numerator and denominator increased by same amount

Let the positive fraction ${\displaystyle \frac{a}{b}}$ be altered by adding a positive number $\delta$ to both $a$ and $b$.  Then

$$\frac{a}{b}>\frac{a+\delta }{b+\delta }\mathit{\hspace{1em}}\text{if}\mathit{\hspace{1em}}a>b.$$

The asserted inequalities follow from the identity

$$\frac{a}{b}-\frac{a+\delta }{b+\delta }=\frac{(a-b)\delta }{b^2+b\delta }.$$

Accordingly, we have for example