How many integers n are there such that r < n < s ?

SOLUTION

How many integers n are there such that r < n< s ?

The question asks: how many integers \(n\) satisfy \(r < n < s\), or how many integers are between \(r\) and \(s\), not inclusive?

Notice that if \(r\) and \(s\) are integers, for example \(s=5\) and \(r=0\) then there will be 4 integers between them: 1, 2, 3, and 4. But if \(r\) and \(s\) are NOT integers for example \(s=5.5\) and \(r=0.5\) then there will be 5, so one more, integers between them: 1, 2, 3, 4 and 5.

(1) s - r = 5. Not sufficient.
(2) r and s are not integers. Not sufficient.

(1)+(2) According to above reasoning since r and s are not integers then there are 5 integers between them (case 2). Sufficient.

Answer: C.

Similar questions to practice:
https://gmatclub.com/forum/how-many-inte ... 15561.html
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(1) s - r = 5

let s= 6.5 and r=1.5 then there are 5 integers between s and r , 2,3,4,5 and 6.

let s=6 and r=1 then there are 4 integers between s and r, 2,3,4, and 5

hence insufficient

(2) r and s are not integers.

Clearly insufficient s=10.5 r=1.5 or s=2.5 r=1.5, etc

1+2

Sufficient as any 2 non integers , such that their difference is 5, will have 5 integers between them.

e.g.

s= 6.5 and r=1.5 then there are 5 integers between s and r , 2,3,4,5 and 6

Answer C
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The "number" of n can be multiple of 4 or 5, says 8 or 10, 12, 15, ... can not be determined...

It seems that you misinterpreted the question.

Question ask about the number of integers between r and s, not inclusive.

For example: how many integers are between 10 and 15, not inclusive? Answer: 4, namely 11, 12, 13, and 14.
Or: how many integers are between 5.3 and 10.3? Answer: 5, namely 6, 7, 8, 9 and 10.

So taken together statements are sufficient to answer that there are 5 integers between r and s.

Hope it's clear.
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Hi Bunuel ,

I understood your point , just wanted to ask one thing about my interpretation...

Can’t I have repetitive “n” ? We don’t know whether the numbers are in any sequence or not .

So , can’t I say this for option (i)


i ) s-r = 5

For eg, r = 1 and “n” can be 2 , 2,3,4,5,5 then s =6

So , s-r = 6-1 = 5 , but “n” can be anything between 1 and 6 ..

Please correct me

I know I am over assuming

chetan2u - plz guide

Posted from my mobile device

The answer to the question how many integers are between 1 and 6 is four: 2, 3, 4, and 5. How would you answer this? Would you include any of the integers more than once?
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Asked: How many integers n are there such that r < n < s ?

(1) s - r = 5
If s & r are integers such that s>r; e.g r=1; s =6 n = {2,3,4,5}; 4 integers
But if s & r are non-integers such that s>r; e.g r =0.5, s = 5.5; n={1,2,3,4,5}; 5 integers
NOT SUFFICIENT

(2) r and s are not integers.
r & s are take multiple values
NOT SUFFICIENT

(1) + (2)
(1) s - r = 5
(2) r and s are not integers.
s>r; e.g r =0.5, s = 5.5; n={1,2,3,4,5}; 5 integers
SUFFICIENT

IMO C

Solution:

Question Stem Analysis:

We need to determine the number of integers between r and s.

Statement One Alone:

Statement one alone is not sufficient. For example, if r = 1 and s = 6, then there are 4 integers between r and s, namely, 2, 3, 4, and 5. However, if r = 1.1 and s = 6.1, then there are 5 integers between r and s, namely 2, 3, 4, 5 and 6.

Statement Two Alone:

Statement two alone is not sufficient. For example, if r = 1.1 and s = 2.1, then there is 1 integer between r and s, namely, 2. However, if r = 1.1 and s = 3.1, then there are 2 integers between r and s, namely 2 and 3.

Statements One and Two Together:

Both statements together are sufficient since there will be 5 integers between r and s when they are not integers themselves as we can see from the analysis from Statement One Alone when r = 1.1 and s = 6.1. We can prove this as follows:

Let [r] be the greatest integer less than or equal to r. Since r is not an integer, [r] < r. Since s - r = 5, s = r + 5, so the integers between r and s (or r + 5) are [r] + 1, [r] + 2, [r] + 3, [r] + 4 and [r] + 5. Therefore, there are 5 integers.

Answer: C
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Target question: How many integers n are there such that r < n < s ?

Statement 1: s - r = 5
The students who (incorrectly) conclude that statement 1 is sufficient make the very common mistake of assuming r and s are integers, even though the question doesn't state this.
Since the question doesn't specifically state whether r and s are integers, there are many different pairs of values but that satisfy the condition that s - r = 5. Here are two:
Case a: r = 2 and s = 7. Here, n can equal 3, 4, 5 or 6, which means the answer to the target question is there are 4 integer values of n such that r < n < s
Case b: r = 2.1 and s = 7.1. Here, n can equal 3, 4, 5, 6 or 7, which means the answer to the target question is there are 5 integer values of n such that r < n < s
Since we can’t answer the target question with certainty, statement 1 is NOT SUFFICIENT

Note: We can generalize the above results as follows:
- If r and s are integers, then there are 4 possible integer values of n such that r < n < s
- If r and s are not integers, then there are 5 possible integer values of n such that r < n < s


Statement 2: r and s are not integers.
Since we aren't told how far apart the values of r and s are, statement 2 is NOT SUFFICIENT

