Examkin

Question

Difficulty: MediumMeasures of Central Tendency (Mean, Median, Mode)

A dataset SS consists of 7 distinct positive integers. The arithmetic mean of the numbers in SS is 20, and the median is 18. If the largest integer in SS is 35, what is the maximum possible value for the second-largest integer in SS?

  1. 34Answer
  2. B
    33
  3. C
    35
  4. D
    62
  5. E
    19

Answer

34
To find the maximum possible value of the second-largest integer, we arrange the 7 distinct positive integers in ascending order: x1<x2<x3<x4<x5<x6<x7x_1 < x_2 < x_3 < x_4 < x_5 < x_6 < x_7. The median x4=18x_4 = 18 and the largest element x7=35x_7 = 35. The total sum of all 7 elements is 7×20=1407 \times 20 = 140. Since x6x_6 must be strictly less than x7=35x_7 = 35, the maximum potential integer value for x6x_6 is 34. If x6=34x_6 = 34, the remaining four elements (x1,x2,x3,x5x_1, x_2, x_3, x_5) must sum to 87−34=5387 - 34 = 53. Choosing x5=19x_5 = 19 (the smallest integer greater than 18) leaves a sum of 34 for x1+x2+x3x_1 + x_2 + x_3, which can be satisfied by distinct positive integers such as 1, 16, and 17. Thus, 34 is achievable.

Step-by-Step Solution

1
Calculate the total sum of the 7 integers in the dataset.
Total sum = 7×20=1407 \times 20 = 140.
The sum of a dataset equals the number of elements multiplied by its arithmetic mean.
2
Identify the positions of known elements when the dataset is ordered in ascending order x1<x2<x3<x4<x5<x6<x7x_1 < x_2 < x_3 < x_4 < x_5 < x_6 < x_7.
Median x4=18x_4 = 18 and largest element x7=35x_7 = 35.
For a 7-element dataset, the median is the 4th element.
3
Calculate the combined sum of the remaining five unknown elements.
x1+x2+x3+x5+x6=140−(18+35)=87x_1 + x_2 + x_3 + x_5 + x_6 = 140 - (18 + 35) = 87.
Subtracting the median and the largest element from the total sum gives the sum of the remaining five elements.
4
Determine the theoretical upper bound for the second-largest integer x6x_6.
Since x6<x7=35x_6 < x_7 = 35 and all integers are distinct, x6≤34x_6 \le 34.
The second-largest integer must be strictly less than the largest integer.
5
Verify if x6=34x_6 = 34 can produce a valid dataset of distinct positive integers.
If x6=34x_6 = 34, then x1+x2+x3+x5=87−34=53x_1 + x_2 + x_3 + x_5 = 87 - 34 = 53. Setting x5=19x_5 = 19, x3=17x_3 = 17, x2=16x_2 = 16, and x1=1x_1 = 1 gives 1+16+17+19=531 + 16 + 17 + 19 = 53, forming the valid set {1,16,17,18,19,34,35}\{1, 16, 17, 18, 19, 34, 35\}.
Since a valid set of distinct positive integers exists satisfying all constraints, 34 is the maximum possible value.

Key Concept

Measures of Central Tendency and Data Constraints
Rate this question

Topics

Measures of Central Tendency (Mean, Median, Mode)

More questions from this topic