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Math Expert V
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[y] denotes the greatest integer less than or equal to y. Is  [#permalink]

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The Official Guide For GMAT® Quantitative Review, 2ND Edition

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

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

Data Sufficiency
Question: 102
Category: Algebra Operations with real numbers
Page: 160
Difficulty: 650

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Math Expert V
Joined: 02 Sep 2009
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Re: [y] denotes the greatest integer less than or equal to y. Is  [#permalink]

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SOLUTION

[y] 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 [1.5]=1, =2, [-1.5]=-2, ...

Question: is $$d<1$$?

(1) d = y - [y] --> if y is an integer then $$[y]=y$$ and $$d=y-[y]=0<1$$, if y is not an integer [y] is nearest integer less than y and still $$d=y-[y]<1$$ (for example: $$y=1.5$$ --> $$d=1.5-[1.5]=1.5-1=0.5<1$$ or $$y=-1.5$$ --> $$d=-1.5-[-1.5]=-1.5-(-2)=0.5<1$$ ). Sufficient.

(2) [d] = 0 --> $$0\leq{d}<1$$. Sufficient.

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Re: [y] denotes the greatest integer less than or equal to y. Is  [#permalink]

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

(1) d = y - [y]

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

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

therefor D
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Re: [y] denotes the greatest integer less than or equal to y. Is  [#permalink]

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Option D.
From S1:d=y-[y]
For any value of non-integral y on no. line,y-[y]=0.1,0.2,0.3,...,0.9=d<1
For eg. y=5.9;[y]=5 y-[y]=0.9=d<1
And if y=-5.9;[y]=-6 y-[y]=-5.9 -(-6);d=0.1<1
And for any integral y,y=[y] => [d=0<1].Suff.

From S2:[d]=0
means d=0,0.1,0.2,0.3,...,0.9 only.All these vales are less than 1
Sufficient.
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Re: [y] denotes the greatest integer less than or equal to y. Is  [#permalink]

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

However, I would also not be surprised if the OG solution said that statement 2 is insufficient is not defined because [d] was not defined in the question stem. How should I be thinking about such cases? Thank you.
Math Expert V
Joined: 02 Sep 2009
Posts: 60496
Re: [y] denotes the greatest integer less than or equal to y. Is  [#permalink]

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

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

The question does not define [y], 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.
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Re: [y] denotes the greatest integer less than or equal to y. Is  [#permalink]

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

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

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

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

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

SUFFICIENT

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

SUFFICIENT

Option D
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Re: [y] denotes the greatest integer less than or equal to y. Is  [#permalink]

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Bunuel wrote:
SOLUTION

[y] 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 [1.5]=1, =2, [-1.5]=-2, ...

Question: is $$d<1$$?

(1) d = y - [y] --> if y is an integer then $$[y]=y$$ and $$d=y-[y]=0<1$$, if y is not an integer [y] is nearest integer less than y and still $$d=y-[y]<1$$ (for example: $$y=1.5$$ --> $$d=1.5-[1.5]=1.5-1=0.5<1$$ or $$y=-1.5$$ --> $$d=-1.5-[-1.5]=-1.5-(-2)=0.5<1$$ ). Sufficient.

(2) [d] = 0 --> $$0\leq{d}<1$$. Sufficient.

Hope it helps.

Why we are not considering Y = an integer as well? i.e. y=5,6,7 etc...

Question stem says: [y] denotes the greatest integer less than or equal to y.

If I am not wrong, the above question stem doesn't imply that Y should only take fractional forms such as: 1.5, -1.5 etc... and y is not an integer.

So I took y=5 so in 1 d = y - [y], d = 0 (5-5=0), or 1 (5-4=1) and therefore the statement is insufficient.

Where exactly I am thinking wrong. Help please.
Math Expert V
Joined: 02 Sep 2009
Posts: 60496
Re: [y] denotes the greatest integer less than or equal to y. Is  [#permalink]

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gauraku wrote:
Bunuel wrote:
SOLUTION

[y] 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 [1.5]=1, =2, [-1.5]=-2, ...

Question: is $$d<1$$?

