\[
\frac{r}{12} \times 200,000 = 716.67 \quad \Rightarrow \quad r = \frac{716.67 \times 12}{200,000} = 0.0430002
\]

Thus, the annual interest rate $ r $ is approximately 4.3%.

- The term of the mortgage is 25 years, meaning there are $ n = 25 \times 12 = 300 $ monthly payments.

### Now, let's compute $ d $ using the formula.

\[
d = 200,000 \times \frac{\frac{0.043}{12} \times (1 + \frac{0.043}{12})^{300}}{(1 + \frac{0.043}{12})^{300} - 1}
\]

I'll calculate this.The fixed monthly payment $ d $ is approximately **\$1,089.08**.

Would you like any further details or have additional questions?

Here are five related questions to deepen your understanding:
1. How is the interest portion of a mortgage payment calculated each month?
2. What changes in the amortization schedule over time?
3. How would a higher interest rate affect the monthly payments?
4. What happens if additional principal payments are made each month?
5. How would a shorter loan term (e.g., 20 years instead of 25) impact the fixed monthly payment?

**Tip:** In a mortgage, the interest portion of each payment decreases over time, while the principal portion increases.

Consider the first payment against a $200,000 mortgage that lasts for 25 years. Fixed repayments are made on a monthly basis. The first row of the amortization schedule is shown below.

Payment #  Payment  Interest  Debt Payment  Balance
1  d  716.67  p1  b1
2  …  …  …  …

Calculate d, the fixed monthly payments that are made. Round your answer to the nearest cent.

Learn how to calculate the fixed monthly payments for a $200,000 mortgage over 25 years. This detailed solution uses the loan amortization formula and shows how the interest rate and loan term affect the payment amount. Ideal for high school or college students studying finance or algebra.

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