Convert the following into a vulgar fraction :

${0.12\overline{34}}$

Let $x = 0.12\overline{34} = 0.12343434…$

Multiplying by 100 to move the decimal past the non-repeating digits:

100x = 12.343434….. (i)

Multiplying by 100 to move the decimal past the first repeating block:

10000x = 1234.343434….. (ii)

Subtracting (i) from (ii):

$9900x = 1222 \\[1em] \Rightarrow x = \dfrac{1222}{9900} = \dfrac{611}{4950}$

Hence, $0.12\overline{34} = \dfrac{611}{4950}$