Respuesta :
Answer:
The yearly interest rate is 5.20%.
Step-by-step explanation:
This is a compound interest problem
The compound interest formula is given by:
[tex]A = P(1 + \frac{r}{n})^{nt}[/tex]
In which A is the amount of money, P is the principal(the initial sum of money), r is the interest rate(as a decimal value), n is the number of times that interest is compounded per unit t and t is the time the money is invested or borrowed for.
In this problem, we have that:
The loan outstanding is the value of the loan that has not been repaid.
Here, it is [tex]$25,044.84[/tex].
To find the interest rate, we first have to find how much money the borrower will have to pay, that will be the value of A in the compound interest formula.
The total he will have to play is [tex]$25,044.84[/tex] plus the $3,568 he has already paid in each of the previous 2 years = 24 months. So:
[tex]A = 25,044.84 + 24*3,568 = 110,676.84[/tex].
P is the value of loan, so [tex]P = 100,000[/tex]
r is the interest rate, the value we have to find.
We have to find the annual interest rate, so [tex]n = 1[/tex].
We found the total amount in 2 years, so [tex]t = 2[/tex].
Solving
[tex]A = P(1 + \frac{r}{n})^{nt}[/tex]
[tex]110,676.84 = 100,000(1 + r)^{2}[/tex]
[tex](1 + r)^{2} = 1.1067684[/tex]
To find r, i will take the square root of both sides of the equation. So
[tex]\sqrt{(1 + r)^{2}} = \sqrt{1.1067684}[/tex]
[tex]1 + r = 1.0520[/tex]
[tex]r = 0.0520[/tex]
The yearly interest rate is 5.20%.
