School rules permit no fewer than 2 teachers per 25 students. There are at least 245 students enrolled. If [tex]$x$[/tex] represents teachers and [tex]$y$[/tex] represents students, which system of linear inequalities can be used to determine the possible number of teachers and students at the school?
A. [tex]$2 y \geq 25 x$[/tex] and [tex]$y \geq 245$[/tex]
B. [tex]$2 y \leq 25 x$[/tex] and [tex]$y \geq 245$[/tex]
C. [tex]$25 y \leq 2 x$[/tex] and [tex]$y \geq 245$[/tex]
D. [tex]$25 y \geq 2 x$[/tex] and [tex]$y \geq 245$[/tex]
Answer (2)
The problem provides two conditions: a minimum teacher-to-student ratio and a minimum number of students.
The first condition translates to the inequality 2 y ≤ 25 x .
The second condition translates to the inequality y g e q 245 .
Combining these, the system of inequalities is 2 y ≤ 25 x and y g e q 245 , so the answer is 2 y ≤ 25 x and y ≥ 245 .
Explanation
Problem Analysis Let's analyze the given problem. We are given two conditions:
School rules permit no fewer than 2 teachers per 25 students. This means for every 25 students, there must be at least 2 teachers. If we have x teachers and y students, this can be written as a ratio: y x ≥ 25 2 .
There are at least 245 students enrolled. This means the number of students, y , must be greater than or equal to 245, so y ≥ 245 .
Our objective is to express these conditions as a system of linear inequalities.
Converting the First Condition to Inequality Now, let's convert the first condition into an inequality. We have y x ≥ 25 2 . To eliminate the fractions, we can multiply both sides by 25 y . Since y represents the number of students, it must be positive, so multiplying by 25 y doesn't change the direction of the inequality:
25 y ⋅ y x ≥ 25 y ⋅ 25 2 25 x ≥ 2 y
We can rewrite this as:
2 y ≤ 25 x
Expressing the Second Condition as Inequality The second condition is straightforward: there are at least 245 students enrolled, which means:
y ≥ 245
Forming the System of Linear Inequalities Combining the two inequalities, we get the system of linear inequalities:
2 y ≤ 25 x y ≥ 245
Comparing this with the given options, we see that the correct system of inequalities is 2 y ≤ 25 x and y ≥ 245 .
Final Answer Therefore, the system of linear inequalities that can be used to determine the possible number of teachers and students at the school is:
2 y ≤ 25 x and y ≥ 245 .
Examples
Understanding the relationship between the number of teachers and students is crucial for school administration. For example, if a school plans to increase its student enrollment, administrators can use these inequalities to determine the minimum number of teachers they need to hire to maintain the required teacher-student ratio. This ensures that the school meets its staffing requirements and provides adequate support for its students. The inequality y ≥ 245 ensures the school meets the minimum student enrollment, while 2 y ≤ 25 x ensures there are enough teachers for the number of students.
The system of inequalities that can be used to determine the possible number of teachers and students at the school is 2 y ≤ 25 x and y ≥ 245 . Therefore, the correct answer is option A. This is based on the requirement of at least 2 teachers for every 25 students and a minimum student count of 245.
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