$\frac{1}{10}+\frac{4}{20}+\frac{9}{30}+\frac{16}{40}+\frac{25}{50}+\frac{36}{60}+\frac{49}{70}+\frac{64}{80}+\frac{81}{90}$
$=\frac{3}{10}+\frac{9}{30}+\frac{16}{40}+\frac{25}{50}+\frac{36}{60}+\frac{49}{70}+\frac{64}{80}+\frac{81}{90}$
$=\frac{3}{5}+\frac{16}{40}+\frac{25}{50}+\frac{36}{60}+\frac{49}{70}+\frac{64}{80}+\frac{81}{90}$
$=1+\frac{25}{50}+\frac{36}{60}+\frac{49}{70}+\frac{64}{80}+\frac{81}{90}$
$=\frac{3}{2}+\frac{36}{60}+\frac{49}{70}+\frac{64}{80}+\frac{81}{90}$
$=\frac{21}{10}+\frac{49}{70}+\frac{64}{80}+\frac{81}{90}$
$=\frac{14}{5}+\frac{64}{80}+\frac{81}{90}$
$=\frac{18}{5}+\frac{81}{90}$
$=\frac{9}{2}$
