The value of $$\frac{3}{{70}} + \frac{1}{{42}} + \frac{1}{{66}} + \frac{3}{{286}} + \frac{1}{{130}} + \frac{1}{{170}}$$       is:

Solution (By Examveda Team)

$$\eqalign{ & \frac{3}{{70}} + \frac{1}{{42}} + \frac{1}{{66}} + \frac{3}{{286}} + \frac{1}{{130}} + \frac{1}{{170}} \cr & = \left( {\frac{3}{{70}} + \frac{1}{{42}}} \right) + \left( {\frac{1}{{66}} + \frac{3}{{286}}} \right) + \left( {\frac{1}{{130}} + \frac{1}{{170}}} \right) \cr & = \left( {\frac{{9 + 5}}{{70 \times 3}}} \right) + \left( {\frac{{13 + 9}}{{286 \times 3}}} \right) + \left( {\frac{{17 + 13}}{{130 \times 17}}} \right) \cr & = \left( {\frac{{14}}{{70 \times 3}}} \right) + \left( {\frac{{22}}{{286 \times 3}}} \right)\left( {\frac{{30}}{{130 \times 17}}} \right) \cr & = \left( {\frac{1}{{15}}} \right) + \left( {\frac{1}{{13 \times 3}}} \right) + \left( {\frac{3}{{13 \times 17}}} \right) \cr & = \left( {\frac{1}{{15}}} \right) + \left( {\frac{{17 + 9}}{{13 \times 3 \times 17}}} \right) \cr & = \left( {\frac{1}{{15}}} \right) + \left( {\frac{{26}}{{13 \times 3 \times 17}}} \right) \cr & = \left( {\frac{1}{{15}}} \right) + \left( {\frac{2}{{3 \times 17}}} \right) \cr & = \left( {\frac{{17 + 10}}{{85 \times 3}}} \right) \cr & = \frac{{27}}{{85 \times 3}} \cr & = \frac{9}{{85}} \cr} $$