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 ?

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 ?