Answer
(a) $0.77777...=\frac{7}{9}$
(b) $0.28888...=\frac{13}{45}$
(c) $0.57575757...=\frac{19}{33}$
Work Step by Step
(a) $$x=0.77777...$$ $$10x=7.77777...$$ $$10x-x=7.77777..-0.77777...$$ $$9x=7.0$$
Hence $x=\frac{7}{9}$
(b) $$x=0.28888...$$ $$100x=28.88888...$$ $$10x=2.88888...$$ $$100x-10x=28.88888..-2.88888...$$ $$90x=26.0$$
Hence $x=\frac{26}{90}=\frac{13}{45}$
(c) $$x=0.57575757...$$ $$100x=57.57575757...$$ $$100x-x=57.57575757-0.57575757...$$ $$99x=57.0$$
Hence $x=\frac{57}{99}=\frac{19}{33}$
