A data set S consists of 11 positive integers. The set has an arithmetic mean of 15, a med… — GMAT

A data set $S$ consists of $11$ positive integers. The set has an arithmetic mean of $15$ , a median of $14$ , and a unique mode of $18$ . If $M$ is the maximum possible value of an element in $S$ and $m$ is the minimum possible value of an element in $S$ , what is the maximum possible value of the range $M - m$ ?

A
$68$
B
$71$
C
$72$
$73$ Answer
E
$74$

Answer

The maximum possible value of the range

M - m

73

To maximize the range

M - m = x_{11} - x_1

, we minimize

x_1

and maximize

x_{11}

. Since elements are positive integers,

m = x_1 = 1

. The sum of all

11

elements is

11 \times 15 = 165

. With median

x_6 = 14

, the unique mode

18

must lie above the median. Setting the frequency of

18

3

allows other numbers to appear up to

2

times. Minimizing

x_1 \dots x_5

gives

1, 1, 2, 2, 3

(sum

9

). Minimizing

x_7 \dots x_{10}

by setting

x_7 = 14

and

x_8 = x_9 = x_{10} = 18

gives sum

68

. The sum of the first

10

terms is

9 + 14 + 68 = 91

, leaving

M = x_{11} = 165 - 91 = 74

. Thus, the maximum range is

74 - 1 = 73

Step-by-Step Solution

Calculate the sum of all elements in the set.

Sum

= 11 \times 15 = 165

The arithmetic mean of

11

elements is

15

Identify the position of the median and order the elements.

Let elements be

x_1 \le x_2 \le \dots \le x_6 \le \dots \le x_{11}

, where

x_6 = 14

For an odd number of elements (

11

), the median is the 6th element.

Determine the mode frequency constraint to maximize

x_{11} - x_1

The unique mode is

18

. If

18

appears

3

times, the maximum frequency of any other value is

2

To maximize

x_{11}

, we minimize

x_1 \dots x_{10}

. Allowing the mode to appear

3

times permits other elements to appear up to

2

times.

Minimize the sum of the lower five elements

x_1, x_2, x_3, x_4, x_5

x_1 = 1, x_2 = 1, x_3 = 2, x_4 = 2, x_5 = 3

, giving sum

= 1 + 1 + 2 + 2 + 3 = 9

The smallest positive integers with maximum frequency

2

are

1, 1, 2, 2, 3

Minimize elements

x_7, x_8, x_9, x_{10}

x_7 = 14

(frequency

2

for

14

), and

x_8 = 18, x_9 = 18, x_{10} = 18

. Sum

= 14 + 18 + 18 + 18 = 68

Since

x_6 = 14

, setting

x_7 = 14

minimizes

x_7

while keeping the frequency of

14

2

, which is less than the mode frequency of

3

Calculate the maximum value

M = x_{11}

and the range

M - m

Sum of first 10 elements

= 9 + 14 + 68 = 91

. Thus

x_{11} = 165 - 91 = 74

. Range

= 74 - 1 = 73

Subtracting the sum of the first 10 elements from the total sum gives

M

, and subtracting

m = 1

yields the range.

Key Concept

Range, Mean, Median, and Mode Constraints in Data Sets

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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