2009 AMC 10A Problem 18

Only percents are given, so pick a handy number of children: $100$.

1. Soccer players: $60\%$ of $100 = 60$.
2. Swimmers: $30$. So the children who do not swim number $100 - 30 = 70$.
3. Soccer players who swim: $40\%$ of the $60$ soccer players, which is $24$.
4. Soccer players who do not swim: $60 - 24 = 36$.
5. The question asks about the $70$ non-swimmers: what part of them play soccer?
$$\frac{36}{70} \approx 0.514,$$
which is $51\%$ to the nearest whole percent.

The answer is $\boxed{\textbf{(D)}\ 51\%}$.

Using $100$ people turns every percent into a plain count. The words "percent of the non-swimmers" tell you what to divide by: the size of that group ($70$), not the whole camp. Read each "of" carefully: "$40\%$ of the soccer players" is a part of the $60$, not of the $30$ swimmers.