Question

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there are 16 junior high students in the brewster town swimming club. a student is chosen at random to be the club vice president. - the probability of choosing a 7th, 8th, or 9th grader is 1. - the probability of choosing a 7th grader is \\(\frac{1}{2}\\). - choosing an 8th grader is more likely than choosing a 9th grader. what is the greatest possible number of 9th graders in the club?

Explanation:

Step1: Calculate number of 7th graders

Number of 7th graders = Total students × Probability of 7th grader
$16 \times \frac{1}{2} = 8$

Step2: Find total of 8th + 9th graders

Total non-7th graders = Total students - 7th graders
$16 - 8 = 8$

Step3: Maximize 9th graders (with 8th > 9th)

Let $n_9$ = number of 9th graders, $n_8$ = number of 8th graders. We need $n_8 > n_9$ and $n_8 + n_9 = 8$. To maximize $n_9$, set $n_8 = n_9 + 1$.
Substitute: $(n_9 + 1) + n_9 = 8$
$2n_9 + 1 = 8$
$2n_9 = 7$
Since $n_9$ must be an integer, the largest integer where $n_8 > n_9$ is $n_9 = 3$ (so $n_8 = 5$, $5+3=8$ and $5>3$).

Answer:

3