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# If |a - b| = |b - c| = 2 , what is |a - c| ?

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If |a - b| = |b - c| = 2 , what is |a - c| ?  [#permalink]

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09 Oct 2010, 04:09
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58% (02:20) correct 42% (02:31) wrong based on 228 sessions

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If |a - b| = |b - c| = 2 , what is |a - c| ?

(1) $$a \lt b \lt c$$

(2) $$c - a \gt c - b$$

(C) 2008 GMAT Club - m16#37
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Re: If |a - b| = |b - c| = 2 , what is |a - c| ?  [#permalink]

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09 Oct 2010, 04:40
2
AndreG wrote:
How do I approach a question like the following:

If $$|a - b| = |b - c| = 2$$ , what is $$|a - c|$$ ?

1. $$a \lt b \lt c$$
2. $$c - a \gt c - b$$

(C) 2008 GMAT Club - m16#37

* Statement (1) ALONE is sufficient, but Statement (2) ALONE is not sufficient
* Statement (2) ALONE is sufficient, but Statement (1) ALONE is not sufficient
* BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient
* EACH statement ALONE is sufficient
* Statements (1) and (2) TOGETHER are NOT sufficient

$$|a - b| = |b - c| = 2$$

Imagine the points on a number line. There is two possibilities, either a & c are on the same side of the line relative to b, or on the opposite sides. Also remember that |x-y| represents distance between x and y on the number line.

So if a & c ar on the same side then a=c. |a-c|=0
If they are on opposite sides, |a-c|=4

(1) $$a \lt b \lt c$$
a and c on opposite sides, answer is 4. Sufficient

(2) $$c - a \gt c - b$$
This only implies $$a \lt b$$
Insufficient to know where c is, same side or opposite side. Insufficient

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Re: If |a - b| = |b - c| = 2 , what is |a - c| ?  [#permalink]

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09 Oct 2010, 05:51
2
1
AndreG wrote:
How do I approach a question like the following:

If $$|a - b| = |b - c| = 2$$ , what is $$|a - c|$$ ?

1. $$a \lt b \lt c$$
2. $$c - a \gt c - b$$

(C) 2008 GMAT Club - m16#37

Generally for $$|x|$$:
When $$x\leq{0}$$, then $$|x|=-x$$;
When $$x\geq{0}$$, then $$|x|=-x$$.

(1) $$a<b<c$$ --> as $$a<b$$ ($$a-b<0$$) then $$|a - b|=2$$ becomes: $$-a+b=2$$, so $$b=2+a$$ and as $$b<c$$ ($${b-c}<0$$) then $$|b-c|=2$$ becomes: $${-b+c}=2$$. Substituting $$b$$ --> $$-2-a+c=2$$ --> $$a-c=-4$$ --> $$|a - c|=|-4|=4$$ . Sufficient.

(2) $$c-a>c-b$$ --> $$b-a>0$$ --> $$|a - b|=2$$ becomes: $$-a+b=2$$, so $$b=2+a$$ --> $$|b - c|=|a-c+2| = 2$$ --> either $$a-c=0$$ or $$a-c=-4$$. Not sufficient.

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Re: If |a - b| = |b - c| = 2 , what is |a - c| ?  [#permalink]

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09 Oct 2010, 10:14
Thanks to the two of you!
While I do understand both solutions, I feel shrouded's is a lot faster, will that always be the case, or is this kind of just lucky for this particular question?
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Re: If |a - b| = |b - c| = 2 , what is |a - c| ?  [#permalink]

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09 Oct 2010, 11:30
Thinking of |x-y| as distance between two points on a number line is a very neat trick and I find it very helpful in a lot of GMAT problems. You should def give it a shot first. Thinking visually is faster than algebraically solving in many cases

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Re: If |a - b| = |b - c| = 2 , what is |a - c| ?  [#permalink]

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09 Oct 2010, 11:38
|a-b| = |b-c| = 2 can be written in 4 ways

