# Number Properties & Theory : GMAT DS

Concept: Terminating decimals, Counter example approach to solve DS

The GMAT DS practice question given below is from Number Properties & Number Theory. Concept tested: Terminating decimals.

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.
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 $\frac{x}{y}\\$ a terminating decimal?

1. x is a multiple of 2
2. y is a multiple of 3

#### What kind of an answer will the question fetch?

The question is an "Is" question. Answer to an "is" questions is either YES or NO.

#### When is the data sufficient?

The data is sufficient if we are able to get DEFINITE YES or DEFINITE NO as an answer from the information in the statements.

If the statements do not have adequate data to provide a CONCLUSIVE answer and we get YES for some instances and NO for others, data is NOT sufficient.

#### What is a terminating decimal?

Numbers that have finite number of digits after the decimal point are called terminating or finite decimals. e.g., 2.5, 0.25, 8.
Note: All integers are terminating decimals.

Numbers that have infinite digits after the decimal point are called non-terminating or infinite decimals. e.g., $\frac{1}{3}\\$ = 0.33333..., $\sqrt{2}\\$ = 1.4142...

#### What is the approach?

Counter Example: We will look for a counter example. i.e., we will look for two examples, both satisfying the information given in the statements - one that gives an answer YES and the other that gives an answer NO.

#### Statement 1: x is a multiple of 2

No information about y has been provided.

Approach: Let us look for a counter example.

Example: When x = 2 and y = 3, $\frac{x}{y}\\$ is non-terminating.

The answer to the question is NO.

Counter Example: When x = 2 and y = 3, $\frac{x}{y}\\$ is non-terminating.

The answer to the question is NO.

We have found a counter example. Therefore, statement 1 is not providing a DEFINITE answer.

Statement 1 ALONE is NOT sufficient.

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

#### Statement 2: y is a multiple of 3

No information about x has been provided.

Approach: Let us look for a counter example.

Example: When x = 3 and y = 3, $\frac{x}{y}\\$ is terminating.

The answer to the question is YES.

Counter Example: When x = 2 and y = 3, $\frac{x}{y}\\$ is non-terminating.

The answer to the question is NO.

We have found a counter example. Therefore, statement 2 is not providing a DEFINITE answer.

Statement 2 ALONE is NOT sufficient.

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

#### Statements Together: x is a multiple of 2 & y is a multiple of 3

Approach: Let us look for a counter example.

Example: When x = 3 and y = 3, $\frac{x}{y}\\$ is terminating.

The answer to the question is YES.

Counter Example: When x = 2 and y = 3, $\frac{x}{y}\\$ is non-terminating.

The answer to the question is NO.

We have found a counter example. Therefore, despite combining the information in the statements we are not able to find a DEFINITE answer.

Statements TOGETHER are NOT SUFFICIENT.