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 ?

To isolate $c$ we can proceed as follow
$$\frac{a}{b} - \frac{a}{c} = 1\iff \frac{a}{c}=\frac{a}{b}-1=\frac{a-b}{b}\iff\frac{c}{a}=\frac{b}{a-b}\iff c=\frac{ab}{a-b}$$
