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}\\quad\\mbox{if}\\quad a\\;<\\;b,

\\frac{a}{b}\\;>\\;\\frac{a\\!+\\!\\delta}{b\\!+\\!\\delta}\\quad\\mbox{if}\\quad 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

\\frac{2}{3}\\;<\\;\\frac{3}{4},\\quad\\frac{200}{99}\\;>\\;\\frac{201}{100}.