Relationship between Growth in Purchasing Power and Real Interest Rate equations
$Growth\ in\ Purchasing\ Power=1+r_r = \displaystyle\frac{1+r}{1+i}$

Also it is given that $real\ interest \ rate=r_r$

So :

$$real\ interest \ rate=(1+r_r)-1=\frac{1+r}{1+i}-1=\frac{1+r}{1+i}-\frac{1+i}{1+i}=\frac{r-i}{1+i}$$