A data set consists of 8 positive integers with an arithmetic mean of 15 and a unique mode… — GMAT

A data set consists of $8$ positive integers with an arithmetic mean of $15$ and a unique mode of $12$ . What is the maximum possible value of an integer in this data set?

Cevap: 81

Cevap

The total sum of the

8

integers is

8 \times 15 = 120

. To maximize the single largest integer, the sum of the remaining

7

integers must be minimized. Since

12

is the unique mode,

12

must appear at least twice, and no other value can appear more than once. The smallest positive integers that can occupy the first

5

positions without creating another mode are

1, 2, 3, 4,

and

5

. Placing

12

in the

6\text{th}

and

7\text{th}

positions minimizes the sum of the first

7

elements to

1 + 2 + 3 + 4 + 5 + 12 + 12 = 39

. Thus, the maximum possible value for the largest integer is

120 - 39 = 81

Adım Adım Çözüm

Calculate the total sum of the 8 positive integers.

Total Sum =

8 \times 15 = 120

The sum of a set of numbers equals the arithmetic mean multiplied by the number of elements.

Minimize the sum of the first 7 integers to maximize the 8th integer.

Minimum sum of the first 7 integers =

1 + 2 + 3 + 4 + 5 + 12 + 12 = 39

To maximize the largest integer, the remaining 7 integers must be as small as possible. Since 12 is the unique mode, 12 must appear at least twice, and no other integer can appear more than once. Placing two 12s at the highest available positions among the 7 terms (

a_6 = 12

and

a_7 = 12

) and selecting the smallest distinct positive integers (

1, 2, 3, 4, 5

) for the first 5 terms minimizes their total sum.

Subtract the minimum sum of the 7 smallest terms from the total sum.

Maximum integer =

120 - 39 = 81

Subtracting the smallest possible sum of 7 terms from the fixed sum of 120 yields the maximum possible value for the 8th term.

Anahtar Kavram

Maximizing an element in a data set under mean and mode constraints

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