To determine which loan is the cheapest for Judy, we need to compare the total costs of the loans after the given durations with the specified interest rates. Let's go through each loan option step-by-step.
### Loan A
Duration: 12 months
Annual Interest Rate: 9.50%
Principal: [tex]$2500
1. Monthly Interest Rate:
\[
\text{Monthly Interest Rate} = \frac{9.50\%}{12} = \frac{9.50}{100 \times 12} = 0.00791667 \text{ (approx.)}
\]
2. Total Cost Calculation:
\[
\text{Total Cost} = \$[/tex]2500 \times (1 + 0.00791667)^{12} \approx \[tex]$2748.12
\]
### Loan B
Duration: 24 months
Annual Interest Rate: 8.75%
Principal: $[/tex]2500
1. Monthly Interest Rate:
[tex]\[
\text{Monthly Interest Rate} = \frac{8.75\%}{12} = \frac{8.75}{100 \times 12} = 0.00729167 \text{ (approx.)}
\][/tex]
2. Total Cost Calculation:
[tex]\[
\text{Total Cost} = \$2500 \times (1 + 0.00729167)^{24} \approx \$2976.23
\][/tex]
### Loan C
Duration: 36 months
Annual Interest Rate: 7.75%
Principal: [tex]$2500
1. Monthly Interest Rate:
\[
\text{Monthly Interest Rate} = \frac{7.75\%}{12} = \frac{7.75}{100 \times 12} = 0.00645833 \text{ (approx.)}
\]
2. Total Cost Calculation:
\[
\text{Total Cost} = \$[/tex]2500 \times (1 + 0.00645833)^{36} \approx \[tex]$3152.02
\]
### Loan D
Duration: 48 months
Annual Interest Rate: 6.60%
Principal: $[/tex]2500
1. Monthly Interest Rate:
[tex]\[
\text{Monthly Interest Rate} = \frac{6.60\%}{12} = \frac{6.60}{100 \times 12} = 0.0055
\][/tex]
2. Total Cost Calculation:
[tex]\[
\text{Total Cost} = \$2500 \times (1 + 0.0055)^{48} \approx \$3252.97
\][/tex]
### Summary of Total Costs
- Loan A: [tex]\(\$2748.12\)[/tex]
- Loan B: [tex]\(\$2976.23\)[/tex]
- Loan C: [tex]\(\$3152.02\)[/tex]
- Loan D: [tex]\(\$3252.97\)[/tex]
Among these, Loan A has the lowest total cost of [tex]\(\$2748.12\)[/tex].
### Conclusion
The cheapest loan for Judy in the long run is:
a. Loan A
