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

Percentage of the total lifetime cost of the system that the original price made up = 59.07%

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%

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