A GMAT DS question in Inequalities. Tests your understanding of rules of indices, positive and negative numbers. Gist of what is highlighted is one's ability to find a counter example to establish that a statement is not sufficient to answer the question.

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.

##### 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 a < b?

- a
^{b}< b^{a} - \\frac{a}{b}\\) > 1

#### Explanatory Answer

Video explanation will be added soon#### What kind of an answer will the question fetch?

The question is an "IS" question. For "is" questions, the answer is "YES" or "NO".

#### When is the data sufficient?

The data is sufficient if we are able to get a CONCLUSIVE YES or a CONCLUSIVE NO from the information in the statements.

If using the information in the statement(s), we arrive at an answer that is sometimes yes and sometimes no, the data is not sufficient.

#### When is the answer yes?

The answer is yes when a < b.

#### When is the answer no?

The answer is no when a ≥ b.

Note that the answer is no both when a > b and when a = b.

#### Statement 1: a^{b} < b^{a}

Approach: Counter Example

** Example**: Now, if a = 1, b = 100 a

^{b}< b

^{a}and a < b.

If we can think of one example where a^{b} < b^{a} such that a > b, we can say this statement is insufficient.

** Counter Example**:If we take "b" as a negative number this could be very easily accomplished (especially if "a" were an even number). Let us say b = -2 and a = 2.

2^{(-2)} < (-2)^{2}, but 2 > -2

Statement 1 ALONE is NOT sufficient.

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

Another counter example could just be 5^{3} < 3^{5}, but 5 > 3

#### Statement 2: \\frac{a}{b}\\) > 1

Approach: Counter Example

If a, b are both positive, then \\frac{a}{b}\\) > 1 implies a > b.

If both are negative, it means the opposite.

** Example**: a = 5, b = 3, a > b and satisfies this condition \\frac{a}{b}\\) > 1.

** Counter Example**: if a = -5, b = -3, a < b and these values also satisfy this condition \\frac{a}{b}\\) > 1.

Statement 2 ALONE is NOT sufficient.

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

#### Statements Together: a^{b} < b^{a} and \\frac{a}{b}\\) > 1

If \\frac{a}{b}\\) is greater than 1, then either both are positive or both are negative.

If both are positive then a has to be greater than b.

An example for a^{b} < b^{a} is easy to find. Say, a = 10 and b = 2 satisfies both statements and a > b.

So, we can establish that the statements together are not sufficient if we can find one example where a, b both are negative and such that a < b and a^{b} < b^{a}.

If we take "a" as an even number and "b" as an odd number, we should be through.

(-4)^{(-3)} < (-3)^{(-4)}

Negative numbers with odd powers are negative and negative numbers with even powers are positive. Negative numbers are lesser than positive numbers.

This example satisfies both the statements and in this case, a < b.

Statements Together are NOT sufficient.

Eliminate choice C. Choice E is the answer.

It is an excellent question, mainly because it makes one think of many possibilities. And THAT skill is very essential for DS. Students should be thinking along the lines of "Can I figure out a counter example for this".

Just to recap.

a = 5, b = 3 satisfies both statements. And a > b

a = -4, b = -3 satisfies both statements. And a < b.

So, we cannot answer the question even when the information in both the statements are used together.

Choice E is the answer.

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