Answer:
[tex]A=500(1.0188)^{12t}[/tex] , APR ≅ 22.56% ⇒ 2nd answer
Step-by-step explanation:
The formula of the compounded interest is [tex]A=P(1+\frac{r}{n})^{nt}[/tex] , where
- A is the future value of the investment/loan, including interest
- P is the principal investment amount
- r is the annual interest rate (decimal)
- n is the number of times that interest is compounded per unit t
- t is the time the money is invested or borrowed for
∵ The equation of one-year loan of $500 with an interest rate
of 25% compounded only once is [tex]A=500(1.25)^{t}[/tex]
∴ t = 1
∴ A = 500(1.25) = 625
∵ A loan that would cost the same amount overall, if it is
compounded monthly, instead of only once
∴ n = 12 ⇒ compounded monthly
∵ t = 1 and n = 12
∵ P = 500
- Use the formula of the compounded interest above
∴ [tex]A=500(1+\frac{r}{12})^{12}[/tex]
- Equate A by 625 (the value of money of the 1st equation)
∵ [tex]500(1+\frac{r}{12})^{12}=625[/tex]
- Divide both sides by 500
∴ [tex](1+\frac{r}{12})^{12}=1.25[/tex]
- Reverse the exponent and take it the the other side
∴ [tex]1+\frac{r}{12}=(1.25)^{\frac{1}{12}}[/tex]
- Subtract 1 from both sides
∴ [tex]\frac{r}{12}=(1.25)^{\frac{1}{12}}-1[/tex]
∴ [tex]\frac{r}{12}=0.01876926512[/tex]
- Multiply both sides by 12
∴ r ≅ 0.2256
- Multiply it by 100%
∴ r = 22.56%
The annual percentage rate (APR) is 22.56%
Substitute the value of r in the equation above
∴ [tex]A=500(1+\frac{0.2256}{12})^{12t}[/tex]
∵ [tex](1+\frac{0.2256}{12})=1.0188[/tex]
∴ [tex]A=500(1.0188)^{12t}[/tex]
He should use the equation [tex]A=500(1.0188)^{12t}[/tex]
