The average points accumulated by 6 contestants is 87.5. They have all attained different points which are whole numbers. The lowest point is 78 while the highest point is 93.
- Find the average points achieved by the 4 contestants whose marks lie between the highest and the lowest.
- Find the smallest possible second lowest mark among these 6 contestants.
- Find the largest possible second lowest mark among these 6 contestants.
(a)
Total points of 6 contestants
= 6 x 87.5
= 525
Total points of the rest of the 4 contestants
= 525 - 93 - 78
= 354
Average points of 4 contestants
= 354 ÷ 4
= 88.5
(b)
To have the smallest possible second lowest mark among the 6 contestants, the 3rd contestant to the 6th contestant must have as high a score as possible and the 2nd contestant must have the second lowest score among the 6 contestants.
So the 6th contestant will have the highest score. The 5th contestant will have 1 mark will less than 6th contestant and this pattern continues to the 3rd contestant.
1st contestant's score = 78
2nd contestant's score = ?
3rd contestant's score = 93 - 3 = 90
4th contestant's score = 93 - 2 = 91
5th contestant's score = 93 - 1 = 92
6th contestant's score = 93
The smallest possible second lowest mark among these 6 contestants
= 354 - 92 - 91 - 90
= 81
(c)
To have the largest possible second lowest mark among these 6 contestants, the 2nd contestant to the 5th contestant must have as high a score as possible but lower than the 6th contestant.
So the 2nd contestant 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 contestant's score = 78
2nd contestant's score = 1 u
3rd contestant's score = 1 u + 1
4th contestant's score = 1 u + 2
5th contestant's score = 1 u + 3
6th contestant's score = 93
1 u + (1 u + 1) + (1 u + 2) + (1 u + 3) = 354
4 u + 6 = 354
4 u = 354 - 6
4 u = 348
1 u = 348 ÷ 4 = 87
87 x 4 + 6 = 354 (Equal to 354)
Largest possible second lowest mark among these 6 contestants = 87
Answer(s): (a) 88.5; (b) 81; (c) 87
