Proving that $\frac{a}{b}>\frac{a-1}{b-1},\; a<b$ where $a$ and $b$ are positive constants

Question

I tried $\frac{a}{b}>\frac{a-1}{b-1}$

$a\left( b-1 \right)>b\left( a-1 \right)$

then this gives $a<b$ which is already given. Any ideas?

Answer 1

$$\dfrac ab-\dfrac{a-1}{b-1}=\dfrac{b-a}{b(b-1)}$$

If $b>0,b>a;$ $$\dfrac ab-\dfrac{a-1}{b-1}\text{ will be }>\text{ or }<0$$

according as $b-1>$ or $<0$