Find the reciprocal of $.\overline{2} + .\overline{6}$ expressed as a decimal

Find the reciprocal of

$$.\overline{2} + .\overline{6}$$ expressed as a decimal

$$\\0.\overline{2} = \dfrac{ 1 } {4.5 } \\\\ 0.\overline{6} = \dfrac{ 1 } { 1.5 } \\\\$$

so wie have:

$$\small{\text{ $ \dfrac{1} { 0.\overline{2} + 0.\overline{6} } = \dfrac{1} { \dfrac{ 1 } { 4.5 } + \dfrac{ 1 } { 1.5 } } =\dfrac{1} { \dfrac{ 1.5+4.5 } { 1.5\cdot 4.5 } } = \dfrac{ 1.5\cdot 4.5 }{ 1.5+4.5 } =\dfrac{ 6.75 }{ 6 } = 1.125 $}}$$