The average points accumulated by 6 students is 82.5. They have all attained different points which are whole numbers. The lowest point is 54 while the highest point is 91.
- Find the average points achieved by the 4 students whose marks lie between the highest and the lowest.
- Find the smallest possible second lowest mark among these 6 students.
- Find the largest possible second lowest mark among these 6 students.
(a)
Total points of 6 students
= 6 x 82.5
= 495
Total points of the rest of the 4 students
= 495 - 91 - 54
= 350
Average points of 4 students
= 350 ÷ 4
= 87.5
(b)
To have the smallest possible second lowest mark among the 6 students, the 3rd student to the 6th student must have as high a score as possible and the 2nd student must have the second lowest score among the 6 students.
So the 6th student will have the highest score. The 5th student will have 1 mark will less than 6th student and this pattern continues to the 3rd student.
1st student's score = 54
2nd student's score = ?
3rd student's score = 91 - 3 = 88
4th student's score = 91 - 2 = 89
5th student's score = 91 - 1 = 90
6th student's score = 91
The smallest possible second lowest mark among these 6 students
= 350 - 90 - 89 - 88
= 83
(c)
To have the largest possible second lowest mark among these 6 students, the 2nd student to the 5th student must have as high a score as possible but lower than the 6th student.
So the 2nd student will have the highest possible second lowest score. The 3rd child must have at least 1 mark higher than the 2nd child and the pattern continues to the 5th child.
1st student's score = 54
2nd student's score = 1 u
3rd student's score = 1 u + 1
4th student's score = 1 u + 2
5th student's score = 1 u + 3
6th student's score = 91
1 u + (1 u + 1) + (1 u + 2) + (1 u + 3) = 350
4 u + 6 = 350
4 u = 350 - 6
4 u = 344
1 u = 344 ÷ 4 = 86
86 x 4 + 6 = 350 (Equal to 350)
Largest possible second lowest mark among these 6 students = 86
Answer(s): (a) 87.5; (b) 83; (c) 86