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 ? 

Good observation, these inequalities are quite useful. But you need a little bit more:


*

*if $a\ge b$, then $\dfrac ab\ge\dfrac{a+c}{b+c}$

*if $a\le b$, then $\dfrac ab\le\dfrac{a+c}{b+c}$


You can prove it by multiplying by the common denominator:


*

*$a(b+c)\ge b(a+c)\Longleftrightarrow ac\ge bc$

*$a(b+c)\le b(a+c)\Longleftrightarrow ac\le bc$

