The arithmetic mean of the list of numbers above is \(4.\) If \(k\) and \(m\) are integers and \(k \ne m,\) what is the

M7MBA wrote: ↑

Wed Jun 24, 2020 6:40 am

\(\{3, k, 2, 8, m, 3\}\)

The arithmetic mean of the list of numbers above is \(4.\) If \(k\) and \(m\) are integers and \(k \ne m,\) what is the median of the list?

(A) 2
(B) 2.5
(C) 3
(D) 3.5
(E) 4

[spoiler]OA=C[/spoiler]

Source: Official Guide

The arithmetic mean of the list of numbers above is 4.
So, (3 + k + 2 + 8 + m + 3)/6 = 4
Multiply both sides by 6 to get: 3 + k + 2 + 8 + m + 3 = 24
Simplify: 16 + k + m = 24
Subtract 16 from both sides to get: k + m = 8

If k and m are integers k ≠ m, what is the median of the list?
Let's assign some values to k and m that satisfy the above condition AND such that k + m = 8
How about k = 1 and m = 7

So, our set of values becomes {3, 1, 2, 8, 7, 3}

What is the median of the list?
Arrange numbers in ASCENDING ORDER to get: { 1, 2, 3, 3, 7, 8}
Since we have an EVEN number of values, the median will equal the AVERAGE of the 2 middlemost values
Median = (3 + 3)/2 = 6/2 = 3

Answer: C

M7MBA wrote: ↑

Wed Jun 24, 2020 6:40 am

\(\{3, k, 2, 8, m, 3\}\)

The arithmetic mean of the list of numbers above is \(4.\) If \(k\) and \(m\) are integers and \(k \ne m,\) what is the median of the list?

(A) 2
(B) 2.5
(C) 3
(D) 3.5
(E) 4

[spoiler]OA=C[/spoiler]

Source: Official Guide

Solution:

Using the average formula: average = sum/number, we see that the sum of these numbers is 24. Thus we have:

3 + k + 2 + 8 + m + 3 = 24

16 + k + m = 24

k + m = 8

Since k ≠ m and assuming that k < m, then the ordered pairs of (k, m) could be (3, 5), (2, 6), (1, 7), (0, 8), etc.

Let’s investigate the possible ordered pairs further:

If (k, m) = (3, 5), then the numbers in ascending order are:

2, 3, 3, 3, 5, 8 --- with median = 3

If (k, m) = (2, 6), then the numbers in ascending order are:

2, 2, 3, 3, 6, 8 --- with median = 3

If (k, m) = (1, 7), then the numbers in ascending order are:

1, 2, 3, 3, 7, 8 --- with median = 3

If (k, m) = (0, 8), then the numbers in ascending order are:

0, 2, 3, 3, 8, 8 --- with median = 3

At this point, we can see that no matter how we “stretch” k and m (e.g., let’s say (k, m) = (-92, 100)), we would still have median = 3.

Answer: C