Let $a$ and $b$ be positive real numbers satisfying
$$a -12b = 11  -\frac{100}{a} \,\,\,\,and \,\,\,\, a -\frac{12}{b}= 4 -\frac{100}{a}.$$
Then $a + b = \frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Let's think step by step and output the final answer within \boxed{}.
user