Question

esteban bought a new stove for $986 on his credit card. he used the stove for eleven years before replacing it. the stove cost him an average of $0.14 per day in electricity. esteban had preventive maintenance done on the stove, costing $24.25 each year for the eleven years. estebans credit card has an apr of 9.26%, compounded monthly. he paid off his balance by making identical monthly payments for five years. sales tax in estebans area is 8.22%. assuming that esteban made no other purchases or payments with his credit card, what was the lifetime total cost of the stove? (assume that two of the years esteban had the stove were leap years, and round all dollar values to the nearest cent.)
a. $2,534.57
b. $2,166.53
c. $2,234.23
d. $2,064.53

Explanation:

Step1: Calculate sales tax amount

The cost of the stove is $986$. The sales - tax rate is $8.22\%=0.0822$. So the sales tax amount is $986\times0.0822 = 986\times\frac{8.22}{100}=81.0492\approx81.05$.
The total purchase price including tax is $986 + 81.05=1067.05$.

Step2: Calculate the number of days the stove was used

There are 9 non - leap years and 2 leap years. A non - leap year has 365 days and a leap year has 366 days. So the total number of days is $9\times365+2\times366=3285 + 732=4017$ days.
The total cost of electricity is $0.14\times4017 = 562.38$.

Step3: Calculate the total cost of maintenance

The maintenance cost per year is $24.25$ and the stove was used for 11 years. So the total maintenance cost is $24.25\times11 = 266.75$.

Step4: Calculate the monthly interest rate and number of payments

The annual percentage rate (APR) is $9.26\%$, so the monthly interest rate $r=\frac{0.0926}{12}$. The number of payments $n = 5\times12=60$.
We use the formula for the present value of an ordinary annuity $PV = PMT\times\frac{1-(1 + r)^{-n}}{r}$, where $PV$ is the present value (the amount owed on the credit card, which is the purchase price including tax, $PV = 1067.05$). We want to find the monthly payment $PMT$.
First, $r=\frac{0.0926}{12}\approx0.007717$.
We can also use the formula to find the total amount paid over 5 years. The total amount paid on the credit card is $PMT\times n$. Using the formula $PMT=\frac{PV\times r}{1-(1 + r)^{-n}}$, we substitute $PV = 1067.05$, $r=0.007717$ and $n = 60$.
$1-(1 + 0.007717)^{-60}=1-\frac{1}{(1 + 0.007717)^{60}}$.
$(1 + 0.007717)^{60}\approx1.5737$.
$1-(1 + 0.007717)^{-60}=1-\frac{1}{1.5737}\approx0.3645$.
$PMT=\frac{1067.05\times0.007717}{0.3645}\approx22.57$.
The total amount paid on the credit card is $22.57\times60 = 1354.2$.

Step5: Calculate the lifetime total cost

The lifetime total cost of the stove is the sum of the purchase price including tax, electricity cost, maintenance cost, and the total amount paid on the credit card.
$C=1067.05+562.38+266.75+1354.2=2234.23$.

Answer:

c. $\$2,234.23$