Is there a general relation between $a/b$ and $(a+c)/(b+c)$ where $a,b,c > 0 $? Is there a general relation between $a/b$ and $(a+c)/(b+c)$ where $a,b> 0$ and $c\geq 0$ ? 
Is there a general proof for that relation ? 

Suppose $a,b,c \in \mathbb N$, $a < b$.
$$\color{red}{\dfrac ab} 
  = \dfrac{a(b+c)}{b(b+c)} 
  = \dfrac{ab+ac}{b(b+c)}
  \color{red}{<} \dfrac{ab+bc}{b(b+c)} 
  = \dfrac{b(a+c)}{b(b+c)} 
  = \color{red}{\dfrac{a+c}{b+c}}
  \color{red}{<} \dfrac{b+c}{b+c} = \color{red}1$$
Suppose $a,b,c \in \mathbb N$, $a > b$.
$$\color{red}{\dfrac ab} 
  = \dfrac{a(b+c)}{b(b+c)} 
  = \dfrac{ab+ac}{b(b+c)}
  \color{red}{>} \dfrac{ab+bc}{b(b+c)} 
  = \dfrac{b(a+c)}{b(b+c)} 
  = \color{red}{\dfrac{a+c}{b+c}}
  \color{red}{>} \dfrac{b+c}{b+c} = \color{red}1$$
