Question

there are 40 junior high students in the environmental club. a student is chosen at random. use the info to solve. the probability of choosing a 7th, 8th, or 9th grader is 1. the probability of choosing a 7th grader is \\(\frac{1}{4}\\). 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: Define variables

Let $x$ = number of 7th graders, $y$ = number of 8th graders, $z$ = number of 9th graders. We know $x + y + z = 48$.

Step2: Use probability constraints

From the second bullet: $\frac{x}{48} = \frac{1}{6}$, so $x = \frac{48}{6} = 8$.

Step3: Set up inequality for grades

We know $y > z$, and $y + z = 48 - 8 = 40$.

Step4: Maximize 9th graders

To maximize $z$, set $y = z + 1$ (smallest possible difference for $y > z$). Substitute into $y + z = 40$:
$(z + 1) + z = 40$
$2z + 1 = 40$
$2z = 39$
Since we need whole numbers, round down $z$ to 19 (so $y = 21$, which is greater than 19, satisfying $y > z$).

Answer:

19