To determine which equation can be used to solve for the principal amount [tex]\(P\)[/tex] such that the total cost using both credit cards over the year is the same, let's analyze the situation step-by-step.
1. Understand the given information:
- Credit Card A has an APR (Annual Percentage Rate) of [tex]\(20.8\%\)[/tex] and an annual fee of [tex]\(\$60\)[/tex].
- Credit Card B has an APR of [tex]\(24.6\%\)[/tex] and no annual fee.
- Interest is compounded monthly for both cards.
2. Set up the equation for Credit Card A:
- The monthly interest rate for Credit Card A is [tex]\(\frac{20.8\%}{12} = 0.208 / 12\)[/tex].
- Over 12 months, the compound interest formula gives us the final amount, [tex]\(F_A\)[/tex], as follows:
[tex]\[
F_A = P \cdot \left(1 + \frac{0.208}{12}\right)^{12} - \text{annual fee}
\][/tex]
- Including the annual fee, we subtract [tex]\(\$60\)[/tex] from the final amount:
[tex]\[
F_A = P \cdot \left(1 + \frac{0.208}{12}\right)^{12} - 60
\][/tex]
3. Set up the equation for Credit Card B:
- The monthly interest rate for Credit Card B is [tex]\(\frac{24.6\%}{12} = 0.246 / 12\)[/tex].
- Over 12 months, the compound interest formula gives us the final amount, [tex]\(F_B\)[/tex], as follows:
[tex]\[
F_B = P \cdot \left(1 + \frac{0.246}{12}\right)^{12}
\][/tex]
- No annual fee needs to be subtracted since Credit Card B does not have one.
4. Equating the final amounts:
- To find the principal [tex]\(P\)[/tex] where the costs for both cards are the same, we set [tex]\(F_A = F_B\)[/tex]:
[tex]\[
P \cdot \left(1 + \frac{0.208}{12}\right)^{12} - 60 = P \cdot \left(1 + \frac{0.246}{12}\right)^{12}
\][/tex]
Based on our analysis, the correct equation that can be used to solve for the principal [tex]\(P\)[/tex] is:
[tex]\[
\boxed{P \cdot(1+0.208 / 12)^{12}-60 = P \cdot(1+0.246 / 12)^{12}}
\][/tex]
Thus, the correct choice is [tex]\(D\)[/tex].
