A data set consists of five positive integers a, b, c, d, and e, written in non-decreasing… — GMAT

A data set consists of five positive integers $a, b, c, d,$ and $e$ , written in non-decreasing order ( $a \le b \le c \le d \le e$ ). The set has a unique mode of $8$ , a median of $10$ , and an arithmetic mean of $12$ . If the range of the set is $14$ , what is the value of the largest integer $e$ ?

A
17
B
20
22Answer
D
24
E
26

Answer

The median of a 5-element ordered set is the middle element, so

c = 10

. Because 8 is the unique mode and is strictly less than 10, the first two numbers must both be 8 (

a = 8, b = 8

). The range is 14, meaning the difference between the maximum element

e

and the minimum element

a

is 14. Therefore,

e = 8 + 14 = 22

Step-by-Step Solution

Determine the median value

c

c = 10

In an ordered 5-element set, the median is the 3rd element.

Determine the values of

a

and

b

using the unique mode

a = 8

and

b = 8

Since 8 is the unique mode and

8 < 10

, 8 must appear at least twice in the first two positions.

Calculate the largest integer

e

using the given range

e = a + 14 = 8 + 14 = 22

The range of a set is the difference between the maximum (

e

) and minimum (

a

) values.

Verify consistency with the mean

Sum

= 5 \times 12 = 60

;

d = 60 - (8 + 8 + 10 + 22) = 12

, which satisfies

10 \le 12 \le 22

Ensures all set conditions (mean of 12, non-decreasing order, unique mode) are satisfied.

Key Concept

Relating Mean, Median, Mode, and Range in Ordered Data Sets

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