Answer
(a) $5.23232323...=\frac{518}{99}$
(b) $1.37777...=\frac{62}{45}$
(c) $2.135353535...=\frac{1057}{495}$
Work Step by Step
(a) $$x=5.23232323...$$ $$100x=523.23232323....$$ $$100x-x=523.23232323...-5.23232323...$$ $$99x=518.0$$
Hence $x=\frac{518}{99}$
(b) $$x=1.37777...$$ $$100x=137.77777...$$ $$10x=13.77777...$$ $$100x-10x=137.77777-13.77777...$$ $$90x=124.0$$
Hence $x=\frac{124}{90}=\frac{62}{45}$
(c) $$x=2.135353535...$$ $$10x=21.35353535...$$ $$1000x=2135.35353535...$$ $$1000x-10x=2135.35353535-21.35353535$$ $$990x=2114.0$$
Hence $990x=\frac{2114}{990}=\frac{1057}{495}$
