Let's go through each part of the problem in detail to determine the best loan option for Karen.
### Part (a)
Calculate the monthly payment for the online lending company's loan:
1. Loan Amount and Terms:
- Principal ([tex]\(P\)[/tex]): \[tex]$85,000
- Annual interest rate: 14.3%
- Loan term: 9 years
2. Monthly Interest Rate:
\[
r = \frac{14.3\%}{12} = \frac{0.143}{12} \approx 0.0119167
\]
3. Total Number of Payments (Months):
\[
n = 9 \times 12 = 108 \text{ months}
\]
4. Monthly Payment Formula:
\[
M = P \cdot \frac{r(1 + r)^n}{(1 + r)^n - 1}
\]
\[
M = 85000 \cdot \frac{0.0119167(1 + 0.0119167)^{108}}{(1 + 0.0119167)^{108} - 1} \approx \$[/tex]1403.33
\]
So, the monthly payment for the online lending company's loan is approximately \[tex]$1403.33.
### Part (b)
Calculate the monthly payment for the bank's loan:
1. Loan Amount and Terms:
- Principal (\(P\)): \$[/tex]85,000
- Annual interest rate: 12.7%
- Loan term: 10 years
2. Monthly Interest Rate:
[tex]\[
r = \frac{12.7\%}{12} = \frac{0.127}{12} \approx 0.0105833
\][/tex]
3. Total Number of Payments (Months):
[tex]\[
n = 10 \times 12 = 120 \text{ months}
\][/tex]
4. Monthly Payment Formula:
[tex]\[
M = P \cdot \frac{r(1 + r)^n}{(1 + r)^n - 1}
\][/tex]
[tex]\[
M = 85000 \cdot \frac{0.0105833(1 + 0.0105833)^{120}}{(1 + 0.0105833)^{120} - 1} \approx \$1254.15
\][/tex]
So, the monthly payment for the bank's loan is approximately \[tex]$1254.15.
### Part (c)
Calculate the total payment for each loan and compare them:
1. Total Payment for Online Lending Company:
\[
\text{Total Payment} = \text{Monthly Payment} \times \text{Number of Payments}
\]
\[
\text{Total Payment} = \$[/tex]1403.33 \times 108 \approx \[tex]$151,559.64
\]
2. Total Payment for Bank:
\[
\text{Total Payment} = \text{Monthly Payment} \times \text{Number of Payments}
\]
\[
\text{Total Payment} = \$[/tex]1254.15 \times 120 \approx \[tex]$150,497.12
\]
Comparison:
Since \$[/tex]150,497.12 (bank) is less than \[tex]$151,559.64 (online lending company), the bank offers a better deal.
Savings:
\[
\text{Savings} = \$[/tex]151,559.64 - \[tex]$150,497.12 \approx \$[/tex]1062.52
\]
Conclusion:
- The bank's loan would have the lowest total amount to pay off.
- The total amount paid to the bank would be approximately \$1062.52 less than to the online lending company.
