Using number line in quants

The biggest challenge on many CAT questions is the strain they put on our working memory. Working memory, or our ability to process information that we hold temporarily, is by definition quite limited. It’s why phone numbers only contain seven digits – any more than that and most people wouldn’t be able to recall them. (Yes, there was a time, in the dark and distant past, when we had to remember phone numbers.) One of the most simple and effective strategies we can deploy to combat our working memory limitations is to simply list out the sample space of scenarios we’re dealing with. If we were told, for example, that x is a prime number less than 20, rather than internalize this information, we can jot down x = 2, 3, 5, 7, 11, 13, 17, or 19. The harder and more abstract the question, the more necessary such a strategy will prove to be. Take this challenging Data Sufficiency question, for example: On the number line, the distance between x and y is greater than the distance between x and z. Does z lie between x and y on the number line? 1) xyz < 0 2) xy <0 So, rather than try to conceptualize this problem mentally, let’s start by actually writing down all the number line configurations that we might have to deal with before even glancing at the statements. We know that x and z are closer than x and y. So we could get the following: x____z_______________________y z____x_______________________y Or we can swap x and y to generate a kind of mirror image y______________________x_____z y______________________z_____x The above number lines are the only four possibilities given the constraints provided in the question stem. Now we have something concrete and visual that we can use when evaluating the statements. Statement 1 tells us that the product of the three variables is negative. If you’ve internalized your number properties – and we heartily encourage that you do – you know that a product is negative if there are an odd number of negative elements in said product. In this case, that means that either one of the variables is negative, or all three of them are. So let’s use say one of the variables is negative. By placing a 0 strategically, we can use any of our above number lines: x__0__z______________________y z__0__x______________________y y__0___________________x_____z y__0___________________z_____x Each of these scenarios will satisfy that first statement. But we only need two. In our first number line, z is between x and y, so we get a YES to the question. In our second number line, z is not between x and y, so we get a NO to the question. Because we can get a YES or a NO to the original question, Statement 1 alone is not sufficient. Eliminate answer choices A and D. Statement 2 tells us that the product of x and y is negative. Thus, we know that one of the variables is positive, and one of the variables is negative. Again, we can simply peruse our number lines and select a couple of examples that satisfy this condition. In our first number line, z is between x and y, so we get a YES to the question. In our third number line, z is not between x and y, so we get a NO to the question. Like with Statement 1, because we can get a YES or NO to the original question, Statement 2 alone is also not sufficient. Eliminate answer choice B. When testing the statements together, we know two pieces of information. Statement 1 tells us that either one variable is negative or all three are. Statement 2 tells us that, between x and y, we have one negative and one positive. Therefore, together, we know that either x or y is negative, and the remaining variables are all positive. Now all we have to do is peruse our sample space and locate these scenarios. It turns out that we can use the same two number lines we used when testing Statement 2: In our first number line, z is between x and y, so we get a YES to the question. In our third number line, z is not between x and y, so we get a NO to the question. So even together, the statements are not sufficient to answer the question – the correct answer is E. Takeaway: on the CAT there’s no reason to strain your brain any more than is necessary. The more concrete you can make the information you’re provided on a given question, the more likely it is that you’ll be able to properly execute whatever math or logic maneuvers you’re asked to perform.