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

A dataset SS consists of 77 distinct positive integers arranged in ascending order: x1,x2,x3,x4,x5,x6,x7x_1, x_2, x_3, x_4, x_5, x_6, x_7. The arithmetic mean of the entire dataset is 2828, and the median is 2525. The arithmetic mean of the 33 smallest integers in SS is 1212. If MM is the maximum possible value of x7x_7 and mm is the minimum possible value of x7x_7, what is the value of M−mM - m?

  1. A
    32
  2. 36Answer
  3. C
    37
  4. D
    40
  5. E
    52

Answer

The value of M−mM - m is 3636.
The sum of all 77 distinct positive integers is 7×28=1967 \times 28 = 196. Since the dataset is ordered and has 77 elements, the median is x4=25x_4 = 25. The sum of the smallest 33 integers is 3×12=363 \times 12 = 36. Therefore, the sum of the remaining three integers x5+x6+x7=196−36−25=135x_5 + x_6 + x_7 = 196 - 36 - 25 = 135.

To maximize x7x_7, x5x_5 and x6x_6 must be as small as possible. Since all elements are distinct integers greater than x4=25x_4 = 25, the smallest possible values are x5=26x_5 = 26 and x6=27x_6 = 27. Thus, M=135−26−27=82M = 135 - 26 - 27 = 82.

To minimize x7x_7, x5x_5, x6x_6, and x7x_7 must be as close together as possible while preserving 25<x5<x6<x725 < x_5 < x_6 < x_7. Dividing 135135 by 33 gives 4545. The consecutive integers centered around 4545 are 44,45,4644, 45, 46, which sum to 135135 and satisfy all inequalities. Thus, m=46m = 46.

The difference M−m=82−46=36M - m = 82 - 46 = 36, which corresponds to the value 3636.

Step-by-Step Solution

1
Calculate the total sum of all 7 integers in dataset S.
Total sum = 7×28=1967 \times 28 = 196.
The arithmetic mean of nn numbers is the total sum divided by nn.
2
Identify the median value and the sum of the smallest 3 integers.
Median x4=25x_4 = 25, and x1+x2+x3=3×12=36x_1 + x_2 + x_3 = 3 \times 12 = 36.
For an odd number of ordered elements (77), the middle term x4x_4 is the median. The mean of the first 3 terms gives their sum.
3
Determine the sum of the top 3 integers (x5+x6+x7)(x_5 + x_6 + x_7).
x5+x6+x7=196−36−25=135x_5 + x_6 + x_7 = 196 - 36 - 25 = 135.
Subtracting x1+x2+x3x_1 + x_2 + x_3 and x4x_4 from the total sum leaves the sum of the remaining three elements.
4
Calculate the maximum possible value MM of x7x_7.
M=82M = 82.
To maximize x7x_7, minimize x5x_5 and x6x_6. Since elements are distinct integers and x4=25x_4 = 25, the minimum values are x5=26x_5 = 26 and x6=27x_6 = 27. Thus x7=135−26−27=82x_7 = 135 - 26 - 27 = 82.
5
Calculate the minimum possible value mm of x7x_7.
m=46m = 46.
To minimize x7x_7, maximize x5x_5 and x6x_6 such that 25<x5<x6<x725 < x_5 < x_6 < x_7 and x5+x6+x7=135x_5 + x_6 + x_7 = 135. Setting x5=44,x6=45,x7=46x_5 = 44, x_6 = 45, x_7 = 46 gives 44+45+46=13544 + 45 + 46 = 135, maintaining strict inequalities.
6
Compute M−mM - m.
M−m=82−46=36M - m = 82 - 46 = 36.
Subtract the minimum possible value of x7x_7 from its maximum possible value.

Key Concept

Measures of Central Tendency with Extreme Value Optimization
Estimated Time:2m 30s
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Topics

Measures of Central Tendency (Mean, Median, Mode)

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