Question

Credit card A has an APR of 26.2% and an annual fee of $30, while credit card B has anAPR of 27.1% and no annual fee. ALl else being equal,at about what balance will the cards offer the same deal over the course of a year?(Assume all interest is compounded monthly.)
A. $261.78
B. $2617.85
C.$26,178.46
D.$26.18

Answer 1

Taking the quiz now. 
The answer is $2617.85.

Answer 2

Let the required balance be $x$.

Thus, for Card A, which has an APR of 26.2% and an annual fee of $30, the amount after a year when compounded monthly will be:

$Amount_A=30+(1+\frac{0.262}{12})^{12}x$........(Equation 1)

Likewise, for Card B, which has an APR of 27.1% and no annual fee, the amount after a year when compounded monthly will be:

$Amount_B=0+(1+\frac{0.271}{12})^{12}x=(1+\frac{0.271}{12})^{12}x$....(Equation 2)

Therefore, all else being equal, the balance, $x$, at which the cards offer the same deal over the course of a year can be found by equating the equations 1 and 2 and solving for x.

Thus we have:

$30+(1+\frac{0.262}{12})^{12}x=(1+\frac{0.271}{12})^{12}x$

$30+(1.0218)^{12}x=(1.0226)^{12}x$

Simplification gives us:

$(1.0226)^{12}x-(1.0218)^{12}x=30$

$0.0122x=30$

$\therefore x\approx2459.02$ dollars

This is the closest to the second option. Thus, option B is the correct option.

Important Note: If we do not round off the intermediate steps and calculate it directly using a calculator then we will get the exact answer of option B which is: $2617.85.