A rational number is called "lucky" if it equals both $a+\frac{b}{c}$ and $a\times\frac{b}{c}$ for some positive integers $a,b,c$. How many lucky numbers are there between $5$ and $10$?
Here's what I have so far: $$a+\frac{b}{c}=a\times\frac{b}{c}$$ $$\frac{ac}{c}+\frac{b}{c}=\frac{ab}{c}$$ $$\frac{ac+b}{c}=\frac{ab}{c}$$ $$ac+b=ab$$ I don't know what to do after this. Any ideas?