S = $\frac{1}{2}$ + $\frac{5}{6}$ + $\frac{11}{12}$ + $\frac{19}{20}$ +$\frac{29}{30}$ + $\frac{41}{42}$ + $\frac{55}{56}$ + $\frac{71}{72}$ + $\frac{89}{90}$
= ( 1 – $\frac{1}{2}$) +( 1 – $\frac{5}{6}$) + ( 1 -$\frac{11}{12}$) + ( 1 – $\frac{19}{20}$) + ( 1 -$\frac{29}{30}$) + ( 1 – $\frac{41}{42}$) + ( 1 – $\frac{55}{56}$) +( 1 – $\frac{71}{72}$) + ( 1 -$\frac{89}{90}$)
= (1+1+1+1+1+1+1+1+1) – ($\frac{1}{2}$ + $\frac{1}{6}$ + $\frac{1}{12}$ + $\frac{1}{20}$ +$\frac{1}{30}$ + $\frac{1}{42}$ + $\frac{1}{56}$ + $\frac{1}{72}$ + $\frac{1}{90}$)
= 8 – ($\frac{1}{1.2}$ + $\frac{1}{2.3}$ + $\frac{1}{3.4}$ + $\frac{1}{4.5}$ +$\frac{1}{5.6}$ + $\frac{1}{6.7}$ + $\frac{1}{7.8}$ + $\frac{1}{8.9}$ + $\frac{1}{9.10}$)
S=$\frac{1}{2}$+$\frac{5}{6}$+$\frac{11}{12}$+$\frac{19}{20}$+$\frac{29}{30}$+$\frac{41}{42}$+$\frac{55}{56}$+$\frac{71}{72}$+$\frac{89}{90}$
S=(1-$\frac{1}{2}$)+(1-$\frac{1}{6}$)+(1-$\frac{1}{12}$ )+(1-$\frac{1}{20}$ )+(1-$\frac{1}{30}$)+(1-$\frac{1}{42}$)+(1-$\frac{1}{56}$ )+(1-$\frac{1}{72}$ )+(1-$\frac{1}{90}$ )
S=(1+1+1+1+1+1+1+1+1)-($\frac{1}{2}$+$\frac{1}{6}$+$\frac{1}{12}$+$\frac{1}{20}$+$\frac{1}{30}$+$\frac{1}{42}$+$\frac{1}{56}$+$\frac{1}{72}$+$\frac{1}{90}$)
S=9-($\frac{1}{1.2}$+$\frac{1}{2.3}$+$\frac{1}{3.4}$+$\frac{1}{4.5}$+$\frac{1}{5.6}$+$\frac{1}{6.7}$+$\frac{1}{7.8}$+$\frac{1}{8.9}$+$\frac{1}{9.10}$
S=9-(1-$\frac{1}{2}$+$\frac{1}{2}$-$\frac{1}{3}$+$\frac{1}{3}$-$\frac{1}{4}$+$\frac{1}{4}$-$\frac{1}{5}$+$\frac{1}{5}$-$\frac{1}{6}$+$\frac{1}{6}$-$\frac{1}{7}$+$\frac{1}{7}$-$\frac{1}{8}$+$\frac{1}{8}$-$\frac{1}{9}$+$\frac{1}{9}$-$\frac{1}{10}$)
S=9-(1-$\frac{1}{10}$ )
S=9-$\frac{9}{10}$
S=$\frac{90-9}{10}$
S=$\frac{81}{10}$