2007 AMC 10B Problem 14

1. Two girls leave and two boys arrive. That is $2$ out and $2$ in, so the size of the group does not change. Call it $n$.
2. At the start, $40\%$ of the group are girls: $0.4n$ girls.
3. After the swap there are $2$ fewer girls, and that is $30\%$ of the same group:
$$0.4n - 2 = 0.3n.$$
4. The girls went from $40\%$ to $30\%$ of the group, a drop of $10\%$, and that drop is the $2$ girls who left:
$$0.1n = 2, \qquad n = 20.$$
5. Girls at the start: $0.4 \times 20 = 8$.
6. Check: $8$ girls out of $20$ is $40\%$. After the swap, $6$ girls out of $20$ is $30\%$.

The answer is $\boxed{\textbf{(C)}\ 8}$.

A percent only makes sense once you know what it is a percent of. Here the group size is the same before and after, so the two percents can be compared directly: $10$ percentage points of the group equals $2$ people. Before writing an equation, look for the quantity that does not change.

Try the answer choices. With $8$ girls making up $40\%$, the group has $20$ people; after the swap $6$ out of $20$ is $30\%$, which works. For comparison, $6$ girls at $40\%$ means $15$ people, and then $4$ out of $15$ is about $27\%$, which fails.