How many odd integers are greater than the integer x and

Question

How many odd integers are greater than the integer x and less than the integer y?

(1) There are 12 even integers greater than x and less than y
(2) There are 24 integers greater than x and less than y

why are the integers assumed consecutive over here?

Bunuel · Accepted Answer

How many odd integers are greater than the integer x and less than the integer y?

(1) Here is the string of 12 even integers and 11 odd integers between them:  eoeoeoeoeoeoeoeoeoeoeoe .

4 cases are possible:

If x and y are both odd then X eoeoeoeoeoeoeoeoeoeoeoe Y then there will be 11 odd integers between them;
If x and y are both even then X o  eoeoeoeoeoeoeoeoeoeoeoe  o Y then there will be 11+2=13 odd integers between them;
If x is even and y is odd then X o  eoeoeoeoeoeoeoeoeoeoeoe Y then there will be 11+1=12 odd integers between them;
If x is odd and y is even then X eoeoeoeoeoeoeoeoeoeoeoe  o Y then again there will be 11+1=12 odd integers between them.

Not sufficient.

(2) Out of 24 consecutive integers greater than X and less than Y in any case half will be odd and another half will be even, so there are 12 odd integers greater than X and less than Y. Sufficient.

Answer: B.

P.S. You can try instead of 12 and 24 some smaller numbers to simplify calculations for example 2 even integers for (1) and 4 integers for (2).

Are you talking about (2)? If we are told that there are 4 integers more than X=1 and less than Y=6, then these integers are 2, 3, 4, and 5 - 4 consecutive integers, how else? Note that half are odd and half are even. Or if X=2 and Y=7 then these integers would be 3, 4, 5 and 6 - 4 consecutive integers: half are odd and half are even.

Hope it's clear.

shrouded1 · Answer

Statement (1) : 12 even integers between x and y ... As an example let x be 20 and y be 46. There are 12 even integers and 13 odd integers; if x is 20 and y is 45, there are 12 even and 12 odd integers ... So clearly (1) alone is not sufficient

Statement (2) : 24 integers between x and y .... if you consider them in order they have to be even, odd, even, odd .... OR odd,even, odd, even ..... : Either case 12 even and 12 odd ... So (2) alone is sufficient

Orange08 · Answer

You are genius. Many thanks Bunuel

satishreddy · Answer

hey bunnel, when you say "4 integers more than X=1 and less than Y=6" why cant it be 1,2,2,2,2,6, or, 1,2,5,5,5,6 this satisfies the statement rather than consecutive integers, right, the question did not say only consecutive or different numbers,,,, ,,iam i missing somethi here,,,,could you explain

Bunuel · Answer

What is "1,2,2,2,2,6, or, 1,2,5,5,5,6"?

Let me ask you a question: how many integers are more than 1 and less than 6? Can the answer be other than 4 (2, 3, 4 and 5)?

Or another way: if 1<n<6 then how many integer values n can take?

whiplash2411 · Answer

The first statement says there are 12 even integers. There are two cases for this:

1. x is an odd integer, in which case there will be 12 even integers but only 11 odd integers.
2. x is an even integer, in which case there will be 12 each of even and odd integers.

Hence this is insufficient.

Statement 2 says there are 24 integers greater than x and lesser than y.

Two cases:

1. x is an odd integers, which means there will be 12 even integers and 12 odd integers.
2. x is an even integer, in which case there will be 12 each of even and odd integers.

In both cases, the answer is the same and hence answer is B. I got tripped up by this question, good one.

imadkho · Answer

Bunuel, I chose E instead of B, since it is not stated that the integers are consecutive, so do you have any explanation on this matter ? 
thank you

Bunuel · Answer

(2) says that: there are 24 integers greater than x and less than y. Naturally those 24 integers between x and y are consecutive, how else? Consider x=1 and y=26: there are following 24 integers between them: 2, 3, 4, ..., 25. Also check this post about the same issue: how-many-odd-integers-are-greater-than-the-integer-x-and-100521.html#p809821 Hope it's clear.

Sachin9 · Answer

I think the best way to cehck 2 would be to reduce the number to 4 from 24 and try out.. eoeoeo.. the result would be in sync with 24 numbers .. same applies to 1 as well.

what say?

PrashantPonde · Answer

Yes that should work for this problem. Even I would have solved this way (reduce the sample size from 12 to 2 and 24 to 4)
1) There are  2  even integers greater than x and less than y
 INSUFFICIENT : XeoeY or XoeoeY - gives 1 or 2 odds
2) There are  4  integers greater than x and less than y
 SUFFICIENT : XoeoeY or XeoeoY - both give same number of odds, 2.  If it works for 4 then it will work for 24.

