Given

\(\displaystyle{\frac{{{5}}}{{{9}}}}\times{\frac{{{7}}}{{{19}}}}+{\frac{{{7}}}{{{19}}}}\div{\frac{{{18}}}{{{19}}}}\)

\(\displaystyle\Rightarrow{\frac{{{5}}}{{{9}}}}\times{\frac{{{1}}}{{{2}}}}+{\frac{{{7}}}{{{19}}}}\times{\frac{{{19}}}{{{18}}}}\)

\(\displaystyle\Rightarrow{\frac{{{5}}}{{{18}}}}+{\frac{{{7}}}{{{18}}}}\)

\(\displaystyle\Rightarrow{\frac{{{5}+{7}}}{{{18}}}}={\frac{{{12}}}{{{18}}}}={\frac{{{2}}}{{{3}}}}\)

\(\displaystyle\Rightarrow{\frac{{{5}}}{{{9}}}}\times{\frac{{{1}}}{{{2}}}}+{\frac{{{7}}}{{{19}}}}\div{\frac{{{18}}}{{{19}}}}={\frac{{{2}}}{{{3}}}}\)

\(\displaystyle{\frac{{{5}}}{{{9}}}}\times{\frac{{{7}}}{{{19}}}}+{\frac{{{7}}}{{{19}}}}\div{\frac{{{18}}}{{{19}}}}\)

\(\displaystyle\Rightarrow{\frac{{{5}}}{{{9}}}}\times{\frac{{{1}}}{{{2}}}}+{\frac{{{7}}}{{{19}}}}\times{\frac{{{19}}}{{{18}}}}\)

\(\displaystyle\Rightarrow{\frac{{{5}}}{{{18}}}}+{\frac{{{7}}}{{{18}}}}\)

\(\displaystyle\Rightarrow{\frac{{{5}+{7}}}{{{18}}}}={\frac{{{12}}}{{{18}}}}={\frac{{{2}}}{{{3}}}}\)

\(\displaystyle\Rightarrow{\frac{{{5}}}{{{9}}}}\times{\frac{{{1}}}{{{2}}}}+{\frac{{{7}}}{{{19}}}}\div{\frac{{{18}}}{{{19}}}}={\frac{{{2}}}{{{3}}}}\)