(1) d = y - [y] --> if y is an integer then $$[y]=y$$ and $$d=y-[y]=0<1$$, if y is not an integer [y] is nearest integer less than y and still $$d=y-[y]<1$$ (for example: $$y=1.5$$ --> $$d=1.5-[1.5]=1.5-1=0.5<1$$ or $$y=-1.5$$ --> $$d=-1.5-[-1.5]=-1.5-(-2)=0.5<1$$ ). Sufficient.

(2) [d] = 0 --> $$0\leq{d}<1$$. Sufficient.

Hope it helps.

Why we are not considering Y = an integer as well? i.e. y=5,6,7 etc...

Question stem says: [y] denotes the greatest integer less than or equal to y.

If I am not wrong, the above question stem doesn't imply that Y should only take fractional forms such as: 1.5, -1.5 etc... and y is not an integer.

So I took y=5 so in 1 d = y - [y], d = 0 (5-5=0), or 1 (5-4=1) and therefore the statement is insufficient.

Where exactly I am thinking wrong. Help please.

Please re-read the explanation. We are NOT considering y as fraction only.

If y = 5, then d = y - [y] = 5 - 5 = 0 < 1. HOW can it be 1?
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Joined: 25 Jul 2012
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[y] denotes the greatest integer less than or equal to y. Is  [#permalink]

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Bunuel wrote:
gauraku wrote:
Bunuel wrote:
SOLUTION

[y] 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 [1.5]=1, =2, [-1.5]=-2, ...

Question: is $$d<1$$?

(1) d = y - [y] --> if y is an integer then $$[y]=y$$ and $$d=y-[y]=0<1$$, if y is not an integer [y] is nearest integer less than y and still $$d=y-[y]<1$$ (for example: $$y=1.5$$ --> $$d=1.5-[1.5]=1.5-1=0.5<1$$ or $$y=-1.5$$ --> $$d=-1.5-[-1.5]=-1.5-(-2)=0.5<1$$ ). Sufficient.

(2) [d] = 0 --> $$0\leq{d}<1$$. Sufficient.

Hope it helps.

Why we are not considering Y = an integer as well? i.e. y=5,6,7 etc...

Question stem says: [y] denotes the greatest integer less than or equal to y.

If I am not wrong, the above question stem doesn't imply that Y should only take fractional forms such as: 1.5, -1.5 etc... and y is not an integer.

So I took y=5 so in 1 d = y - [y], d = 0 (5-5=0), or 1 (5-4=1) and therefore the statement is insufficient.

Where exactly I am thinking wrong. Help please.

Please re-read the explanation. We are NOT considering y as fraction only.

If y = 5, then d = y - [y] = 5 - 5 = 0 < 1. HOW can it be 1?

Statement says that [Y] denotes either the greatest integer less than or equal to y. So if y =5, [Y] = either 5 (equal to y) or 4 (greatest integer less than y) So d will be equal to either 5-5 =0 or 5-4=1.
Math Expert V
Joined: 02 Sep 2009
Posts: 60496
Re: [y] denotes the greatest integer less than or equal to y. Is  [#permalink]

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gauraku wrote:
Bunuel wrote:
gauraku wrote:

Why we are not considering Y = an integer as well? i.e. y=5,6,7 etc...

Question stem says: [y] denotes the greatest integer less than or equal to y.

If I am not wrong, the above question stem doesn't imply that Y should only take fractional forms such as: 1.5, -1.5 etc... and y is not an integer.

So I took y=5 so in 1 d = y - [y], d = 0 (5-5=0), or 1 (5-4=1) and therefore the statement is insufficient.

Where exactly I am thinking wrong. Help please.

Please re-read the explanation. We are NOT considering y as fraction only.

If y = 5, then d = y - [y] = 5 - 5 = 0 < 1. HOW can it be 1?

Statement says that [Y] denotes either the greatest integer less than or equal to y. So if y =5, [Y] = either 5 (equal to y) or 4 (greatest integer less than y) So d will be equal to either 5-5 =0 or 5-4=1.

[y] can take only one value, which will be the greatest integer less than or equal to y. Function [] rounds DOWN a number to the nearest integer. For example [1.5]=1, =2, [-1.5]=-2, ...

Check more questions in Rounding Functions Questions.
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Re: [y] denotes the greatest integer less than or equal to y. Is  [#permalink]

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Thanks a lot for the explanation. I got it now and made a note of it.
Current Student S
Joined: 18 Aug 2014
Posts: 323
Re: [y] denotes the greatest integer less than or equal to y. Is  [#permalink]

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Bunuel wrote:

[y] can take only one value, which will be the greatest integer less than or equal to y. Function [] rounds DOWN a number to the nearest integer. For example [1.5]=1, =2, [-1.5]=-2, ...

