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QR #49: How many integers n are there such that r<n<s

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QR #49: How many integers n are there such that r<n<s

by saintforlife » Sun Nov 04, 2012 12:59 pm

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

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

OA = C

I read the answer explanation in the Quant Review #2 book, but I don't understand something.

By combining both, if you say that the number of integers n=5 between r and s, doesn't the example they give in Statement 1 solution violate the inequality?

They say if r=6.5 and s=11.5, then n=5. Doesn't this violate r<n<s inequality because clearly n=5 < r=6.5?

I know it works for cases such as r=0.1 and s=5.1, then n=5. Here the inequality r<n<s holds, but for the above case, it doesn't.

Can somebody weigh-in? What am I missing?
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by pemdas » Sun Nov 04, 2012 5:36 pm

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did you understand the logic behind this question?
why to ask about this gibberish by official explanation

if we knew about s and r (integers or not integers) our task would be simplified to st(1) Suff. Since we know nothing about s and r, there could be always four integers (case when s and r are integers) or five integers (when s or r is not integer)

st(2) helps with assigning type for our numbers (s and r), but we know nothing about values

combined st(1&2) supports our previously inquired data, i.e. s and r are not integers, hence the number of integers in the given interval will always be 5.

answer c)
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by kullayappayenugula » Sun Nov 04, 2012 10:15 pm

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hi...you got the question wrong. The n is not the number of integers but the integers themsleves.

How many integers n are there such that r < n < s?
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by Brent@GMATPrepNow » Sun Nov 04, 2012 10:19 pm

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saintforlife wrote:How many integers n are there such that r < n < s?
(1) s - r = 5
(2) r and s are not integers
Target question: How many integers n are there such that r < n < s? [/color]

Statement 1: s - r = 5
There are two cases we need to consider.
Case a: s and r are integers.
For example, s=6 and r=1, in which case there are 4 integers between r and s (2, 3, 4 and 5)
Case b: s and r are not integers.
For example, s=6.1 and r=1.1, in which case there are 5 integers between r and s (2, 3, 4, 5, and 6)

Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: r and s are not integers
This is definitely not enough information here to answer the target question.
Consider these 2 cases.
Case a: r=1.1 and s=2.1, in which case there is 1 integer between r and s (2)
Case a: r=1.1 and s=3.1, in which case there are 2 integers between r and s (2 and 3)

Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined:
From statement 1, we know that there are either 4 or 5 integers between r and s, depending on whether or not r and s are integers.
Statement 2 rules out the possibility that r and s are integers.
If r and s are non integers, then there must be 5 integers between r and s

Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer = C

Cheers,
Brent
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by vsri1987 » Sun Sep 22, 2013 1:47 am

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Hi Brent,

s - r = 5

If s=3 and r=-2 then we will have 0 as well in the integer list(count will be 5), isn't it?
Also, it is nowhere mentioned that these are positive integers or do we just assume this in GMAT, please suggest?


Regards,
Vishal
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by Brent@GMATPrepNow » Sun Sep 22, 2013 5:49 am

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vsri1987 wrote:Hi Brent,

s - r = 5

If s=3 and r=-2 then we will have 0 as well in the integer list(count will be 5), isn't it?
Also, it is nowhere mentioned that these are positive integers or do we just assume this in GMAT, please suggest?


Regards,
Vishal
Good question.

If s=3 and r=-2, then -1, 0, 1, and 2 (4 values) are the only integers between r and s.

To answer your main question, there is nothing in the original question that tells us that r and s must be positive.
In fact there are MANY (an infinite number actually) different values for r and s (both positive and negative) that we can use to demonstrate that statement 1 is not sufficient.
I happened to use positive numbers to demonstrate this, but I could have also used negative numbers.
The bottom line is that, for statement 1, there will be either 4 or 5 integers that are between r and s.
As such, we cannot answer the target question with certainty.

Cheers,
Brent
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by theCodeToGMAT » Sun Sep 22, 2013 6:41 am

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s > n > r

1. s - r = 5
if r & s are integers .. then 4 integers.
(0.1) and (5.1) --> 5 integers..
INSUFFICIENT

2. r & s are not integers
INSUFFiCIENT since there is not relation..


Combining...

SUFFICIENT

ANSWER [spoiler][C][/spoiler]
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Re: QR #49: How many integers n are there such that r<n<s

by Scott@TargetTestPrep » Mon Jul 05, 2021 5:20 am

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saintforlife wrote: ↑
Sun Nov 04, 2012 12:59 pm
 How many integers n are there such that r < n < s?

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

OA = 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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