In a spelling bee $50\%$ of the students were eliminated after the first round. Only $\frac{1}{3}$ of the remaining students were still in the contest after the second round. If 24 students were still in the contest after the second round, how many students began the contest?
I need to find the number of students who began the contest, let me call that x. After the first round, half of them were eliminated, so $\frac{x}{2}$ were still in the contest. After the second round, only one-third of the remaining students were still in the contest, so $\frac{1}{3} \times \frac{x}{2} = \frac{x}{6}$ were still in the contest. I know that this number is 24, so I can set up an equation: $\frac{x}{6} = 24$. To solve for x, I can multiply both sides by 6: $x = 6 \times 24 = 144$. Therefore, 144 students began the contest. 