[y] denotes the greatest integer less than or equal to y. Is d

Question

denotes the greatest integer less than or equal to y. Is d < 1?

(1) d = y -  
(2)  = 0

Bunuel · Accepted Answer

SOLUTION denotes the greatest integer less than or equal to y. Is d < 1? Some function rounds DOWN a number to the nearest integer. For example =1, =2, =-2, ... Question: is d<1 ? (1) d = y - --> if y is an integer then =y and d=y- =0<1 , if y is not an integer is nearest integer less than y and still d=y- <1 (for example: y=1.5 --> d=1.5- =1.5-1=0.5<1 or y=-1.5 --> d=-1.5- =-1.5-(-2)=0.5<1 ). Sufficient. (2) = 0 --> 0\leq{d}<1 . Sufficient. Answer: D. Similar questions to practice: if-denotes-the-least-integer-greater-than-or-equal-to-x-132223.html if-denotes-the-greatest-integer-less-than-or-equal-to-x-94687.html for-all-numbers-t-let-t-be-defined-as-the-greatest-105416.html is-x-an-odd-integer-1-x-2-2x-99856.html for-all-z-denotes-the-least-integer-greater-than-or-110561.html denotes-to-be-the-least-integer-no-less-than-x-is-130841.html for-all-numbers-t-let-t-be-defined-as-the-greatest-105416.html if-is-the-greatest-integer-less-than-or-equal-to-x-what-144277.html how-x-become-fraction-if-its-been-said-an-integer-94687.html awkward-ds-105416.html help-data-sufficiency-equation-and-inequalities-99856.html if-denotes-the-greatest-integer-less-than-or-equal-to-x-94687.html denotes-to-be-the-least-integer-no-less-than-x-is-130841.html is-the-greatest-integer-less-than-or-equal-to-the-real-n-154006.html if-denotes-the-greatest-integer-less-than-or-equal-to-z-162078.html if-denotes-the-least-integer-greater-than-or-equal-to-x-132223.html is-the-greatest-integer-less-than-or-equal-to-the-real-n-154006.html for-all-z-denotes-the-least-integer-greater-than-or-equ-167640.html Hope it helps.

BrentGMATPrepNow · Answer

First, let's take a moment to get a good idea of what this strange notation means. 
A few examples:
  = 5
  = 3
  = 8
  = -2
  = -14

IMPORTANT FACT #1:     <  y 
IMPORTANT FACT #2: The difference between y and   is always less than 1. In other words,  y -   < 1

Target question :    Is d < 1?

Statement 1 : d = y -    
Take IMPORTANT FACT #1 from above:     <  y 
Subtract   from both sides to get:  0  <  y -   
Now take IMPORTANT FACT #2 and add it to our inequality to get:  0  <  y -   < 1 
Statement 1 tells us that d = y -  , so let's replace y -   with d to get:  0  <  d < 1 
PERFECT, we can now be certain that   d < 1 
Since we can answer the  target question  with certainty, statement 1 is SUFFICIENT

Statement 2 :   = 0  
We're going to use IMPORTANT FACT #2 in our solution.
Take   = 0 and add d to both sides to get:   + d = d
Subtract   from both sides to get: d = d -  
From IMPORTANT FACT #2, we know that  d -   < 1 
Since we just showed that d = d -  , we can be certain that   d < 1 
Since we can answer the  target question  with certainty, statement 2 is SUFFICIENT

Answer: D

Cheers,
Brent

manpreetsingh86 · Answer

denotes the greatest integer less than or equal to y. Is d < 1?

(1) d = y -

if y is an integer than y- =0, if y is non-integer say 4.9 then y- = 4.9-4= .9 in both of these cases d is less than 1 hence sufficient
(2)  = 0

=0 means 0<= d< 1. hence sufficient

therefor D

AKG1593 · Answer

Option D.
From S1:d=y- 
For any value of non-integral y on no. line,y- =0.1,0.2,0.3,...,0.9=d<1
For eg. y=5.9; =5 y- =0.9=d<1
And if y=-5.9; =-6 y- =-5.9 -(-6);d=0.1<1
And for any integral y,y=  =>  .Suff.

From S2: =0
means d=0,0.1,0.2,0.3,...,0.9 only.All these vales are less than 1
Sufficient.

mattdnb · Answer

I have a question on this.   is not defined in the question stem, only   is. Now, I understand how one can infer that statement 2 is sufficient based on the definition of   given.

