To determine which loan offers Sam the lowest monthly payment, we need to calculate the monthly payment for each loan option. The formula for the monthly payment on a loan (with monthly compounding interest) is:
[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 amount,
- [tex]\( r \)[/tex] is the monthly interest rate,
- [tex]\( n \)[/tex] is the number of monthly payments (loan term in months).
Given the loan amount [tex]\( P = \$8,900 \)[/tex], let's calculate the monthly payments for each loan option one by one.
### Loan A:
- Annual interest rate: [tex]\( 9.5\% \)[/tex]
- Monthly interest rate: [tex]\(\frac{9.5}{100 \times 12} = 0.00791667 \)[/tex]
- Loan term: [tex]\( 12 \)[/tex] months
Using the formula:
[tex]\[ M_A = 8900 \times \frac{0.00791667 \times (1 + 0.00791667)^{12}}{((1 + 0.00791667)^{12} - 1)} \][/tex]
This results in:
[tex]\[ M_A = 780.38 \][/tex]
### Loan B:
- Annual interest rate: [tex]\( 8.75\% \)[/tex]
- Monthly interest rate: [tex]\(\frac{8.75}{100 \times 12} = 0.00729167 \)[/tex]
- Loan term: [tex]\( 24 \)[/tex] months
Using the formula:
[tex]\[ M_B = 8900 \times \frac{0.00729167 \times (1 + 0.00729167)^{24}}{((1 + 0.00729167)^{24} - 1)} \][/tex]
This results in:
[tex]\[ M_B = 405.57 \][/tex]
### Loan C:
- Annual interest rate: [tex]\( 7.75\% \)[/tex]
- Monthly interest rate: [tex]\(\frac{7.75}{100 \times 12} = 0.00645833 \)[/tex]
- Loan term: [tex]\( 36 \)[/tex] months
Using the formula:
[tex]\[ M_C = 8900 \times \frac{0.00645833 \times (1 + 0.00645833)^{36}}{((1 + 0.00645833)^{36} - 1)} \][/tex]
This results in:
[tex]\[ M_C = 277.87 \][/tex]
### Loan D:
- Annual interest rate: [tex]\( 6.60\% \)[/tex]
- Monthly interest rate: [tex]\(\frac{6.60}{100 \times 12} = 0.0055 \)[/tex]
- Loan term: [tex]\( 48 \)[/tex] months
Using the formula:
[tex]\[ M_D = 8900 \times \frac{0.0055 \times (1 + 0.0055)^{48}}{((1 + 0.0055)^{48} - 1)} \][/tex]
This results in:
[tex]\[ M_D = 211.47 \][/tex]
### Comparison of monthly payments:
- Loan A: [tex]\( 780.38 \)[/tex]
- Loan B: [tex]\( 405.57 \)[/tex]
- Loan C: [tex]\( 277.87 \)[/tex]
- Loan D: [tex]\( 211.47 \)[/tex]
The lowest monthly payment is given by Loan D ([tex]\( 211.47 \)[/tex]).
Therefore, Sam should choose Loan D for the lowest monthly payment.
_Answer: d. Loan D_
