How many integers are there between, but not including,

[BTGmoderatorDC]

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.

OA D

Source: Official Guide

[GMATGuruNY]

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.

Statement 1:
Case 1: r=0 and s=10
Integers between r and s:
1, 2, 3, 4, 5, 6, 7, 8, 9 --> 9 integers

Case 2: r=1 and s=11
Integers between r and s:
2, 3, 4, 5, 6, 7, 8, 9, 10 --> 9 integers

Case 3: r=2 and s=12
Integers between r and s:
3, 4, 5, 6, 7, 8, 9, 10, 11 --> 9 integers

In every case, the number of integers between r and s = 9.
SUFFICIENT.

Statement 2:
On the number line:
...r...r+1...8 integers...s...s+1
Here, the 9 integers in color lie between r+1 and s+1.
As the number line illustrates, between r and s there are a total of 9 integers:
r+1 and the 8 integers in blue.
SUFFICIENT.

The correct answer is D.

[deloitte247]

Number of integers between two numbers r and s without including r and s can be stated as
$$\left(s-r\right)+\left(1-2\right)=s-r-1$$
Find the value of $$s-r-1$$
$$statement\ 1=\ s-r=10$$
$$from\ s-r-1$$ we have 10-1=9 integers between r and s not including r and s, hence statement 1 is INSUFFICIENT

Statement 2
There are integers between but nut including $$\ r+1\ and\ s+1$$
$$\left(s+1\right)-\left(r+1\right)-1=9$$
$$s+1-r-1-1=9$$
$$s-r-1=9$$
$$hence\ statement\ 2\ is\ INSUFFICIENT$$

[fskilnik@GMATH]

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.

Source: Official Guide

$$?\,\, = \,\,s - r - 1\,\,\,\,\left( {{\rm{the\,\, fingers}}\,\,{\rm{counting}}\,\,{\rm{technique}}} \right)$$
$$\left( 1 \right)\,\,s - r = 10\,\,\,\,\,\, \Rightarrow \,\,\,\,\,? = 9\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{SUFF}}.\,\,\,\,\,$$
$$\left( 2 \right)\,\,\left( {s + 1} \right) - \left( {r + 1} \right) - 1 = 9\,\,\,\left( {{\rm{same}}\,\,{\rm{technique}}} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,? = 9\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{SUFF}}.\,\,\,\,$$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.

[Brent@GMATPrepNow]

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.

OA D

Source: Official Guide

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