Bar Notation Problem

Question

everyone! I came across a problem in math that dealt with bar notation. Does anyone know how, for instance, 1.234(with a bar notation over the 34) is expressed as a fraction? I know already how 1.22(with a bar notation over the 22) is expressed as 1 and 2/9. Any answer would be helpful. Thanks!

Answer 1

HINT:

Let $x=1.2\overline{34}$

$\implies 100x=123.4\overline{34}$

So, $100x-x=123.4\overline{34}-1.2\overline{34}=122.2$

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Alternatively, $1.2\overline{34}=1.2+0.01\cdot 0.\overline{34}$

Now, let $S=0.\overline{34}=0.343434\cdots$

$100S=34.3434\cdots$

$\implies 100S-S=34.3434\cdots-0.343434\cdots=34$

$\implies 1.2\overline{34}=1.2+0.01\cdot\frac{34}{99}$

Answer 2

$1.2\overline{34}= 1\frac{2\frac{34}{99}}{10}=1\frac{232}{990}=1\frac{116}{495}$