A set of 7 positive integers has an arithmetic mean of 16, a median of 15, a unique mode o… — GMAT

A set of $7$ positive integers has an arithmetic mean of $16$ , a median of $15$ , a unique mode of $12$ , and a range of $14$ . What is the maximum possible value of the largest integer in this set?

Cevap: 26

Cevap

The maximum possible value of the largest integer in the set is 26.

The total sum of the 7 positive integers is

7 \times 16 = 112

. Ordering the terms as

a \le b \le c \le d \le e \le f \le g

, the median constraint gives

d = 15

. The range constraint gives

g - a = 14

, or

g = a + 14

. To maximize

g

, we need to maximize

a

. Because 12 is the unique mode of the set, 12 must appear at least twice. Since

d = 15

, the number 12 can only occupy positions

a, b,

c

, which means

a \le 12

. Setting

a = 12

gives the maximum value

g = 12 + 14 = 26

. A valid set achieving this is

\{12, 12, 12, 15, 15, 20, 26\}

, which sums to 112 and meets all statistical constraints.

Adım Adım Çözüm

Determine the sum of the set

Sum = 112

The mean of 7 positive integers is 16, so the sum is 7 times 16.

Apply median and range constraints

d = 15 and a = g - 14

In a sorted set of 7 integers, the 4th element is the median (15), and range is the difference between the largest element g and smallest element a.

Bound the smallest element using the mode constraint

a <= 12, so max g = 12 + 14 = 26

Since 12 is the unique mode, it must occur at least twice. Because elements are sorted and median is 15, 12 must be among the first three terms, so the smallest element a cannot exceed 12.

Verify existence of a valid set with g = 26

Set {12, 12, 12, 15, 15, 20, 26} satisfies all conditions

The sum is 112, median is 15, mode is 12 (appears 3 times), and range is 26 - 12 = 14.

Anahtar Kavram

Extremal problems involving mean, median, mode, and range constraints

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