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

.2222......= 22/99 = 2/9

.6666......= 66/99 = 2/3

So.....   2/3 + 2/9   =  [18 + 6] / 27  = 24/27  = 8/9  . and the reciprocal is...

9/8  = 1 + 1/8 =  1.125

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:

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