An integer then a decimal?

For any two integers $x$ and $y$ , the notation $x.y$ represents a number whose integral part is composed of the digits of $x$ and its fractional part is composed of the digits of $y$ . For example, if $x=123$ and $y=456$ then $x.y=123.456$ . Find the sum of all possible solutions of the equation $\dfrac{a}{b}=b.a$ , where $a$ and $b$ are relatively prime positive integers.

• This is from a reviewer I received. So I take no credit.

• "The sum of all possible solutions" means that if $(a_1,b_1), (a_2, b_2), \dots, (a_n, b_n)$ are all the possible solutions, then we are asked to find $\displaystyle \sum_{k=1}^{n} a_k+b_k$ .