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Is s between r and t

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DS inequalities and Modulus  [#permalink]

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New post 02 Sep 2012, 03:52
1
1
00:00
A
B
C
D
E

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(N/A)

Question Stats:

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

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Please help to solve the following:-

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|
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Is s between r and t  [#permalink]

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New post Updated on: 03 Sep 2012, 04:08
1
6
Is s between r and t

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

Official answer E


Is s between r and t

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

Official answer C

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.
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Re: DS inequalities and Modulus  [#permalink]

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New post 02 Sep 2012, 08:38
1
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.
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Re: Is s between r and t  [#permalink]

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New post 03 Sep 2012, 14:39
1
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|




Official answer C



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).

Answer E


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 \(t.\)

Answer C
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Re: DS inequalities and Modulus  [#permalink]

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New post Updated on: 06 Sep 2012, 23:16
pnf619 wrote:
Please help to solve the following:-

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
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
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.
Answer (E)
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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.
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Re: DS inequalities and Modulus  [#permalink]

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New post 06 Sep 2012, 23:13
pnf619 wrote:
Please help to solve the following:-

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
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
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.

Answer (C)
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Re: DS inequalities and Modulus  [#permalink]

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New post 07 Sep 2012, 03:49
pnf619 wrote:
Please help to solve the following:-

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).

Answer E


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 \(t.\)

Answer C
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Re: Is s between r and t  [#permalink]

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New post 07 Oct 2012, 16:36
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?
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Re: Is s between r and t  [#permalink]

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New post 07 Oct 2012, 22:18
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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New post 08 Oct 2012, 05:20
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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