How Many Integers n are there such that r < n

bySayantani Barman Experta en el extranjero

Question: How many integers n are there such that r < n < s?

s – r = 5
r and s are not integers

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
EACH statement ALONE is sufficient.
Statements (1) and (2) TOGETHER are NOT sufficient.

Answer:

Solution with Explanation:
Approach Solution (1):

Notice that if r and s are integers, for example s = 5 and r = 0 then there will be 4 integers between them: 1, 2, 3 and 4. But if r and s are NOT integers
For example s = 5.5 and r = 0.5 then there will be 5, so one more, integers between them: 1, 2, 3, 4, and 5.

(1) s – r = 5
Not sufficient

(2) r and s are not integers.
Not sufficient

(1) + (2) According to above reasoning since r and s are not integers the there are 5 integers between them (case 2)
Sufficient

Correct Option: C

Approach Solution (2):

(1) s – r = 5

Let s = 6.5 and r = 1.5 then there are 5 integers between s and r 2, 3, 4, 5, and 6.
Let s = 6 and r = 1 then there are 4 integers between s and r 2, 3, 4, and 5.
Hence insufficient

(2) Clearly insufficient s = 10.5 r = 1.5 or s = 2.5 r = 1.5, etc
(1) + (2)
Sufficient as any 2 non-integers, such that their difference is 5, will have 5 integers between them.
Example: s = 6.5 and r = 1.5 then there are 5 integers between s and r 2, 3, 4, 5, and 6

Correct Option: C

Approach Solution (3):

(1) s – r = 5
If s and r are integers such that s > r; example: r = 1, s = 6, n = {2, 3, 4, 5}; 4 integers
But if s and r are non-integers such that s > r; example r = 0.5, s = 5.5; n = {1, 2, 3, 4, 5}; 5 integers
Not sufficient

(2) r and s are not integers
r and s are take multiple values
Not sufficient
(1) + (2)
(1) s – r = 5
(2) r and s are not integers
s > r; example r = 0.5, s = 5.5; n = {1, 2, 3, 4, 5}; 5 integers

Correct Option: C

“How many integers n are there such that r < n < s?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Quantitative Review". GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement 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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How Many Integers n are there such that r < n < s? GMAT Data Sufficiency

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