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A Tricky DS problem on inequality

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Source: — Data Sufficiency |
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by amitmj » Thu Apr 10, 2014 8:49 pm
I have taken this question from Vertias Prep for DS, and I found the answer to be wrong. Hence asking this question.
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by [email protected] » Thu Apr 10, 2014 9:33 pm
Hi amitmj,

This DS question is built around a test of your thoroughness. Notice how you are NOT told anything about A and B.....?

We're asked how many integers are between A and B. This is a perfect question for TESTing Values

Fact 1: B - A = 7

If B = 8 and A = 1, then the integers between them are 2, 3, 4, 5, 6, and 7 (a total of 6 integers)
If B = 8.5 and A = 1.5, then the integers between them are 2, 3, 4, 5, 6, 7 and 8 (a total of 7 integers)
Fact 1 is INSUFFICIENT.

Fact 2: A and B are integers.
Since we don't know what A and B are, there's really no way to answer this question.....but if you're looking for proof....
If B = 8 and A = 1, then the answer to the question is 6 integers
If B = 2 and A = 1, then the answer to the question is 0 integers.
Fact 2 is INSUFFICIENT

Combined, we know that B - A = 7 AND A and B are integers. This means that there must be exactly 6 integers between them. You can TEST any set of values that fit these rules and you'll get the same answer every time.
Combined, SUFFICIENT.

Final Answer: C

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by amitmj » Thu Apr 10, 2014 9:48 pm
So let me tell you why I posted this, I was confused when I read this statment
"How many integers x exist such that a < x < b? "

I had two choices in mind when I read this:
1)x should be a value between a and b, and since b-a=7, x could have infinite values. As x , could be 1, or 11, 23 ,37 etc.
2)x is the no. of integer or decimal value between b and a

How do I select the right choice (which as per your answer is 2)?

Thanks,
Amit
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by ajaysingh24 » Fri Apr 11, 2014 12:03 pm
amitmj wrote:So let me tell you why I posted this, I was confused when I read this statment
"How many integers x exist such that a < x < b? "

I had two choices in mind when I read this:
1)x should be a value between a and b, and since b-a=7, x could have infinite values. As x , could be 1, or 11, 23 ,37 etc.
2)x is the no. of integer or decimal value between b and a

How do I select the right choice (which as per your answer is 2)?

Thanks,
Amit
It is asking how many integer x must exist .... it is just asking about the count of integer .. it does not have any bearing on the value of integer
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by [email protected] » Fri Apr 11, 2014 4:46 pm
Hi amitmj,

Every GMAT question is carefully worded, so you have to put in the extra effort to recognize when specific information is presented. Furthermore, GMAT questions are carefully worded to ask you for a specific goal. If you're not clear on the question, you'll likely get it wrong.

Here, we're asked "HOW MANY INTEGERS are between A and B?"

This means that there is an actual number of integers; we have to figure out HOW MANY there are. The rest of the work is based on answering the question that's asked and considering the possibilities.

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by ceilidh.erickson » Sat Apr 12, 2014 1:45 pm
amitmj wrote:x could have infinite values. As x , could be 1, or 11, 23 ,37 etc.
Hi Amit,

I think I understand your confusion. You're right - even with both statements together, these numbers could fall anywhere on the number line, and thus there are infinite possibilities for what a, b, and x could be.

However - we have to work with the assumption that integers a and b have knowable values. There are infinite possibilities for what those values could be, but for any a and b that we pick, such that b - a = 7, there will only be 6 integers between them. Thus, there are 6 potential x's that exist between each possible a and b.

Since a and b are presumed to have real values, we can say that the answer is C.

Does that help?
Ceilidh Erickson
EdM in Mind, Brain, and Education
Harvard Graduate School of Education
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by Matt@VeritasPrep » Sat Apr 12, 2014 2:02 pm
Put in a friendlier way (and, of course, the GMAT itself will not be friendly!), the original stem could read "How many integers are there between the numbers a and b on the number line?"

Two important things:

1:: The specific values of a and b don't really matter, just (i) how far apart they are on the number line and (ii) whether they're integers or not.

For instance, there are as many integers between and 1 and 6 {2, 3, 4, 5} as there are between 10 and 15 {11, 12, 13, 14}. Likewise, there are as many integers between 1.5 and 6.5 {2, 3, 4, 5, 6} as there are between 10.5 and 15.5 {11, 12, 13, 14, 15}. So all we care about is how far apart a and b are.

2:: x isn't a value here, it's just a way of naming the type of integer we want (integers between a and b). When we say "How many integers x are there such that a < x < b?", we're really saying "How many integer members are there of the set of all numbers greater than a and less than b?" x here is a property that a number has (it's an integer between a and b) rather than an actual number.
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