<s>[INST] Increasing numerator o decreasing denominator? Which fraction is bigger? How can i demonstrate it?
$$\frac{a+1}{b}$$ or $$\frac{a}{b-1}$$
 [/INST] If $1<b<a+1$ then
\begin{eqnarray*}
ab+b-a-1 < ab \\
(a+1)(b-1) < ab \\
\frac{a+1}{b} < \frac{a}{b-1}. 
\end{eqnarray*}
Similarly if $b>a+1>0$
\begin{eqnarray*}
\frac{a+1}{b} > \frac{a}{b-1}. 
\end{eqnarray*}
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