\[2\frac{2}{9}\ :1\frac{7}{9} = \frac{20}{9}\ :\frac{16}{9} = \frac{20 \cdot 9}{9 \cdot 16} = \frac{20}{16} =\]
\[= \frac{5}{4} = 1\frac{1}{4}\ \]