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Infinite geometric interpretation

Authors
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    Name
    Vu Hung
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Problem Statement

Evaluate

0.42‾=0.42222…0.4\overline{2}=0.42222\ldots

as a fraction.

Hints

Write as 0.4+0.02222…0.4+0.02222\ldots and convert the recurring part using a GP.

Solutions

0.02222…=290=145.0.02222\ldots=\frac{2}{90}=\frac{1}{45}.

So

0.42‾=25+145=1845+145=1945.0.4\overline{2}=\frac{2}{5}+\frac{1}{45}=\frac{18}{45}+\frac{1}{45}=\frac{19}{45}.

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