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

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]=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.

Answer: D.

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Hope it helps.
_________________

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

Answer: D

Cheers,
Brent

[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

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.

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.
_________________

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

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.
_________________

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

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

Statement 2
[d]=0
0≤d<1
Sufficient

OFFICIAL GMAT EXPLANATION

Algebra Operations with real numbers

1. It is given d = y − [y]. If y is an integer, then y = [y], and thus y − [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 [y]. Since each of these two consecutive integers is at a distance of less than 1 from y, it follows that [y] is at a distance of less than 1 from y, or y − [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 [d] = 0, which is equivalent to 0 ≤ d < 1. This can be inferred by examining a few representative examples, such as [−0.1] = −1, [0] = 0, [0.1] = 0, [0.9] = 0, and [1.1] = 1. From 0 ≤ d < 1, it follows that d < 1; SUFFICIENT.

Each statement alone is sufficient.

Solution:

We need to determine whether d is less than 1.

Statement One Alone:

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

Statement Two Alone:

Since [d] = 0, 0 ≤ d < 1. Statement two alone is sufficient.

Answer: D
_________________

BrentGMATPrepNow
Thank you for your helpful explanation.

When I read this question I interpreted it as [Y] is equal to any number less than or equal to Y

Statement 1:
d=y - a number less than or equal to y
So, you could have 3-2=1 or 3-3=0, insufficient

Statement 2:
I was very confused by this statement, as I was confused whether or not you could apply the same rules of the brackets that surround Y to another variable (d) in this case.

I would be so appreciative for your thoughts on the points that I mentioned above. Thank you again.