Incorrect in solving $\frac{a}{b} - \frac{a}{c} = 1$ for $c$ I have this:
$$\frac{a}{b} - \frac{a}{c} = 1$$ Solve for $c$. Then,
$$\frac{a}{b} - \frac{a}{c} = 1 \cdot bc$$
$$ = ac - ab = bc$$
$$ = a(c - b) = bc$$
$$ c = \frac{bc}{a} + b$$     This is my final result.
But the correct result is:
$$c=\frac{ab}{a-b}$$
What I development wrong in this equation ?

Your algebra is correct but $$c = \frac{bc}{a} + b$$ is solving $c$ in terms of $a$, $b$, and $c$ which is sort of cyclic. We want to see $c$ in terms of only $a$ and $b$.
You have to isolate $c$ by keeping all terms involving $c$ on the left side and other terms on the right side. 
Note that $$ ac - ab = bc\implies c(a-b)=ab \implies c= \frac {ab}{a-b}$$