1) a-b = b-c = 2 => a>b>c with a diff of 2
2) a-b = c-b = 2 => a=c
3) b-a = b-c = 2 => a=c
4) b-a = c-b = 2 => a<b<c with a diff of 2

A) a<b<c : based on the 4th statement above , we can say that |a-c| = 4
Sufficient
B) c-a > c-b : we can understand that a<b but there is no relationship with C.
Hence Insufficient

Ans : A

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Re: If |a - b| = |b - c| = 2 , what is |a - c| ?  [#permalink]

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16 Oct 2010, 05:38
1
I always find these type of questions to be killer..
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Re: If |a - b| = |b - c| = 2 , what is |a - c| ?  [#permalink]

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06 Jun 2014, 08:09
I don't think that even the options are required to answer this question

From question itself

|<-------------2------------->|<--------------2-------------->|
A------------------------------B--------------------------------C

|<----------------------------4--------------------------------->|

The question is badly framed and is definitely not GMAT.
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Re: If |a - b| = |b - c| = 2 , what is |a - c| ?  [#permalink]

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06 Jun 2014, 08:36
If |a - b| = |b - c| = 2 , what is |a - c| ?

(1) $$a \lt b \lt c$$
(2) $$c - a \gt c - b$$

I don't think that even the options are required to answer this question

From question itself

|<-------------2------------->|<--------------2-------------->|
A------------------------------B--------------------------------C

|<----------------------------4--------------------------------->|

The question is badly framed and is definitely not GMAT.

There is noting wrong with the question. The second statement is NOT sufficient. Consider:
b=3, a=1, and c=1 --> |a - c| = 0.
b=3, a=1, and c=5 --> |a - c| = 4.

Hope it helps.
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Re: If |a - b| = |b - c| = 2 , what is |a - c| ?  [#permalink]

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30 Jul 2015, 03:18
|a-b|=|b-c|=2 means that distance between a and b equal to that between b and c

Four possibilities in number line:
----a------b------c-------->
----c------b------a-------->
----a,c--------b----------->
----b----------a,c--------->

What is the |a-c|?

So, if b between a and c, the answer is 4; if a=c, the answer is 0

St.1 a<b<c, b between a and c, so answer is 4. SUFF

St.2 c-a>c-b => -a>-b => a<b. But we do not know where c. INSUFF

A
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Re: If |a - b| = |b - c| = 2 , what is |a - c| ?  [#permalink]

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01 Jul 2017, 16:03
AndreG wrote:
If |a - b| = |b - c| = 2 , what is |a - c| ?

(1) $$a \lt b \lt c$$

(2) $$c - a \gt c - b$$

(C) 2008 GMAT Club - m16#37

Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

Question

In the original question, there are 3 variables and 2 equations. Thus D is most likely.

For the condition 1), $$| a - b | = b - a = 2$$ and $$| b - c | = c - b = 2$$.
When we add two equations $$c - a = ( b - a ) + ( c - b ) = c - a = 4$$.
Thus $$| a - c | = 4$$.
The condition 1) is sufficient.

For the condition 2) $$c - a > c - b$$ is equivalent to $$-a > -b$$ or $$a < b$$.
However, we don't know anything about c from the condition 2).

There two cases that satisfy $$| a - b | = | b - c | = 2$$.

----a----b----c---->

---a,c----b--------->

Thus $$|a-c| = c - a = ( c - b ) + ( b - a ) = 2 + 2 = 4$$ or $$| a - c | = 0$$.
The condition 2) is sufficient.

For cases where we need 1 more equation, such as original conditions with “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 59 % chance that D is the answer, while A or B has 38% chance and C or E has 3% chance. Since D is most likely to be the answer using 1) and 2) separately according to DS definition. Obviously there may be cases where the answer is A, B, C or E.
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Re: If |a - b| = |b - c| = 2 , what is |a - c| ?   [#permalink] 01 Jul 2017, 16:03
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