The batting scores of a batsman in three consecutive test matches are given below. The score in the first test was a two-digit number. The score in the second test was relatively better, and the increase was exactly equal to the sum of the digits of the first test score. The score in the third test was even better than the second, and the increase was equal to half the sum of the digits of the first test score. If the batsman's average score was $15$ more than the first score, then the square root of the sum of the three scores can be expressed in the form $a\sqrt{b}$, where $a$ and $b$ are coprime. Determine the value of $a + b$.