Respuesta :
Let the required balance be [tex] x [/tex].
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:
[tex] Amount_A=30+(1+\frac{0.262}{12})^{12}x [/tex]........(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:
[tex] Amount_B=0+(1+\frac{0.271}{12})^{12}x=(1+\frac{0.271}{12})^{12}x [/tex]....(Equation 2)
Therefore, all else being equal, the balance, [tex] x [/tex], 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:
[tex] 30+(1+\frac{0.262}{12})^{12}x=(1+\frac{0.271}{12})^{12}x [/tex]
[tex] 30+(1.0218)^{12}x=(1.0226)^{12}x [/tex]
Simplification gives us:
[tex] (1.0226)^{12}x-(1.0218)^{12}x=30 [/tex]
[tex] 0.0122x=30 [/tex]
[tex] \therefore x\approx2459.02 [/tex] 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.
