Discover a world of knowledge and get your questions answered at IDNLearn.com. Join our interactive community and access reliable, detailed answers from experienced professionals across a variety of topics.
Sagot :
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.
### 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.
Thank you for contributing to our discussion. Don't forget to check back for new answers. Keep asking, answering, and sharing useful information. Your questions deserve reliable answers. Thanks for visiting IDNLearn.com, and see you again soon for more helpful information.