If you're not convinced, here are two conflicting pairs of values for r and s that satisfy statement 2:
Case a: r = 2.1 and s = 4.1. Here, n can equal 3 or 4, which means the answer to the target question is there are 2 integer values of n such that r < n < s
Case b: r = 2.1 and s = 3.4. Here, n can equal 3 only, which means the answer to the target question is there is 1 integer value of n such that r < n < s
Since we can’t answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Now that we know s - r = 5 AND r and s are not integers, we can apply our generalization above, which means the answer to the target question is there are 5 integer values of n such that r < n < s
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer: C

Cheers,
Brent

I did r-r<n-r<s-r ---------0<n-r<5 and I got both choices as insufficient. Please tell me what have I done wrong. WHy is this not working BrentGMATPrepNow ThatDudeKnows JeffTargetTestPrep GMATNinja

I'm a huge fan of using number lines on inequality questions...something about visualizing makes it easier. Check out the attachment. The blue and red brackets are two of infinitely many possible brackets for which the range is five. The blue one has four integers between r and s. The red one has five integers between r and s.

Statement (1) allows for either the blue or the red, so we don't know whether n is 4 or 5. BCE.
Statement (2) allows for n to be anything. CE.
Statements (1) and (2) together remove the blue bracket from possibility and we left with only the red. n must be 5. C.

Pls see what I had asked.
i am specifically talking about the approach I have used and hence please tell me in that context ThatDudeKnows

Okayyyy, but I think it's probably more useful to add to your skillset a way of thinking about inequalities questions, generally, than to focus on this specific problem...

Looking at statement (1) alone: 0<n-r<5. We are told that n is an integer. If r is an integer, how many values can you pick for n? 4. If you don't see this intuitively, try picking different integer values for r and then seeing how many possible n's there are. If r=0, n could be 1, 2, 3, 4. If r=1, n could be 2, 3, 4, 5. If r=62, n could be 63, 64, 65, 66. Whenever we pick an integer for r, n=4. If r is not an integer, how many values can you pick for n? 5. You could again try picking different values for r. If r=0.5, n could be 1, 2, 3, 4, 5. if r=-37.2, n could be -37, -36, -35, -34, -33. n can be either 4 or 5. Do we have enough information to know the value of n? Nope. BCE.

Looking at statement (2) alone: Definitely not enough information to know the value of n. CE.

Looking at both statements together: From statement (1) we had two possibilities: n=4 or n=5. Statement (2) removed one of those from contention. We are left with only one. Do we have enough information to know the value of n? Yep. C.

Thanks for the great answer Bunuel. However, I believe you made a minor typo and meant to say "Answer: 5, namely 6, 7, 8, 9 and 10," not "namely 5, 6, 7, 8, 9 and 10," since 5 is not between 5.3 and 10.3.

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Fixed the typo. Thank you!
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Hi ThatDudeKnows - i am struggling to undertand what "n" should be, when I read the question stem specifically. Attached is the question stem :



Attached is your visualization

Lets focus on the red line specifically :

-- Is "n" - the number of tick marks between 1.5 and 6.5 ?

OR

-- Is "n" - an actual NUMBER/ integer BETWEEN 1.5 and 6.5 ?

Hi jabhatta2
Thanks for your query.


As far as I can understand from your query, you are struggling with two main process skills – translation and visualization - ‘TRANSLATION’ of the question stem and then ‘VISUALIZATION’ of numbers on a number line.

Let me first help you with the translation, so that we at least understand the question completely. Then, we will move forward to a number line representation.


TRANSLATION:
Question stem: “How many integers n are there such that r < n < s?”.


Let’s try to translate this sentence piece-by-piece into mathematical terms. This will ensure that we don’t miss even a single word.

• “How many integers n”
  
  • Integers n – This itself makes it clear that ‘n’ is an integer.
  • This is all you need to know about ‘n’. (Spend time here and see why you were confused about the nature of n. Build the correct skills right here!)
• “are there such that r < n < s”
  
  • This part clarifies that we have to consider only those INTEGRAL (integers) values of ‘n’ which are STRICTLY less than ‘s’ and STRICTLY more than ‘r’.

So, from the question stem, we understand that we need to find the NUMBER of integers (n) that lie between r and s, exclusive (r < n < s).

Now, let’s try to understand it’s visualization on a number line.


VISUALIZATION:
This is the number line representation you shared:

The main thing you need to understand is that in every basic representation of a number line, the tick represents the exact integer written below it.

Let’s try to understand that by just considering the red portion above as an example. Note that the starting mark represents ‘r’, and the end represents ‘s’.


Red portion:
Since the red line starts from some point between integers 1 and 2, we are considering ‘r’ to be something between 1 and 2. Say r = 1.5.
Similarly, since the red line ends at some point between 6 and 7, we are considering ‘s’ to be something between 6 and 7. Say 6.5.
Then, all the tick marks that you see between r and s are the integer values that n can take. If you count the ticks under the red line from the number line, you will find exactly 5 ticks. This means that there are exactly 5 integers between 1.5 and 6.5.
You can easily verify that by actually listing all the integers between 1.5 and 6.5. These are: 2, 3, 4, 5, and 6. See how it is exactly 5 as we got by counting the ticks also!


Now, just try going to ThatDudeKnows’ explanation again. You should get it! 😊


Hope this helps!


Best,
Aditi Gupta
Quant expert, e-GMAT
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