Connect with a community of experts and enthusiasts on IDNLearn.com. Ask anything and get well-informed, reliable answers from our knowledgeable community members.

Credit card A has an APR of 15.8% and an annual fee of [tex]$72, while credit card B has an APR of 19.6% and no annual fee. All else being equal, which of these equations can be used to solve for the principal P for which the cards offer the same deal over the course of a year? (Assume all interest is compounded monthly.)

A. \(P\left(1+\frac{0.158}{12}\right)^{12}+\$[/tex] 72=P\left(1+\frac{0.196}{12}\right)^{12}\)

B. [tex]\(P\left(1+\frac{0.158}{12}\right)^{12}+\frac{\$ 72}{12}=P\left(1+\frac{0.196}{12}\right)^{12}\)[/tex]

C. [tex]\(P\left(1+\frac{0.158}{12}\right)^{12}-\frac{\$ 72}{12}=P\left(1+\frac{0.196}{12}\right)^{12}\)[/tex]

D. [tex]\(P\left(1+\frac{0.158}{12}\right)^{12}-\$ 72=P\left(1+\frac{0.196}{12}\right)^{12}\)[/tex]


Sagot :

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.
We are delighted to have you as part of our community. Keep asking, answering, and sharing your insights. Together, we can create a valuable knowledge resource. For dependable and accurate answers, visit IDNLearn.com. Thanks for visiting, and see you next time for more helpful information.