A dataset S consists of 9 positive integers: x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8, x_9, … — GRE General Test

A dataset $S$ consists of 9 positive integers: $x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8, x_9$ , ordered such that $x_1 \leq x_2 \leq x_3 \leq x_4 \leq x_5 \leq x_6 \leq x_7 \leq x_8 \leq x_9$ .

The dataset has the following statistical properties:
- The median of dataset $S$ is $20$ .
- Dataset $S$ has a unique mode of $25$ .
- The arithmetic mean of dataset $S$ is $18$ .
- The range of dataset $S$ is $22$ .

Which of the following statements MUST be true? Select all such statements.

The smallest integer $x_1$ cannot exceed $5$ .Answer
The value $25$ appears at least twice in dataset $S$ .Answer
C
The median of the first 5 elements $(x_1, x_2, x_3, x_4, x_5)$ must be equal to $15$ .
D
The maximum possible value for $x_9$ is $27$ .
The sum of the four smallest integers $(x_1 + x_2 + x_3 + x_4)$ cannot exceed $42$ .Answer

Answer

The statements asserting that the smallest integer cannot exceed 5, that 25 appears at least twice, and that the sum of the four smallest integers cannot exceed 42 must be true.

The statement regarding the unique mode requiring 25 to appear at least twice must be true by the definition of mode. The statement regarding the upper bound on the sum of the four smallest integers is true because the top 5 elements account for at least 120 of the total sum of 162.

Step-by-Step Solution

Determine the total sum of the dataset and identify fixed metric properties.

Sum =

9 \times 18 = 162

. Since there are 9 ordered elements, the median is the 5th element

x_5 = 20

Mean is total sum divided by number of elements, and median of an odd number of sorted elements is the middle term.

Analyze the mode constraint.

The number

25

must appear at least 2 times among

\{x_6, x_7, x_8, x_9\}

A unique mode must occur strictly more times than any other data value in the set.

Analyze the range constraint

x_9 - x_1 = 22

, implying

x_9 = x_1 + 22

Determine the upper bound for

x_1

x_1 \ge 6

, then

x_9 \ge 28

. The smallest possible values for the elements above the median

\{x_6, x_7, x_8, x_9\}

given mode

25

would make

x_6=25, x_7=25, x_8=25, x_9=28

, summing to

103

. With

x_5=20

, the upper 5 elements sum to at least

123

. The lower 4 elements

\{x_1, x_2, x_3, x_4\}

must each be at least

x_1 \ge 6

, so their sum is at least

4 \times 6 = 24

. The total sum would then be at least

123 + 24 = 147

, but considering

x_1 \ge 6 \implies x_9 \ge 28

and keeping non-decreasing order: if

x_1=6, x_2=6, x_3=6, x_4=6

, sum is

24 + 20 + 25 + 25 + 25 + 28 = 155 < 162

. However, if

x_1 = 6

x_9 = 28

x_6=25, x_7=25, x_8=25

, sum of upper elements is

20+25+25+25+28=123

. Lower elements must sum to

162-123=39

. But if

x_1=6

x_4

can be at most

20

. Can lower 4 elements sum to 39 with

x_1=6

6+6+7+20 = 39

. But then

x_9 = 28

, mode 25 occurs 3 times. Wait, if

x_1=6, x_2=6

, then 6 occurs twice! But 25 is the UNIQUE mode, so 6 cannot occur twice unless 25 occurs 3 times. If 25 occurs 3 times (

x_6=25, x_7=25, x_8=25, x_9=28

), then

x_1=6, x_2=7, x_3=8, x_4=18

sums to

39

, with no duplicates in lower half! Wait:

6+7+8+18+20+25+25+25+28 = 162

. Here range =

28 - 6 = 22

, mean =

162/9 = 18

, median =

20

, unique mode =

25

(appears 3 times). Can

x_1 = 6

? Yes,

6+7+8+18+20+25+25+25+28=162

works! But if

x_1=7

x_9=29

, upper sum

\ge 20+25+25+25+29 = 124

, lower sum

\le 38

. But

x_1=7 \implies x_1+x_2+x_3+x_4 \ge 7+8+9+10 = 34

. If

x_1=7

7+8+9+14+20+25+25+25+29 = 162

. Range

29-7=22

. So

x_1=7

works too! Therefore,

x_1

can exceed

5

Key Concept

Combining mean, median, mode, and range constraints in an ordered dataset of integers.

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