How many odd integers are greater than the integer x and

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 XeoeoeoeoeoeoeoeoeoeoeoeY then there will be 11 odd integers between them;
If x and y are both even then XoeoeoeoeoeoeoeoeoeoeoeoeoY then there will be 11+2=13 odd integers between them;
If x is even and y is odd then XoeoeoeoeoeoeoeoeoeoeoeoeY then there will be 11+1=12 odd integers between them;
If x is odd and y is even then XeoeoeoeoeoeoeoeoeoeoeoeoY 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.
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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
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You are genius. Many thanks Bunuel

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

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?
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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.

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

(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.
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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?
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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).

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?

(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.
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Hi Bunuel
If in statement-2, there is odd number ..like 5 integers between x and y then it will be insuff ?
Thanks.

Yes, in that case the statement would be insufficient. We could have 2 odd, 3 even or 2 even, 3 odd.
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Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

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

There are 2 variables (x,y) and 2 equations from the question and the 2 conditions, so there is high chance (C) will be our answer.
Looking at the conditions together, if there are 12 even numbers out of the 24 integers, there are of course 12 odd integers, so the answer becomes (C). But this is an integer question which is one the the key questions, if we apply 4(A) mistake types,
Looking at condition 1, the number of odd integers becomes (35-11)/2+1=13 when x=10, y=36, and (33-13)/2+1=11 when x=11, y=35; this does not give unique answer, so this is insufficient.
From condition 2, if there are 24 integers, there has to be 12 even and 12 odd. This is sufficient, making the answer (B).

For cases where we need 2 more equation, such as original conditions with “2 variables”, or “3 variables and 1 equation”, or “4 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 70% chance that C is the answer, while E has 25% chance. These two are the majority. In case of common mistake type 3,4, the answer may be from A, B or D but there is only 5% chance. Since C is most likely to be the answer using 1) and 2) separately according to DS definition (It saves us time). Obviously there may be cases where the answer is A, B, D or E.

I might sound dumb but the fact that consecutive is confusing. we're only told 24 integers are greater than x :/
I was confused between B and E

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

What if;

1<N<1000----> Any 24 numbers between 1 and 1000 will also satisfy the second argument of the question.

Please help me to understand how the numbers are in consecutive order.

Hello

If you are taking X=1 and Y=1000, then there are not just '24' integers between them. Between 1 and 1000 there are '998' integers (2, 3, 4, ...999). So this example (of 1 and 1000) cannot be taken considering the second statement.

When the statement says, "there are 24 integers between X and Y", it means there are '24 integers only'. That is only possible if Y = X+25. And out of those 24 integers, 12 will be odd and 12 will be even (no matter whether X is odd or even, or whether Y is even or odd)