To solve for the principal [tex]\( P \)[/tex] where both credit cards result in the same cost over the course of a year, we need to consider the annual percentage rate (APR), which affects the final amount due to monthly compounding interest, and any annual fees associated with the cards.
Given the APR for both cards are compounded monthly, we will use the formula for monthly compounding interest to find the effective annual rate (EAR) for both cards. The formula for the effective annual rate considering monthly compounding is:
[tex]\[
\text{EAR} = \left(1 + \frac{\text{APR}}{12}\right)^{12}
\][/tex]
For Credit Card A:
- APR = [tex]\( 27.2\% = 0.272 \)[/tex]
- Annual Fee = $96
For Credit Card B:
- APR = [tex]\( 30.3\% = 0.303 \)[/tex]
- No Annual Fee
First, we calculate the effective annual rate for both cards.
1. Calculate the effective annual rate (EAR) for Credit Card A:
[tex]\[
\text{EAR}_A = \left(1 + \frac{0.272}{12}\right)^{12}
\][/tex]
Given the result, the calculation of [tex]\(\left(1 + \frac{0.272}{12}\right)^{12}\)[/tex] is approximately [tex]\(1.308607\)[/tex].
2. Calculate the effective annual rate (EAR) for Credit Card B:
[tex]\[
\text{EAR}_B = \left(1 + \frac{0.303}{12}\right)^{12}
\][/tex]
Given the result, the calculation of [tex]\(\left(1 + \frac{0.303}{12}\right)^{12}\)[/tex] is approximately [tex]\(1.34883\)[/tex].
Next, for the two cards to offer the same deal over the course of a year, the total cost considering the interest and annual fee must be the same. For Credit Card A, we subtract the annual fee from the product of the principal and the EAR.
The equation becomes:
[tex]\[
P \times 1.308607 - 96 = P \times 1.34883
\][/tex]
From the given options, the corresponding equation is:
[tex]\[
P\left(1+\frac{0.272}{12}\right)^{12}-96=P\left(1+\frac{0.303}{12}\right)^{12}
\][/tex]
So, the correct answer is:
[tex]\[
\boxed{A}
\][/tex]
