jmc

If $x$ is the number of people, the first caterer charges $100+15x$ dollars, while the second charges $200+12x$ dollars. We want $$100+15x>200+12x.$$ Subtracting $100$ from both sides gives $$15x>100+12x,$$ then subtracting $12x$ from both sides gives $$3x>100.$$ Finally, dividing both sides by $3$ gives $x=\frac{100}{3}=33\frac{1}{3}$, so the smallest number of people is $\boxed{34}$.