Question

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. melissa’s 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%

Explanation:

Step1: Calculate the total cost of electricity

• Calculate the number of days in 9 years: \(7\times365 + 2\times366=3287\) days
• Cost of electricity: \(0.19\times3287 = 624.53\) dollars

Step2: Calculate the monthly payment for the credit - card debt

• \(n = 4\times12=48\) months, \(r=\frac{0.1684}{12}\), \(P = 745\)
• Use the formula for the monthly payment of a loan \(M=\frac{P\times r\times(1 + r)^{n}}{(1 + r)^{n}-1}\)
• \(M=\frac{745\times\frac{0.1684}{12}\times(1+\frac{0.1684}{12})^{48}}{(1+\frac{0.1684}{12})^{48}-1}\)
• First, calculate \((1+\frac{0.1684}{12})^{48}\approx(1 + 0.014033)^{48}\approx2.037\)
• \(M=\frac{745\times0.014033\times2.037}{2.037 - 1}\approx\frac{745\times0.014033\times2.037}{1.037}\approx20.97\)
• Total payment for the credit - card debt: \(20.97\times48 = 1006.56\) dollars

Step3: Calculate the lifetime cost of the refrigerator

• Lifetime cost \(C=1006.56+624.53 = 1631.09\) dollars

Step4: Calculate the percentage of the electricity cost

• Percentage \(=\frac{624.53}{1631.09}\times100\%\approx37.77\%\)

Answer:

d. \(37.77\%\)