If [x] Denotes The Greatest Integer Less Than or Equal to x GMAT Data Sufficiency

Question: If [x] denotes the greatest integer less than or equal to x, is [x] = 0 ?

(1) 5x + 1 = 3 + 2x
(2) 0 < x < 1

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

4. EACH statement ALONE is sufficient.

5. Statements (1) and (2) TOGETHER are not sufficient.

Correct Answer: D

Solution and Explanation
Approach Solution 1:
This question has only one approach.

It is given in the question that [x] denotes the greatest integer less than equal to x.
It has asked whether we can say that [x] = 0 from the given two conditions
This question involved the use of the greatest integer function. Some important points related to the greatest integer function are given below:
This function rounds down the real number to the nearest integer less than the given number.
Examples –
[1.25] = 1
[4.767655] = 4
[50] = 50
It has given us two condition which we have to check and tell whether the question can be answered from them.
Given in the first statement –
5x + 1 = 3 + 2x
=> 5x - 2x = 3-1
=> 3x = 2
x= \(\frac{2}{3}\)
Here we get the value of x to be \(\frac{2}{3}\).
Now to find the [x] first we have to convert \(\frac{1}{3}\)to decimal form.
\(\frac{1}{3}\)= 0.67
[x] = [0.67] = 0
The nearest integer smaller than 0.67 is 0
Hence this condition is sufficient to say that [x] = 0
In the second condition it is given that 0 < x < 1
Here it is given that x lies in between the numbers 0 < x < 1
For all the values lying in range (0,1) the value of [x] = 0
Therefore this statement is sufficient to say that [x] = 0
As both statements individually are sufficient to state that [x] = 0. Therefore this is sufficient. 

“If [x] denotes the greatest integer less than or equal to x”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "The Official Guide for GMAT Review". GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

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