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# Is s between r and t

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Is s between r and t [#permalink]  02 Sep 2012, 08:03
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Is s between r and t

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

[Reveal] Spoiler:

Is s between r and t

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

[Reveal] Spoiler:

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]  02 Sep 2012, 08:38
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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.
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Re: Is s between r and t [#permalink]  03 Sep 2012, 14:39
1
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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|

[Reveal] Spoiler:

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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Re: Is s between r and t [#permalink]  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]  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]  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
Re: Is s between r and t   [#permalink] 08 Oct 2012, 05:20
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