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505-555 Level|   Number Properties|                  
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How many integers are there between, but not including, integers r and 15 Oct 2012, 04:51
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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.
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SOLUTION

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.
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The best way to solve such kind of problem is to use Number line (Sorry i am not including the same)
1) Two integers are 10 units apart, thus the no of integers between 2 integers will be 9---->Sufficient
2) The no of integers between 2 integers r+1 & s+1 is 9. If we move 1 unit in the same direction, no of integers will remain same, which will be 9--->Sufficient

Answer D
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statement 1

s=r+10

so basically the numbers we have are r,r+1,r+2,r+3,r+4,r+5,r+6,r+7,r+8,r+9,r+10( r+10 is S)

so there are 9 integers in between another way to solve would be s-r+1 = 10+1 = 11 total number of integers
since we need the numbers in between 11 - 2 ( the 2 numbers we subtract are r and s) hence we get 9

statement 2

there are 9 integers so we have a total of 11 integers at the moment.

s+1 - (r+1) + 1 = 11
s+1-r-1=10
s-r=10 which is same as statement 1 sufficient

so answer is D
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From statement 1,
s-r=10
Sufficient to conclude number that s is 9 integers after r.

Statment 2,
s+1 is 9 integers after r+1
or s is 9 integers after r (same as S1)
Sufficient.

Hence D

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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.

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How many integers are there between, but not including, integers r and 21 Jan 2013, 18:57
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Bunuel
SOLUTION

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.

Kudos points given to everyone with correct solution. Let me know if I missed someone.
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I still do not understand how (1) is sufficient. My train of thought on it being insufficient is as follows with and example
s-r=10

12-2= 10 if s= 12 and r= 2 but the consecutive set could be consecutive multiples 2,4,6,8,10,12. There would only be 4 integer in between.
Do we just assume they are a consecutive set of integers?
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How many integers are there between, but not including, integers r and 22 Jan 2013, 04:40
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Bunuel
SOLUTION

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.

Kudos points given to everyone with correct solution. Let me know if I missed someone.

I still do not understand how (1) is sufficient. My train of thought on it being insufficient is as follows with and example
s-r=10

12-2= 10 if s= 12 and r= 2 but the consecutive set could be consecutive multiples 2,4,6,8,10,12. There would only be 4 integer in between.
Do we just assume they are a consecutive set of integers?
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Let me ask you a question: how many integers are there between, but not including, 2 and 12?
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How many integers are there between, but not including, integers r and 22 Jan 2013, 06:34
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Let me ask you a question: how many integers are there between, but not including, 2 and 12?
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BINGO- I get it now. The lights just went on. There are 9. The question specifically states how many integers between the two numbers. It does not mention intervals. The question states "integers between"
The good thing is that I am learning that I am sometimes reading way too much into a question, thinking it can not be this easy.
This is part of my learning "what to watch out for, and what kind of mistakes I am making"
Thanks so much for your help
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How many integers are there between, but not including, integers r and 31 Mar 2013, 06:40
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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.

Hope it's clear.
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Hi Bunuel , if i am not mistaken , why are we assuming that r and s belongs to an evenly spaced set of integers ??
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How many integers are there between, but not including, integers r and 31 Mar 2013, 08:21
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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.

Hope it's clear.

Hi Bunuel , if i am not mistaken , why are we assuming that r and s belongs to an evenly spaced set of integers ??
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Not sure I understand what you mean.

s and r are just two integers. The question asks: how many integers are there between, but not including r and s.

Ask yourself, how many integers are there between, but not including, 0 and 10? The answer is 9: 1, 2, 3, 4, 5, 6, 7, 8, and 9.
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How many integers are there between, but not including, integers r and 22 May 2013, 04:51
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Bunuel
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.

Hope it's clear.
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But, Bunuel - Considering your example in the first case in which r=0 and s=10, the number of integers between them could be maximum 9 or it could be any number less than that. Because, there is no mention of the word "consecutive" in the question. But, the second one clearly states that there are 9 integers. Hence B is sufficient, but A is not! Can you please explain where I am going wrong? Since this is an official problem and also you had solved it, I am 100% confident that D is the answer, but, I want to know where I am going wrong.
Thanks!
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Bunuel
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.

Hope it's clear.
But, Bunuel - Considering your example in the first case in which r=0 and s=10, the number of integers between them could be maximum 9 or it could be any number less than that. Because, there is no mention of the word "consecutive" in the question. But, the second one clearly states that there are 9 integers. Hence B is sufficient, but A is not! Can you please explain where I am going wrong? Since this is an official problem and also you had solved it, I am 100% confident that D is the answer, but, I want to know where I am going wrong.
Thanks!
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You don't need the word "consecutive".

How many integers are there between, but not including 1 and 3? Only one integer: 2.
How many integers are there between, but not including 1 and 5? Three: 2, 3, and 4.

Similarly: how many integers are there between, but not including, 0 and 10? The answer is 9: 1, 2, 3, 4, 5, 6, 7, 8, and 9.

Hope it's clear.
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How many integers are there between, but not including, integers r and 27 Jan 2015, 09:09
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Bunuel , why can't s and r be negative?s could be -10 and r could be 20 so s-r would still be 10 but the number of integers would be 30?I'm sure i'm missing something stupid here.
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Bunuel , why can't s and r be negative?s could be -10 and r could be 20 so s-r would still be 10 but the number of integers would be 30?I'm sure i'm missing something stupid here.
:/
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If s = -10 and r = 20, then s - r = -10 - 20 = -30, not 10.
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How many integers are there between, but not including, integers r and 27 Jan 2015, 11:08
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Oh.i knew its something stupid.This warrants a break.Thanks a ton Bunuel . You rock.
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The number of integers between two numbers r and s (including both numbers) is s - r +1
The number of integers between two numbers r and s (excluding both numbers) is s - r - 1

Basically the question is asking the value of (s - r - 1)?

(1) s - r = 10.
10 - 1 = 9 Sufficient

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

[(s + 1) - (r + 1) - 1] = 9
or (s - r - 1 ) = 9 Sufficient

Hence D.
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Bunuel
SOLUTION

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.
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Hi Bunuel, how to know that the integers between r and s are consecutive integers?
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Bunuel
SOLUTION

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.

Hi Bunuel, how to know that the integers between r and s are consecutive integers?
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Integers between r and s consists of all the integers between r and s, so they ARE consecutive. For example, integers between 5 and 10 are 6, 7, 8, and 9, which are naturally consecutive.
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one small doubt in the statement 1. Why shouldn't we consider s=-10 and r=20 in which case the sum is 10 but the number of integers between them is not 9. Kindly explain
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How many integers are there between, but not including, integers r and 12 Apr 2018, 21:11
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longhaul123
one small doubt in the statement 1. Why shouldn't we consider s=-10 and r=20 in which case the sum is 10 but the number of integers between them is not 9. Kindly explain
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(1) says that s – r = 10.

If s = -10 and r = 20, then s - r = -10 - 20 = -30, not 10.
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