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 ?

Solution 1:

If $\underline{a\ge b}$ then $ac\ge bc$ hence $ab+ac=a(b+c)\ge ab+bc=b(a+c)$ so $$\frac ab\ge \frac{a+c}{b+c}$$

Solution 2:

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:

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