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DS inequalities and Modulus
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02 Sep 2012, 03:52
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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) rs < rt b) rs < st
Is s between r and t
a) rt > rs b) rt > ts



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Is s between r and t
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Updated on: 03 Sep 2012, 04:08
Is s between r and t (1) rs < rt (2) rs < st Is s between r and t (1) rt > rs (2) rt > ts
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
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02 Sep 2012, 08:38
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 rs>rt violating statement (1). If s<r<t, then ts>rt violating statement (2). If r<t<s, then rs>rt violating statement (1). If t<r<s, then ts>rt 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 rs is the smallest of the three possible differences (rs, rt, and st). 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 rt 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
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03 Sep 2012, 14:39
pnf619 wrote: Is s between r and t (1) rs < rt (2) rs < st Is s between r and t (1) rt > rs (2) rt > ts 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
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Updated on: 06 Sep 2012, 23:16
pnf619 wrote: Please help to solve the following:
Is s between r and t
a) rs < rt b) rs < st
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 ... edoredid/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) rs < rt 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) rs < st 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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Re: DS inequalities and Modulus
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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) rt > rs b) rt > ts The question stem is the same so your initial thought process will be the same. Let's look at the statements. a) rt > rs Again, this is same as statement 1 above (rs < rt) 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) rt > ts 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. Answer (C)
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Re: DS inequalities and Modulus
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07 Sep 2012, 03:49
pnf619 wrote: Please help to solve the following:
Is s between r and t
a) rs < rt b) rs < st
Is s between r and t
a) rt > rs b) rt > ts 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
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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
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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: dsinequalitiesandmodulus138237.html#p1119436
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Re: Is s between r and t
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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: dsinequalitiesandmodulus138237.html#p1119436



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