However, I would also not be surprised if the OG solution said that statement 2 is insufficient is not defined because   was not defined in the question stem. How should I be thinking about such cases? Thank you.

Bunuel · Answer

The question does not define  , it defines function denoted as  , which rounds DOWN a number to the nearest integer.

Check other function questions in our  Special Questions Directory :

Operations/functions defining algebraic/arithmetic expressions 
 Symbols Representing Arithmetic Operation 
 Rounding Functions 
 Various Functions

Hope it helps.

TeamGMATIFY · Answer

Any number x can be written in the form of I + f
Here I is the integral part or  , example: 1 , 2, 3 etc.
and f is the fractional part: 0.1, 0.2, 0.001 etc.

Given  :   is the integral part of a number
  Required  : Is d < 1

Statement 1  : d = y -  
We can say y = I + f
Where I =  
So, y -  means the fractional part of a number.
This would always be between 0 and 1

SUFFICIENT

Statement 2  :  = 0
The integral part of a number is ) means number is greated than 0 and less than 1.

SUFFICIENT

Option D

redfield · Answer

I shared a similar confusion as Garauku; it saying "greatest integer less than or equal to" seemed to me it was saying any integer <= y, which could be as garauku suggested y = 5   = 5, 4, etc. How would it have been phrased if this was the intended meaning rather than that the function rounds down?

ENGRTOMBA2018 · Answer

Couple of points:

1. This is an official question, so do not question the OA or the language, it will only be a waste of time and energy.

2. The question as such is completely clear. If y=5, then   can only be = 5 and NOT 4 as you need the GREATEST integer less or equal to y. What this means is that you must take the greatest integer of <y or =y to be that particular integer. There is no ambiguity here.

For y=1.5,   = 1, for y=4.1,  =4 etc.

Hope this helps.

nick1816 · Answer

Statement 1-
y= +{y}
where {y} is fractional part and   is integral part

d=y- 
d= +{y}- 
d={y}
or 0≤d<1
Sufficient

Statement 2
 =0
0≤d<1
Sufficient

Hovkial · Answer

OFFICIAL GMAT EXPLANATION

Algebra Operations with real numbers

1. It is given d = y −  . If y is an integer, then y =  , and thus y −   = 0, which is less than 1. If y is not an integer, then y lies between two consecutive integers, the smaller of which is equal to  . Since each of these two consecutive integers is at a distance of less than 1 from y, it follows that   is at a distance of less than 1 from y, or y −   < 1. Thus, regardless of whether y is an integer or y is not an integer, it can be determined that d < 1; SUFFICIENT.

2. It is given that   = 0, which is equivalent to 0 ≤ d < 1. This can be inferred by examining a few representative examples, such as   = −1,   = 0,   = 0,   = 0, and   = 1. From 0 ≤ d < 1, it follows that d < 1; SUFFICIENT.

Each statement alone is sufficient.

ScottTargetTestPrep · Answer

Solution:

We need to determine whether d is less than 1.

Statement One Alone:
 
The difference between y and   is always between 0 (inclusive) and 1 (exclusive). For example, if y = 3.14, then   = 3 and y -   = 0.14. If y = -3.14, then   = -4 and y -   = 0.86. Since d = y -  , 0 ≤ d < 1. Statement one alone is sufficient.

Statement Two Alone:
 
Since   = 0, 0 ≤ d < 1. Statement two alone is sufficient.

Answer: D

GMATinsight · Answer

Question: Is d < 1?

Statement 1  : d = y -  
now, imagine y in the range 5 to 6 so I am taking values like {5, 5.1, 5.4, 5.6, 5.9, 6}

@y = 5,   = 5, d = y -   = 5-5 = 0  YES 
@y = 5.1,   = 5, d = y -   = 5.1-5 = 0.1  YES 
@y = 5.4,   = 5, d = y -   = 5.4-5 = 0.4  YES 
@y = 5.6,   = 5, d = y -   = 5.6-5 = 0.6  YES 
@y = 5.9,   = 5, d = y -   = 5.9-5 = 0.9  YES 
@y = 6,   = 6, d = y -   = 6-6= 0  YES

SUFFICIENT

Statement 2  :  = 0
i.e. 0 ≤ d < 1  YES 
 SUFFICIENT
  
Answer: Option D

[y] denotes the greatest integer less than or equal to y. Is d < 1?

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GMAT Data Sufficiency (DS) Questions

[y] denotes the greatest integer less than or equal to y. Is d < 1?

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