Problem Solving Question: Descriptive Statistics

This question combines core concepts of statistics including mean and median. It is a typical example of a question which requires you to iterate to get to the correct answer.

Question

  • PS
  • Three positive integers a, b, and c are such that their average is 20 and a ≤ b ≤ c. If the median is (a + 11), what is the least possible value of c? A. 23
    B. 21
    C. 25
    D. 26
    E. 24

 

  • Correct Answer
    Choice C. The least possible value for c is 25.

Explanatory Answer

Detailed Solution

Given Data

  1. 3 positive integers
  2. Average of a, b, and c = 20.
    So, \\frac{a + b + c}{3}\\) = 20 or a + b + c = 60
  3. a ≤ b ≤ c
  4. Median = (a + 11)

Median of these 3 numbers a, b, and c is ‘b’ because a ≤ b ≤ c
Therefore, b = a + 11

Objective To find the least value of 'c'

Theoretically, the least value of c is when c = b.
Therefore, a + (a + 11) + (a + 11) = 60 (b and c are equal and b, the median, is a + 11)
Or 3a = 38 or a = 12.66
So, b = c = 12.66 + 11 = 23.66

However, we know that these numbers are all integers.
Therefore, a, b, and c cannot take these values.
So, the least value for c with this constraint is NOT likely to be when c = b

Let us increment c by 1. Let c = (b + 1)
In this scenario, a + (a + 11) + (a + 12) = 60
Or 3a = 37. The value of the numbers is not an integer in this scenario as well.

Let us increment c again by 1. i.e., c = b + 2
Now, a + (a + 11) + (a + 13) = 60
Or 3a = 36 or a = 12.
If a = 12, b = 23 and c = 25.
The least value for c that satisfies all these conditions is 25.

Correct answer is choice C.