Convert the following into a vulgar fraction :

${0.57\overline{4}}$

Let $x = 0.57\overline{4} = 0.57444…$

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

100x = 57.444….. (i)

Multiplying by 10 to move the decimal past the first repeating block:

1000x = 574.444….. (ii)

Subtracting (i) from (ii):

$900x = 517 \\[1em] \Rightarrow x = \dfrac{517}{900}$

Hence, $0.57\overline{4} = \dfrac{517}{900}$