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16 Sep 2014, 01:00
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If \(|a - b| = |b - c| = 2\), what is the value of \(|a - c|\)?
(1) \(a < b < c\)
(2) \(c - a > c - b\)
(1) \(a < b < c\)
(2) \(c - a > c - b\)
16 Sep 2014, 01:00
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Official Solution:
If \(|a - b| = |b - c| = 2\), what is the value of \(|a - c|\)?
(1) \(a < b < c\).
Given that \(a < b\), we can deduce that \(|a-b| = -(a-b) = 2\), which simplifies to \(b-a = 2\). Similarly, since \(b < c\) is given, we can deduce that \(|b-c| = -(b-c) = 2\), which simplifies to \(c-b = 2\). By adding these two equations: \((b-a) + (c-b) = 2 + 2\), we get \(c-a = 4\). Therefore, \(|a-c| = 4\). Sufficient.
(2) \(c - a > c - b\).
If we rearrange this inequality and cancel out \(c\)s on both sides, we get: \(a < b\). However, this is not sufficient to determine the value of \(|a - c|\). For instance, if \(a=0\), \(b=2\), and \(c=4\), then \(|a-c| = 4\). On the other hand, if \(a=0\), \(b=2\), and \(c=0\), then \(|a-c| = 0\).
Answer: A
If \(|a - b| = |b - c| = 2\), what is the value of \(|a - c|\)?
(1) \(a < b < c\).
Given that \(a < b\), we can deduce that \(|a-b| = -(a-b) = 2\), which simplifies to \(b-a = 2\). Similarly, since \(b < c\) is given, we can deduce that \(|b-c| = -(b-c) = 2\), which simplifies to \(c-b = 2\). By adding these two equations: \((b-a) + (c-b) = 2 + 2\), we get \(c-a = 4\). Therefore, \(|a-c| = 4\). Sufficient.
(2) \(c - a > c - b\).
If we rearrange this inequality and cancel out \(c\)s on both sides, we get: \(a < b\). However, this is not sufficient to determine the value of \(|a - c|\). For instance, if \(a=0\), \(b=2\), and \(c=4\), then \(|a-c| = 4\). On the other hand, if \(a=0\), \(b=2\), and \(c=0\), then \(|a-c| = 0\).
Answer: A
General Discussion
22 Nov 2015, 16:22
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Bunuel
my approach:
We can also write this question as:"if the distance between a and b, and, b and c is 2 then what is the distance between a and c ?"
1) if a < b < c then this is an arithmetic series with common difference 2. Which is a,b,c = a, a+2, a+4 thus the distance between a and c is 4. ===sufficient
2) if c-a > c-b (which is a < b) then c could be c= b+2 = a+4 or c= b-2 = a) ===insufficient
Answer A
