This practice question is a GMAT data sufficiency question from descriptive statistics. It tests your understanding of the concepts relating to medians of positive numbers, relation between median and geometric mean of 3 numbers in different scenarios - (i) when all the numbers are positive and (ii) when some numbers are negative.

This data sufficiency problem consists of a question and two statements, labeled (1) and (2), in which certain data are given. You have to decide whether the data given in the statements are sufficient for answering the question. Using the data given in the statements, plus your knowledge of mathematics and everyday facts (such as the number of days in a leap year or the meaning of the word counterclockwise), you must indicate whether -

- Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
- Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
- BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
- EACH statement ALONE is sufficient to answer the question asked.
- Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

All numbers used are real numbers.

A figure accompanying a data sufficiency question will conform to the information given in the question but will not necessarily conform to the additional information given in statements (1) and (2)

Lines shown as straight can be assumed to be straight and lines that appear jagged can also be assumed to be straight

You may assume that the positions of points, angles, regions, etc. exist in the order shown and that angle measures are greater than zero.

All figures lie in a plane unless otherwise indicated.

In data sufficiency problems that ask for the value of a quantity, the data given in the statement are sufficient only when it is possible to determine exactly one numerical value for the quantity.

Question 11: Is 'b' the median of 3 numbers a, b, and c?

- \\frac {b} {a}) = \\frac {c} {b})
- ab < 0

The Question: We have to determine whether 'b' is the median of 3 numbers a, b, and c.

__Median__ of 3 numbers is the __second number__ when the numbers are written in ascending or descending order.

In general, for a very large population, the median denotes the 50th percentile value. i.e., 50% of the data points are less than or equal to this value and the remaining 50% are more than or equal to this value. The median, essentially, is the middle number when the numbers are written in ascending or descending order.

As with any data sufficiency question, let us evaluate each of the statements independently to see if the statement is sufficient to answer the question

Evaluate Statement (1) ALONE: \\frac {b} {a}) = \\frac {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 geometric sequence or for 3 negative numbers a, b, and c, that are in geometric sequence 'b' will be the geometric mean and the median.

i.e., If the common ratio of a geometric sequence is positive, the terms of the sequence will be all positive or all negative.

However, 'b' will not be the median or the middle term if all 3 terms are not of the same sign. In other words, if the common ratio is negative 'b' will not be the median.

For e.g., let 'a', the first term of a GP, be 2 and let the common ratio be -2 (negative 2).

The 3 terms of the geometric sequence are 2, -4 and 8.

Writing the 3 terms in ascending order, we get -4, 2, and 8.

In this case, the median is 2 - which is the first term 'a'

Because, we do not know whether the 3 terms a, b, and c are all positive, we cannot determine whether 'b' is the median.

**Statement 1 ALONE is NOT sufficient.**

Eliminate choices A and D. Choices narrow down to B, C or E.

Evaluate Statement (2) ALONE: 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 and the other is positive.

However, this information alone 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.

**Statement 2 ALONE is NOT sufficient.**

Eliminate choice B. Choices narrow down to C or E.

Evaluate Statements (1) & (2) Together: \\frac {b} {a}) = \\frac {c} {b}) & ab < 0

*If either statement 1 alone or statement 2 alone had provided us with a definitive answer, we should never venture to combine the two statements.*

Because neither statements provided us with a definitive answer, let us combine the two statements.

For the 3 numbers a, b, and c - from the two statement we know that \\frac {b} {a}) = \\frac {c} {b}) and ab < 0

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 the three numbers - a, b, and c are not all positive nor all negative.

We can further conclude that the common ratio of the geometric sequence is negative.

'b' will be median only if the common ratio of the geometric progression is positive.

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

The information given in the two statements taken together is sufficient to answer the question.

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