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How many integers are there between, but not including

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How many integers are there between, but not including [#permalink] New post 30 Jun 2009, 13:00
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How many integers are there between, but not including, integers r and s?

(1) s - r = 10
(2) There are 9 integers between, but not including, r + 1 and s + 1.

[Reveal] Spoiler:
Hi there. I searched for this problem in the board but could not find it, so if I'm reposting, I apologize. Here is my question.

When, if ever, do you ASSUME that a list of numbers between to variables are consecutive integers, when it is not explicitly stated? One of my problems in Data Sufficiency is that I don't know when to assume things and when not to. For example:

My answer: E
Insufficient A: If the integers between r and s are odd, and we assume s=11 and r=1, then we there are 4 odd integers (3,5,7,9). If integers were even, there are 5 even integers (2,4,6,8,10). If integers are consecutive, then there are 9 integers (2,3,4,5,6,7,8,9,10). Therefore, A is INSUFFICIENT

Insufficient to B: If there are 9 integers between s and r, but do not include r+1 and s+1, then using the same r=1 and s=11, then it would seem that the numbers are consecutive since B states there are 9 integers between s and r, but then it states that the integers do not include r+1. So B states, using the sample variable amounts, that the integers are (3,4,5,6,7,8,9,10). s+1=12 does not matter because it was never in the set of integers to begin with.

Insufficient to C: Since A states there are different numbers of integers depending of if the integers are odd, even or consecutive, and the question never states what time of group of integers they are, and B gives what seems to be a paradox without further information. C is Insufficient.

D is already been ruled out.

Only answer is E.

However, in the answer guide, it states:
1) Although the difference between s and r is 10, there are not 10 integers between them. For example, if s is 24 and r is 14, their difference is 10, but there are only 9 integers between them: 15,16,17,18,19,20,21,22, and 23. This holds true for any two integers whose difference is 10; SUFFICIENT.

2) Since r and s are the same distance apart as r+1 and s+1, there would still be 9 integers between r and s in this case, although the integers themselves would change; SUFFICIENT.

The Correct answer is D; each statement alone is sufficient.


This explanation shows that my mistake was that I didn't assume that the integers were consecutive and that I misinterpreted statement 2.

So can someone advise me as to whether my explanations are correct in my mistakes? Am I suppose to ASSUME consecutive integers? When am I suppose to assume this and when do I not? I always thought you never assume in Data Sufficiency questions (i.e. do not assume integers unless stated because numbers can be fractions or decimals) Also, can someone please explain statement 2 to me? I've read the explanation several times, and I still cannot understand how the second statement is referring to a shift in the integers themselves, when the statement clearly states, "There are 9 integers between them, but not including r+1...)

Thanks.

Chris


OPEN DISCUSSION OF THIS QUESTION IS HERE: how-many-integers-are-there-between-but-not-including-140737.html
[Reveal] Spoiler: OA

Last edited by Bunuel on 21 Sep 2013, 13:40, edited 1 time in total.
Edited the question and added the OA
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Re: Assumption of Consecutive Integers. [#permalink] New post 30 Jun 2009, 18:26
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You are right, you don't need to assume. Especially in ds questions, you just have to verify whether the ques can be answered using the information GIVEN... no scope for assumptions.

Perhaps you didnt get the question right, or you are not clear with numbers. :)

the question is asking "how many integers in between two integers"
it has been specified that you need to deal only with integers, so no question of assuming fractions or decimals here, comfort.

If you find it difficult to understand what is asked when there are variables, use numbers,
Say - how many integers are there between the ingeters 3 and 6, not inclusive.
ans is 4 and 5, (2 integers)
question DID NOT ask how many odd or even integers. I would suggest that each time you practice questions, make sure you are very clear with what is being asked. Many people make mistakes in interpreting the question wrongly.

As for statement 2.

Consider the number line, its an imaginary line that stretches to infinity on either sides, with positive numbers on the right side of zero and negative on the left.

infinity....-10..-9..-8............-1..0...1...2...3....4...5...........9.....10.......20......................infinity

now let s and r be anywhere on the number line, (we are not concerned with values of s and r, only the distance between them, ie, the number of integers between them)

for example 3 and 5 are 1 integer between them, 23 and 16 have 6 integers between them and so on.

now 16 and 23 have the same number of integers between them as 17 and 24. (add 1 to each 16 and 23)
what did we do here? shifted our range from 16-23 to 17-24, ie we shifted our range 1 unit towards right.

Its like, your car is 5 feet long, and you drive it 1 meter. the distance between the front and back of your car will always be constant ie 5 feet, not matter how much you shift the car. Same applies to shifting two integers on the number line by adding or subtracting a fixed number from them (just like shifting the car front or back)

Hope this helps :)
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Re: Assumption of Consecutive Integers. [#permalink] New post 01 Jul 2009, 13:15
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well said, rashminet84.

I was about to mention the same. If we subtract two numbers, the difference is the number of integers between the two numbers. For this problem,we don't have to bring the consecutive integer concept.

10-5 would always be 5.
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Re: Assumption of Consecutive Integers. [#permalink] New post 20 Aug 2011, 07:03
Thank you all for a very good explanation, but I am still not sure as to how do you know that you have to consider positive numbers only. When I used positive numbers, I picked D as an answer and agreed that both statements are suffeciant.

But what if you take -1 and 11.. when you do -1+11 = 10, but now the number of integers in between just changed.

Again, at what point should I assume or know for sure that negative numbers are not even an option for a given question?

Any help, clarification is very much appreciated..Thanks
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Re: Assumption of Consecutive Integers. [#permalink] New post 22 Aug 2011, 23:44
The answer is B...

from the first statement the numbers could be 1 dnd 11 or 1.5 and 11.5.. the range which is S - R is 10 in both the cases . Between 1 and 11 there are 9 numbers . betweeen 1.5 and 10.5 there are 10 numbers..
from statement two it is clearly staing that there are 9 numbers bwteen S and R because S+1 and R+ 1 would contain the same number is intigers between them ..
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Re: Assumption of Consecutive Integers. [#permalink] New post 23 Aug 2011, 11:37
The Answer is D.

We can rephrase the question as, what is the value of s-r-1? (As number of integers b/w two numbers not including the two numbers is s-r+1-2)
So any options which answers this is sufficient.

Statement 1. s-r=10, Hence sufficient.
from Stament two, (s+1)-(r+1)+1-2 = s-r-1= 9. Hence sufficient.

Hence each statement alone is sufficient.(D)

Please corect me if I am wrong.
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Re: Assumption of Consecutive Integers. [#permalink] New post 23 Aug 2011, 21:15
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shikari wrote:
Thank you all for a very good explanation, but I am still not sure as to how do you know that you have to consider positive numbers only. When I used positive numbers, I picked D as an answer and agreed that both statements are suffeciant.

But what if you take -1 and 11.. when you do -1+11 = 10, but now the number of integers in between just changed.

Again, at what point should I assume or know for sure that negative numbers are not even an option for a given question?

Any help, clarification is very much appreciated..Thanks


We are not assuming that the integers are positive. r and s could be negative too.
It doesn't matter where r and s lie on the number line; s is 10 steps to the right of r (s is greater than r since (s - r) is a positive number). So between them there will be 9 integers (not including r and s)

How many integers are between -5 and 5?
5 - (-5) = 10
Number of integers = 9

If you take r = -1 and s = 11, s is 12 steps to the right of r.
s - r should be 10 but 11 - (-1) is not. So you cannot take these values.
-1 + 11 is incorrect because here their sum is 10, not their difference.

