What is the value of $$99\frac{{11}}{{99}} + 99\frac{{13}}{{99}} + 99\frac{{15}}{{99}} + ........ + 99\frac{{67}}{{99}}?$$

Solution(By Examveda Team)

$$\eqalign{ & 99\frac{{11}}{{99}} + 99\frac{{13}}{{99}} + 99\frac{{15}}{{99}} + ........ + 99\frac{{67}}{{99}} \cr & {\text{Number of term}} = \frac{{67 - 11}}{2} + 1 = 29 \cr & = 99 \times 29 + \frac{1}{{99}}\left( {11 + 13 + 15 + ........ + 67} \right) \cr & = 2871 + \frac{1}{{99}}\left[ {\frac{{29}}{2}\left( {2 \times 11 + \left( {29 - 1} \right)2} \right)} \right] \cr & = 2871 + \frac{1}{{99}}\left[ {\frac{{99}}{2}\left( {22 + 56} \right)} \right] \cr & = 2871 + \frac{1}{{99}}\left( {29 \times 39} \right) \cr & = 2871 + \frac{{1131}}{{99}} \cr & = 2871 + \frac{{377}}{{33}} \cr & = \frac{{95120}}{{33}} \cr} $$