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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%.

Sagot :

By applying formula of Present value of annuity we got that after eight years, percentage of the total lifetime cost of the system did the original price make up is 38.25%

What is Present value of annuity?

The current value of future payments from an annuity, given a specified rate of return, or discount rate is known as Present value of annuity.

We know that formula of Present value of annuity is

[tex]\text{PV}=P\left[\frac{1-(1+r)^{-n}}{r}\right][/tex]

Where

P= Payment per period (monthly)

r= Rate of interest per period =10.31 % annually  =0.1031/12 monthly

n= Number of periods =4 years =48 months

So  Payment per period can be calculated as :

[tex]$\begin{aligned}&\Rightarrow 1874=P\left[\frac{1-\left(1+\frac{0.1031}{12}\right)^{-48}}{\frac{0.1031}{12}}\right] \\\\&\Rightarrow P=\frac{1874}{\left[\frac{\left.1-\left(1+\frac{0.9331}{12}\right)^{-4 .}\right]}{\frac{0.121}{12}}\right]} \\\\&\Rightarrow P=\$ 47.81\end{aligned}$[/tex]

Total payment =47.81[tex]\times[/tex]48=2294.88

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

So total amount  [tex]$=2.11 \times 52 \times 8$=\$ 877.76$[/tex]

The percentage of the total lifetime cost of the system did the original price make up can be calculated as

[tex]$\begin{aligned}&=\frac{877.76}{2294.88} \times 100 \% \\\\&=38.25 \%\end{aligned}$[/tex]

By applying formula of Present value of annuity we got that after eight years, percentage of the total lifetime cost of the system did the original price make up is 38.25%

To learn more about  Present value of annuity visit:

https://brainly.com/question/25792915

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