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$frac{5}{6} + 2 frac{3}{4} : 4 frac{7}{9} * 2 frac{2}{3} -1 frac{2}{3}=frac{181}{258}$

$1) 2 frac{3}{4} : 4 frac{7}{9} = frac{11}{4} : frac{43}{9} = frac{11}{4} * frac{9}{43} = frac{99}{172}$

$2) frac{99}{172} * 2 frac{2}{3} = frac{99}{172} * frac{8}{3} = frac{33}{43} * frac{2}{1} = frac{66}{43} = 1 frac{23}{43}$

$3) frac{5}{6} + 1 frac{23}{43} = 1 + frac{5}{6} + frac{23}{43} = 1 + frac{215}{258} + frac{138}{258} = 1 + frac{353}{258} = 1 + 1 frac{95}{258} = 2 frac{95}{258}$

$4) 2 frac{95}{258} - 1 frac{2}{3} = 2-1+ frac{95}{258} - frac{2}{3} = 1 + frac{95}{258} - frac{172}{258} = 1 + frac{-77}{258} = frac{1}{1} + frac{-77}{258} = frac{258}{258} - frac{77}{258} = frac{181}{258}$