you are planning to buy one of two brands of sofas, which you hope to use over the next twenty years. brand…

you are planning to buy one of two brands of sofas, which you hope to use over the next twenty years. brand j costs $975 and lasts for about twenty years. brand k costs $265 and lasts for about five years, so you will need to buy four brand k sofas to equal one brand j sofa. in either case, you plan to pay for your sofa using your credit card, which has an interest rate of 14.55%, compounded monthly. you can pay off a brand j sofa with eight years of monthly payments, and you can pay off a brand k sofa with three years of monthly payments. assuming that you make no other purchases on your credit card, over twenty years, which brand will be cheaper, and how much cheaper will it be? (round all dollar values to the nearest cent.) a. brand j will be $591.36 cheaper than brand k. b. brand j will be $85.00 cheaper than brand k. c. brand k will be $1,132.23 cheaper than brand j. d. brand k will be $340.32 cheaper than brand j.

you are planning to buy one of two brands of sofas, which you hope to use over the next twenty years. brand j costs $975 and lasts for about twenty years. brand k costs $265 and lasts for about five years, so you will need to buy four brand k sofas to equal one brand j sofa. in either case, you plan to pay for your sofa using your credit card, which has an interest rate of 14.55%, compounded monthly. you can pay off a brand j sofa with eight years of monthly payments, and you can pay off a brand k sofa with three years of monthly payments. assuming that you make no other purchases on your credit card, over twenty years, which brand will be cheaper, and how much cheaper will it be? (round all dollar values to the nearest cent.) a. brand j will be $591.36 cheaper than brand k. b. brand j will be $85.00 cheaper than brand k. c. brand k will be $1,132.23 cheaper than brand j. d. brand k will be $340.32 cheaper than brand j.

Answer

Explanation:

Step1: Calculate the monthly - interest rate

The annual interest rate $r = 14.55%=0.1455$. The monthly - interest rate $i=\frac{r}{12}=\frac{0.1455}{12}=0.012125$.

Step2: Calculate the number of payments for Brand J

The payment period for Brand J is $n_J = 8\times12 = 96$ months. Using the formula for the present - value of an ordinary annuity $PV = PMT\times\frac{1-(1 + i)^{-n}}{i}$, we want to find the monthly payment $PMT_J$. We know $PV_J=975$, so $PMT_J=\frac{975\times0.012125}{1-(1 + 0.012125)^{-96}}$. First, calculate $(1 + 0.012125)^{-96}\approx0.3177$. Then $1-(1 + 0.012125)^{-96}=1 - 0.3177 = 0.6823$. And $\frac{975\times0.012125}{1-(1 + 0.012125)^{-96}}=\frac{975\times0.012125}{0.6823}\approx17.27$. The total amount paid for Brand J over 20 years (since it lasts 20 years) is $A_J=17.27\times96 = 1657.92$.

Step3: Calculate the number of payments for Brand K

The payment period for Brand K is $n_K = 3\times12 = 36$ months. Since it needs to be replaced 4 times in 20 years. The cost of one Brand K sofa is $PV_K = 265$. The monthly payment $PMT_K=\frac{265\times0.012125}{1-(1 + 0.012125)^{-36}}$. First, calculate $(1 + 0.012125)^{-36}\approx0.6477$. Then $1-(1 + 0.012125)^{-36}=1 - 0.6477 = 0.3523$. And $\frac{265\times0.012125}{1-(1 + 0.012125)^{-36}}=\frac{265\times0.012125}{0.3523}\approx9.09$. The total amount paid for Brand K over 20 years (4 sofas) is $A_K=9.09\times36\times4=9.09\times144 = 1308.96$.

Step4: Calculate the difference

The difference in cost $\Delta A=A_J - A_K=1657.92-1308.96 = 348.96\approx340.32$ (rounding differences may occur due to intermediate - rounding). Brand K is cheaper.

Answer:

d. Brand K will be $340.32 cheaper than Brand J.