Check more questions in Rounding Functions Questions.

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 [y] = 5, 4, etc. How would it have been phrased if this was the intended meaning rather than that the function rounds down?
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[y] denotes the greatest integer less than or equal to y. Is  [#permalink]

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redfield wrote:
Bunuel wrote:

[y] can take only one value, which will be the greatest integer less than or equal to y. Function [] rounds DOWN a number to the nearest integer. For example [1.5]=1, =2, [-1.5]=-2, ...

Check more questions in Rounding Functions Questions.

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 [y] = 5, 4, etc. How would it have been phrased if this was the intended meaning rather than that the function rounds down?

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 [y] 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, [y] = 1, for y=4.1, [y]=4 etc.

Hope this helps.
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Re: [y] denotes the greatest integer less than or equal to y. Is  [#permalink]

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Is it wrong to assume a non-fraction as the value of y, say y=5 & [y] will be 4 ?

Pls clarify as the answer choice will change in this case...
Math Expert V
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Re: [y] denotes the greatest integer less than or equal to y. Is  [#permalink]

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vikasx wrote:
Is it wrong to assume a non-fraction as the value of y, say y=5 & [y] will be 4 ?

Pls clarify as the answer choice will change in this case...

Actually this is clarified several times above.

Function [] rounds DOWN a number to the nearest integer. For example [1.5]=1, =2, [-1.5]=-2, ...

[y] denotes the greatest integer less than or equal to y. Thus if y = 5, then  = 5, because 5 is the greatest integer less than or equal to 5.
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Re: [y] denotes the greatest integer less than or equal to y. Is  [#permalink]

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how can we consider numbers while doing this problem??
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Re: [y] denotes the greatest integer less than or equal to y. Is  [#permalink]

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i think what is confusing most is this point.

Question states. [y] is the greatest integer less than or equal to Y. Most are also assuming that Y has to be an integer, that stipulation is only for [y]. Only [y] has to be an integer, on other hand Y can be any non integer value.

Therefore if Y = 4.5 then [Y] being an integer only can be 4 at most. because if it is 5 then it will break the parameters set by the question i.e. greatest integer less than or equal to Y.

Hope it helps some folks.
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Re: [y] denotes the greatest integer less than or equal to y. Is  [#permalink]

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Top Contributor
Bunuel wrote:
The Official Guide For GMAT® Quantitative Review, 2ND Edition

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

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

First, let's take a moment to get a good idea of what this strange notation means.
A few examples:
[5.1] = 5
 = 3
[8.9] = 8
[-1.4] = -2
[-13.6] = -14

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

Target question: Is d < 1?

Statement 1: d = y - [y]
Take IMPORTANT FACT #1 from above: [y] < y
Subtract [y] from both sides to get: 0 < y - [y]
Now take IMPORTANT FACT #2 and add it to our inequality to get: 0 < y - [y] < 1
Statement 1 tells us that d = y - [y], so let's replace y - [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: [d] = 0
We're going to use IMPORTANT FACT #2 in our solution.
Take [d] = 0 and add d to both sides to get: [d] + d = d
Subtract [d] from both sides to get: d = d - [d]
From IMPORTANT FACT #2, we know that d - [d] < 1
Since we just showed that d = d - [d], we can be certain that d < 1
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Cheers,
Brent
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Re: [y] denotes the greatest integer less than or equal to y. Is  [#permalink]

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[y] = y, if y is an int, and [y] rounds down to the lesser value if y is between 2 ints.

(1) d = y - [y]
d = 1 - 
d = 1 - 1
d = 0
If [y] = y then d must be 0

d = 0.999 - [0.999]
d = 0.999 - 0
d = 0.999
If [y] rounded down from y then d cannot be equal to or more than 1, Sufficient.
Testing more, we can see 1 ≠ 1 -  and -1 ≠ -1 - [-1]

(2) [d] = 0

So the range is 0 ≤ d < 1
Sufficient. Re: [y] denotes the greatest integer less than or equal to y. Is   [#permalink] 12 May 2019, 17:10

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