Fractions or decimals?

$0.12331233\dots = \frac{1233}{9999} = \frac{137}{1111} = 137 + 1111 = \boxed{1248}$

One can create any periodic decimal like this simply by taking the desired periodic string of length $r$ (in this case $1233$ ) and dividing it by a string of $r$ $9$ 's. So, here, our periodic string has a length of $4$ , so $n = \dfrac{1233}{9999}$ . We are told that the denominator is $1111$ , so we just divide both the numerator and denominator by $9$ and reach $n= \dfrac{137}{1111}$ . So, $a+b = 1248$ .