# Relationship between Growth in Purchasing Power and Real Interest Rate equations

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: $$r_r = \displaystyle\frac{1+r}{1+i}-1$$$$r_r = \displaystyle\frac{r-i}{1+i}$$ ?

## 1 Answer

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

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

So :

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