Convert the following into a vulgar fraction :

${1.\overline{243}}$

Let $x = 1.\overline{243}$. Then,

x = 1.243243243… (i)

⇒ 1000x = 1243.243243… (ii)

On subtracting (i) from (ii), we get:

$999x = 1242 \\[1em] \Rightarrow x = \dfrac{1242}{999} \\[1em] \Rightarrow x = 1\dfrac{243}{999} \\[1em]$

Hence, $1.\overline{243} = 1\dfrac{243}{999}$