A data set consists of seven integers written in ascending order: 10, 12, 15, x, 26, 29, 3… — GMAT

A data set consists of seven integers written in ascending order: $10, 12, 15, x, 26, 29, 34$ . If the arithmetic mean of the data set is equal to its median, what is the value of $x$ ?

A
$18$
B
$20.5$
$21$ Answer
D
$23$
E
$25.2$

Answer

The value of

x

21

In an ordered set of seven numbers, the median is the 4th number, which is

x

. The arithmetic mean of the seven numbers is

\frac{10 + 12 + 15 + x + 26 + 29 + 34}{7} = \frac{126 + x}{7}

. Equating the mean to the median gives

\frac{126 + x}{7} = x

. Solving this linear equation yields

126 + x = 7x \implies 6x = 126 \implies x = 21

. Since

21

lies between

15

and

26

, it maintains the ascending order of the set.

Step-by-Step Solution

Identify the median of the ordered set

Median =

x

For an ordered set with an odd number of elements (

n = 7

), the median is the middle element, which is the

\frac{7 + 1}{2} = 4

th element. Since the elements are in ascending order, the 4th element is

x

Calculate the sum of all elements in terms of

x

Sum =

126 + x

The sum of the known six numbers is

10 + 12 + 15 + 26 + 29 + 34 = 126

. Adding

x

gives a total sum of

126 + x

Set up the equation equating mean and median

\frac{126 + x}{7} = x

The arithmetic mean is defined as the total sum divided by the total number of elements (

7

). We are given that Mean = Median.

Solve for

x

and verify ordering bounds

x = 21

Multiplying both sides by

7

gives

126 + x = 7x

, so

6x = 126

, yielding

x = 21

. Checking the ascending order condition

15 \le x \le 26

21

satisfies the condition

15 \le 21 \le 26

Key Concept

Mean and Median of an Ordered Set

Estimated Time:1m 30s

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