Question

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? 3 the probability of choosing a 9th grader is greater than 0. what is the least possible number of 9th graders in the club?

Explanation:

Step1: Find number of 7th graders

Total students = 16. Probability of 7th grader is $\frac{1}{2}$, so number of 7th graders:
$16 \times \frac{1}{2} = 8$

Step2: Calculate remaining students

Total non-7th graders (8th + 9th):
$16 - 8 = 8$
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For greatest 9th graders:

Step3: Minimize 8th graders (rule: 8th > 9th)

Let number of 9th graders = $n$, 8th graders = $n+1$.
$n + (n+1) = 8$
$2n + 1 = 8$
$2n = 7$
Since counts are integers, round down: $n=3$ (8th graders = 5, $5>3$ and $3+5=8$)
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For least 9th graders:

Step4: Maximize 8th graders (rule: 8th > 9th, 9th >0)

Minimum positive integer for 9th graders is 1, and 8th graders = $8-1=7$, which satisfies $7>1$.

Answer:

Greatest possible number of 9th graders: 3
Least possible number of 9th graders: 1