`A = 1/1.2 + 1/3.4 + ... + 1/99.100`

`A=  1- 1/2 + 1/3 - 1/4 + ... + 1/99 -1/100`

`A=  (1+1/3+ ... + 1/99) - (1/2 + 1/4 + ... + 1/100)`

`A=  (1+1/2+ 1/3+1/4+  ... + 1/99+1/100) - 2 . (1/2 + 1/4 + ... + 1/100)`

`A = (1+1/2+ 1/3+1/4+  ... + 1/99+1/100) - ( 1 + 1/2 + ... + 1/50)`

`A = 1/51 + 1/52 + ... + 1/100`

`B = 1/51.100 + 1/52.99 + ... + 1/99.52 + 1/100.51`

`B = 1/(51 . (151-51) ) + 1/(52 . (151-52) ) + ... + 1/( 99 . (151 - 99) ) + 1/(100 . ( 151 -100) )`

`B = 1/51 . 1/(151 -51) + 1/52 . 1/(151-52) + ... + 1/99 . 1/(151-99) + 1/100  . 1/(151-100)`

`B = 1/151(1/51 +1/100) + 1/151(1/52 +1/99)  + ... + 1/151(1/99+1/52) + 1/151(1/100 + 1/51)`

`B = 1/151 . [ ( 1/51 +1/100) + (1/52+1/99) + ... + (1/99+1/52) + (1/100+1/51) ]`

`B = 1/151 . [ 2 . ( 1/51 + 1/52 + ... + 1/99 + 1/100 ) ]`

`B = 2/151 . ( 1/51 + 1/52 + ... + 1/99 + 1/100 )`

`A/B = (1/51 + 1/52 + ... + 1/100)/(2/151 . ( 1/51 + 1/52 + ... + 1/99 + 1/100 ) )`

`A/B = 1/(2/151) = 151/2`