Spotting How Easy Math Can Solve Certain Types of GMAT Quant Questions

To properly follow along with this post, it would help if you owned a recent copy of the GMAC Official Guide. Assuming that you have one, the question that I’ll be discussing can be found in the GMAT2015, GMAT2016 or GMAT2017 – it’s question number 229 in the Problem Solving section of the book (on page 184 or 185, depending on which version you own). The question has also been posted at GMATClub.com:
http://gmatclub.com/forum/how-many-of-the-integers-that-satisfy-the-inequality-x-2-x-134194.html

One of the great ‘design aspects’ of most GMAT questions is that they were built so that they could be solved using more than one approach. This is a way to reward someone for being a strong critical thinker, pattern-recognizer, and pen-to-pad worker. Being a brilliant mathematician isn’t required.

Many Test Takers face pacing problems in the Quant section – they simply cannot reasonably answer all 37 questions in 75 minutes. The cause of their pacing problem is almost always rooted in how they choose to go about answering questions. “Their way” takes too long when other methods would be faster (and likely easier) to implement. In this case, the faster approach is what’s called “brute force” – you don’t need any special knowledge nor advanced math skills – you just need to be willing to perform enough small calculations to prove what the correct answer actually is.

In the prompt, we’re told that a given fraction is greater than, or equal to, zero. The prompt asks us for the number of INTEGER solutions to this inequality that are less than 5.
Looking at the answer choices, we can see that there are either 1, 2, 3, 4 or 5 possible integer solutions. That is a big ‘clue’ that we can use brute force here. There aren’t that many integers that fit the given inequality, so how hard could it really be to find them all?
The most obvious integer to start with is the one that is closest to 5: the number 4. If you plug X=4 into the prompt, you’ll end up with:

(6)(7)/(2) = 42/2 = 21

This is clearly greater than, or equal to, 0. So we have our first solution – and I bet that it didn’t take you more than 5-10 seconds to prove it. We know that there are no more than 4 other solutions, so now you have to go find them…

Instead of just doing those little calculations for you, I’m going to give you a couple of hints so that you can finish this question on your own:

1) What happens when you try X=3?
2) X cannot equal 2 (since that would create an undefined number), but what happens if you try X=1 or X=0?
3) How about negative integers such as X = -1 or X = -2?
4) If you’re writing everything down on the pad, there should be a point at which you’ll KNOW when you can stop working.

All things considered, how long did those little calculations take you to perform? I bet you could do them all in under 2 minutes. Ultimately, this shows how a strong thinker who’s willing to do some basic work can answer complex-looking GMAT questions relatively quickly. Training to take advantage of these opportunities will take some effort, but on Test Day, it’s helpful to know that there are some easy approaches to questions that you too can use.