fraction simplication, resolution with one variable

I'm having some problems to understand a simplification and I know the correct answer but I'd like to understand why my method is wrong or what rule I'm breaking.

I have an equation like the following:

$$Fr \cdot \left( \frac1{Sr} + \frac1{Sp} \right) = \frac{Ft}{Sp}$$

Where the variable I'm looking for is $Fr$. My tendency was to do:

$$Fr = \frac{Ft}{Sp} \cdot ( Sr + Sp )$$

But it seems to be wrong, I should have do:

$$Ft \cdot \left( \frac{Sp + Sr}{Sr \cdot Sp} \right) = \frac{Ft}{Sp}$$

and then:

$$Ft = \frac{Sr \cdot Sp \cdot Ft}{Sp + Sr}$$

Why I can not simply pass & inverse the fraction to the other side? Thanks so much!!

Answer

Because

$$\frac1a + \frac1b \neq \frac1{a+b}$$

This is already the case for $a=b=1$, where one the left side you get $2$ and on the right side you get $\frac12$.