91Ó°ÊÓ

Given, a business purchased for $650,000 in 2005 is sold in 2008 for $850,000. The annual rate of return for this investment is to be determined.

According to the compound interest formula, the amount A after t years for the principal P with an annual rate of interest r compounded n times per year is $A=P{\left(1+\frac{r}{n}\right)}^{nt}$

Here the amount invested is $650,000 i.e. $P=650000$

The business is purchased in 2005 and sold in 2008 so the time for the investment is 3 years i.e. $t=3$

Since the annual rate of return is to be determined, considering the compounding as yearly i.e. $n=1$

The business is finally sold for $850,000 i.e.$A=850000$

Plugging the values:

$$\begin{gathered} A=P{\left(1+\frac{r}{n}\right)}^{nt} \\ 850000=650000{\left(1+\frac{r}{1}\right)}^{\left(1\right)\left(3\right)} \\ {\left(1+r\right)}^3=\frac{850000}{650000} \\ 1+r=\sqrt[3]{\frac{850000}{650000}} \\ r=\sqrt[3]{\frac{850000}{650000}}-1 \\ râ‰\dots 0.09354 \end{gathered}$$

Multiplying the decimal by 100 to get the annual rate of interest:$0.09354=9.354\%$

Hence, the annual rate of return for the investment is 9.354%