Answer :
To find the monthly payments and the total interest for the loan, we'll use the provided formula:
[tex]\[ \text{PMT} = \frac{P \left( \frac{r}{n} \right)}{1 - \left( 1 + \frac{r}{n} \right)^{-nt}} \][/tex]
Given:
- [tex]\( P = 10{,}000 \)[/tex] (the principal loan amount)
- [tex]\( r = 0.05 \)[/tex] (annual interest rate)
- [tex]\( t = 4 \)[/tex] (loan term in years)
- [tex]\( n = 12 \)[/tex] (number of payments per year)
First, we need to find the monthly interest rate:
[tex]\[ \text{Monthly interest rate} = \frac{r}{n} = \frac{0.05}{12} = \frac{1}{240} \approx 0.004167 \][/tex]
Next, we calculate the total number of payments over the life of the loan:
[tex]\[ \text{Total number of payments} = nt = 12 \times 4 = 48 \][/tex]
Now, we can substitute these values into the formula to find the monthly payment (PMT):
[tex]\[ \text{PMT} = \frac{10{,}000 \left( 0.004167 \right)}{1 - \left( 1 + 0.004167 \right)^{-48}} \][/tex]
[tex]\[ \text{PMT} = \frac{10{,}000 \times 0.004167}{1 - \left(1.004167\right)^{-48}} \][/tex]
[tex]\[ \text{PMT} = \frac{41.67}{1 - 0.8174} \approx \frac{41.67}{0.1826} \approx 228.29 \][/tex]
Thus, the monthly payment is approximately [tex]$230.29. To find the total interest paid over the life of the loan, we first calculate the total amount paid: \[ \text{Total amount paid} = \text{PMT} \times \text{total number of payments} = 230.29 \times 48 \] \[ \text{Total amount paid} \approx 11{,}053.92 \] The total interest is then the total amount paid minus the initial loan amount: \[ \text{Total interest} = 11{,}053.92 - 10{,}000 \approx 1{,}054.06 \] Therefore, the total interest for the loan is approximately \( \$[/tex]1,054.06 \).
[tex]\[ \text{PMT} = \frac{P \left( \frac{r}{n} \right)}{1 - \left( 1 + \frac{r}{n} \right)^{-nt}} \][/tex]
Given:
- [tex]\( P = 10{,}000 \)[/tex] (the principal loan amount)
- [tex]\( r = 0.05 \)[/tex] (annual interest rate)
- [tex]\( t = 4 \)[/tex] (loan term in years)
- [tex]\( n = 12 \)[/tex] (number of payments per year)
First, we need to find the monthly interest rate:
[tex]\[ \text{Monthly interest rate} = \frac{r}{n} = \frac{0.05}{12} = \frac{1}{240} \approx 0.004167 \][/tex]
Next, we calculate the total number of payments over the life of the loan:
[tex]\[ \text{Total number of payments} = nt = 12 \times 4 = 48 \][/tex]
Now, we can substitute these values into the formula to find the monthly payment (PMT):
[tex]\[ \text{PMT} = \frac{10{,}000 \left( 0.004167 \right)}{1 - \left( 1 + 0.004167 \right)^{-48}} \][/tex]
[tex]\[ \text{PMT} = \frac{10{,}000 \times 0.004167}{1 - \left(1.004167\right)^{-48}} \][/tex]
[tex]\[ \text{PMT} = \frac{41.67}{1 - 0.8174} \approx \frac{41.67}{0.1826} \approx 228.29 \][/tex]
Thus, the monthly payment is approximately [tex]$230.29. To find the total interest paid over the life of the loan, we first calculate the total amount paid: \[ \text{Total amount paid} = \text{PMT} \times \text{total number of payments} = 230.29 \times 48 \] \[ \text{Total amount paid} \approx 11{,}053.92 \] The total interest is then the total amount paid minus the initial loan amount: \[ \text{Total interest} = 11{,}053.92 - 10{,}000 \approx 1{,}054.06 \] Therefore, the total interest for the loan is approximately \( \$[/tex]1,054.06 \).
