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

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Merged. Thank you.

Target question: How many integers are there between, but not including, integers r and s ?

Statement 1: s - r = 10
First of all, this tells us that s is greater than r
So, on the number line, we have: ------r------------s----

Also notice that, if we take the given equation and add r to both sides, we get s = r+10
So, we can replace s with r+10 to get: ------r------------(r+10)----

Since r is an integer, we know that r+1 is an integer, and r+2 is an integer, and r+3 is an integer, etc.
If we add all of these values to our number line we get: ------r--(r+1)--(r+2)--(r+3)--(r+4)--(r+5)--(r+6)--(r+7)--(r+8)--(r+9)--(r+10)----
We can see that there are 9 integers between r and s
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: There are 9 integers between, but not including, r + 1 and s + 1.
The key here is to recognize that the number of integers between r + 1 and s + 1 IS THE SAME AS the number of integers between r and s

For example, we know that there are three integers between 5 and 9 (the integers are 6, 7 and 8)
If we add one to 5 and 9, we get 6 and 10
Notice that there are also three integers between 6 and 10 (the integers are 7, 8 and 9)

So, if there are 9 integers between r + 1 and s + 1, then we can also conclude that there are 9 integers between r and s
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer: D

Cheers,
Brent

If s=-10 and r=-20 then it makes sense. s-r=10 and the integers between s and r is also 9.
Hope it helps.

Hello ammuseeru

You can consider whatever you like in the highlighted part until you consider both statements. But, if you take statement 1 then you have to be specific in your thinking.
In statement 1, s-r=10 means s=r+10.

r--(r+1)---(r+2)---(r+3)----(r+4)---(r+5)---(r+6)---(r+7)---(r+8)--(r+9)----s (r+10)-->The are just 9 integers.
So, it has to be in ''continuous serious'' definitely.
If you're asked that how many integers between 97 and 100? can you skip any digit? definitely not. The series are continuous (97, 98, 99, 100).

Here, the number of integers=(s-r)+1
=10+1 (in statement 1, s-r=10)
=11
=11-2(excluding s and r)
=9 integers
Hope it helps.

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