1. **State the problem:** There are 16 students in the swimming club, all from grades 7, 8, or 9. The probability of choosing a 7th, 8th, or 9th grader is 1, meaning all students are from these grades. 2. **Given:** - Probability of choosing a 7th grader is $\frac{1}{2}$. - Probability of choosing an 8th grader is greater than the probability of choosing a 9th grader. - Total students = 16. 3. **Translate probabilities to numbers:** - Number of 7th graders = $16 \times \frac{1}{2} = 8$. 4. **Let:** - Number of 8th graders = $x$. - Number of 9th graders = $y$. 5. **Total students equation:** $$8 + x + y = 16$$ $$x + y = 8$$ 6. **Probability condition:** - Probability of choosing an 8th grader $> $ Probability of choosing a 9th grader - $$\frac{x}{16} > \frac{y}{16} \implies x > y$$ 7. **Find greatest possible $y$:** - Since $x + y = 8$ and $x > y$, the greatest integer $y$ satisfying this is when $x = y + 1$ (smallest integer greater than $y$). - Substitute $x = y + 1$ into $x + y = 8$: $$y + 1 + y = 8$$ $$2y + 1 = 8$$ $$2y = 7$$ $$y = \frac{7}{2} = 3.5$$ - Since $y$ must be an integer number of students, the greatest integer less than 3.5 is 3. 8. **Check:** - If $y = 3$, then $x = 8 - 3 = 5$. - Check $x > y$: $5 > 3$ is true. **Final answer:** The greatest possible number of 9th graders is **3**.