Even if s = -5 and r = -15,
s - r = -5 - (-15) = 10
Number of integers between them = 9
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Re: Assumption of Consecutive Integers. [#permalink] New post 23 Aug 2011, 21:23
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chrisahn112 wrote:
Hi there. I searched for this problem in the board but could not find it, so if I'm reposting, I apologize. Here is my question.

When, if ever, do you ASSUME that a list of numbers between to variables are consecutive integers, when it is not explicitly stated? One of my problems in Data Sufficiency is that I don't know when to assume things and when not to. For example:

How many integers are there between, but not including, integers r and s?
1) s - r = 10
2) There are 9 integers between, but not including, r + 1 and s + 1.

My answer: E
Insufficient A: If the integers between r and s are odd, and we assume s=11 and r=1, then we there are 4 odd integers (3,5,7,9). If integers were even, there are 5 even integers (2,4,6,8,10). If integers are consecutive, then there are 9 integers (2,3,4,5,6,7,8,9,10). Therefore, A is INSUFFICIENT

Insufficient to B: If there are 9 integers between s and r, but do not include r+1 and s+1, then using the same r=1 and s=11, then it would seem that the numbers are consecutive since B states there are 9 integers between s and r, but then it states that the integers do not include r+1. So B states, using the sample variable amounts, that the integers are (3,4,5,6,7,8,9,10). s+1=12 does not matter because it was never in the set of integers to begin with.

Insufficient to C: Since A states there are different numbers of integers depending of if the integers are odd, even or consecutive, and the question never states what time of group of integers they are, and B gives what seems to be a paradox without further information. C is Insufficient.

D is already been ruled out.

Only answer is E.

However, in the answer guide, it states:
1) Although the difference between s and r is 10, there are not 10 integers between them. For example, if s is 24 and r is 14, their difference is 10, but there are only 9 integers between them: 15,16,17,18,19,20,21,22, and 23. This holds true for any two integers whose difference is 10; SUFFICIENT.

2) Since r and s are the same distance apart as r+1 and s+1, there would still be 9 integers between r and s in this case, although the integers themselves would change; SUFFICIENT.

The Correct answer is D; each statement alone is sufficient.


This explanation shows that my mistake was that I didn't assume that the integers were consecutive and that I misinterpreted statement 2.

So can someone advise me as to whether my explanations are correct in my mistakes? Am I suppose to ASSUME consecutive integers? When am I suppose to assume this and when do I not? I always thought you never assume in Data Sufficiency questions (i.e. do not assume integers unless stated because numbers can be fractions or decimals) Also, can someone please explain statement 2 to me? I've read the explanation several times, and I still cannot understand how the second statement is referring to a shift in the integers themselves, when the statement clearly states, "There are 9 integers between them, but not including r+1...)

Thanks.

Chris


Let me give you an example here:

Say, there are 5 people standing in a line:

Chris, Tom, Christina, Beth, John

My question is: How many people are between Chris and John?
Can you say there are 3 people?
Would you start thinking - well, there is one boy and 2 girls. I don't know whether I should give the number of girls or the number of boys or the number of all people. So, I cannot give the answer.
Would you?

The question is simply how many people are there. You need to give the total number of people i.e. 3.

Similarly say, between 10 and 20, how many integers are there? Just give the total number of integers between 10 and 20. They just want the total number of integers. It doesn't matter how many are odd, how many are even, whether they are consecutive etc.
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Re: Assumption of Consecutive Integers. [#permalink] New post 24 Aug 2011, 08:07
Karishma,

you ROCK. Thank you so much..It makes sense to me now. I rushed and didnt realized that even with a negative number, the integers between still stays 9...

I really liked your explanation to another student as well, where you used names to get your point across. Very well said..

Thank you
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Re: Assumption of Consecutive Integers. [#permalink] New post 14 Jan 2012, 22:07
Thank you Karishma and rashminet84 for the convincing explanations.

I had the same kind of understanding as chrisahn112 had, although I was sure of the option 2.

