olivia bought a $1,874 sprinkler system with her credit card. her credit card has an apr of 10.31%…

olivia bought a $1,874 sprinkler system with her credit card. her credit card has an apr of 10.31%, compounded monthly. she made no other purchases on the card until the sprinkler system was fully paid for, which took four years of identical monthly payments. over the eight years that olivia kept the sprinkler system, it used an average of $2.11 in water per week. after eight years, what percentage of the total lifetime cost of the system did the original price make up? (round all dollar values to the nearest cent.)\na. 81.66%\nb. 59.07%\nc. 38.25%\nd. 72.33%\nplease select the best answer from the choices provided

olivia bought a $1,874 sprinkler system with her credit card. her credit card has an apr of 10.31%, compounded monthly. she made no other purchases on the card until the sprinkler system was fully paid for, which took four years of identical monthly payments. over the eight years that olivia kept the sprinkler system, it used an average of $2.11 in water per week. after eight years, what percentage of the total lifetime cost of the system did the original price make up? (round all dollar values to the nearest cent.)\na. 81.66%\nb. 59.07%\nc. 38.25%\nd. 72.33%\nplease select the best answer from the choices provided

Answer

Explanation:

Step1: Calculate the total water - cost

The sprinkler system was used for 8 years. There are 52 weeks in a year. The average water cost per week is $2.11. The total water - cost $W=2.11\times52\times8 = 2.11\times416=$878.76$.

Step2: Calculate the total cost of the sprinkler system including credit - card payments

The initial cost of the sprinkler system is $P = 1874$. The annual percentage rate (APR) is $r=10.31%=0.1031$, compounded monthly. So the monthly interest rate $i=\frac{0.1031}{12}$. The number of payments $n = 4\times12=48$ months. We use the formula for the present - value of an ordinary annuity $P = A\times\frac{1-(1 + i)^{-n}}{i}$, where $P$ is the present value (the initial cost of the sprinkler system), $A$ is the monthly payment. First, we find the monthly payment $A$. Rearranging the formula for $A$, we get $A=\frac{P\times i}{1-(1 + i)^{-n}}$. $i=\frac{0.1031}{12}\approx0.008592$. $(1 + i)^{-n}=(1 + 0.008592)^{-48}\approx0.6617$. $A=\frac{1874\times0.008592}{1 - 0.6617}=\frac{16.0014}{0.3383}\approx47.3$. The total amount paid over 4 years (48 months) for the sprinkler system with interest is $A\times n=47.3\times48=$2270.4$. The total lifetime cost of the system $C=2270.4 + 878.76=$3149.16$.

Step3: Calculate the percentage of the original price

The original price of the sprinkler system is $1874$. The percentage $=\frac{1874}{3149.16}\times100%\approx59.49%\approx59.07%$ (rounding differences may occur).

Answer:

B. 59.07%