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Intern  Joined: 29 Jul 2012
Posts: 14
DS inequalities and Modulus  [#permalink]

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Question Stats: 82% (01:41) correct 18% (01:16) wrong based on 187 sessions

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Is s between r and t

a) |r-s| < |r-t|
b) |r-s| < |s-t|

Is s between r and t

a) |r-t| > |r-s|
b) |r-t| > |t-s|
Intern  Joined: 29 Jul 2012
Posts: 14
Is s between r and t  [#permalink]

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6
Is s between r and t

(1) |r-s| < |r-t|
(2) |r-s| < |s-t|

Is s between r and t

(1) |r-t| > |r-s|
(2) |r-t| > |t-s|

Originally posted by pnf619 on 02 Sep 2012, 08:03.
Last edited by Bunuel on 03 Sep 2012, 04:08, edited 1 time in total.
Renamed and edited the topic.
Intern  Joined: 28 Aug 2012
Posts: 39
Location: Austria
GMAT 1: 770 Q51 V42 Re: DS inequalities and Modulus  [#permalink]

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First question:
Statement (1) + (2) together: r=1, s=2, t=4 OR s=1, r=2, t=4 Both satisfy both statements.
So we do not know, whether s is between r and t. --> not sufficient

Second question:
Statement (1) + (2) together: If s<t<r, then |r-s|>|r-t| violating statement (1).
If s<r<t, then |t-s|>|r-t| violating statement (2).
If r<t<s, then |r-s|>|r-t| violating statement (1).
If t<r<s, then |t-s|>|r-t| violating statement (2).
Therefore, s must be between r and t to satisfy both statements at once. --> sufficient

Or with other words:
In the first question, we know that |r-s| is the smallest of the three possible differences (|r-s|, |r-t|, and |s-t|).
But that's not sufficient, as shown above. r and s can switch, while t remains the same, simply being far away from r and s.

In the second question, we know that |r-t| is the greatest of the three possible diffences.
So we know that r and t are the extreme values, being minimum and maximum.
The only place left for s is in the middle.
Director  Joined: 22 Mar 2011
Posts: 588
WE: Science (Education)
Re: Is s between r and t  [#permalink]

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1
pnf619 wrote:
Is s between r and t

(1) |r-s| < |r-t|
(2) |r-s| < |s-t|

Is s between r and t

(1) |r-t| > |r-s|
(2) |r-t| > |t-s|

Use the property of absolute value, $$|a - b|$$ is the distance between $$a$$ and $$b,$$ and visualization on the number line.

Q1:
(1) t- - - - s - - r - - s - - - - t
$$s$$ and $$t$$ can be on either side of $$r.$$
Not sufficient.
(2) t - - r - - - s - - - r - - t
Now $$r$$ and $$t$$ can be on either side of $$s.$$
Not sufficient.
(1) and (2): Still not sufficient, as one can see from the above situation for (1).

Q2:
(1) t - - - s - - r - - s - - - t
$$s$$ and $$t$$ can be on either side of $$r.$$
Not sufficient.
(2) r - - s - - - t - - - s - - r
Now $$r$$ and $$s$$ can be on either side of $$t.$$
Not sufficient.
(1) and (2): Sufficient, because in (1) now $$s$$ must be between $$r$$ and $$t.$$

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Love GMAT Quant questions and running.
Veritas Prep GMAT Instructor V
Joined: 16 Oct 2010
Posts: 9701
Location: Pune, India
Re: DS inequalities and Modulus  [#permalink]

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pnf619 wrote:

Is s between r and t

a) |r-s| < |r-t|
b) |r-s| < |s-t|

Use the number line to solve such questions. Don't get lost in algebra here.
If you are not comfortable with the distance approach of mods, check this post first: http://www.veritasprep.com/blog/2011/01 ... edore-did/

When you read "Is s between r and t", think of the following diagram:
Attachment: Ques5.jpg [ 2.75 KiB | Viewed 2051 times ]

s can be in any one of the three regions - 'between r and t' or 'to the left of r' or 'to the right of t' (r and t can switch places too).
You need to find out whether s lies in the green line region.

a) |r-s| < |r-t|
This implies that distance between r and s is less than the distance between r and t. Look at the diagram below. This can happen in 2 ways. s can be to the left of r or it can be between r and t. Hence not sufficient.