I was thinking interms of a set of integers
When (1) says s-r = 10, the set could contain {-30, 1, 2, 4, 12} or {-30,1,2,3,55,,4,5,-1,6,7,1000,8,20,9,10,33,11,12}.
So I dont know for sure how many integers are there between s and r.

When (2) says 9 integers between...., I can surely that there are 9 integers no matter what.

So I chose answer B..

Am I being dense or could this also be a line of thinking?
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Re: Assumption of Consecutive Integers. [#permalink] New post 14 Jan 2012, 22:48
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shm0401 wrote:
Thank you Karishma and rashminet84 for the convincing explanations.

I had the same kind of understanding as chrisahn112 had, although I was sure of the option 2.

I was thinking interms of a set of integers
When (1) says s-r = 10, the set could contain {-30, 1, 2, 4, 12} or {-30,1,2,3,55,,4,5,-1,6,7,1000,8,20,9,10,33,11,12}.
So I dont know for sure how many integers are there between s and r.

When (2) says 9 integers between...., I can surely that there are 9 integers no matter what.

So I chose answer B..

Am I being dense or could this also be a line of thinking?


I think you are thinking too much. The use of variables is confusing you.

Let's look at the question again:

"How many integers are there between, but not including, integers r and s?"

r and s are integers. Let me substitute r and s with actual integers.

"How many integers are there between, but not including, integers 10 and 20?"

Now, forget everything else. Just focus on the question above in bold. How will you answer it? Would you say there are 9 integers - 11, 12, 13, 14, 15, 16, 17, 18, 19?
Would you instead think, "I don't know. It could be {10, 12, 20} or {10, 879, 76, 17, 20} etc."
You wouldn't assume that 'between' means 'placed between in a set'. It means the numbers lying between these two numbers on the number line.
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Re: Assumption of Consecutive Integers. [#permalink] New post 16 Jan 2012, 05:54
Thank you Karishma for your reply.
Kudos to you!!!
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Re: Assumption of Consecutive Integers. [#permalink] New post 16 Jan 2012, 07:28
Good Explanation.
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Re: How many integers are there between, but not including [#permalink] New post 21 Sep 2013, 13:34
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Re: How many integers are there between, but not including [#permalink] New post 21 Sep 2013, 13:41
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How many integers are there between, but not including, integers r and s ?

Notice that we are told that r and s are integers.

(1) s – r = 10 --> since r and s are integers and s – r = 10 then there will be 9 integers between them. For example take s=10 and r=0, then there are following integers between them: 1, 2, 3, 4, 5, 6, 7, 8, and 9. Sufficient.

(2) There are 9 integers between, but not including, r + 1 and s + 1 --> the distance between r and s is the same as the distance between r+1 and s+1, so if there are 9 integers between, but not including, r+1 and s+1 then there will be 9 integers between, but not including, r and s too. For example consider s+1=11 and r+1=1 (9 integers between them: 2, 3, 4, 5, 6, 7, 8, 9, and 10) --> s=10 and r=0 the same as above. Sufficient.

Answer: D.

OPEN DISCUSSION OF THIS QUESTION IS HERE: how-many-integers-are-there-between-but-not-including-140737.html

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Re: How many integers are there between, but not including [#permalink] New post 18 May 2014, 10:03
In statement 1, we have s=10+r, let's say r=1, then s=11, the number of integers between 1 and 11 is 9 excluding 1 and 11 itself. No matter what integer you pick for r, the number of integers between r and s excluding will be fixed. Sufficient

In statement 2, we have to find the number of integers between r+1 and s+1. We are told that the number of integers between r+1 and s+1 is 9 excluding r+1 and s+1. Let's say r=1,r+1=2, so s+1 must be 9 integers away from r+1 in order to satisfy the condition.i,e, s+1=12 Sufficient

Hence D
Re: How many integers are there between, but not including   [#permalink] 18 May 2014, 10:03
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