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\(\frac{1}{2}+\frac{1}{2}=\frac{1+1}{2}=\frac{2}{2}=1\)
\(\frac{3}{4}+\frac{1}{4}=\frac{3+1}{4}=\frac{4}{4}=1\)
\(\frac{3}{9}+\frac{6}{9}=\frac{3+6}{9}=\frac{9}{9}=1\)

\(\frac{1}{2}+\frac{1}{6}+\frac{1}{9}=\frac{9}{18}+\frac{3}{18}+\frac{2}{18}=\frac{14}{18}=\frac{7}{9}\)

\(\frac{1}{2}:\frac{3}{2}:\frac{5}{4}:\frac{6}{5}:\frac{7}{6}:\frac{8}{7}\)
\(=\frac{1}{2}\cdot\frac{2}{3}\cdot\frac{4}{5}\cdot\frac{5}{6}\cdot\frac{6}{7}\cdot\frac{7}{8}\)
\(=\frac{1\cdot\left(2\cdot5\cdot6\cdot7\right)}{8\cdot3\cdot\left(2\cdot5\cdot6\cdot7\right)}\)
\(=\frac{1}{24}\)
\(\frac{1}{2}\cdot\frac{2}{3}\cdot\frac{3}{4}\cdot\frac{4}{5}\cdot\frac{5}{6}\cdot\frac{6}{7}\cdot\frac{7}{8}\cdot\frac{8}{9}\cdot\frac{9}{10}\)
\(=\frac{1\cdot\left(2\cdot3\cdot4\cdot5\cdot6\cdot7\cdot8\cdot9\right)}{\left(2\cdot3\cdot4\cdot5\cdot6\cdot7\cdot8\cdot9\right)\cdot10}\)
\(=\frac{1}{10}\)

\(S=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+\frac{1}{5}-\frac{1}{6}+\frac{1}{6}-\frac{1}{7}+\frac{1}{7}-\frac{1}{8}+\frac{1}{8}-\frac{1}{9}\)
\(S=1-\frac{1}{9}=\frac{8}{9}\)

\(\frac{1}{6}+\frac{5}{6}=\frac{6}{6}=1\)
\(\frac{25}{5}+\frac{2}{6}=\frac{150}{30}+\frac{10}{30}=\frac{160}{30}=\frac{32}{6}=\frac{16}{3}\)
\(\frac{12}{5}+\frac{6}{9}=\frac{108}{45}+\frac{30}{45}=\frac{138}{45}\)


9x1=9
9x2=18
9x3=27
9x4=36
9x5=45
9x6=54
9x7=63
9x8=73
9x9=81
9x10=90
2/5+3/6=2/5+1/2=4/10+5/10=9/10
\(\frac{2}{5}+\frac{3}{6}=\frac{12}{30}+\frac{15}{30}=\frac{27}{30}=\frac{9}{10}\)
\(\dfrac{1}{2}-\dfrac{9}{6}+\dfrac{2}{2}\)
\(=\dfrac{3}{2}-\dfrac{3}{2}\)
=0
= 0