Twenty-seven students averages 70 on their midterm. Could 21 of them have scored about 90?

27 * 70 = 1890

21 * 90 = 1890

That would be possible only if the other 6 students scored zeroes.

To determine whether it is possible for 21 out of 27 students to have scored about 90 on their midterm, we can follow these steps:

Step 1: Calculate the total score of all the students on the midterm:
Total score = Number of students × Average score
Total score = 27 × 70
Total score = 1890

Step 2: Calculate the maximum score the remaining 6 students could have achieved:
Maximum score of the remaining 6 students = Total score - Total score of 21 students
Maximum score of the remaining 6 students = 1890 - (21 × 90)
Maximum score of the remaining 6 students = 1890 - 1890
Maximum score of the remaining 6 students = 0

Step 3: Determine the average score the remaining 6 students could have achieved:
Average score of the remaining 6 students = Maximum score / Number of remaining students
Average score of the remaining 6 students = 0 / 6
Average score of the remaining 6 students = 0

Since the maximum score the remaining 6 students could have achieved is 0, it is not possible for them to have scored about 90. Therefore, it is not possible for 21 out of 27 students to have scored about 90 on their midterm.

To determine whether 21 students could have scored about 90 on their midterm while the remaining students averaged 70, we can use the concept of weighted averages.

Here's how we can calculate if it is possible:

Step 1: Calculate the total number of points earned by the 27 students.
Total points = Number of students × Average score
Total points = 27 × 70 = 1890

Step 2: Calculate the maximum possible points earned by those 21 students.
Maximum possible points = Number of students × Desired score
Maximum possible points = 21 × 90 = 1890

Step 3: Compare the total points with the maximum possible points.
If the total points are equal to or greater than the maximum possible points, it is possible that 21 students scored about 90.

In this case, since the total points and the maximum possible points are equal (both 1890), it is indeed possible that 21 students scored about 90 on their midterm while the remaining students averaged 70.