Question

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.)
a. 81.66%
b. 59.07%
c. 38.25%
d. 72.33%

Answer 1

Answer:

C. 38.25%

Step-by-step explanation:

We know that,

$\text{PV of annuity}=P\left[\dfrac{1-(1+r)^{-n}}{r}\right]$

here,

PV = Present value of annuity = $1,874

P = Payment per period (monthly)

r = Rate of interest per period = 10.31% annually = $\dfrac{0.1031}{12}$ monthly

n = Number of periods = 4 years = 48 months

Putting the values,

$\Rightarrow 1874=P\left[\dfrac{1-(1+\frac{0.1031}{12})^{-48}}{\frac{0.1031}{12}}\right]$

$\Rightarrow P=\dfrac{1874}{\left[\frac{1-\left(1+\frac{0.1031}{12}\right)^{-48}}{\frac{0.1031}{12}}\right]}$

$\Rightarrow P=\ 47.81$

So the monthly payment is $47.81, then the total payment will be,

$=47.81\times 48=\ 2294.88$

Over the eight years that Olivia kept the sprinkler system, it used an average of $2.11 in water per week.

The total amount will be,

$=2.11\times 52\times 8\\\\=\ 877.76$

Then the percentage of the total lifetime cost of the system did the original price make up is,

$=\dfrac{877.76}{2294.88}\times 100\%$

$=38.25\%$