2011 AMC 10B Problem 6

Work backward. Before giving $4$ candies to his sister, Casper had $12$ candies, because he ended the second day with $8$. Those $12$ candies were $\dfrac23$ of what he had after the first day, so after the first day he had $18$ candies. Before giving $2$ candies to his brother, he had $20$ candies. This was $\dfrac23$ of his original amount, so he began with $30$ candies. Thus, A is the correct answer.