Credit card A has an APR of [tex]$27.2 \%$[/tex] and an annual fee of [tex]$96[/tex], while credit card B has an APR of [tex]$30.3 \%$[/tex] and no annual fee. All else being equal, which of these equations can be used to solve for the principal [tex]$P$[/tex] for which the cards offer the same deal over the course of a year? (Assume all interest is compounded monthly.)
A. [tex]$P\left(1+\frac{0.272}{12}\right)^{12}+\frac{\$ 96}{12}=P\left(1+\frac{0.303}{12}\right)^{12}$[/tex]
B. [tex]$P\left(1+\frac{0.272}{12}\right)^{12}-\frac{\$ 96}{12}=P\left(1+\frac{0.303}{12}\right)^{12}$[/tex]
C. [tex]$P\left(1+\frac{0.272}{12}\right)^{12}+\$ 96=P\left(1+\frac{0.303}{12}\right)^{12}$[/tex]
D. [tex]$P\left(1+\frac{0.272}{12}\right)^{12}-\$ 96=P\left(1+\frac{0.303}{12}\right)^{12}$[/tex]
Answer (2)
To find the principal amount for which two credit cards offer the same deal, we equate the total costs, including interest and any fees. The equation to use is P ( 1 + 12 0.272 ) 12 + 96 = P ( 1 + 12 0.303 ) 12 . The correct answer is option C.
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The total cost of credit card A is calculated as P ( 1 + 12 0.272 ) 12 + 96 .
The total cost of credit card B is calculated as P ( 1 + 12 0.303 ) 12 .
Set the total cost of credit card A equal to the total cost of credit card B: P ( 1 + 12 0.272 ) 12 + 96 = P ( 1 + 12 0.303 ) 12 .
The equation that can be used to solve for the principal P is P ( 1 + 12 0.272 ) 12 + $96 = P ( 1 + 12 0.303 ) 12 .
Explanation
Understanding the Problem We are given two credit card options and asked to find the equation that can be used to solve for the principal P for which the cards offer the same deal over the course of a year. Credit card A has an APR of 27.2% and an annual fee of $96 , while credit card B has an APR of 30.3% and no annual fee. We assume that the interest is compounded monthly.
Calculating Total Cost for Card A The total cost of credit card A after one year includes the principal, interest, and annual fee. The formula for the total cost with monthly compounding is given by:
Total cost of A = P ( 1 + 12 A P R A ) 12 + annual fee A
where A P R A = 0.272 and annual fee A = $96 .
Calculating Total Cost for Card B The total cost of credit card B after one year includes the principal and interest. The formula for the total cost with monthly compounding is given by:
Total cost of B = P ( 1 + 12 A P R B ) 12
where A P R B = 0.303 and there is no annual fee.
Equating Total Costs We want to find the principal P for which the total cost of credit card A is equal to the total cost of credit card B. Therefore, we set the total cost of credit card A equal to the total cost of credit card B:
P ( 1 + 12 0.272 ) 12 + 96 = P ( 1 + 12 0.303 ) 12
Identifying the Correct Equation Comparing the derived equation with the given options, we see that option C matches our equation:
P ( 1 + 12 0.272 ) 12 + $96 = P ( 1 + 12 0.303 ) 12
Examples
This type of problem is useful when deciding between different credit card offers. By setting up an equation that equates the total cost of two different credit cards, you can determine the principal amount for which the two cards would cost the same over a year. This helps you make an informed decision based on your spending habits and expected balance.
