To determine which loan is the cheapest for Judy in the long run, we need to calculate the total payment for each loan package using the compound interest formula. For each loan, the total amount to be paid, including interest, is calculated using the formula:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
where:
- [tex]\( P \)[/tex] is the principal amount (initial loan amount),
- [tex]\( r \)[/tex] is the annual interest rate,
- [tex]\( n \)[/tex] is the number of times interest is compounded per year (monthly compounding means [tex]\( n = 12 \)[/tex]),
- [tex]\( t \)[/tex] is the time the money is invested or borrowed for, in years.
Given Judy's loan amount [tex]\( P = \$ 2,500 \)[/tex], we now calculate the total payments for each loan package:
1. Loan A:
- Duration: 12 months ([tex]\( t = 1 \)[/tex] year)
- Annual interest rate: [tex]\( 9.50\% \)[/tex]
- Monthly interest rate: [tex]\( \frac{0.095}{12} \)[/tex]
- Total payment:
[tex]\[
A_A = 2500 \left(1 + \frac{0.095}{12}\right)^{12 \times 1}
\][/tex]
This results in [tex]\( A_A ≈ 2748.12 \)[/tex].
2. Loan B:
- Duration: 24 months ([tex]\( t = 2 \)[/tex] years)
- Annual interest rate: [tex]\( 8.75\% \)[/tex]
- Monthly interest rate: [tex]\( \frac{0.0875}{12} \)[/tex]
- Total payment:
[tex]\[
A_B = 2500 \left(1 + \frac{0.0875}{12}\right)^{24}
\][/tex]
This results in [tex]\( A_B ≈ 2976.23 \)[/tex].
3. Loan C:
- Duration: 36 months ([tex]\( t = 3 \)[/tex] years)
- Annual interest rate: [tex]\( 7.75\% \)[/tex]
- Monthly interest rate: [tex]\( \frac{0.0775}{12} \)[/tex]
- Total payment:
[tex]\[
A_C = 2500 \left(1 + \frac{0.0775}{12}\right)^{36}
\][/tex]
This results in [tex]\( A_C ≈ 3152.02 \)[/tex].
4. Loan D:
- Duration: 48 months ([tex]\( t = 4 \)[/tex] years)
- Annual interest rate: [tex]\( 6.60\% \)[/tex]
- Monthly interest rate: [tex]\( \frac{0.066}{12} \)[/tex]
- Total payment:
[tex]\[
A_D = 2500 \left(1 + \frac{0.066}{12}\right)^{48}
\][/tex]
This results in [tex]\( A_D ≈ 3252.97 \)[/tex].
Comparing the total payments:
- Loan A: [tex]\( 2748.12 \)[/tex]
- Loan B: [tex]\( 2976.23 \)[/tex]
- Loan C: [tex]\( 3152.02 \)[/tex]
- Loan D: [tex]\( 3252.97 \)[/tex]
The cheapest loan for Judy is Loan A with the total payment of approximately \$2748.12.
Therefore, the answer is:
a. loan A
