Recently, in a GMAT forum, someone asked: “I am facing this problem with plugging numbers on the DS section. I realized this happens when I miss plugging in some specific numbers that just turn around the answer of the question. When I plug in, say, 2, the statement is sufficient, but plug in -1/2 and it’s not sufficient. Please help me identify what is the best way to plug numbers: is there a specific set of numbers I should test?”

In some ways, this is a *very hard* question to answer in the abstract, because often the nature of the question itself will dictate something about the choices we make. Given that, here’s the best I can suggest for a rough–and–ready guide for what numbers to plug in, since number picking is a great GMAT Data Sufficiency strategy.:

1. First of all, it’s usually good to try 1, 0, and –1, if for no other reason than: they are usually very easy to plug in.

2. You almost always have to consider: a positive number greater than 1, a positive decimal less than 1, a negative number less than –1, and a negative decimal between zero and –1.

3. Notice that the quartet outlined in (b) is centered around zero. If there’s some prominent algebraic factor that has a obvious root ––– e.g. the factor (x – 5) has a root at x = 5 –––– then it may well be pertinent to test both that root and a quartet analogous to the one in (b) centered around that root. For example, if the factor (x – 5) were prominent, in the problem, then I would test the root x = 5 as well as x = 7, x = 5.1, x = 4.9, x = 3.

Now, that is a hyper–thorough list, and it would be very time–consuming to go through that entire procedure each and every time. The nature of the problem often will allow you to streamline things. For example, for any value that only appears in squared form, you don’t have to consider both positive and negative values, since both square to positive.

Also, remember: for Data Sufficiency questions, when you are trying to prove a statement ** insufficient**,

*all you need are two possibilities*– each one yields a different answer. Once you have two different conclusions, and the statement is insufficient, you can stop picking. It’s only when you suspect a statement is

**that**

*sufficient**you have to exhaust every possibility*to try to disprove its sufficiency.

You want to give the statements a glance, so you don’t pick conditions that won’t be permitted by either statement, but after that, it’s often fruitful to play around with the prompt a bit, just enough to see if you can pick choices that indicate a conclusion each way. Those choices will help you narrow things down in the statements, and you may even discover patterns that you can extend when applying the statements. (If you had to sum up all of advanced mathematics in only three words, those three words most likely would be “extend known patterns.” In other words, when you are extending a mathematical pattern, you are really thinking like a mathematician.)

Below are a couple DS problems, and in the solutions, I talk more specifically about the number–picking strategies I employ.

## Practice Problems: Data Sufficiency with Variables

1. Given that x ≠ 5, is >

(1) x > 0

(2) x > 10

- (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.

(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.

(C) BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.=

(D) EACH statement ALONE is sufficient to answer the question asked.

(E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

2. Given that x ≠ 0, y ≠ 0,and (x + y) ≠ 0, is >

(1) x > -1

(2) y > -1

- (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.

(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.

(C) BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.

(D) EACH statement ALONE is sufficient to answer the question asked.

(E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

## Answers and Explanations

1. **B**

2. **E**

1. First of all, notice that one prominent factor in the prompt is (x – 5), and it’s squared. The fact that (x – 5) is prominent makes us suspect that four possible choices should be

- 5 + (number bigger than one)
- 5 + (decimal between one and zero)
- 5 – (decimal between one and zero)
- 5 – (number bigger than one)

BUT, since the factor is squared, we don’t need the final two; when we subtract 5, they will result in negatives, which will duplicate the positives when squared.

This analysis suggests that x = 7 and x = 5.1 might be good numbers for starters. Indeed, we find

which is smaller than 7

which is larger than 5.1

Right there, two numbers lead to two different conclusions. Any condition that allows us to plug in both 5.1 and 7 would be insufficient.

Also, notice: I am not as concerned with what would happen with negative values of x, because I’ve already scanned the answer choices and see that negatives values of x won’t be possible with either condition. I will point out, though, that 1/(x – 5)^2 is something squared, so it will always be positive, and if x is negative, any positive number is always bigger than any negative number: < for all negative values of x.

**Statement #1:** x > 0

This statement allows both x = 5.1 and x = 7, so we already know it’s insufficient.

**Statement #2**: x > 10

This statement doesn’t allow either x = 5.1 or x = 7. We have to think about what’s possible now. We know that when x = 10, it must be true that (x – 5) > 5. The right side of the original inequality would become

We know that 1/25 is much smaller than one, and as the denominator gets bigger and bigger, those fractions will just get smaller and smaller. They will always be smaller than one, so x > 1 >1/(number bigger than 25). This statement is enough to conclude that x is always greater. Statement #2 is sufficient.

**Answer = B**

2. Given that x ≠ 0 and (x + y) ≠ 0, is >

Start with what’s true for the most obvious numbers. Clearly, for both x and y equal to positive integers greater than one, this inequality is true: for x = 2 and y = 3, > = 2/5. Making the denominator bigger makes the fraction smaller.

Strange things often happen with negatives and inequalities, so I’ll also try x = –2 and y = –3. Negative over negative is positive, so this repeats the previous answer > , and the inequality is still true.

What is, of x and y, one is positive and one is negative? With and , the right fraction is , and the left fraction is , which is more negative, so now the inequality is false.

Notice that I didn’t even consider fractions, because, up until this point in the problem, I didn’t have to. I was able to find different integer values that led to different conclusions.

**Statement #1**: x >–1

First of all, notice that our old friends and are still valid choices, so the inequality is true for them.For the reverse, notice that we have already check and , which is also permitted, and the inequality is false for that choice. Two choices, two different conclusions. Statement #1 by itself is insufficient.

