Sure, I'll walk you through solving this problem step-by-step.
First, let's define a few things:
1. APR (Annual Percentage Rate): This is the annual interest rate charged for borrowing which is different for each card. For credit card A, the APR is [tex]\(12.5\%\)[/tex] or [tex]\( 0.125 \)[/tex] in decimal form. For credit card B, the APR is [tex]\(15.4\%\)[/tex] or [tex]\(0.154 \)[/tex] in decimal form.
2. Annual Fee: This is an extra cost that credit card A has annually, which is \[tex]$48.
3. Compounding: Both cards have monthly compounding. This means interest is calculated monthly.
Next, let's break down how to calculate the effective amount owed at the end of the year for each card.
1. Monthly Interest Rate:
- For credit card A: \( \text{Monthly Rate} = \frac{0.125}{12} \)
- For credit card B: \( \text{Monthly Rate} = \frac{0.154}{12} \)
2. Compounded Annual Rate can be calculated using the formula for compound interest over 12 months:
- For credit card A:
\[
(1 + \text{Monthly Rate}_A)^{12} = (1 + \frac{0.125}{12})^{12}
\]
- For credit card B:
\[
(1 + \text{Monthly Rate}_B)^{12} = (1 + \frac{0.154}{12})^{12}
\]
3. Total Cost Analysis:
- For credit card A: The total cost also includes the annual fee of \$[/tex]48. Therefore, the effective annual cost for credit card A would be:
[tex]\[
P \times (1 + \frac{0.125}{12})^{12} + 48
\][/tex]
- For credit card B: The total cost would simply be the compound interest with no additional fees:
[tex]\[
P \times (1 + \frac{0.154}{12})^{12}
\][/tex]
We want to find the principal amount, [tex]\( P \)[/tex], at which these two costs are equal. Therefore, we set the total costs for both cards equal to each other:
[tex]\[
P \times (1 + \frac{0.125}{12})^{12} + 48 = P \times (1 + \frac{0.154}{12})^{12}
\][/tex]
None of the other equations take into account the conditions required for this problem correctly (like [tex]\( B \cdot \)[/tex] [tex]$48 \cdot 12 \) would imply 48 dollars a month, not per year).
Hence, the correct choice is:
\[
A. \quad P \cdot(1+0.125 / 12)^{12}+\$[/tex] 48=P \cdot(1+0.154 / 12)^{12}
\]
