Answer:
Percentage of the total lifetime cost of the system that the original price made up = 59.07%
Step-by-step explanation:
The monthly payments for the sprinkler system can be calculated using the formula for calculating the present value of an ordinary annuity as follows:
PV = P * ((1 - (1 / (1 + r))^n) / r) …………………………………. (1)
Where;
PV = Present value or original price of the sprinkler = $1,874
P = Monthly payment = ?
r = Monthly interest rate = APR / 12 = 10.31% / 12 = 0.1031 / 12 = 0.00859166666666667
n = number of months = number of years of payment * 12 = 4 * 12 = 48
Substitute the values into equation (1) and solve P, we have:
$1,874 = P * ((1 - (1 / (1 + 0.00859166666666667))^48) / 0.00859166666666667)
$1,874 = P * 39.1976732321759
P = $1,874 / 39.1976732321759
P = $47.81
Therefore, we have:
Total payment for the sprinkler = Monthly payments * Number of months = P * n = $47.81 * 48 = $2,294.88
Total cost in water = Cost in water per week * Number of weeks in a year * Number of years that Olivia kept the sprinkler system = $2.11 * 52 * 8 = $877.76
Total lifetime cost of the system = Total payment for the sprinkler + Total cost in water = $2,294.88 + $877.76 = $3,172.64
Percentage of the total lifetime cost of the system that the original price made up = (Original price of the sprinkler / Total lifetime cost of the system) * 100 = ($1,874 / $3,172.64) * 100 = 59.07%