there are 16 junior high students in the brewster town swimming club. a student is chosen at random to be…

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 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?

Answer

Explanation:

Step1: Determine number of 7th - graders

Since the probability of choosing a 7th - grader is $\frac{1}{2}$ and there are 16 students in total, the number of 7th - graders is $16\times\frac{1}{2}=8$.

Step2: Find number of 8th and 9th - graders combined

The probability of choosing a 7th, 8th, or 9th - grader is 1. After finding 8 7th - graders, the number of 8th and 9th - graders combined is $16 - 8=8$.

Step3: Set up inequality for 8th and 9th - graders

Let the number of 9th - graders be $x$ and the number of 8th - graders be $y$. We know that $x + y=8$ and $y>x$.

Step4: Solve for the maximum value of $x$

From $y = 8 - x$ and $y>x$, we substitute $y$ to get $8 - x>x$. Adding $x$ to both sides gives $8>2x$. Dividing both sides by 2, we have $x < 4$. Since $x$ is a non - negative integer, the greatest possible value of $x$ is 3.