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prernamalhotra: Senior | Next Rank: 100 Posts; Posts: 44; Joined: Wed Apr 02, 2014 2:01 amFollowed by: 1 member; Thanked: 2 times

by prernamalhotra » Fri May 23, 2014 9:14 pm

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

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

Thank you,
Prerna

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Source: Beat The GMAT — Data Sufficiency | Fri May 23, 2014 9:14 pm

GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

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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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

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

Brent Hanneson - Creator of GMATPrepNow.com

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ceilidh.erickson: GMAT Instructor; Posts: 2095; Joined: Tue Dec 04, 2012 3:22 pmFollowed by: 247 members; Thanked: 1443 times

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.

Ceilidh Erickson
EdM in Mind, Brain, and Education
Harvard Graduate School of Education

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kamalakarthi: Senior | Next Rank: 100 Posts; Posts: 80; Joined: Mon Jan 31, 2011 10:08 pmFollowed by: 2 members

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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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

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

Brent Hanneson - Creator of GMATPrepNow.com

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ceilidh.erickson: GMAT Instructor; Posts: 2095; Joined: Tue Dec 04, 2012 3:22 pmFollowed by: 247 members; Thanked: 1443 times

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.

Ceilidh Erickson
EdM in Mind, Brain, and Education
Harvard Graduate School of Education

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