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Source: — Data Sufficiency |
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by GMATGuruNY » Fri May 23, 2014 10:38 pm
prernamalhotra wrote:[y] denotes the greatest integer less than or equal to y. Is d<1?

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

Thank you,
Prerna
Statement 1:
If y=1, then d = 1 - [the greatest integer less than or equal to 1] = 1-1 = 0.
If y=-1, then d = -1 - [the greatest integer less than or equal to -1] = -1 - (-1) = 0.
If y=1/2, then d = 1/2 - [the greatest integer less than or equal to 1/2] = 1/2 - 0 = 1/2.
If y=-1/2, then d = -1/2 - [the greatest integer less than or equal to -1/2] = -1/2 - (-1) = 1/2.
In every case, d < 1.
SUFFICIENT.

Statement 2:
If d=1, then [d] = the greatest integer less than or equal to 1 = 1.
Not possible, since [d] must be equal to 0.
If d=1.5, then [d] = the greatest integer less than or equal to 1.5 = 1.
Not possible, since [d] must be equal to 0.
If d=2, then [d] = the greatest integer less than or equal to 2 = 2.
Not possible, since [d] must be equal to 0.

Implication:
It is not possible for d to be greater than or equal to 1.
Thus, d < 1.
SUFFICIENT.

The correct answer is D.
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by Brent@GMATPrepNow » Sat May 24, 2014 8:19 am
prernamalhotra wrote:[y] denotes the greatest integer less than or equal to y. Is d < 1?

1) d = y - [y]
2) [d] = 0
Here's an algebraic approach...

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
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by ceilidh.erickson » Mon May 26, 2014 7:47 am
The most important thing to do with this type of formula question is to TRANSLATE:

"the greatest integer less than or equal to y" really means round down to the nearest integer.

"the least integer greater than or equal to y" would mean round up to the nearest integer.

If we want to know if d<1, then we really want to know: would [d] round down to 0 or lower?

1) d = y - [y]

If d is the difference between a number and the integer that the number would round down to, that difference must always be less than 1. Therefore d must be less than 1. Sufficient.

2) [d] = 0

This answers our target question exactly - if rounds down to 0, it must be less than 1.
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by kamalakarthi » Wed Aug 02, 2017 4:11 pm
"the greatest integer less than or equal to y" really means round down to the nearest integer.

Say if Y =2.6 then greatest integer less than Y or Y = 2
if Y=-2.6 then the greatest integer less than Y = 3

Am I right ?

"the least integer greater than or equal to y" would mean round up to the nearest integer.

Say if Y =2.6 then least integer less than Y or Y <= 3
if Y=-2.6 then the least integer less than Y <=- 3

Am I right ?
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by Brent@GMATPrepNow » Wed Aug 02, 2017 4:15 pm
kamalakarthi wrote:"the greatest integer less than or equal to y" really means round down to the nearest integer.

Say if Y =2.6 then greatest integer less than Y or Y = 2
if Y=-2.6 then the greatest integer less than Y = 3

Am I right ?

"the least integer greater than or equal to y" would mean round up to the nearest integer.

Say if Y =2.6 then least integer less than Y or Y <= 3
if Y=-2.6 then the least integer less than Y <=- 3

Am I right ?
Close
if Y=-2.6 then the greatest integer less than Y = -3
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by ceilidh.erickson » Wed Aug 02, 2017 4:46 pm
kamalakarthi wrote:"the greatest integer less than or equal to y" really means round down to the nearest integer.

Say if Y =2.6 then greatest integer less than Y or Y = 2
if Y=-2.6 then the greatest integer less than Y = 3

Am I right ?

"the least integer greater than or equal to y" would mean round up to the nearest integer.

Say if Y =2.6 then least integer less than Y or Y <= 3
if Y=-2.6 then the least integer less than Y <=- 3

Am I right ?
And by the same logic:
if Y = -2.6 then the least integer greater than Y = -2

Be careful with your negatives! When you ROUND UP with negatives, you get closer to 0. When you ROUND DOWN, you get further from 0.
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