M21-02

Official Solution:


What is the value of the median of data set \(S =\{a - b, b - a, a + b\}\) ?

(1) The average (arithmetic mean) of set \(S\) is equal to \(a + b\).

The above implies that \(\frac{(a - b)+(b - a)+(a + b)}{3}=a+b\), which leads to \(a+b=0\). In this case, \(a - b = a - (-a) = 2a\) and \(b - a = (-a) - a = -2a\). Hence, the data set becomes \(S =\{2a, -2a, 0\}\).

• If \(a < 0\), then \(2a\) will be negative and \(-2a\) will be positive, making data set S in ascending order \(\{negative, 0, positive\}\).

• If \(a = 0\), then both \(2a\) and \(-2a\) will be 0, making data set S \(\{0, 0, 0\}\).

• If \(a > 0\), then \(2a\) will be positive and \(-2a\) will be negative, making data set S in ascending order \(\{negative, 0, positive\}\).

In any case, the median of \(S\) is 0. Sufficient.

(2) The range of set \(S\) is equal to \(2b\).

If \(a=b=0\), then the median of \(S\) is 0, but if \(a=0\) and \(b=1\), then the median of \(S\) is 1. Not sufficient.


Answer: A