Data Sufficiency Pack 3, Question 4 If |a - b| = 6...

Hi All,

This is a high-difficulty level DS prompt that requires thoroughness and the willingness to prove whether more than one answer to the question exists or not. Since we're dealing with several absolute value calculations, we'll have to consider multiple possibilities (positive and negative) to get to the correct answer.

From the outset, we're given two absolute value equations to work with (|a - b| = 6 and |b - c| = 15). We're asked for the value of |c|.

From the two Facts, it appears that Fact 2 will be considerably easier to deal with (since it limits b to just two possible values), so I'm going to start there.

2) |b| = 9

This Fact tells us that b can equal 9 or -9, so we should do a bit of work to see how those two values would impact the other two variables...

When b = - 9, c can equal two possibilities:

-9 - c = 15
c = - 24
So |c| = |-24| = 24

or

-9 - c = -15
c = 6
So |c| = |6| = 6
Fact 1 is clearly INSUFFICIENT

Before moving on to Fact 1 though, I'm going to do a bit more work to figure out what the variable a could equal....

When b = -9...

a - (-9) = 6
a = -3

or

a - (-9) = -6
a = -15

When b = +9...

a - (9) = 6
a = 15

or

a - (9) = -6
a = 3

1) |a - c| = 9

Since we've done a number of different calculations in Fact 2, we should look to see if any of them will also 'fit' Fact 1.

IF...
a = -3
b = -9
c = 6
Then all 3 of the absolute value equations (the two in the prompt and the one in Fact 1) are satisfied and the answer to the question is |6| = 6.

IF...
a = -15
b = -9
c = -24
Then all 3 of the absolute value equations (the two in the prompt and the one in Fact 1) are satisfied and the answer to the question is |-24| = 24.
Fact 1 is INSUFFICIENT.

Combined, we already have two sets of values that fit ALL of the given information and provide two different solutions (see above).
Combined, INSUFFICIENT

Final Answer:

EMPOWERgmatRichC, can you please check the question as it seems to me that the given values are contradictory.
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Hi Engr2012,

Thanks for catching the error - there was one typo in the prompt. It's since been fixed.

GMAT assassins aren't born, they're made,
Rich

If |a – b| = 6 and |b – c| = 15, then what is the value of |c|?

(1) |a – c| = 9
(2) |b| = 9

IMO E is correct

I]|a-c|= 9
this doesn't give any value for |c|

II] |b|=9
gives two different values for C

Joining I] and II]

doesn't reveal any unique value

SO E is correct

EMPOWERgmatRichC thank you for updating the question.

As for the question, this question uses the definition of the absolute values ---> distance between 2 variables.

Given, |a – b| = 6 and |b – c| = 15 ---> distance between a and b is 6 units , while the distance between b and c is 15 units. Clearly if you are given the distance between a and c as 9 units, it will be sufficient to answer for |c| or distance of c from the 'zero' on the number line.

Statement 1 gives you that exact statement making it sufficient. |a-c| + |a-b| = |b-c| and from this relation you can get a unique 'distance' value of c , thus giving you a unique value of |c|.

Statement 2 does not give any new information for |c|. Not sufficient.

A is the correct answer.
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Hi Rich,
Really good question. Can you please detail your way in answering this question?Is Algebra only way to solve it? Using Test IT could not help me? I reached the answer A only as there are 3 variables with 3 equations but I could not prove it because I do not know how to solve 3 equations under modulus.

Hi Engr2012,

Even I worked on the question on the similar way
considering these values on number line
But Statement 1 only provides value /magnitude of difference among these three points , but none provides value for the exact point.

As you mentioned it would be sufficient to answer for |c| or distance of c from the 'zero' on the number line.

But where does zero exist or reference of points in terms of zero is unknown.

ITS statement B which provides that |B|=9
thats also provides us with two options of value of B as 9 or -9
according to which value of C changes

Thats why I picked E

Please let me know flaw in my reasoning

Hi Rich,
Thanks for detailing your explanation. I have come with a question. What is the fact or condition which could be added to make the solution sufficient? It will help see how the variation of the question can happen.

Thanks

1 way will be to have b=9 instead of |b|=9 for statement 2, and for statement 1, you can eliminate some cases by adding the fact that a and/or c have to be >0 etc. Many different ways to make the statements sufficient on their own and even when combined.

EMPOWERgmatRichC , thank you for the solution. Statement 1 is not sufficient. I interpreted the sufficiency incorrectly above. Statement 1 and the question stem mention the same thing, making this statement not sufficient.
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Very nice question.

\(|a-b|=6\)
\(|b-c|=15\) or \(|c-b|=15\)

On the real line:

\(c---------a------b------a---------c\)

each - represents 1 unit

(1) |a-c|=9. This is just repeating the above. Can't help answering the question (otherwise this would not be a DS...). Don't need even to consider letter C, its B or E.

(2)

if b=-9 then \((c=-24)----------(a=-15)------(b=-9)------(a=-3)---------(c=6)\)

if b=9 then \((c=-6)----------(a=3)------(b=9)------(a=15)---------(c=24)\)

|c| has two values 6 or 24 , insufficient. Letter E

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.

If |a – b| = 6 and |b – c| = 15, then what is the value of |c|?

(1) |a – c| = 9
(2) |b| = 9

There are 3 variables (a,b,c) and 2 equations (|a – b| = 6 and |b – c| = 15), so we need one more equation. There are 2 equations given from the 2 conditions, so there is high chance (D) will be our answer.
From condition 1, a-b=-6, 6/ b-c=-15, 15/ a-c=-9,9. There are too many possible answers, so this is insufficient.
From condition 2, a-b=-6, 6/ b-c=-15, 15/ b=-9,9; this is insufficient for the same reason
Looking at them together, c=-24, a=-15, b=-9 or c=-6, a=3, b=9. This also gives no unique answer,so this is insufficient as well.

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.

Responding to a PM.

If we can visualise these points on a line, it becomes an easy question.

If |a – b| = 6 and |b – c| = 15, then what is the value of |c|?
So a is 6 units away from b, which in turn is 15 units away from c.
Two cases
a) …….a…..b…..c OR …….c……b…..a: In this case, a and c will be 6+15 or 21 units away
b) …….c…..a…..b OR …….b……a…..c: In this case, a and c will be 15-6 or 9 units away
We are looking for the value of |c|

(1) |a – c| = 9
This just confirms one of the above cases, that is case (b), but we do not know the location of c on the number line.
Nothing to move ahead.
Insufficient

(2) |b| = 9
When b=9: |b-c|=15….|9-c|=15……c could be 24 or -6
When b=-9: |b-c|=15….|-9-c|=15……c could be -24 or 6
So |c| could be 24 or 6.
Insufficient


Combined
Nothing new added to statement II when combined with statement I.
Let us see
|a-c|=9
Case I: ……c……a…….9 or …..-6……3……9 => |c|=6
Case II: ……9……a…….c or …..9……15……24 => |c|=24
Insufficient


E

rocky620
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