Respuesta :
Using simple interest, it is found that $13,872 was loaned at 9%.
Simple Interest
The amount of money after t years in simple interest is modeled by:
[tex]A(t) = P(1 + rt)[/tex]
In which:
- P is the initial amount.
- r is the interest rate, as a decimal.
The interest earned is:
[tex]I(t) = Prt[/tex]
A bank loaned out $14000, in two parts, hence:
[tex]P_2 = 14000 - P_1[/tex]
Part of it at the rate of 9% per year and the rest at 17% per year, hence:
[tex]r_1 = 0.09, r_2 = 0.17[/tex]
The interest received in one year totaled $2000, hence:
[tex]I_2 = 2000 - I_1[/tex]
Then:
[tex]I_1 = 0.09P_1[/tex]
[tex]I_2 = 0.17I_2[/tex]
[tex]14000 - P_1 = 0.17(2000 - I_1)[/tex]
Isolating [tex]I_1[/tex] as a function of [tex]P_1[/tex], then we can replace on the equation for the first interest.
[tex]14000 - P_1 = 340 - 0.17I_1[/tex]
[tex]0.17I_1 = -13660 + P_1[/tex]
[tex]I_1 = \frac{P_1 - 13660}{0.17}[/tex]
Then:
[tex]I_1 = 0.09P_1[/tex]
[tex]\frac{P_1 - 13660}{0.17} = 0.09P_1[/tex]
[tex]P_1 - 13660 = 0.0153P_1[/tex]
[tex]0.9847P_1 = 13660[/tex]
[tex]P_1 = \frac{13660}{0.9847}[/tex]
[tex]P_1 = 13872[/tex]
$13,872 was loaned at 9%.
To learn more about simple interest, you can take a look at https://brainly.com/question/25296782
