Find them all

From the LHS, we have $-1 \le \sin(\pi x) \le 1$ . Therefore, $-1 \le \dfrac x{100} \le 1$ , $\implies -100 \le x \le 100$ . Since both the LHS and RHS are odd functions. The number of solutions in $0 \le x \le 100$ and $-100 \le x \le 0$ are the same and they share one solution at $x=0$ (see figure below). We only need to consider $0 \le x \le 100$ .

From the figure, we note for every $x = (2n-1)\pi$ , where $n \in \mathbb N$ , there are 2 solutions. From $\{1, 3, 5 ... 99\}$ , there are 50 odd numbers and therefore $100$ solutions. And from $-100 \le x \le 100$ , there are $100 \times 2 - 1 = \boxed{199}$ solutions.