$\begin{align}
& \frac{108}{120}+\frac{104}{130}+\frac{98}{140}+\frac{90}{150}+\frac{80}{160}+\frac{68}{170}+\frac{54}{180}+\frac{38}{190}+\frac{1}{10} \\
& =\frac{9}{10}+\frac{4}{5}+\frac{7}{10}+\frac{3}{5}+\frac{1}{2}+\frac{2}{5}+\frac{3}{10}+\frac{1}{5}+\frac{1}{10} \\
& =\left( \frac{9}{10}+\frac{1}{10} \right)+\left( \frac{4}{5}+\frac{1}{5} \right)+\left( \frac{7}{10}+\frac{3}{10} \right)+\left( \frac{3}{5}+\frac{2}{5} \right)+\frac{1}{2} \\
& =1+1+1+1+\frac{1}{2}=4+\frac{1}{2}=\frac{9}{2} \\
\end{align}$