How to sum up these fractions? Found this question in a competitive math test for elementary students. The long way is to add all the decimal values but is there a pattern/trick to solve this question (or these types)? I don't know how to solve this except by the long method of adding all the decimal equivalents.
The Answer is $1$

Compute: 
  $$\frac17 + \frac18 + \frac19 + \frac1{10} + \frac1{11} + \frac1{12} + \frac1{14} + \frac1{15} + \frac1{18} + \frac1{22} + \frac1{24} + \frac1{28} + \frac1{33} = ? $$


Not sure if this is what you are looking for but this is what I did:
$$S = \frac17 + \frac18 + \frac19 + \frac1{10} + \frac1{11} + \frac1{12} + \frac1{14} + \frac1{15} + \frac1{18} + \frac1{22} + \frac1{24} + \frac1{28} + \frac1{33}$$
I observed the following groupings:


*

*7, 14, 28

*8, 12, 24

*11, 22, 33

*I did grunt work on the remaining four numbers and found that the LCM is $90$


$$S = \underbrace{\frac17 + \frac1{14} + \frac1{28}}_{\frac14} + \underbrace{\frac18 + \frac1{12} + \frac1{24}}_{\frac14} + \underbrace{\frac1{11} + \frac1{22} +  \frac1{33}}_{\frac16} + \underbrace{\frac19 + \frac1{10} + \frac1{15} + \frac1{18}}_{\frac13}$$
$$S = \frac14 + \frac14 + \frac16 + \frac13$$
$$ = \frac12 + \frac12 = 1$$
