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]
