A group of friends went to an ice-cream parlour and ordered only two

In Statement 1, the ratio of the total number of people to the number who ate neither must be greater than 1, since the people who ate neither are part (but not all) of the total. So Statement 1 tells us "the ratio of the number who ate chocolate to the number who ate strawberry is greater than 1" which means more people ate chocolate, and Statement 1 is sufficient.

If you let b = people who ate both, c = people who only ate chocolate, and s = people who only ate strawberry, Statement 2 tells us that c + s > b + s, or c > b, so Statement 2 tells us more people only ate chocolate than ate both types of ice cream. We know nothing about the size of s from that, so Statement 2 is not sufficient, and the answer is A.

The wording of the question is deeply problematic, though: when Statement 1 begins "The ratio of the number of people who ate chocolate ice-cream and people who ate strawberry ice-cream" it sounds like it's talking about the people who ate both types of ice cream. I'm expecting the word "to" to follow, and then to learn what the ratio is comparing that group to. But that's not what Statement 1 is trying to say. When stating a ratio, the word "to" always must separate the two parts of the ratio, and that's not the case in either Statement here.