melissa used her credit card to buy a $745 refrigerator. she kept the refrigerator for exactly nine years…

melissa used her credit card to buy a $745 refrigerator. she kept the refrigerator for exactly nine years, during which time it consumed an average of $0.19 of electricity every day. melissas credit card has an apr of 16.84%, compounded monthly. melissa paid off her refrigerator by making identical monthly payments for four years. if she made no other purchases with her card, what percentage of the lifetime cost of the refrigerator did the electricity make up? (assume that two of the years were leap years, and round all dollar values to the nearest cent.) a. 59.17% b. 18.48% c. 66.99% d. 37.77%

melissa used her credit card to buy a $745 refrigerator. she kept the refrigerator for exactly nine years, during which time it consumed an average of $0.19 of electricity every day. melissas credit card has an apr of 16.84%, compounded monthly. melissa paid off her refrigerator by making identical monthly payments for four years. if she made no other purchases with her card, what percentage of the lifetime cost of the refrigerator did the electricity make up? (assume that two of the years were leap years, and round all dollar values to the nearest cent.) a. 59.17% b. 18.48% c. 66.99% d. 37.77%

Answer

Explanation:

Step1: Calculate the total electricity cost

The refrigerator was kept for 9 years. There are 2 leap - years and 7 non - leap years. A non - leap year has 365 days and a leap year has 366 days. The total number of days $d=7\times365 + 2\times366=2555+732 = 3287$ days. The daily electricity cost is $0.19$. So the total electricity cost $E = 0.19\times3287=$624.53$.

Step2: Calculate the monthly interest rate

The APR is $r = 16.84%=0.1684$. The monthly interest rate $i=\frac{0.1684}{12}$.

Step3: Calculate the number of payments

She paid for $n = 4\times12 = 48$ months.

Step4: Use the present - value of an ordinary annuity formula $PV = PMT\times\frac{1-(1 + i)^{-n}}{i}$ to find the monthly payment $PMT$. Here, $PV = 745$.

$745=PMT\times\frac{1-(1+\frac{0.1684}{12})^{-48}}{\frac{0.1684}{12}}$. First, calculate $(1+\frac{0.1684}{12})^{-48}$. Let $x=\frac{0.1684}{12}\approx0.014033$. Then $(1 + x)^{-48}=(1.014033)^{-48}\approx0.5137$. $1-(1 + x)^{-48}=1 - 0.5137 = 0.4863$. $\frac{1-(1 + x)^{-n}}{i}=\frac{0.4863}{\frac{0.1684}{12}}\approx34.63$. $PMT=\frac{745}{34.63}\approx21.51$. The total amount paid on the credit card is $PMT\times n=21.51\times48=$1032.48$.

Step5: Calculate the lifetime cost of the refrigerator

The lifetime cost $C$ of the refrigerator is the sum of the total amount paid on the credit card and the total electricity cost, $C=1032.48 + 624.53=$1657.01$.

Step6: Calculate the percentage of the lifetime cost that is electricity

The percentage $p=\frac{624.53}{1657.01}\times100%\approx37.77%$.

Answer:

d. 37.77%