The toughest GMAT Quant problems are usually Data Sufficiency questions involving numbers systems and inequalities.

These problems tend to pose a lot of difficulties for test-takers because they seem to invite test-takers to plug in numbers and once they take that road, test-takers find themselves taking an inordinate amount of time. Once test-takers become aware of the of the ticking of the timer, panic sets in and even if they do manage to solve it, test-takers are still unsure of their answer.

Answering these type of questions correctly is key to moving from a scaled score of 48 to a scaled score of 50 on the Quant.

Let us look at the GMAT question below to see how these questions need to be approached.

How many odd integers are greater than integer x and less than 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

The usual tendency of test-takers is to read the question and quickly move on to the statements to make more sense of it. A better way is to first identify whether the question is one that needs pre-work?

The question discussed in the previous post — If n is an integer between 10 and 99, is n< 80? — is one that needs no pre-work since the information given in the question can lead to no deductions.

The same applies to this question — What is the GCD of m and n?

But the question under discussion is one that needs pre-work since there are some deductions that can be made from the information given in the question.

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

Now we know that this depends on the nature of x and y, whether they are odd or even. So before even going to the statements we can deduce what pattern emerges for different combinations of x and y.

The first case to consider is both x and y are either odd or even. The deduction in both cases — both odd or both even — will be same since it both even and both odd are mirror reflections of each other.

If we take both even, say 2 and 8, the integers between them are 3,4,5,6,7 — odd number of integers in between with one extra odd integer. If both are odd it has to be that the number of integers in between will still be odd but with one extra even integer.

The second case is when one of them is even and the other odd — say 3 and 8. In this case, the integers between them are 4,5,6 and 7 — an even number of integers. If the number of integers between them is even then half of them will be even and half odd!

So before even going to the statements we know what information we need to answer this question.

If you start reading the statements after doing this pre-work, from there on the question will take barely 30 seconds.

Statement (1) says there are 12 even integers in between x and y since we do not know the nature of x and y we cannot determine how many odd integers are there between them.

Statement (2) says there are 24 integers between x and y. This means that x and y are belong to the second case, one of them even and one odd, and there will be the same number of odd and even integers between them — 12 each.

Hence, (2) alone is sufficient to answer this question.

The only way to avoid this pre-work is if you can visualize the question on a number line.

Statement (1) says that there are 12 even integers between x and y. This can be visualized in this way — X E E E E E E E E E E E E Y

Between every pair of E’s there will be one O (odd) for sure, so there will be a minimum of 11 Os. In addition there can be an O immediately after X and/or immediately before Y. Hence, we cannot determine the exact number of odd numbers.

Statement (2) says there are 24 numbers between X and Y. If there is an even number of integers between two numbers on a number line, the number of odd and even integers between them has to be equal.

More often than not the tougher and trickier questions on GMAT quant often involve pre-work as you can see in this post.

Consciously incorporating this approach on Data Sufficiency questions will result in a marked improvement in both the speed as well as accuracy.

We will take up a few more problems of this type in upcoming posts.