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}.

Title	numerator and denominator increased by same amount
Canonical name	NumeratorAndDenominatorIncreasedBySameAmount
Date of creation	2013-03-22 19:36:29
Last modified on	2013-03-22 19:36:29
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	9
Author	pahio (2872)
Entry type	Result
Classification	msc 26D07
Classification	msc 11A99
Related topic	AdjacentFraction