Attachment: Ques6.jpg [ 2.64 KiB | Viewed 2047 times ]

b) |r-s| < |s-t|
Distance between r and s is less than the distance between s and t. The same diagram as above can be used for this statement too. This can happen in 2 ways. s can be to the left of r or it can be between r and t. Hence not sufficient.

Since we get same two cases from both the statements, both together will not be sufficient.
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Originally posted by VeritasKarishma on 06 Sep 2012, 22:59.
Last edited by VeritasKarishma on 06 Sep 2012, 23:16, edited 1 time in total.
Veritas Prep GMAT Instructor V
Joined: 16 Oct 2010
Posts: 9701
Location: Pune, India
Re: DS inequalities and Modulus  [#permalink]

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pnf619 wrote:

If you are having trouble understanding the 'distance concept' of mods, check out this post first:

Is s between r and t

a) |r-t| > |r-s|
b) |r-t| > |t-s|

The question stem is the same so your initial thought process will be the same. Let's look at the statements.

a) |r-t| > |r-s|

Again, this is same as statement 1 above (|r-s| < |r-t|) so the diagram will also be the same with the same 2 cases. Notice that s cannot be to the right of t because distance between r and t must be less than the distance between r and s. Not sufficient.
Attachment: Ques6.jpg [ 2.64 KiB | Viewed 2050 times ]

b) |r-t| > |t-s|
Distance between s and t is less than the distance between r and t. Look at the diagram. s can be between r and t or to the right of t. It cannot be to the left of r anymore because then the distance between s and t will become more than the distance between r and t. Since 2 cases are possible, the statement is not sufficient.
Attachment: Ques7.jpg [ 2.6 KiB | Viewed 2048 times ]

Using both together, from statement 1, s cannot be to the right of t and from statement 2, s cannot be to the left of r. There is only one region left now - "between r and t". So s must be between r and t.

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Director  Joined: 22 Mar 2011
Posts: 588
WE: Science (Education)
Re: DS inequalities and Modulus  [#permalink]

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pnf619 wrote:

Is s between r and t

a) |r-s| < |r-t|
b) |r-s| < |s-t|

Is s between r and t

a) |r-t| > |r-s|
b) |r-t| > |t-s|

Use the property of absolute value, $$|a - b|$$ is the distance between $$a$$ and $$b,$$ and visualization on the number line.

Q1:
(1) t- - - - s - - r - - s - - - - t
$$s$$ and $$t$$ can be on either side of $$r.$$
Not sufficient.
(2) t - - r - - - s - - - r - - t
Now $$r$$ and $$t$$ can be on either side of $$s.$$
Not sufficient.
(1) and (2): Still not sufficient, as one can see from the above situation for (1).

Q2:
(1) t - - - s - - r - - s - - - t
$$s$$ and $$t$$ can be on either side of $$r.$$
Not sufficient.
(2) r - - s - - - t - - - s - - r
Now $$r$$ and $$s$$ can be on either side of $$t.$$
Not sufficient.
(1) and (2): Sufficient, because in (1) now $$s$$ must be between $$r$$ and $$t.$$

_________________
PhD in Applied Mathematics
Love GMAT Quant questions and running.
Intern  Joined: 01 Jun 2012
Posts: 18
Location: United States
Concentration: Nonprofit
GMAT 1: 720 Q48 V43 GPA: 3.83
Re: Is s between r and t  [#permalink]

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For Q2, I just plugged in numbers and came up with the following:

r = 3, s= 5

a) |3 -5| > |3 - s| = 2 > |3 - s| -> therefore, s can equal 2, 3, or 4 to make the expression true. Only 4 is between t and r, but we can't be sure that is the answer. No sufficient.

b) |3 - 5| > |5 - s| = 2 > |5 - s| -> therefore, s can equal 4, 5, or 6 to make the expression true. Only 4 is between t and r, but we can't be sure that is the answer. Not sufficient.

Taking both together, 4 is the only answer that both statements have in common so that must be the solution to the problem. Since, 4 is between t and r, both statements together is sufficient. Answer is C.

