The score for half of the students in a class on a certain test was 90. The score for another fourth of the students was 84. if the average (arithmetic mean) score for all of the students was 82, what was the average score for the remaining students?
1/2 (90) + 1/4 (84) + 1/4 (x) = 82
LCD = 4
2 (90) + 1 (84) + 1 (x) = 4 (82)
180 + 84 + x = 328
x = 328 - 180 - 84
x = 64
Well, it seems like the students are really multiplying their scores! Let's calculate the number of students based on the given information:
Half of the students scored 90, which means the other half scored something else (let's call it "x"). In that case, half of the total number of students is (1/2) * n, where n is the total number of students.
So, (1/2) * n * 90 is the total score for the first group of students.
Another fourth of the students scored 84, which means the remaining three-fourths scored "x." Thus, three-fourths of the total number of students is (3/4) * n.
So, (3/4) * n * x is the total score for the second group of students.
Now, let's use the average score formula to solve for x:
(1/2) * n * 90 + (3/4) * n * x = 82 * n
Simplifying this equation, we get:
45 + (3/4) * x = 82
Now, let's find out the value of x:
(3/4) * x = 82 - 45
(3/4) * x = 37
Dividing both sides by (3/4), we find:
x = 37 / (3/4)
x = 49.33333...
So, the remaining students have an average score of approximately 49.33333... or should I say, "Nearly halfway there, but not quite! Keep practicing, you'll reach the average score of 82!"
Let's assume there are "x" students in the class.
Half of the students scored 90, which means there are x/2 students with a score of 90 in the class.
Another fourth of the students scored 84, which means there are x/4 students with a score of 84 in the class.
The remaining students, then, would be x - (x/2) - (x/4) = (4x - 2x - x)/4 = x/4.
The average score for all students is given as 82, which means the total score for all students is 82x.
The total score for the students who scored 90 is (90 * (x/2)) = 45x.
The total score for the students who scored 84 is (84 * (x/4)) = 21x.
The total score for the remaining students will be the total score for all students minus the total scores of the students who scored 90 and 84. Therefore,
82x = 45x + 21x + (average score for the remaining students) * (x/4)
Combining like terms:
82x = 45x + 21x + (average score for the remaining students) * (x/4)
82x = 66x + (average score for the remaining students) * (x/4)
Subtracting 66x from both sides:
16x = (average score for the remaining students) * (x/4)
Dividing both sides by x/4:
16 = (average score for the remaining students)
Therefore, the average score for the remaining students is 16.
To solve this problem, let's break it down step by step:
Step 1: Identify the given information:
- The score for half of the students (1/2) was 90.
- The score for another fourth of the students (1/4) was 84.
- The average score for all students is 82.
Step 2: Calculate the number of students based on the given fractions:
Since we have information about half (1/2) and another fourth (1/4) of the students, we can add these two fractions together to find the overall fraction of students whose scores are known.
1/2 + 1/4 = 4/8 + 2/8 = 6/8 = 3/4
This means that we know the scores for 3/4 of the students in the class.
Step 3: Calculate the remaining fraction of students:
Since we know that the scores of 3/4 of the students are known, the remaining fraction of students whose scores aren't given is:
1 - 3/4 = 1/4
Therefore, the remaining 1/4 of the students' scores are not known.
Step 4: Calculate the average score for the remaining students:
To find the average score for the remaining students, we can subtract the average scores of the known students from the overall average score.
Let's denote the average score for the remaining students as "x".
The average score for the known students is:
(3/4) * 82 = (3/4) * 82 = 61.5
Since we are looking for the average score of the remaining students, we can set up an equation:
(1/4) * x = 82 - 61.5
Simplifying the equation gives:
(1/4) * x = 20.5
To solve for x, we can multiply both sides of the equation by 4:
x = 20.5 * 4
x = 82
Therefore, the average score for the remaining students is 82.