To determine which credit card offers the same deal over the course of a year, we need to set up equations that account for the interest and fees for each card and then find the value of the principal [tex]\( P \)[/tex] where they are equal.
Credit card A has an APR of [tex]\( 15.8\% \)[/tex] and an annual fee of [tex]\( \$ 72 \)[/tex]. Since the interest is compounded monthly, we convert the APR to a monthly interest rate by dividing by 12. The monthly interest rate for card A is:
[tex]\[ \frac{15.8\%}{12} = \frac{0.158}{12} \][/tex]
Using compound interest formula, the amount owed after one year, including the annual fee, is:
[tex]\[ P \left(1 + \frac{0.158}{12}\right)^{12} + \$ 72 \][/tex]
Credit card B has an APR of [tex]\( 19.6\% \)[/tex] and no annual fee. The monthly interest rate for card B is:
[tex]\[ \frac{19.6\%}{12} = \frac{0.196}{12} \][/tex]
Using compound interest formula, the amount owed after one year is:
[tex]\[ P \left(1 + \frac{0.196}{12}\right)^{12} \][/tex]
To find the principal [tex]\( P \)[/tex] for which the total annual costs including interest are the same for both credit cards, we equate the two expressions:
[tex]\[ P \left(1 + \frac{0.158}{12}\right)^{12} + \$ 72 = P \left(1 + \frac{0.196}{12}\right)^{12} \][/tex]
Thus, the correct equation is:
A.
[tex]\[ P\left(1 + \frac{0.158}{12}\right)^{12} + \$ 72 = P\left(1 + \frac{0.196}{12}\right)^{12} \][/tex]
So, the answer is:
1
This equation matches with option A, which is derived in a straightforward manner considering both the compounding of interest and the additional annual fee of card A.
