# 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 NumeratorAndDenominatorIncreasedBySameAmount 2013-03-22 19:36:29 2013-03-22 19:36:29 pahio (2872) pahio (2872) 9 pahio (2872) Result msc 26D07 msc 11A99 AdjacentFraction