# GMAT DS - Inequalities, Descriptive Statistics

This data sufficiency question tests your understanding of Inequalities and descriptive statistics.

Question

Is 'b' the median of 3 numbers a, b, and c?
1. b/a = c/b
2. ab < 0

The correct answer is C. Both the statements together are sufficient to answer the question.

When the numbers a, b, and c are arranged in an ascending order, the middle number is the median. We need to determine if 'b' is the median of these 3 numbers.

Statement 1:
b/a = c/b
i.e., b^2 = ac
So, we can conclude that a, b and c are in a geometric progression with 'b' as their geometric mean.

For 3 positive numbers a, b and c that are in a geometric progression, b will be the geometric mean and the median.

However, we do not know if all 3 numbers a, b and c are positive
Hence, we cannot determine if 'b' is the median of these 3 numbers.
Hence, Statement 1 alone is NOT sufficient.

Statement 2:
ab < 0

The product of two numbers is negative if one of the numbers is negative and the other is positive. So, from this statement we can conclude that one of a or b is negative.
However, that is not sufficient to determine whether b is the median of the 3 numbers.

For instance, a = -4, b = 5 and c = 10, then b will be the median.
Conversely, a = -4 , b = 5 and c = -15, then a will be the median.

Hence, statement 2 alone is NOT sufficient.

Combining the two statements

We know from statement 1 that b is the geometric mean of a, b and c.
We know from statement 2 that one of a or b is negative.
Therefore, we can conclude that not all three numbers - a, b and c are positive.

If one or more of the 3 numbers happen to be negative numbers, then b will not be the median of these numbers.

We can therefore, answer conclusively using the two statements that 'b' is not the median.

Hence, combining the information given in the two statements is SUFFICIENT to answer the question.