The value of $$\left( {2.\overline 4 \times 2.\overline 6 \times 3 \times 0.1\overline 6 } \right) \times \left[ {0.\overline {27} \times \left( {0.8\overline 3 \div 0.1\overline 6 } \right)} \right]{\text{is:}}$$

$$\eqalign{ & \left( {2.\overline 4 \times 2.\overline 6 \times 3 \times 0.1\overline 6 } \right) \times \left[ {0.\overline {27} \times \left( {0.8\overline 3 \div 0.1\overline 6 } \right)} \right] \cr & = \left( {\frac{{22}}{9} \times \frac{6}{9} \times 3 \times \frac{{15}}{{90}}} \right) \times \left[ {\frac{{27}}{{99}} \times \left( {\frac{{75}}{{90}} \div \frac{{15}}{{90}}} \right)} \right] \cr & = \frac{{22}}{{27}} \times \left[ {\frac{{27}}{{99}} \times \left( {\frac{{75}}{{90}} \times \frac{{90}}{{15}}} \right)} \right] \cr & = \frac{{22}}{{27}} \times \frac{{27}}{{99}} \times 5 \cr & = \frac{{10}}{9} \cr & = 1.\overline 1 \cr} $$