You can consider x to be a negative number say -10 and y to be a positive number. Now statement 2 states that there are 24 numbers greater than x and lesser than y. Hence, the value of y is 15. Now, calculate the number of odd numbers between these numbers. It will be 12
-9,-7,-5,-3,-1,1,3,5,7,9,11,13.

Even if x is a negative odd number and y a positive even numbers with 24 numbers between them, the result would be same. I am unsure of how you found statement 2 to be INSUFFICIENT.

It would be great if you could let me know about your understanding of this statement.

Statement 2 states that There are 24 integers greater than x and less than y. There are totally 24 integers between these two numbers excluding these two numbers. The statement does not mention anything about the sign or values of this number. There is no other way for 24 integers to exist between 2 integers without being consecutive.

Consider the number of integers between 1 and 10 exclusive. The answer has to be 8(2,3,4,5,6,7,8,9). Is there any other possible answer?

Hope this helps!

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?
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thanks bunnel... i guess i am understanding
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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.
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(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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Great explanation as always Buunel! Happy to see the consecutive integers doubt clarified as well.

A little off topic, but first look at eoeoeoeoeoeoeoeoeoeoeoe reminded me of the minion from Despicable Me

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

1.
let just say there are 3 even integers greater than x and less than y

if x =even
x odd e odd e odd e odd => so 4 odd.
if X =odd
o e o e o e o => 2 odd

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

let just say there are 4 integers greater than x and less than y

e o e o e 0 => 2 odd
o e o e 0 o => 2 odd

THis is working only if There are 24 integers greater than x and less than y, the value in red is even. if it is odd. will not work. (untill unless we know what x is even /odd)

Answer would be insufficient - eg. take smaller nos.
if x=1 & y=5, so we have 1 <2,3,4<5 so only 1 odd no
if x=2 & y=6, so we have 2<3,4,5<6 , so now we have 2 odd nos.

I have posted a similar query to bunuel for clarification regarding generalization of this counting principle.But from what I can gather, when we have an even no of consecutive integers, we will have have an equal no of odd & equal no of even nos between them. & when we have even no or odd number of integers to count, then it will depend on the 2 numbers encapsulating them.