1/2 + 5/6 + 11/12 + 19/20 +... + 89/90
= 1 - 1/2 + 1 - 1/6 + 1 - 1/20 + .... + 1 - 1/90
= 9 - (1/2 + 1/6 + 1/12 + 1/20 +... + 1/90)
= 9 - (1/1.2 + 1/2.3 + 1/3.4 + 1/4.5 +... + 1/9.10)
= 9 - (1 - 1/2 + 1/2 - 1/3 + 1/3 - 1/4 + 1/4 -... + 1/9 - 1/10)
= 9 - (1 - 1/10)
= 9 - 9/10
= 81/10