To solve this problem, we use the formula for calculating the monthly payment for a loan, known as the amortization formula:
[tex]\[ M = P \frac{r(1+r)^n}{(1+r)^n - 1}, \][/tex]
where:
- [tex]\( M \)[/tex] is the monthly payment,
- [tex]\( P \)[/tex] is the loan principal,
- [tex]\( r \)[/tex] is the monthly interest rate (annual interest rate divided by 12),
- [tex]\( n \)[/tex] is the number of monthly payments (loan term in years multiplied by 12).
Given:
- Principal [tex]\( P = 58000 \)[/tex] dollars,
- Annual interest rate [tex]\( r = 4.5\% \)[/tex] (or 0.045 as a decimal),
- Loan term [tex]\( t = 5 \)[/tex] years.
First, we convert the annual interest rate to the monthly interest rate:
[tex]\[ r_{\text{monthly}} = \frac{0.045}{12} \approx 0.00375 \][/tex]
Next, we calculate the number of monthly payments:
[tex]\[ n = 5 \text{ years} \times 12 \text{ months/year} = 60 \text{ months} \][/tex]
Now, substitute these values into the amortization formula:
1. Calculate the numerator:
[tex]\[ P \times r_{\text{monthly}} \times (1 + r_{\text{monthly}})^n \][/tex]
[tex]\[ 58000 \times 0.00375 \times (1 + 0.00375)^{60} \][/tex]
2. Calculate the denominator:
[tex]\[ (1 + r_{\text{monthly}})^n - 1 \][/tex]
[tex]\[ (1 + 0.00375)^{60} - 1 \][/tex]
3. Compute the monthly payment [tex]\( M \)[/tex]:
[tex]\[ M = \frac{58000 \times 0.00375 \times (1 + 0.00375)^{60}}{(1 + 0.00375)^{60} - 1} \][/tex]
[tex]\[ M \approx 1081.30 \][/tex] dollars
Therefore, her monthly payment for the loan will be approximately \[tex]$1081.30.
To find the total finance charge, we need to calculate the total amount paid over the term of the loan and then subtract the principal:
4. Calculate the total payment:
\[ \text{Total Payment} = M \times n \]
\[ \text{Total Payment} = 1081.30 \times 60 \]
\[ \text{Total Payment} \approx 64877.70 \] dollars
5. Find the finance charge:
\[ \text{Finance Charge} = \text{Total Payment} - P \]
\[ \text{Finance Charge} = 64877.70 - 58000 \]
\[ \text{Finance Charge} \approx 6877.71 \] dollars
Thus, the total finance charge that she will pay on the loan is approximately \$[/tex]6877.71.
Her monthly payment for the loan will be \[tex]$1081.30. The total finance charge that she will pay on the loan is \$[/tex]6877.71.
