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Sagot :
To calculate the monthly payment and the total interest for Joe Sisneros's mortgage on a vacation home, follow these steps:
1. Determine the down payment amount:
- The home price is \[tex]$345,000. - The down payment percentage is 20%. Calculation: \[ \text{Down Payment Amount} = \text{Home Price} \times \text{Down Payment Percentage} = 345,000 \times 0.20 = 69,000 \] 2. Calculate the loan amount: - The loan amount is the home price minus the down payment amount. Calculation: \[ \text{Loan Amount} = \text{Home Price} - \text{Down Payment Amount} = 345,000 - 69,000 = 276,000 \] 3. Convert the annual interest rate to a monthly interest rate: - The annual interest rate is 6.0%. - To get the monthly interest rate, divide the annual rate by 12. Calculation: \[ \text{Monthly Interest Rate} = \frac{\text{Annual Interest Rate}}{12} = \frac{0.06}{12} = 0.005 \] 4. Calculate the number of monthly payments: - The loan term is 30 years. - To get the number of monthly payments, multiply the number of years by 12. Calculation: \[ \text{Number of Payments} = \text{Loan Term in Years} \times 12 = 30 \times 12 = 360 \] 5. Calculate the monthly mortgage payment using the formula for a fixed-rate mortgage: \[ M = P \frac{r(1+r)^n}{(1+r)^n - 1} \] where: - \( M \) is the monthly payment, - \( P \) is the loan amount (\$[/tex]276,000),
- [tex]\( r \)[/tex] is the monthly interest rate (0.005),
- [tex]\( n \)[/tex] is the number of payments (360).
Numerator calculation:
[tex]\[ \text{Numerator} = P \times r \times (1 + r)^n = 276,000 \times 0.005 \times (1 + 0.005)^{360} \][/tex]
Denominator calculation:
[tex]\[ \text{Denominator} = (1 + r)^n - 1 = (1.005)^{360} - 1 \][/tex]
Therefore, the monthly payment calculation is:
[tex]\[ M = \frac{\text{Numerator}}{\text{Denominator}} \][/tex]
6. Determine the monthly payment:
- Substituting the calculated numerators and denominator gives the monthly payment.
The monthly payment is:
[tex]\[ \text{Monthly Payment} = \$1,654.76 \][/tex]
7. Calculate the total payment over the loan term:
- Multiply the monthly payment by the number of payments.
Calculation:
[tex]\[ \text{Total Payment} = \text{Monthly Payment} \times \text{Number of Payments} = 1,654.76 \times 360 = 595,713.6 \][/tex]
8. Calculate the total interest paid:
- Subtract the loan amount from the total payment.
Calculation:
[tex]\[ \text{Total Interest} = \text{Total Payment} - \text{Loan Amount} = 595,713.6 - 276,000 = 319,713.4 \][/tex]
Therefore, the monthly payment Joe Sisneros needs to make is \[tex]$1,654.76, and the total interest paid over the life of the loan is \$[/tex]319,713.40. Let's summarize this in a tabular format:
[tex]\[ \begin{array}{|l|l|} \hline \text{Monthly payment} & \$1,654.76 \\ \hline \text{Total interest} & \$319,713.40 \\ \hline \end{array} \][/tex]
1. Determine the down payment amount:
- The home price is \[tex]$345,000. - The down payment percentage is 20%. Calculation: \[ \text{Down Payment Amount} = \text{Home Price} \times \text{Down Payment Percentage} = 345,000 \times 0.20 = 69,000 \] 2. Calculate the loan amount: - The loan amount is the home price minus the down payment amount. Calculation: \[ \text{Loan Amount} = \text{Home Price} - \text{Down Payment Amount} = 345,000 - 69,000 = 276,000 \] 3. Convert the annual interest rate to a monthly interest rate: - The annual interest rate is 6.0%. - To get the monthly interest rate, divide the annual rate by 12. Calculation: \[ \text{Monthly Interest Rate} = \frac{\text{Annual Interest Rate}}{12} = \frac{0.06}{12} = 0.005 \] 4. Calculate the number of monthly payments: - The loan term is 30 years. - To get the number of monthly payments, multiply the number of years by 12. Calculation: \[ \text{Number of Payments} = \text{Loan Term in Years} \times 12 = 30 \times 12 = 360 \] 5. Calculate the monthly mortgage payment using the formula for a fixed-rate mortgage: \[ M = P \frac{r(1+r)^n}{(1+r)^n - 1} \] where: - \( M \) is the monthly payment, - \( P \) is the loan amount (\$[/tex]276,000),
- [tex]\( r \)[/tex] is the monthly interest rate (0.005),
- [tex]\( n \)[/tex] is the number of payments (360).
Numerator calculation:
[tex]\[ \text{Numerator} = P \times r \times (1 + r)^n = 276,000 \times 0.005 \times (1 + 0.005)^{360} \][/tex]
Denominator calculation:
[tex]\[ \text{Denominator} = (1 + r)^n - 1 = (1.005)^{360} - 1 \][/tex]
Therefore, the monthly payment calculation is:
[tex]\[ M = \frac{\text{Numerator}}{\text{Denominator}} \][/tex]
6. Determine the monthly payment:
- Substituting the calculated numerators and denominator gives the monthly payment.
The monthly payment is:
[tex]\[ \text{Monthly Payment} = \$1,654.76 \][/tex]
7. Calculate the total payment over the loan term:
- Multiply the monthly payment by the number of payments.
Calculation:
[tex]\[ \text{Total Payment} = \text{Monthly Payment} \times \text{Number of Payments} = 1,654.76 \times 360 = 595,713.6 \][/tex]
8. Calculate the total interest paid:
- Subtract the loan amount from the total payment.
Calculation:
[tex]\[ \text{Total Interest} = \text{Total Payment} - \text{Loan Amount} = 595,713.6 - 276,000 = 319,713.4 \][/tex]
Therefore, the monthly payment Joe Sisneros needs to make is \[tex]$1,654.76, and the total interest paid over the life of the loan is \$[/tex]319,713.40. Let's summarize this in a tabular format:
[tex]\[ \begin{array}{|l|l|} \hline \text{Monthly payment} & \$1,654.76 \\ \hline \text{Total interest} & \$319,713.40 \\ \hline \end{array} \][/tex]
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