**Statement #2**: y > –1

Again, our old friends and are still valid choices, so the inequality is true for them. Above, we also tried and , and the inequality is false for these. Two choices, two different conclusions. Statement #2 by itself is insufficient.

**Combined Statements #1 and #2**: x >–1 and y >–1

Yet again, our old friends x = 2 and y = 3 are still valid choices, so the inequality is true for them. We need some new values, and the positive/negative mix seemed to be working well for us, so we will try: and , which are still allowed under the combined condition. Now,

left side = = ==

right side = ====

These choices make the inequality false. Two different choices for the variables, two different conclusions. Even together, Statements #1 and #2 are insufficient.

Answer = **E**

Hi Mike,

I have a small doubt: in #2, can we use algebra to solve the inequality such that when we solve, we arrive at– is y>0?

Here’s how:

is

y(x+y) > xy

xy + y (squared) > xy

y (squared)>0

y>0?

I can take this one, Mike!

Akansha, the problem with that approach is that it’s only true if we also hold true that y/x > y(x+y). And we can’t actually know this is true, because the question itself is asking

ifthis is true. Does that make sense? 🙂Hi Mike,

In your solution to problem #1, there is a typo error. Right at the end, it states “Statement #1 is sufficient. Answer = B”

Jen

Dear Jen,

Thank you for catching that typo. I just fixed that. Thanks for pointing this out.

Mike 🙂

Hey Mike!

Thanks for the explanation.

I have a doubt in the second question. I don’t seem to get how the inequality y/x > y/x+y stands true when you plug in x= -2 and y=3.

Thanks!

Neha,

I’m very sorry. That was a mistake on my part. When we plug in x= -2 and y=3, the inequality is indeed FALSE. Thank you very much for pointing this out, my friend! I just rewrote the solution to that problem: the answer is the same, but a few things in the solution needed to be changed. Thanks again!

Mike 🙂

Oh my Gosh…………….Gmat is really evil. Totally insane.

Thanks for your effort Mike 🙂 Is priceless

You are quite welcome. Mike 🙂

Okay yeah, this page is a toughie if I ever saw one; I have looked at it twice on two difference occasions and can only seem to get through half of Q#2 each time. An I still don’t fully get Question 1- your explanation of why you choose 7 and 5.1, even though there are details.

Also, despite my confusion, when you were substituting x & y into th original prompt equation for Q2, did you invert the two? If so, this is furthering my difficulty here. Not sure.

RT,

To some extent, what you are asking about is Number Sense. See this blog:

https://magoosh.com/gmat/2012/number-sense-for-the-gmat/

It’s the kind of intuition you get by playing around with numbers for fun. If you are the kind of person who only deals with numbers when you absolutely have to, then this intuition might seem puzzling.

For example, for #1, I would say an excellent exercise for you would be to plug in all the numbers from 1 to 10 (except 5), just to get a sense of the patterns. You will see that the right side is biggest between 4 and 6, so plug in decimals in that range as well. Forget about practicing these two question in an “efficient” GMAT-solving mode. Each one of these would be an excellent one to spend 30+ minutes on, just plugging in a ton of numbers and understanding

whythey follow the patterns they do.You might also think about graphing it — say, in #1, making a graph of y = x and of y = 1/(x-5)^2. If you have any kind of graphing calculator or graphical software available, plug these in. The visual perspective can be a powerful complement for the numerical perspective. You really understand if you understand how both the picture and the patterns among the numbers follow directly from the algebraic expressions.

In general, I’m going to guess that, because math is a challenge area for you, you haven’t taken the time to play with numbers in an open-ended way. People usually avoid the things they don’t like, and that compounds the frustration. It would be very helpful for you, whenever you come across a totally puzzling GMAT problem for you, to forget about solving it in a time-constrained GMAT mode, and just spend 30+ minutes exploring the numerical patterns with every possible choice. It’s only by playing with numbers and observing the patterns that you will develop the intuition that makes choices like the ones I have made here relatively clear. It’s nothing that anyone else can give you, and there’s no substitute for doing the hard word, the time-consuming work, of playing with the numbers for yourself.

Does all this make sense?

Mike 🙂

Hi Mike,

Me again.

I have the same confusion as RT when reviewing Q#2. I read your reply but I didn’t see where you addressed his question about plugging in the numbers for x and y. You chose x=2 and y=3, but then for y/x >y/x+y plugged in 2/3>2/2+3. Am I missing some rule with inequalities?

(yes, I’ve read the number sense blog, watched the video, haven’t gotten to the inequalities yet)

Thanks!!

Jen

Jen,

I’m happy to respond. 🙂 As you may appreciate, plugging in numbers on the DS is a very effective way to demonstrate a lack of sufficiency. The trick is to plug in two different sets of numbers that produce two different answer.

When I plugged in x = 2 and y = 3, I got that (2/3) and (2/5), and indeed, (2/3) > (2/5), so we get a “YES” answer to the prompt question. Fine, we got our YES, which is value for each statement individually and for the combined statements. In fact, any two positive integers would produce this same YES answer. That’s the very easy answer to get.

The hard answer to get is the NO, the case in which (y/x) < y/(x + y). As it turns out, for this we need to dip into opposite signed fractions, a region that the more lily-livered GMAT takers will avoid on principle.

Here's what I will say. Don't believe me. Plug in all the numbers for yourself. Math is not a spectator sport. You only understand it by doing it yourself. Plug in two positive fractions, then plug in two negative fractions, and then plug in one positive and one negative. See for yourself whether the prompt question has a YES or NO answer in each case. It's by playing with number that you build number sense, and there's no substitute for doing it yourself.

Does all this make sense?

Mike 🙂