To determine Frank's total monthly credit card payment over 24 months for each of his credit cards, we need to calculate the monthly payment for each individual credit card balance given its APR (Annual Percentage Rate) and then sum the payments.
Here's a step-by-step breakdown:
1. Convert APR to a Monthly Rate:
- Monthly rate [tex]\( r \)[/tex] is obtained by dividing the APR by 12.
For Credit Card A:
[tex]\[ \text{APR} = 19\% \rightarrow \text{monthly rate} = \frac{19\%}{12} = 0.0158333 \][/tex]
For Credit Card B:
[tex]\[ \text{APR} = 15\% \rightarrow \text{monthly rate} = \frac{15\%}{12} = 0.0125 \][/tex]
For Credit Card C:
[tex]\[ \text{APR} = 17.5\% \rightarrow \text{monthly rate} = \frac{17.5\%}{12} = 0.0145833 \][/tex]
For Credit Card D:
[tex]\[ \text{APR} = 21\% \rightarrow \text{monthly rate} = \frac{21\%}{12} = 0.0175 \][/tex]
2. Calculate the Monthly Payment Using the Formula:
The formula to determine the monthly payment [tex]\( P \)[/tex] for an installment loan is:
[tex]\[ P = \frac{B \times r}{1 - (1 + r)^{-n}} \][/tex]
where:
- [tex]\( B \)[/tex] is the balance,
- [tex]\( r \)[/tex] is the monthly interest rate,
- [tex]\( n \)[/tex] is the number of payments (months).
Let's calculate each credit card's monthly payment:
For Credit Card A:
[tex]\[ B = \$2380, r = 0.0158333, n = 24 \][/tex]
Monthly payment, [tex]\( P_A = 119.97 \)[/tex]
For Credit Card B:
[tex]\[ B = \$4500, r = 0.0125, n = 24 \][/tex]
Monthly payment, [tex]\( P_B = 218.19 \)[/tex]
For Credit Card C:
[tex]\[ B = \$1580, r = 0.0145833, n = 24 \][/tex]
Monthly payment, [tex]\( P_C = 78.50 \)[/tex]
For Credit Card D:
[tex]\[ B = \$900, r = 0.0175, n = 24 \][/tex]
Monthly payment, \( P_D = 46.25 \]
3. Sum the Monthly Payments:
Summing these monthly payments gives the total monthly payment.
[tex]\[ \text{Total monthly payment} = P_A + P_B + P_C + P_D = 119.97 + 218.19 + 78.50 + 46.25 = 462.91 \][/tex]
Hence, the correct answer is:
d. \$462.91
