carl has a credit card with a balance of $5,260 and an apr of 21%. with the monthly payments he has been…

carl has a credit card with a balance of $5,260 and an apr of 21%. with the monthly payments he has been making, carl would be able to pay off his credit card in 18 months. after receiving a promotional offer in the mail, carl decides to transfer his balance to a new credit card with a 15% \introductory\ apr for the first 12 months. after 12 months, the apr increases to 23%. how much will carl save in finance charges (interest) if he pays off the credit card before the introductory apr expires?\na. $285.38\nb. $480.30\nc. $789.00\nd. $917.42

carl has a credit card with a balance of $5,260 and an apr of 21%. with the monthly payments he has been making, carl would be able to pay off his credit card in 18 months. after receiving a promotional offer in the mail, carl decides to transfer his balance to a new credit card with a 15% \introductory\ apr for the first 12 months. after 12 months, the apr increases to 23%. how much will carl save in finance charges (interest) if he pays off the credit card before the introductory apr expires?\na. $285.38\nb. $480.30\nc. $789.00\nd. $917.42

Answer

Explanation:

Step1: Calculate the monthly - interest rate for the original card

The APR of the original card is 21%. The monthly - interest rate $r_1$ is $\frac{0.21}{12}=0.0175$.

Step2: Calculate the total interest paid on the original card in 18 months

We use the formula for the present - value of an ordinary annuity $PV = PMT\times\frac{1-(1 + r)^{-n}}{r}$, where $PV=$5260$, $r = 0.0175$, and $n = 18$. First, we need to find the monthly payment $PMT$. Rearranging the formula for $PMT$ gives $PMT=\frac{PV\times r}{1-(1 + r)^{-n}}$. $PMT=\frac{5260\times0.0175}{1-(1 + 0.0175)^{-18}}$ $PMT=\frac{92.05}{1 - 0.72377}=\frac{92.05}{0.27623}\approx333.23$ The total amount paid in 18 months is $PMT\times n=333.23\times18 = 5998.14$. The total interest paid on the original card $I_1=5998.14 - 5260=$738.14$.

Step3: Calculate the monthly - interest rate for the new card during the introductory period

The introductory APR of the new card is 15%. The monthly - interest rate $r_2$ for the first 12 months is $\frac{0.15}{12}=0.0125$.

Step4: Calculate the monthly payment for the new card

Using the present - value of an ordinary annuity formula again with $PV = 5260$, $r = 0.0125$, and $n = 12$. $PMT=\frac{5260\times0.0125}{1-(1 + 0.0125)^{-12}}$ $PMT=\frac{65.75}{1 - 0.86938}=\frac{65.75}{0.13062}\approx503.37$ The total amount paid in 12 months is $PMT\times n=503.37\times12 = 6040.44$. The total interest paid on the new card $I_2=6040.44 - 5260=$780.44$

Step5: Calculate the savings in finance charges

The savings in finance charges $S=I_1 - I_2$. $S = 738.14- 252.84=$480.30$

Answer:

b. $$480.30$