A dataset D consists of 11 distinct positive integers. The median of D is 40, and the arit… — GRE General Test

A dataset $D$ consists of $11$ distinct positive integers. The median of $D$ is $40$ , and the arithmetic mean of $D$ is $45$ . A new dataset $D'$ is created by increasing each of the $5$ largest integers in $D$ by $10$ and decreasing each of the $5$ smallest integers in $D$ by a positive integer $x$ . If the median of $D'$ is strictly less than the arithmetic mean of $D'$ , what is the maximum possible integer value of $x$ ?

$20$ Answer
B
$21$
C
$19$
D
$25$
E
$10$

Answer

The maximum possible integer value of

x

20

The sum of the original

11

values is

11 \times 45 = 495

. When the

5

largest integers are each increased by

10

, the sum increases by

+50

. When the

5

smallest integers are each decreased by

x

, the sum decreases by

-5x

. The new sum is

545 - 5x

, making the new mean

\frac{545 - 5x}{11}

. Because

D

has

11

elements, the median is the

6^{\text{th}}

element. Changing the smallest

5

and largest

5

elements does not alter the value of the

6^{\text{th}}

element, so the median remains

40

. Requiring the median to be strictly less than the new mean gives

40 < \frac{545 - 5x}{11} \implies 440 < 545 - 5x \implies 5x < 105 \implies x < 21

. The greatest integer less than

21

20

Step-by-Step Solution

Calculate the total sum of the original dataset

D

Since

D

has

11

elements with a mean of

45

, the sum of elements is

11 \times 45 = 495

The mean formula is

\text{Mean} = \frac{\text{Sum}}{n}

, so

\text{Sum} = n \times \text{Mean}

Determine the median and sum of the modified dataset

D'

The median remains the

6^{\text{th}}

element, which is

40

. The sum of

D'

495 + 5(10) - 5(x) = 545 - 5x

Modifying only the

5

smallest and

5

largest elements leaves the

6^{\text{th}}

central element unchanged.

Set up and solve the inequality comparing the median to the mean of

D'

Solve

40 < \frac{545 - 5x}{11} \implies 440 < 545 - 5x \implies 5x < 105 \implies x < 21

The problem specifies that the median must be strictly less than the mean.

Identify the maximum integer value satisfying the inequality.

The largest integer strictly less than

21

20

x

must be an integer.

Key Concept

Effect of data modifications on mean and median

Estimated Time:2m 30s

Question

Answer

Step-by-Step Solution

Key Concept

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