A dataset W consists of 10 distinct positive integers. The arithmetic mean of the dataset … — GRE General Test

A dataset $W$ consists of $10$ distinct positive integers. The arithmetic mean of the dataset is $24$ , and its median is $22$ . A new dataset $V$ is created by replacing every integer $x$ in $W$ that is strictly less than the median of $W$ with $2x + 1$ , while leaving all other integers in $W$ unchanged. If the sum of the $5$ smallest integers in $W$ is $60$ , what is the arithmetic mean of the integers in dataset $V$ ?

A
28.5
B
30.0
30.5Answer
D
31.0
E
36.5

Answer

30.5

The total sum of the original dataset

W

10 \times 24 = 240

. Since

W

consists of

10

distinct integers, exactly

5

elements are strictly less than the median. The sum of these

5

elements is given as

60

, which leaves the sum of the remaining

5

elements as

240 - 60 = 180

. When each of the

5

lower elements

x

is replaced with

2x + 1

, their new sum becomes

2(60) + 5(1) = 125

. The sum of the new dataset

V

125 + 180 = 305

, and its mean is

305 / 10 = 30.5

Step-by-Step Solution

Calculate the total sum of the original dataset

W

Since dataset

W

has

10

elements with a mean of

24

, its total sum is

10 \times 24 = 240

The sum of elements in any set is equal to the number of elements multiplied by the arithmetic mean.

Determine how many elements are strictly less than the median of

W

Because all

10

elements in

W

are distinct positive integers, ordering them as

x_1 < x_2 < x_3 < x_4 < x_5 < x_6 < x_7 < x_8 < x_9 < x_{10}

places the median between

x_5

and

x_6

. Thus, exactly

5

elements (

x_1

through

x_5

) are strictly less than the median.

In an even-sized set of distinct values, exactly half of the elements lie strictly below the median position.

Find the sum of the remaining

5

elements (the upper half) in

W

Given that the sum of the

5

smallest elements is

60

, the sum of the remaining

5

elements is

240 - 60 = 180

The total sum of the dataset is the sum of its lower

5

elements plus the sum of its upper

5

elements.

Calculate the new sum of the transformed

5

smallest elements in dataset

V

Each of the

5

elements

x_i

is replaced by

2x_i + 1

. The new sum is

\sum_{i=1}^5 (2x_i + 1) = 2 \sum_{i=1}^5 x_i + 5(1) = 2(60) + 5 = 125

Linear transformations applied to individual elements scale their sum by the multiplier and add the constant term multiplied by the number of elements.

Compute the total sum and arithmetic mean of dataset

V

The total sum of

V

125 + 180 = 305

. The new mean is

\frac{305}{10} = 30.5

The mean of

V

is its total sum divided by the number of elements (

10

Key Concept

Linear transformation of dataset subsets and median position in distinct ordered sets

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