Hence choice(B).

NGGMAT · Answer

hi bunnel

st 1 can still be written and tried...
but st 2 will be time consuming... how do we know for sure that when there r 24 even I between x & y then there will be 24 odd I, without manually doing this? as doing it manually time consuming...

as you suggest we should take smaller number to try but how to decide which numbers will gave same result as 12 & 24..
any double? like i can take 4 for case 1 and 8 for case 2?

Bunuel · Answer

(2) says that  there are 24 ( even ) integers greater than integer x and less than integer y . The important part is that the number of integers between x and y is  even . In this case half of them must be odd and another half must be even. How else? Can there be 11 odd integers and 13 odd integer greater than x and less than y?

If we were told that  there are 3 ( odd ) integers greater than integer x and less than integer y , then this would be insufficient, because there could be 1 odd and 2 evens or 2 odds and 1 even.

BrentGMATPrepNow · Answer

Target question :    How many odd integers are greater than the integer x and less than the integer y?

Statement 1 : There are 12 even integers greater than x and less than y  
There are many scenarios that satisfy statement 1. Here are two: 
 Case a : x = 1 and y = 25. In this case, the answer to the target question is  there are  11  odd integers greater than the integer x and less than the integer y 
 Case b : x = 1 and y = 26. In this case, the answer to the target question is  there are  12  odd integers greater than the integer x and less than the integer y 
Since we can’t answer the  target question  with certainty, statement 1 is NOT SUFFICIENT

Statement 2 : There are 24 integers greater than x and less than y 
  ASIDE: Many students will read the statement 2 and conclude that, since we aren't told whether those 24 integers are CONSECUTIVE integers, there's no way to tell how many how many odd integers there are. 
However, the integers between x and y will  always be consecutive . 
For example, the integers between 5 and 11 are 6, 7, 8, 9 and 10 (consecutive)
Similarly, the integers between -2 and 8 are -1, 0, 1, 2, 3, 4, 5, 6 and 7 (consecutive)   
 
Since even and odd integers alternate in consecutive integers (e.g., 1 is odd, 2 is even, 3 is odd, 4 is even,... etc.) , we know that 12 of the 24 integers must be even, and the other   12 must be odd  
Since we can answer the  target question  with certainty, statement 2 is SUFFICIENT

Answer: B

Cheers,
Brent

GMATinsight · Answer

Solve the Official Questions more productively Click here for Timed Sectional Tests with Video solutions of each question Also explore Dedicated Data Sufficiency (DS) Course Answer: Option B Step-by-Step Video solution by GMATinsight https://www.youtube.com/watch?v=tEshNldqOQQ BEST On-Demand QUANT Course: Topicwise Modules | 2000+Qns | Official Qns. | 100% Video solution . Two MUST join YouTube channels : GMATinsight (1000+ FREE Videos) and GMATclub

avigutman · Answer

Video solution from Quant Reasoning: Subscribe for more: https://www.youtube.com/QuantReasoning? ... irmation=1 https://www.youtube.com/watch?v=unyEbEzuTqQ

Dip050701 · Answer

IanStewart  please explain this question? In a simpler way

IanStewart · Answer

I haven't read the other explanations, but looking first at Statement 2:

(2) There are 24 integers greater than x and less than y

Since even and odd integers alternate, 12 of these integers must be even, and the other 12 must be odd. So Statement 2 is sufficient. If instead Statement 2 said "There are 25 integers between x and y" (or any other odd number), it would not be sufficient, because it might be that 13 are even, 12 are odd, or it might be that 13 are odd and 12 are even.

Looking at the harder Statement, Statement 1:

(1) There are 12 even integers greater than x and less than y

It doesn't matter which integers these are, so we can just imagine they are the twelve integers 2, 4, 6, 8, ..., 22, 24. These need to be the only even integers between the two integers x and y. So it might be that x = 1 and y = 25, and then there are eleven odd integers between x and y. But it might be that x = 1 and y = 26, in which case we'll have one more odd integer in our range, namely 25, so we could have twelve odd integers in total. Or it might even be that x = 0 and y = 26, in which case we'll also have '1' in our range, and we'll have thirteen odd integers between x and y. So from Statement 1, all we know is that there are 11, 12 or 13 odd integers between x and y, and Statement 1 is not sufficient. So the answer is B.

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

How many odd integers are greater than the integer x and

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Data Sufficiency (DS) Questions

How many odd integers are greater than the integer x and

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards