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Answer :
To determine which equation can be used to solve for the principal [tex]\( P \)[/tex] where the two credit cards offer the same deal over the course of a year, let's break down the problem into simple steps:
1. Understand the Scenario:
- Credit Card A has an APR (Annual Percentage Rate) of 12.5% and an annual fee of [tex]$48.
- Credit Card B has an APR of 15.4% with no annual fee.
- We are looking to find the principal amount \( P \) for which the costs of holding each card are the same after one year, with interest being compounded monthly.
2. Gather the Formulas for Both Cards:
- For Credit Card A:
- The interest is compounded monthly, so the monthly interest rate is \(\frac{0.125}{12}\).
- The final amount after one year (compounded monthly) for card A is given by:
\[ P \left(1 + \frac{0.125}{12}\right)^{12} \]
- We need to add the annual fee of $[/tex]48 to this, so the total cost for card A after one year is:
[tex]\[ P \left(1 + \frac{0.125}{12}\right)^{12} + 48 \][/tex]
- For Credit Card B:
- The monthly interest rate is [tex]\(\frac{0.154}{12}\)[/tex].
- The final amount after one year (compounded monthly) for card B is:
[tex]\[ P \left(1 + \frac{0.154}{12}\right)^{12} \][/tex]
3. Setting the Equations Equal:
- We want the total costs for both cards to be equal, since they offer the same deal. Therefore, we equate the total costs:
[tex]\[ P \left(1 + \frac{0.125}{12}\right)^{12} + 48 = P \left(1 + \frac{0.154}{12}\right)^{12} \][/tex]
4. Conclusion:
- The correct equation that you can use to find the principal [tex]\( P \)[/tex] where both cards end up costing the same is:
[tex]\[ P \left(1 + \frac{0.125}{12}\right)^{12} + 48 = P \left(1 + \frac{0.154}{12}\right)^{12} \][/tex]
Therefore, the choice that matches this equation is choice B.
1. Understand the Scenario:
- Credit Card A has an APR (Annual Percentage Rate) of 12.5% and an annual fee of [tex]$48.
- Credit Card B has an APR of 15.4% with no annual fee.
- We are looking to find the principal amount \( P \) for which the costs of holding each card are the same after one year, with interest being compounded monthly.
2. Gather the Formulas for Both Cards:
- For Credit Card A:
- The interest is compounded monthly, so the monthly interest rate is \(\frac{0.125}{12}\).
- The final amount after one year (compounded monthly) for card A is given by:
\[ P \left(1 + \frac{0.125}{12}\right)^{12} \]
- We need to add the annual fee of $[/tex]48 to this, so the total cost for card A after one year is:
[tex]\[ P \left(1 + \frac{0.125}{12}\right)^{12} + 48 \][/tex]
- For Credit Card B:
- The monthly interest rate is [tex]\(\frac{0.154}{12}\)[/tex].
- The final amount after one year (compounded monthly) for card B is:
[tex]\[ P \left(1 + \frac{0.154}{12}\right)^{12} \][/tex]
3. Setting the Equations Equal:
- We want the total costs for both cards to be equal, since they offer the same deal. Therefore, we equate the total costs:
[tex]\[ P \left(1 + \frac{0.125}{12}\right)^{12} + 48 = P \left(1 + \frac{0.154}{12}\right)^{12} \][/tex]
4. Conclusion:
- The correct equation that you can use to find the principal [tex]\( P \)[/tex] where both cards end up costing the same is:
[tex]\[ P \left(1 + \frac{0.125}{12}\right)^{12} + 48 = P \left(1 + \frac{0.154}{12}\right)^{12} \][/tex]
Therefore, the choice that matches this equation is choice B.
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