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.\nthe probability of choosing a 7th, 8th, or 9th grader is 1.\nthe probability of choosing a 7th grader is $\frac{1}{2}$.\nchoosing an 8th grader is more likely than choosing a 9th grader.\nwhat 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: Analyze remaining students
The number of 8th and 9th - graders together is $16 - 8=8$. Let the number of 8th - graders be $x$ and the number of 9th - graders be $y$, so $x + y=8$ and $x>y$.
Step3: Find maximum value of $y$
Since $x + y = 8$ and $x>y$, when $x$ is just 1 more than $y$, we can solve the system of equations $\begin{cases}x + y=8\x=y + 1\end{cases}$. Substitute $x=y + 1$ into $x + y=8$, we get $(y + 1)+y=8$, $2y+1 = 8$, $2y=7$, $y = 3.5$. But the number of students must be an integer. So the greatest integer value of $y$ for which $x>y$ and $x + y=8$ is 3.
Answer:
3