In a math class, each student who does not turn in a homework assignment receives a 0 for that assignment. Of the 10 students in the math class, what is the maximum possible number who did not turn in Homework assignment #1?
We are given that one student's score on Homework assignment #1 is 96 out of 100 possible points and that the class average is 86.7 out of 100 possible points.
ten students with an 86.7 average means 867 total points
if more than one student receives a zero
... then the average can't be above 80
If 8 students got 100, and one student got a 96, and one got zero, that means that the maximum total points are 896, and the class average is 89.6
With two zeroes, the average is 79.6 max, so only 1 student could have scored a zero.
To find the maximum possible number of students who did not turn in Homework assignment #1, we need to consider two scenarios:
Scenario 1: The student with a score of 96 turned in the assignment.
In this case, the maximum possible class average excluding this student's score would be when all other students receive a score of 0. Since the class average is 86.7, we can calculate the total points obtained by the class without considering this student's score.
Let x be the maximum possible number of students who did not turn in Homework assignment #1.
Since there are 10 students in total, the maximum total points excluding the student with a score of 96 would be (86.7 * 10) - 96.
Therefore, the equation would be 86.7 * (10 - x) = (86.7 * 10) - 96.
Simplifying the equation, we have 867 - 86.7x = 867 - 96.
This simplifies further to 86.7x = 96.
Solving for x, we divide both sides of the equation by 86.7.
x = 96 / 86.7.
x ≈ 1.107.
Since we can't have a fraction of a student, we round this number down to the nearest whole number.
Therefore, in Scenario 1, the maximum possible number of students who did not turn in Homework assignment #1 is 1.
Scenario 2: The student with a score of 96 did not turn in the assignment.
In this case, the maximum possible class average could be obtained by giving all other students the maximum possible score, which is 100. Since the class average is 86.7, we can calculate the number of students who did not turn in the assignment by subtracting the number of points this student received from the maximum possible class average.
Let y be the maximum possible number of students who did not turn in Homework assignment #1.
The equation would be (86.7 * (10 - y) + 96) / 10 = 86.7.
Solving for y, we can start by multiplying both sides of the equation by 10:
86.7 * (10 - y) + 96 = 86.7 * 10.
Next, we subtract 96 from both sides of the equation:
86.7 * (10 - y) = 86.7 * 10 - 96.
Simplifying the equation, we have 867 - 86.7y = 867 - 96.
This further simplifies to 86.7y = 96.
Solving for y, we divide both sides of the equation by 86.7.
y = 96 / 86.7.
y ≈ 1.107.
Again, since we can't have a fraction of a student, we round this number down to the nearest whole number.
Therefore, in Scenario 2, the maximum possible number of students who did not turn in Homework assignment #1 is 1.
Since both scenarios have the same maximum number of students who did not turn in the assignment, we can conclude that the maximum possible number of students who did not turn in Homework assignment #1 is 1, regardless of whether the student with a score of 96 turned in the assignment or not.