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Answer :
To find the principal amount [tex]\( P \)[/tex] at which both credit cards offer the same deal over the course of a year, we need to consider the annual percentage rates (APR) and any fees associated with each card. We'll compare credit card A with an APR of 12.5% and an annual fee of [tex]$48, and credit card B with an APR of 15.4% and no annual fee. Assume that interest is compounded monthly for both cards.
Here’s how to set up the equations:
1. Calculate Amount for Credit Card A:
- Card A has an APR of 12.5%. Since this is compounded monthly, the monthly interest rate is \(\frac{0.125}{12}\).
- Over a year, we compound this rate for 12 months: \(\left(1 + \frac{0.125}{12}\right)^{12}\).
- The total amount for card A also includes the annual fee of $[/tex]48.
- Therefore, the expression for the amount with card A over a year is:
[tex]\[
P \left(1 + \frac{0.125}{12}\right)^{12} - 48
\][/tex]
2. Calculate Amount for Credit Card B:
- Card B has an APR of 15.4%, with a monthly interest rate of [tex]\(\frac{0.154}{12}\)[/tex].
- Over a year, this rate compounds monthly: [tex]\(\left(1 + \frac{0.154}{12}\right)^{12}\)[/tex].
- There is no annual fee for card B.
- The expression for the amount with card B over a year is:
[tex]\[
P \left(1 + \frac{0.154}{12}\right)^{12}
\][/tex]
3. Set Up the Equation for Same Total Cost:
- We want the total cost (including fees and interest) for both cards to be equal over the year:
[tex]\[
P \left(1 + \frac{0.125}{12}\right)^{12} - 48 = P \left(1 + \frac{0.154}{12}\right)^{12}
\][/tex]
4. Determine the Correct Choice:
- We compare these expressions with the given options. The correct equation representing this scenario is:
[tex]\[
P \left(1 + \frac{0.125}{12}\right)^{12} - 48 = P \left(1 + \frac{0.154}{12}\right)^{12}
\][/tex]
- This matches with the option provided in the question as choice A.
Therefore, the answer is that choice A represents the correct equation to solve for the principal [tex]\( P \)[/tex] where both credit cards offer the same deal over the course of a year.
Here’s how to set up the equations:
1. Calculate Amount for Credit Card A:
- Card A has an APR of 12.5%. Since this is compounded monthly, the monthly interest rate is \(\frac{0.125}{12}\).
- Over a year, we compound this rate for 12 months: \(\left(1 + \frac{0.125}{12}\right)^{12}\).
- The total amount for card A also includes the annual fee of $[/tex]48.
- Therefore, the expression for the amount with card A over a year is:
[tex]\[
P \left(1 + \frac{0.125}{12}\right)^{12} - 48
\][/tex]
2. Calculate Amount for Credit Card B:
- Card B has an APR of 15.4%, with a monthly interest rate of [tex]\(\frac{0.154}{12}\)[/tex].
- Over a year, this rate compounds monthly: [tex]\(\left(1 + \frac{0.154}{12}\right)^{12}\)[/tex].
- There is no annual fee for card B.
- The expression for the amount with card B over a year is:
[tex]\[
P \left(1 + \frac{0.154}{12}\right)^{12}
\][/tex]
3. Set Up the Equation for Same Total Cost:
- We want the total cost (including fees and interest) for both cards to be equal over the year:
[tex]\[
P \left(1 + \frac{0.125}{12}\right)^{12} - 48 = P \left(1 + \frac{0.154}{12}\right)^{12}
\][/tex]
4. Determine the Correct Choice:
- We compare these expressions with the given options. The correct equation representing this scenario is:
[tex]\[
P \left(1 + \frac{0.125}{12}\right)^{12} - 48 = P \left(1 + \frac{0.154}{12}\right)^{12}
\][/tex]
- This matches with the option provided in the question as choice A.
Therefore, the answer is that choice A represents the correct equation to solve for the principal [tex]\( P \)[/tex] where both credit cards offer the same deal over the course of a year.
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