A data set consists of 7 positive integers. The smallest integer in the set is 4, the rang… — GMAT

A data set consists of $7$ positive integers. The smallest integer in the set is $4$ , the range of the set is $19$ , the median is $12$ , and the set has a unique mode of $8$ . What is the maximum possible arithmetic mean of the $7$ integers?

A
13
B
13.5
14Cevap
D
14.5
E
15

Cevap

The maximum possible arithmetic mean of the 7 integers is 14.

The correct answer of 14 is obtained by setting up the 7 ordered integers as

x_1, x_2, x_3, x_4, x_5, x_6, x_7

. We know

x_1 = 4

, median

x_4 = 12

, and

x_7 = 4 + 19 = 23

. Since 8 is the unique mode and

8 < 12

, 8 must fill

x_2

and

x_3

, giving it a frequency of 2. For 8 to remain the unique mode, no other element can appear more than once. Thus,

x_5, x_6, x_7

must all be distinct. To maximize the mean, we maximize

x_5

and

x_6

under

12 < x_5 < x_6 < 23

, yielding

x_5 = 21

and

x_6 = 22

. The maximum sum is

4 + 8 + 8 + 12 + 21 + 22 + 23 = 98

, giving a mean of

98 / 7 = 14

Adım Adım Çözüm

Determine the fixed values in the ordered 7-element data set.

Let the 7 integers in non-decreasing order be

x_1, x_2, x_3, x_4, x_5, x_6, x_7

. Given

x_1 = 4

, median

x_4 = 12

, and range

x_7 - x_1 = 19

, we find

x_7 = 4 + 19 = 23

The median of a 7-element set is the 4th element, and range is maximum minus minimum.

Determine the frequency and values of the mode.

Since the unique mode is

8

and

8 < 12

, the value

8

must occupy positions

x_2

and

x_3

, so

x_2 = 8

and

x_3 = 8

. Thus,

8

appears exactly twice.

Position 1 is 4 and position 4 is 12, leaving only positions 2 and 3 for the mode value of 8.

Apply the unique mode constraint to maximize remaining terms.

Because

8

appears twice and is the unique mode, no other number can appear 2 or more times. Hence, all other elements must be distinct. To maximize the sum, we choose the largest distinct integers for

x_5

and

x_6

such that

12 < x_5 < x_6 < 23

, giving

x_5 = 21

and

x_6 = 22

Allowing any other value to repeat would create a second mode or a new unique mode, violating the problem conditions.

Calculate the maximum sum and arithmetic mean.

The maximal set is

\{4, 8, 8, 12, 21, 22, 23\}

. The sum is

4 + 8 + 8 + 12 + 21 + 22 + 23 = 98

. The maximum mean is

98 / 7 = 14

Dividing the maximum possible sum by the total number of elements yields the maximum mean.

Anahtar Kavram

Mean, Median, and Mode Constraints

Soru

Cevap

Adım Adım Çözüm

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