# GMAT® Practice : Median : GMAT DS

Concept: Median of 3 numbers in geometric sequence.

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 when all the numbers are positive and when some are negative.

Directions for Data Sufficiency

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 -

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

All numbers used are real numbers.

##### Figures

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.

##### Note

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

1. $\frac {b} {a}$ = $\frac {c} {b}$
2. ab < 0

#### Video Explanation

Scroll down for explanation text

#### Understanding Median

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, the median of a set of numbers is 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 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

#### Statement 1: $\frac {b} {a}$ = $\frac {c} {b}$

i.e., b2 = 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, 'b' will be the geometric mean and the median.

Essentially, if the The ratio between any two consecutive terms of a GP will always be the same and is known as the common ratio.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 positive.

For e.g., let 'a' be 2 and let the common ratio be -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. We can eliminate choices A and D.

We are down from 5 to 3 choices - B, C or E.

#### 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 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 also not sufficient. We can eliminate choice B as well.

We are down from 3 to 2 choices - C or E.

#### Combining the 2 statements

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.

Choice C is the correct answer.

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