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Sagot :
Let's explore what would happen if you paid an extra [tex]$100 each month on your mortgage and the effect on the total cost of the mortgage, given the purchase price, down payment, mortgage terms, and interest rate.
### Step-by-Step Solution:
1. Loan Amount Calculation:
- Purchase Price: $[/tex]169,000
- Down Payment: [tex]$30,000 - Loan Amount: \( \text{Purchase Price} - \text{Down Payment} = \$[/tex]169,000 - \[tex]$30,000 = \$[/tex]139,000 \)
2. Mortgage Parameters:
- Number of Years: 30 years
- Interest Rate: 5.9% per year (or 0.059 as a decimal)
3. Monthly Interest Rate:
To find the monthly interest rate, we divide the annual interest rate by 12.
- Monthly Interest Rate: [tex]\( \frac{0.059}{12} \approx 0.00492 \)[/tex]
4. Number of Monthly Payments:
- Number of Payments (n): [tex]\( 30 \, \text{years} \times 12 \, \text{months/year} = 360 \, \text{payments} \)[/tex]
5. Original Monthly Payment Calculation:
The original monthly payment can be calculated using the loan formula which factors in the loan amount, interest rate, and the number of payments.
- Original Monthly Payment: \[tex]$824.46 (approximately) 6. New Monthly Payment with Extra $[/tex]100:
If you pay an extra [tex]$100 each month, the new monthly payment is: - New Monthly Payment: \( \$[/tex]824.46 + \[tex]$100 = \$[/tex]924.46 \)
7. Total Cost Without Extra Payments:
To calculate the total cost without any extra payments, multiply the original monthly payment by the total number of payments (360):
- Total Cost Without Extra Payments: \[tex]$296,805.51 (approximately) 8. Calculation with Extra Payments: When you pay an extra $[/tex]100 each month, you need to determine the new schedule for the mortgage:
- Remaining Balance: Initially, \[tex]$139,000 - Payments: $[/tex]924.46 each month (new payment)
- Track the remaining balance as you make each payment, applying part of each payment to interest and the remaining part to the principal loan amount.
The mortgage would be paid off in 275 months instead of the original 360 months.
9. Total Cost With Extra Payments:
- Total Cost With Extra Payments: [tex]\( \$924.46 \times 275 \approx \$254,226.43 \)[/tex]
10. Savings:
- Savings: [tex]\( \text{Total Cost Without Extra Payments} - \text{Total Cost With Extra Payments} \)[/tex]
- Savings: [tex]\( \$296,805.51 - \$254,226.43 = \$42,579.08 \)[/tex]
### Summary:
#### a) What would happen if you paid an extra [tex]$100 each month? - The term of the mortgage would be reduced from 360 months to 275 months. - You would pay off the mortgage approximately 7 years and 1 month earlier. #### b) What would be the effect on the total cost of the mortgage? - The total cost of the mortgage would decrease. - You would save approximately \$[/tex]42,579.08 over the life of the loan.
By making an extra $100 payment each month, you significantly reduce the amount of interest paid over the life of the loan and save a substantial amount of money.
- Down Payment: [tex]$30,000 - Loan Amount: \( \text{Purchase Price} - \text{Down Payment} = \$[/tex]169,000 - \[tex]$30,000 = \$[/tex]139,000 \)
2. Mortgage Parameters:
- Number of Years: 30 years
- Interest Rate: 5.9% per year (or 0.059 as a decimal)
3. Monthly Interest Rate:
To find the monthly interest rate, we divide the annual interest rate by 12.
- Monthly Interest Rate: [tex]\( \frac{0.059}{12} \approx 0.00492 \)[/tex]
4. Number of Monthly Payments:
- Number of Payments (n): [tex]\( 30 \, \text{years} \times 12 \, \text{months/year} = 360 \, \text{payments} \)[/tex]
5. Original Monthly Payment Calculation:
The original monthly payment can be calculated using the loan formula which factors in the loan amount, interest rate, and the number of payments.
- Original Monthly Payment: \[tex]$824.46 (approximately) 6. New Monthly Payment with Extra $[/tex]100:
If you pay an extra [tex]$100 each month, the new monthly payment is: - New Monthly Payment: \( \$[/tex]824.46 + \[tex]$100 = \$[/tex]924.46 \)
7. Total Cost Without Extra Payments:
To calculate the total cost without any extra payments, multiply the original monthly payment by the total number of payments (360):
- Total Cost Without Extra Payments: \[tex]$296,805.51 (approximately) 8. Calculation with Extra Payments: When you pay an extra $[/tex]100 each month, you need to determine the new schedule for the mortgage:
- Remaining Balance: Initially, \[tex]$139,000 - Payments: $[/tex]924.46 each month (new payment)
- Track the remaining balance as you make each payment, applying part of each payment to interest and the remaining part to the principal loan amount.
The mortgage would be paid off in 275 months instead of the original 360 months.
9. Total Cost With Extra Payments:
- Total Cost With Extra Payments: [tex]\( \$924.46 \times 275 \approx \$254,226.43 \)[/tex]
10. Savings:
- Savings: [tex]\( \text{Total Cost Without Extra Payments} - \text{Total Cost With Extra Payments} \)[/tex]
- Savings: [tex]\( \$296,805.51 - \$254,226.43 = \$42,579.08 \)[/tex]
### Summary:
#### a) What would happen if you paid an extra [tex]$100 each month? - The term of the mortgage would be reduced from 360 months to 275 months. - You would pay off the mortgage approximately 7 years and 1 month earlier. #### b) What would be the effect on the total cost of the mortgage? - The total cost of the mortgage would decrease. - You would save approximately \$[/tex]42,579.08 over the life of the loan.
By making an extra $100 payment each month, you significantly reduce the amount of interest paid over the life of the loan and save a substantial amount of money.
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