Find expert answers and community support for all your questions on IDNLearn.com. Discover prompt and accurate responses from our experts, ensuring you get the information you need quickly.
Sagot :
To determine which equation correctly represents the principal amount [tex]\( P \)[/tex] at which both credit cards offer the same deal over the course of a year, we need to break down the details of each card's costs and how they accumulate interest.
### Credit Card A
- Annual Percentage Rate (APR): [tex]\( 14.3\% = 0.143 \)[/tex]
- Annual Fee: [tex]\( \$36 \)[/tex]
- Interest is compounded monthly, so we have 12 compounding periods per year.
The formula to calculate the future value ( [tex]\( FV \)[/tex] ) of an investment compounded monthly is:
[tex]\[ FV = P \cdot \left(1 + \frac{r}{n}\right)^{n \cdot t} \][/tex]
where:
- [tex]\( P \)[/tex] is the principal,
- [tex]\( r \)[/tex] is the annual interest rate,
- [tex]\( n \)[/tex] is the number of compounding periods per year,
- [tex]\( t \)[/tex] is the time in years.
For Credit Card A, over one year ( [tex]\( t = 1 \)[/tex] ), the compounded interest is:
[tex]\[ FV_A = P \cdot \left(1 + \frac{0.143}{12}\right)^{12} \][/tex]
Since there is an annual fee of [tex]\( \$36 \)[/tex], the total cost for Credit Card A over one year including the fee becomes:
[tex]\[ \text{Total Cost}_A = P \cdot \left(1 + \frac{0.143}{12}\right)^{12} - 36 \][/tex]
### Credit Card B
- Annual Percentage Rate (APR): [tex]\( 17.1\% = 0.171 \)[/tex]
- No annual fee.
- Interest is compounded monthly, with 12 compounding periods per year.
Similarly, for Credit Card B, the compounded value over one year is:
[tex]\[ FV_B = P \cdot \left(1 + \frac{0.171}{12}\right)^{12} \][/tex]
Since there are no additional fees for Credit Card B, the total cost over one year is:
[tex]\[ \text{Total Cost}_B = P \cdot \left(1 + \frac{0.171}{12}\right)^{12} \][/tex]
### Finding the Principal [tex]\( P \)[/tex]
To find the principal [tex]\( P \)[/tex] at which both credit cards would offer the same deal over the course of a year, we set the total costs equal to each other:
[tex]\[ \text{Total Cost}_A = \text{Total Cost}_B \][/tex]
Substituting the expressions derived for each card's total cost:
[tex]\[ P \cdot \left(1 + \frac{0.143}{12}\right)^{12} - 36 = P \cdot \left(1 + \frac{0.171}{12}\right)^{12} \][/tex]
Therefore, the correct equation that solves for the principal [tex]\( P \)[/tex] is:
[tex]\[ \boxed{C. \ P \cdot \left(1 + \frac{0.143}{12}\right)^{12} - 36 = P \cdot \left(1 + \frac{0.171}{12}\right)^{12}} \][/tex]
### Credit Card A
- Annual Percentage Rate (APR): [tex]\( 14.3\% = 0.143 \)[/tex]
- Annual Fee: [tex]\( \$36 \)[/tex]
- Interest is compounded monthly, so we have 12 compounding periods per year.
The formula to calculate the future value ( [tex]\( FV \)[/tex] ) of an investment compounded monthly is:
[tex]\[ FV = P \cdot \left(1 + \frac{r}{n}\right)^{n \cdot t} \][/tex]
where:
- [tex]\( P \)[/tex] is the principal,
- [tex]\( r \)[/tex] is the annual interest rate,
- [tex]\( n \)[/tex] is the number of compounding periods per year,
- [tex]\( t \)[/tex] is the time in years.
For Credit Card A, over one year ( [tex]\( t = 1 \)[/tex] ), the compounded interest is:
[tex]\[ FV_A = P \cdot \left(1 + \frac{0.143}{12}\right)^{12} \][/tex]
Since there is an annual fee of [tex]\( \$36 \)[/tex], the total cost for Credit Card A over one year including the fee becomes:
[tex]\[ \text{Total Cost}_A = P \cdot \left(1 + \frac{0.143}{12}\right)^{12} - 36 \][/tex]
### Credit Card B
- Annual Percentage Rate (APR): [tex]\( 17.1\% = 0.171 \)[/tex]
- No annual fee.
- Interest is compounded monthly, with 12 compounding periods per year.
Similarly, for Credit Card B, the compounded value over one year is:
[tex]\[ FV_B = P \cdot \left(1 + \frac{0.171}{12}\right)^{12} \][/tex]
Since there are no additional fees for Credit Card B, the total cost over one year is:
[tex]\[ \text{Total Cost}_B = P \cdot \left(1 + \frac{0.171}{12}\right)^{12} \][/tex]
### Finding the Principal [tex]\( P \)[/tex]
To find the principal [tex]\( P \)[/tex] at which both credit cards would offer the same deal over the course of a year, we set the total costs equal to each other:
[tex]\[ \text{Total Cost}_A = \text{Total Cost}_B \][/tex]
Substituting the expressions derived for each card's total cost:
[tex]\[ P \cdot \left(1 + \frac{0.143}{12}\right)^{12} - 36 = P \cdot \left(1 + \frac{0.171}{12}\right)^{12} \][/tex]
Therefore, the correct equation that solves for the principal [tex]\( P \)[/tex] is:
[tex]\[ \boxed{C. \ P \cdot \left(1 + \frac{0.143}{12}\right)^{12} - 36 = P \cdot \left(1 + \frac{0.171}{12}\right)^{12}} \][/tex]
Thank you for being part of this discussion. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Find clear answers at IDNLearn.com. Thanks for stopping by, and come back for more reliable solutions.