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Math Expert V
Joined: 02 Sep 2009
Posts: 57087
How many integers n are there such that r < n < s ?  [#permalink]

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3
32 00:00

Difficulty:   75% (hard)

Question Stats: 52% (01:34) correct 48% (01:29) wrong based on 1330 sessions

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

(1) s - r = 5
(2) r and s are not integers.

Data Sufficiency
Question: 49
Category: Arithmetic Properties of numbers
Page: 156
Difficulty: 650

The Official Guide For GMAT® Quantitative Review, 2ND Edition

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Math Expert V
Joined: 02 Sep 2009
Posts: 57087
How many integers n are there such that r < n < s ?  [#permalink]

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6
7
SOLUTION

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

The question asks: how many integers $$n$$ satisfy $$r < n < s$$, or how many integers are between $$r$$ and $$s$$, not inclusive?

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 then there are 5 integers between them (case 2). Sufficient.

Answer: C.

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Re: How many integers n are there such that r < n < s ?  [#permalink]

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1
The "number" of n can be multiple of 4 or 5, says 8 or 10, 12, 15, ... can not be determined...
Math Expert V
Joined: 02 Sep 2009
Posts: 57087
Re: How many integers n are there such that r < n < s ?  [#permalink]

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ezinis wrote:
The "number" of n can be multiple of 4 or 5, says 8 or 10, 12, 15, ... can not be determined...

It seems that you misinterpreted the question.

Question ask about the number of integers between r and s, not inclusive.

For example: how many integers are between 10 and 15, not inclusive? Answer: 4, namely 11, 12, 13, and 14.
Or: how many integers are between 5.3 and 10.3? Answer: 5, namely 5, 6, 7, 8, 9 and 10.

So taken together statements are sufficient to answer that there are 5 integers between r and s.

Hope it's clear.
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Re: How many integers n are there such that r < n < s ?  [#permalink]

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

The question asks: how many integers $$n$$ satisfy $$r < n < s$$, or how many integers are between $$r$$ and $$s$$, not inclusive?

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 then there are 5 integers between them (case 2). Sufficient.

Answer: C.

Hope it's clear.

can you please eleborate the concept behind this ?? indeed is true but I didn't notice this before.

Thanks
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Re: How many integers n are there such that r < n < s ?  [#permalink]

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carcass wrote:
Bunuel wrote:
How many integers n are there such that r < n< s ?

The question asks: how many integers $$n$$ satisfy $$r < n < s$$, or how many integers are between $$r$$ and $$s$$, not inclusive?

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 then there are 5 integers between them (case 2). Sufficient.

Answer: C.

Hope it's clear.

can you please eleborate the concept behind this ?? indeed is true but I didn't notice this before.

Thanks

There isn't some special concept behind it. It's just common sense based on observation.
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Re: How many integers n are there such that r < n < s ?  [#permalink]

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I had and still continue to have trouble interpreting this question.

how many integers n are there such that r<n<s
r and s are 0 or 5 -> n can be 1,2,3,4
r and s are 110 or 115-> n can be another set of 4 no.s

i thought n can have any number of infinite values for r-s=5 or for r and s are not integers.....can some one pls make it clear...

i understand some of the explanations are towards n=no. of integers between two given values, but question ask how many integers n are there that meet the criteria...
Math Expert V
Joined: 02 Sep 2009
Posts: 57087
Re: How many integers n are there such that r < n < s ?  [#permalink]

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1
nanz236 wrote:
I had and still continue to have trouble interpreting this question.

how many integers n are there such that r<n<s
r and s are 0 or 5 -> n can be 1,2,3,4
r and s are 110 or 115-> n can be another set of 4 no.s

i thought n can have any number of infinite values for r-s=5 or for r and s are not integers.....can some one pls make it clear...

i understand some of the explanations are towards n=no. of integers between two given values, but question ask how many integers n are there that meet the criteria...

How many integers n are there such that 0 < n < 5? Answer: 4.
How many integers n are there such that 110 < n < 115? Answer: 4.

