Let's break down the process step by step to answer each part of the question.
### Part (a): Monthly Payment Calculation
1. Purchase Price and Down Payment:
- Purchase Price: \[tex]$559,000
- Down Payment: \$[/tex]70,000
2. Principal Loan Amount:
- The loan amount (principal) is calculated as:
[tex]\[
\text{Principal} = \text{Purchase Price} - \text{Down Payment} = \$559,000 - \$70,000 = \$489,000
\][/tex]
3. Annual Interest Rate:
- The annual interest rate is [tex]\(6\frac{3}{8}\%\)[/tex].
- Convert [tex]\(6\frac{3}{8}\%\)[/tex] to a decimal form:
[tex]\[
6\frac{3}{8}\% = 6 + \frac{3}{8} = 6.375\%
\][/tex]
- In decimal form, this is:
[tex]\[
6.375\% = 0.06375
\][/tex]
4. Monthly Interest Rate:
- Convert the annual interest rate to a monthly interest rate by dividing by 12:
[tex]\[
\text{Monthly Interest Rate} = \frac{0.06375}{12} \approx 0.0053125
\][/tex]
5. Number of Payments:
- The mortgage duration is 25 years.
- Number of monthly payments is:
[tex]\[
\text{Number of Payments} = 25 \times 12 = 300
\][/tex]
6. Monthly Payment Calculation:
- Using the mortgage payment formula:
[tex]\[
M = P \times \frac{r(1 + r)^n}{(1 + r)^n - 1}
\][/tex]
where:
- [tex]\(M\)[/tex] is the monthly payment
- [tex]\(P\)[/tex] is the principal loan amount (\[tex]$489,000)
- \(r\) is the monthly interest rate (0.0053125)
- \(n\) is the number of monthly payments (300)
- Substituting the values:
\[
M = 489000 \times \frac{0.0053125(1 + 0.0053125)^{300}}{(1 + 0.0053125)^{300} - 1}
\]
7. Result:
- The monthly payment is \$[/tex]3263.67.
So, the monthly payment is:
[tex]\[
\boxed{\$3263.67}
\][/tex]
### Part (b): Balance After the First Payment
1. Interest and Principal Portions of the First Payment:
- The first interest portion is calculated as:
[tex]\[
\text{Interest Portion} = \text{Principal} \times \text{Monthly Interest Rate} = 489000 \times 0.0053125 = \$2597.8125
\][/tex]
- The principal portion of the first payment is:
[tex]\[
\text{Principal Portion} = \text{Monthly Payment} - \text{Interest Portion} = 3263.67 - 2597.8125 = \$665.8578253973474
\][/tex]
2. Balance After the First Payment:
- The remaining balance after the first payment is:
[tex]\[
\text{Balance After First Payment} = \text{Principal} - \text{Principal Portion} = 489000 - 665.8578253973474 = 488334.14
\][/tex]
Thus, after the first payment:
[tex]\[
\begin{array}{|c|c|c|c|}
\hline
\text{Payment number} & \text{Portion to Interest} & \text{Portion to Principal} & \text{Balance of loan outstanding} \\
\hline
1 & \$2597.81 & \$665.86 & \$488334.14 \\
\hline
\end{array}
\][/tex]
So, the balance of the principal after the first payment is:
[tex]\[
\boxed{\$488334.14}
\][/tex]
Each value rounded to the nearest cent as needed.
