To solve this problem, we want to determine the principal amount [tex]\( P \)[/tex] at which both credit cards would result in the same cost over the course of a year when interest is compounded monthly.
Given:
- Credit card A has an APR (Annual Percentage Rate) of [tex]\( 14.3\% \)[/tex] and an annual fee of \[tex]$36.
- Credit card B has an APR of \( 17.1\% \) and no annual fee.
Step-by-Step Solution:
1. Convert APR to Monthly Interest Rate:
- For credit card A:
\[
\text{monthly_rate_A} = \frac{14.3\%}{12} = \frac{0.143}{12}
\]
- For credit card B:
\[
\text{monthly_rate_B} = \frac{17.1\%}{12} = \frac{0.171}{12}
\]
2. Compute the Effective Annual Rate (EAR):
- The effective annual rate for a card with a monthly compounding rate can be calculated using:
\[
\text{effective_rate} = (1 + \text{monthly_rate})^{12}
\]
- For credit card A:
\[
\text{effective_rate_A} = \left(1 + \frac{0.143}{12}\right)^{12}
\]
- For credit card B:
\[
\text{effective_rate_B} = \left(1 + \frac{0.171}{12}\right)^{12}
\]
3. Formulate the Total Cost Equations:
- The total annual cost for card A, including the interest and the annual fee:
\[
\text{Total cost for card A} = P \cdot \text{effective_rate_A} -\$[/tex] 36
\]
- The total annual cost for card B, which only includes the interest:
[tex]\[
\text{Total cost for card B} = P \cdot \text{effective_rate_B}
\][/tex]
4. Set the Costs Equal to Each Other to Find the Principal:
- To find the principal [tex]\( P \)[/tex] where the costs are the same, set the two equations equal:
[tex]\[
P \cdot \text{effective_rate_A} - 36 = P \cdot \text{effective_rate_B}
\][/tex]
Based on these steps, the correct equation that can be used to solve for the principal [tex]\( P \)[/tex] is:
[tex]\[
\boxed{P \cdot(1+0.143 / 12)^{12}-36=P \cdot(1+0.171 / 12)^{12}}
\][/tex]
Thus, the correct option is B.
