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

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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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 ..

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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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

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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Good Explanation.
D