My textbook gives these two equations including the real interest rate and says that the second equation (b) can be arrived at from the first (a), but doesn't explain how.

rr = Real Interest Rate, r = Nominal Interest Rate, i = Inflation

Equation (a): $$Growth\ in\ Purchasing\ Power=1+r_r = \displaystyle\frac{1+r}{1+i}$$

Equation (b):

$$Real\ Interest\ Rate = r_r = \displaystyle\frac{r-i}{1+i}$$

I'm feeling slightly obtuse but I've been away from math quite a while, and can't figure out how to rearrange the first formula into the second one.

What are the exact equation steps necessary to get from: $ r_r = \displaystyle\frac{1+r}{1+i}-1$$r_r = \displaystyle\frac{r-i}{1+i}$ ?


$Growth\ in\ Purchasing\ Power=1+r_r = \displaystyle\frac{1+r}{1+i}$

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

So :

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


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