Sure, let's solve this step-by-step.
### Step 1: Identify Key Information
- Monthly payment: [tex]$98,000
- Annual interest rate: 4.1% (which is 0.041 as a decimal)
- Number of payments per year (compounding frequency): 12 (monthly)
- Total duration of the loan: 35 years
### Step 2: Convert Annual Interest Rate to Monthly Interest Rate
The annual interest rate needs to be converted to a monthly rate since the payments are made monthly.
\[ \text{Monthly interest rate} = \frac{\text{Annual interest rate}}{12} \]
\[ \text{Monthly interest rate} = \frac{0.041}{12} \approx 0.0034166667 \]
### Step 3: Calculate the Total Number of Payments
The number of monthly payments over the entire loan period is:
\[ \text{Total number of payments} = \text{Years} \times \text{Payments per year} \]
\[ \text{Total number of payments} = 35 \times 12 = 420 \]
### Step 4: Use the Present Value of Annuities Formula
To find the loan amount (present value), we use the formula for the present value of annuities:
\[ P = \frac{PMT \times (1 - (1 + r)^{-n})}{r} \]
Where:
- \( P \) is the loan amount (present value)
- \( PMT \) is the monthly payment ($[/tex]98,000)
- [tex]\( r \)[/tex] is the monthly interest rate (0.0034166667)
- [tex]\( n \)[/tex] is the total number of payments (420)
### Step 5: Calculate the Loan Amount
Substitute the given values into the formula:
[tex]\[ P = \frac{98,000 \times (1 - (1 + 0.0034166667)^{-420})}{0.0034166667} \][/tex]
### Step 6: Compute the Value
When you compute the expression, you will get:
[tex]\[ P \approx 21,836,355.78 \][/tex]
### Rounded to the Nearest Cent
The loan amount, rounded to the nearest cent, is:
[tex]\[ \$21,836,355.78 \][/tex]
So, the loan amount is $21,836,355.78.