Is this the right way to go about solving it?
Veritas Prep GMAT Instructor V
Joined: 16 Oct 2010
Posts: 9701
Location: Pune, India
Re: Is s between r and t  [#permalink]

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egiles wrote:
For Q2, I just plugged in numbers and came up with the following:

r = 3, s= 5

a) |3 -5| > |3 - s| = 2 > |3 - s| -> therefore, s can equal 2, 3, or 4 to make the expression true. Only 4 is between t and r, but we can't be sure that is the answer. No sufficient.

b) |3 - 5| > |5 - s| = 2 > |5 - s| -> therefore, s can equal 4, 5, or 6 to make the expression true. Only 4 is between t and r, but we can't be sure that is the answer. Not sufficient.

Taking both together, 4 is the only answer that both statements have in common so that must be the solution to the problem. Since, 4 is between t and r, both statements together is sufficient. Answer is C.

Is this the right way to go about solving it?

Not entirely though the logic is fine.

"a) |3 -5| > |3 - s| = 2 > |3 - s| -> therefore, s can equal 2, 3, or 4 to make the expression true. Only 4 is between t and r, but we can't be sure that is the answer. No sufficient."

Actually s can take any value in this range 1 < s < 5 (since it needn't be an integer)
Some values will lie between 3 and 5 and some will not. Not sufficient.

"b) |3 - 5| > |5 - s| = 2 > |5 - s| -> therefore, s can equal 4, 5, or 6 to make the expression true. Only 4 is between t and r, but we can't be sure that is the answer. Not sufficient."

s can take any value in this range 3 < s < 7
Some values will lie between 3 and 5 and some will not. Not sufficient.

The overlap in the two cases is only of 3 < s < 5 and that lies between 3 and 5. So sufficient.

But generally speaking, taking numbers is not a good idea especially in DS questions. You don't know whether you have considered all relevant cases or not.

The same questions have been put up here and I have discussed how to solve them using number line:
ds-inequalities-and-modulus-138237.html#p1119436
_________________
Karishma
Veritas Prep GMAT Instructor

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Intern  Joined: 01 Jun 2012
Posts: 18
Location: United States
Concentration: Nonprofit
GMAT 1: 720 Q48 V43 GPA: 3.83
Re: Is s between r and t  [#permalink]

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You rock. Thank you so much for the help!

VeritasPrepKarishma wrote:
egiles wrote:
For Q2, I just plugged in numbers and came up with the following:

r = 3, s= 5

a) |3 -5| > |3 - s| = 2 > |3 - s| -> therefore, s can equal 2, 3, or 4 to make the expression true. Only 4 is between t and r, but we can't be sure that is the answer. No sufficient.

b) |3 - 5| > |5 - s| = 2 > |5 - s| -> therefore, s can equal 4, 5, or 6 to make the expression true. Only 4 is between t and r, but we can't be sure that is the answer. Not sufficient.

Taking both together, 4 is the only answer that both statements have in common so that must be the solution to the problem. Since, 4 is between t and r, both statements together is sufficient. Answer is C.

Is this the right way to go about solving it?

Not entirely though the logic is fine.

"a) |3 -5| > |3 - s| = 2 > |3 - s| -> therefore, s can equal 2, 3, or 4 to make the expression true. Only 4 is between t and r, but we can't be sure that is the answer. No sufficient."

Actually s can take any value in this range 1 < s < 5 (since it needn't be an integer)
Some values will lie between 3 and 5 and some will not. Not sufficient.

"b) |3 - 5| > |5 - s| = 2 > |5 - s| -> therefore, s can equal 4, 5, or 6 to make the expression true. Only 4 is between t and r, but we can't be sure that is the answer. Not sufficient."

s can take any value in this range 3 < s < 7
Some values will lie between 3 and 5 and some will not. Not sufficient.

The overlap in the two cases is only of 3 < s < 5 and that lies between 3 and 5. So sufficient.

But generally speaking, taking numbers is not a good idea especially in DS questions. You don't know whether you have considered all relevant cases or not.

The same questions have been put up here and I have discussed how to solve them using number line:
ds-inequalities-and-modulus-138237.html#p1119436
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Re: Is s between r and t  [#permalink]

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_________________ Re: Is s between r and t   [#permalink] 28 Sep 2018, 20:15
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