How Many Integers are There Between, But Not Including, Integers r and s ? GMAT Data Sufficiency

Question: 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.

1. Statement 1 alone is sufficient , but statement 2 alone is not sufficient
2. Statement 2 alone is sufficient , but statement 1 alone is not sufficient
3. Statement 1 and 2 both are insufficient
4. Statement 1 and 2 both are sufficient
5. Both statements together are sufficient but neither is sufficient alone

Correct Answer: D

Solution and Explanation:
Approach Solution 1:
To find the integers between r and s, let’s evaluate the number of integers (including both numbers) as s - r +1.
The number of integers (excluding both numbers) between r and s is s - r - 1.
The question basically asks what the value of (s - r - 1) is.
In statement 1: s - r = 10.
10 - 1 = 9 which is Sufficient.
In statement 2: There are 9 integers between r + 1 and s + 1 but not including them.
[(s + 1) - (r + 1) - 1] = 9
Solving further,
(s - r - 1 ) = 9 which is Sufficient
Therefore, both statements are sufficient.

Approach Solution 2:
The target question is, How many integers exist between, but not including, the integers r and s?
1st Statement is, s - r = 10
First and foremost, this indicates that s is greater than r.
So we have on the number line: ———r——————s——
Also, if we take the given equation and multiply it by r, we get s = r+10.
So, if we replace s with r+10, we get: ———r—————— (r+10) ——
Because r is an integer, we know that r+1 is an integer, r+2 is an integer, r+3 is an integer, and so on.
Adding all of these values to our number line yields: ———r —(r+1)—(r+2)—(r+3)—(r+4)—(r+5) —(r+6)—(r+7)—(r+8)—(r+9)—(r+10) ——
There are 9 integers between r and s, as we can see.
Statement 1 is SUFFICIENT because we can answer the target question with certainty.
In Statement 2, There are 9 integers between r + 1 and s + 1 but not including them.
The important thing to remember here is 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 ranging from 5 to 9. (the integers are 6, 7 and 8)
We get 6 and 10 when we add one to 5 and 9.
There are also three integers between 6 and 10. (the integers are 7, 8 and 9)
So, if there are nine integers between r + 1 and s + 1, we can conclude that there are nine integers between r and s.
Statement 2 is SUFFICIENT because we can answer the target question with certainty.
Therefore, both statements are sufficient.

Approach Solution 3:
The problem statement asks to find the number of integers between integers r and s, but not including r and s.

1.  s – r = 10
   This implies that there will be 9 integers between r and s since s-r=10 and r and s are integers. 
   For example, let s=10 and r=0, the following integers between them are 1, 2, 3, 4, 5, 6, 7, 8, and 9. Hence, Sufficient.
    
2. There are 9 integers between, but not including, r + 1 and s + 1.
   This implies that the distance between r and s is identical to the distance between r+1 and s+1.
   Therefore, if there exist 9 integers between, but not including, r+1 and s+1, there will be 9 integers between, but not including, r and s too.
   For example, let’s consider s+1=11 and r+1=1, the 9 integers between them are 2, 3, 4, 5, 6, 7, 8, 9, and 10. 
   => s=10 and r=0, same as above. Hence, Sufficient.

Therefore, both statements are sufficient.

“How many integers are there between, but not including, integers r and s”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review 2022”. The GMAT Quant section includes 31 questions that need to be solved within 62 minutes. GMAT Data Sufficiency questions consist of a problem statement that is followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

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