The SimplestRational( a, b ) command computes the "simplest" rational number $\frac{p}{q}$ such that $a\le \frac{p}{q}$ and $\frac{p}{q}\le b$.  It is "simplest" in the sense that $p$ and $q$ are as small as possible.  (Note that an integer is considered to be "simpler" than a non-integral rational.)

If Maple is able to determine that $b\le a$, then an exception is raised.

$\mathrm{with}\left(\mathrm{NumberTheory}\right)\:$

$\mathrm{SimplestRational}\left(\frac{1}{2}\,\frac{3}{4}\right)$

$\frac{1}{2}$

$\mathrm{SimplestRational}\left(\frac{1}{2}\,\frac{4}{3}\right)$

$1$

$\mathrm{SimplestRational}\left(\mathrm{\pi }\,\mathrm{\pi }+{\mathrm{\pi }}^{-10}\right)$

$\frac{355}{113}$

$\mathrm{SimplestRational}\left(\frac{9}{10}\,\frac{15}{2}\right)$

$1$

$\mathrm{SimplestRational}\left(\mathrm{sqrt}\left(1000\right)\,\mathrm{sqrt}\left(1001\right)\right)$

$\frac{253}{8}$

$\mathrm{SimplestRational}\left(\mathrm{\pi }\,\mathrm{\gamma }\right)$

Error, (in NumberTheory:-SimplestRational) first argument must be strictly less than the second

The NumberTheory[SimplestRational] command was introduced in Maple 2017.

For more information on Maple 2017 changes, see Updates in Maple 2017.