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Difficulty: MediumData Distributions and Measures

A set of 5 positive integers has a mean of 12, a median of 10, and a unique mode of 8. What is the maximum possible value of the largest integer in this set?

Answer: 23

Answer

The maximum possible value of the largest integer in the set is 23.
To find the maximum possible value of the largest integer in a set of 5 positive integers with a mean of 12, a median of 10, and a unique mode of 8: First, calculate the total sum of the integers, which is 12×5=6012 \times 5 = 60. Let the sorted integers be x1≤x2≤x3≤x4≤x5x_1 \le x_2 \le x_3 \le x_4 \le x_5. The median is the middle term, so x3=10x_3 = 10. Since 8 is the unique mode and is less than the median, it must appear at least twice in the first two slots, so x1=8x_1 = 8 and x2=8x_2 = 8. The sum of the remaining two integers is 60−(8+8+10)=3460 - (8 + 8 + 10) = 34. To maximize the largest integer x5x_5, we must minimize x4x_4. Since the integers are sorted, x4≥x3=10x_4 \ge x_3 = 10. However, if x4=10x_4 = 10, then 10 would appear twice, making it a second mode alongside 8, which violates the unique mode condition. Thus, the smallest possible integer value for x4x_4 is 11, which gives a maximum possible value of 34−11=2334 - 11 = 23 for the largest integer.

Step-by-Step Solution

1
Calculate the sum of the five integers.
The sum of the five integers is 60.
Since the mean of 5 numbers is 12, their sum is 12×5=6012 \times 5 = 60.
2
Set up the ordered list of integers and identify the median.
For integers x1≤x2≤x3≤x4≤x5x_1 \le x_2 \le x_3 \le x_4 \le x_5, the median is x3=10x_3 = 10.
The median of 5 numbers in ordered sequence is the third number.
3
Determine the values of the first two integers using the mode constraint.
x1=8x_1 = 8 and x2=8x_2 = 8.
The mode must be unique and equal to 8. Since 8 is less than the median 10, 8 must occupy the first two spots to appear more than once.
4
Write the sum equation for the remaining unknown integers.
x4+x5=34x_4 + x_5 = 34.
Since the total sum is 60, we subtract the known values: 60−(8+8+10)=3460 - (8 + 8 + 10) = 34.
5
Minimize the fourth integer to maximize the fifth integer while keeping the unique mode of 8.
The minimum value for x4x_4 is 11, which gives the maximum value for x5x_5 as 23.
We must have x4≥10x_4 \ge 10 due to ordering. If x4=10x_4 = 10, 10 would be a mode, so the minimum valid integer for x4x_4 is 11.

Key Concept

Using measures of center (mean, median) and measures of frequency (mode) to determine constraints on individual data values in a data distribution.

Alternative Method

Instead of setting up inequalities, one can test integers starting from the maximum mathematical limit if there were no mode constraint (which is 24, since x4≥10x_4 \ge 10, giving 60−26−10=2460 - 26 - 10 = 24). Testing 24 yields the set {8, 8, 10, 10, 24}, which has two modes. The next highest value to test is 23, which yields the valid set {8, 8, 10, 11, 23}.
Estimated Time:1m 30s
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Topics

Data Distributions and Measures

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