Bài giải:

( Tính nhanh $10-\frac{1}{2}-\frac{5}{6}-\frac{11}{12}-\frac{19}{20}-.....-\frac{89}{90}$ ? )

$10-\frac{1}{2}-\frac{5}{6}-\frac{11}{12}-\frac{19}{20}-.....-\frac{89}{90}$

= $10-(\frac{1}{2}+\frac{5}{6}+\frac{11}{12}+\frac{19}{20}+....+\frac{89}{90})$      ( 1)

Đặt $\frac{1}{2}+\frac{5}{6}+\frac{11}{12}+\frac{19}{20}+....+\frac{89}{90}$ = A

A = $1-\frac{1}{2}+1-\frac{1}{6}+1-\frac{1}{12}+1-\frac{1}{20}+...+1-\frac{1}{90}$

A = ( 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 ) - $\frac{1}{2}-\frac{1}{6}-\frac{1}{12}-\frac{1}{20}-....-\frac{1}{90}$

A = 9 – ($\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+...+\frac{1}{90}$ )

A = 9 – ( $1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{9}-\frac{1}{10}$ )

A = 9 – ( 1 - $\frac{1}{10}$ )

A = 9 - $\frac{9}{10}$

Thay A = 9 - $\frac{9}{10}$vào  ( 1)  ta được

10 – 9 - $\frac{9}{10}$ = 1 -$\frac{9}{10}$ = $\frac{1}{10}$

Đáp số: $\frac{1}{10}$