Does this make sense?
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Re: How many integers n are there such that r < n < s ?  [#permalink]

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2
Imp. Point- its given n is integer not r and s. So we can assume them to be integer .

Possibility 1 - r and s are integer
s=10, r=5
n can be 6,7,8,9

Possibility 2 - r and s are not integer
s=9.9, r=4.9
n can be 5,6,7,8,9

Condition 1 : s-r =5 (not sufficient by logic above)

Condition 2 : r and s are non-integer (not sufficient)

Combine 1 and 2 and value of n will be 5 always.

so answer is (C)
Director  V
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Posts: 838
Re: How many integers n are there such that r < n < s ?  [#permalink]

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2
1
(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) r and s are not integers.

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.

e.g.

s= 6.5 and r=1.5 then there are 5 integers between s and r , 2,3,4,5 and 6

Answer C
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Re: How many integers n are there such that r < n < s ?  [#permalink]

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1
if we knew about s and r (integers or not integers) our task would be simplified to st(1) Suff. Since we know nothing about s and r, there could be always four integers (case when s and r are integers) or five integers (when s or r is not integer)

st(2) helps with assigning type for our numbers (s and r), but we know nothing about values

combined st(1&2) supports our previously inquired data, i.e. s and r are not integers, hence the number of integers in the given interval will always be 5.

answer c)
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Re: How many integers n are there such that r < n < s ?  [#permalink]

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

(1) s - r = 5
(2) r and s are not integers.

Data Sufficiency
Question: 49
Category: Arithmetic Properties of numbers
Page: 156
Difficulty: 650

The Official Guide For GMAT® Quantitative Review, 2ND Edition

s - r = 5

If s = 6 and r = 1 then 1 < n < 6 so n can be 5,4,3,2. I almost fell trap for picking an integer but luckily the second statement made me consider the option of fractions.

if s = 21/2 and n = 11/2 then 5.5 < n < 10.5 so n can be 10,9,8,7,6 (5 integers)

Insufficient.

Statement 2 tells us that s and r are not integers. We already tested one case above so we try another one.

s = 3/2 and r = 1/6

1/6 < n < 1.5 then n can be 1 (1 integer)
Insufficient.

Combined we get they are not integers and they subtract to 5

21/2 -11/2 = 5 and gives us that n has 5 integers that satisfy the equation.

This can also be the case if we test 21/4 - 1/4 = 20/4 = 5

0.25 < n < 5.25 how many possible integers? (5,4,3,2,1) total 5.

Sufficient.

C is the answer choice
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Joined: 19 Aug 2018
Posts: 31
Re: How many integers n are there such that r < n < s ?  [#permalink]

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Bunuel wrote:
nanz236 wrote:
I had and still continue to have trouble interpreting this question.

how many integers n are there such that r<n<s
r and s are 0 or 5 -> n can be 1,2,3,4
r and s are 110 or 115-> n can be another set of 4 no.s

i thought n can have any number of infinite values for r-s=5 or for r and s are not integers.....can some one pls make it clear...

i understand some of the explanations are towards n=no. of integers between two given values, but question ask how many integers n are there that meet the criteria...

How many integers n are there such that 0 < n < 5? Answer: 4.
How many integers n are there such that 110 < n < 115? Answer: 4.

Does this make sense?

Excuse me, Bunuel, why in this case you haven't considered negatives?

Thank you.
Intern  B
Joined: 29 Oct 2018
Posts: 3
How many integers n are there such that r < n < s ?  [#permalink]

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

(1) s - r = 5
(2) r and s are not integers.

Let's say we consider negative values like you suggested for r < n < s => possible values for s - r = 5 would be s=4 and r = -1 or s=3 and r=-2 ...

-2 < n < 3 => -2 < -1 < 0 < 1 < 2 < 3 => 4 integers

The only condition where you have 5 integers is when you have fractions =>

-2.5 < n < 2.5 => -2.5 < -2 < -1 < 0 < 1 < 2 < 2.5 => 5 integers

You will need to use condition 2 to eliminate the fractions. Hence C How many integers n are there such that r < n < s ?   [#permalink] 22 Dec 2018, 14:43
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