A list of 15 positive integers is sorted in non-decreasing order. The minimum value in the… — ACT

A list of $15$ positive integers is sorted in non-decreasing order. The minimum value in the list is $10$ , the maximum value is $50$ , and the median is $30$ . If the unique mode of the list is $20$ , what is the maximum possible sum of the $15$ integers in the list?

A
488
B
503
C
506
508Answer
E
518

Answer

508

The maximum sum is achieved by setting the frequency of the unique mode

20

to its maximum limit of

6

, occupying positions

2

through

7

(since the first element is

10

and the median at position

8

30

). This allows any other value to appear at most

5

times. To maximize the sum, we assign the maximum value

50

5

positions (positions

11

15

), and the next highest value

49

to the remaining

2

positions in the upper half (positions

9

and

10

). Summing these values gives

10 + 6(20) + 30 + 2(49) + 5(50) = 508

Step-by-Step Solution

Identify fixed elements based on the median, minimum, and maximum constraints.

For

15

sorted integers, the median is the 8th term:

x_8 = 30

. The minimum is

x_1 = 10

, and the maximum is

x_{15} = 50

Establishing these boundary points helps constrain the values of the other elements in the sorted list.

Determine the maximum possible frequency of the unique mode

20

Since

x_8 = 30

and the list is sorted,

20

can only occupy positions

2

through

7

(a maximum of

6

occurrences).

To maximize the overall sum, we want to maximize the frequency of the mode so that other large values (like

50

) can also appear as many times as possible.

Determine the maximum allowed frequency of other values under the unique mode constraint.

With the frequency of

20

set to

6

, any other number can appear at most

5

times.

Since

20

is the unique mode, no other value in the list can have a frequency equal to or greater than

6

Assign the largest possible values to the remaining positions in the upper half (

x_9

x_{14}

Set

x_{11} = x_{12} = x_{13} = x_{14} = x_{15} = 50

(frequency of

5

). The remaining two positions,

x_9

and

x_{10}

, are set to the next largest integer,

49

This maximizes the sum of the upper half of the list without violating the maximum frequency limit of

5

for any non-mode value.

Sum the elements of the optimal list.

Sum =

10 + 6(20) + 30 + 2(49) + 5(50) = 10 + 120 + 30 + 98 + 250 = 508

Adding all fifteen optimal values together gives the absolute maximum possible sum.

Key Concept

Using central tendency constraints (median, mode) and range limits to optimize a dataset sum.

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