### What Does the GRE Test? Calculation versus Principle

Some of you may have already read an excellent post discussing how you should study for the GRE, differentiating the application of skill as opposed to the application of knowledge. (Hint: you need both, but many people struggle to progress past pure knowledge!) If you have not read that post, you can find it here.

Today (or whenever you may be reading this) I would like to “riff” on that concept inside the quantitative section. Many, many students that I work with want to treat the GRE quantitative section as a math test: there’s an equation I should use, and a number I should solve for.

And sometimes, yes, that’s exactly what the test wants you to do. But there are other questions. Questions that don’t feel quite so … “math-y”. If you’ve taken a practice test, you probably know what I’m talking about, even if you can’t put your finger on an exact definition. You saw some questions that didn’t have an equation, or questions that had an equation but no definitive “x = 243” final answer. If you had a gut reaction of “This doesn’t feel like math?!?” to these questions, congratulations! You are well on your way to a more nuanced understanding of what the GRE quantitative section wants from you!

This is what I mean in the title “Calculation versus Principle”. Some GRE quant questions are best approached through the application of various math principles; running calculations on these questions is often too time-consuming.

(As an aside, when I use the term “calculation” I am **not** referring to questions you would plug into a calculator. Any questions that require mathematic manipulations to find a definitive numerical result are calculation questions.)

If I were teaching a class, this is about the point where I would get tired of talking. I’m tired of talking, let’s see an example!

Ah, yes, a lovely quant comparison question. What follows is a transcription of a hypothetical test-taker’s calculation approach. Feel free to skim the next two paragraphs; the purpose here is NOT for you to know the calculation approach, but instead to compare this approach to a principle-based approach.

***Begin hypothetical calculation test-taker.***

“I need to compare the area of a triangle to the area of a square. Well that’s easy! Area = ½ b*h , and Area = side*side. Ok, what’s the …. Uh-oh. What am I supposed to do with this? They haven’t given me numbers. No wait, when they don’t give me numbers, I’m allowed to choose numbers that fit the problem. Ok, a triangle and a square have the same perimeter. Let’s make the perimeter 12, so I can easily make a 3-sided and 4-sided figure. Ok, square with sides = 3, area is 3*3 = 9. All right, quantity B is 9. Let’s get quantity A.”

“What triangle should I make? Right triangles are easy, could I make a right triangle? Hey, a 3-4-5 right triangle has a perimeter of 12! Ok, so it’s ½ b*h, and that’s ½ (3)(4) so the triangle has an area of 6 – that’s definitely less than 9. But the problem didn’t tell me it was a right triangle; am I allowed to assume that? No, I should probably try another triangle. Well, I could make an equilateral triangle – 4-4-4. What would the area of this triangle equal? The base is 4, but what’s the height? Ok, I’ll have to draw the height. Ah, I have a 30-60-90 triangle inside here, and the 60 side is going to be . This will have an area of ½ (4)(3.7) – that will be 2*3.7, which is 7.4. Still less than 9. Ok, the answer is B.”

***End hypothetical calculation test-taker.***

Well, this person is correct. The answer is B, quantity B is always larger. But wow, that was a **lot** of work, and in all honesty, I tried to make this hypothetical test-taker an extremely accomplished GRE quant test-taker. The immediate jump to number testing, the recognition that we need to actively try to find the maximum area triangle to correctly compare that to the square area, the immediate recognition of easy right triangles and the immediate ability to calculate the area of the equilateral triangle, the quick estimation of … these are all possible, but to do them **all** in the same problem, and do them correctly? I would prefer an easier way.

So let’s see what happens when we apply a general principle to this problem.

***Begin hypothetical principle approach test-taker***

I’m comparing the area of a square to the area of a triangle. The perimeters have to be the same. Ok, I know that all else being equal, if I want to maximize the area of a shape, I want it to be symmetrical. A square has more area than a rectangle with the same perimeter. What’s the most symmetrical shape? A circle. So the closer my shape gets to a circle – the more sides I put in it – the more I’m maximizing my area. Ok, the square has more sides, and therefore the larger area. B.

***End hypothetical principle approach test-taker***

Hopefully you agree that the principle-based approach is far simpler, just as accurate, and requires much less time.

So now comes the fun part – how do we learn the principles, and how do we know when to apply them?

**Learning the Principles**

There is no easy answer to this, but I can provide some guidelines. Look through your GRE study sources. If they look anything like mine (which are, of course the Manhattan Strategy Guides), there are certain concepts that are in boldface. Compare the following options, all of which at least partly appear in bold in my strategy guides:

1) “Sides correspond to their opposite angles…. The longest side is opposite the largest angle, and the smallest side is opposite the smallest angle.”

2) “The internal angles of a triangle must add up to 180 degrees.”

3) “Rate x Time = Distance”

4) “For some grouping problems, you may want to think about the most or least evenly distributed arrangements of the items.”

Items 1 and 4 are what I would call principle statements. They give relationships or strategies, but don’t readily lend themselves to equations. Items 2 and 3 are calculation statements. They either state clearly defined numerical quantities (and therefore easily lend themselves to equation creation, a la “a+b+c = 180”) or literally state an equation.

Look through your study materials. **The more the content seems to address relationships or ideas that don’t correspond to exact numbers or exact equations, the more you should consider applying these ideas as large principles.**

There is one particular area that I feel deserves special mention: number properties. GRE questions that revolve around positive vs. negative, even vs. odd, prime vs. composite numbers are more often than not principle based. There are broad principles that define specific relationships across these types of numbers. Similarly, the GRE often asks questions that either revolve around or take advantage of what I call “trick” numbers: -1, 0, and 1; and proper fractions, either positive or negative. These numbers have special properties; learning these properties, as opposed to needing to do exact calculations, can save you much heartache on the test.

**Applying the Principles**

When should we apply the principles? **This question relies on you closely reviewing your work**. Whenever a question asks for a relationship between items without providing solid numbers, perhaps you could apply a broad principle. Whenever a question seems to rely less on solving for a specific quantity, and more on identifying what kind of quantity will result – “which of the following must be odd” – perhaps you could apply a general principle. And finally, if a question permits trick numbers, there may be a principle you could apply.

As you review your work, ask yourself the following question: **“Is there a way I could have answered this question without doing any actual math?”** If the answer is yes, you have found a principle question.

Good luck, and happy studying!!

### The Math Beast Challenge Problem of the Week – December 9, 2013

If

xandyare integers such thatx<yandxy= 4, which of the following could be the value of 2x+ 4y?

** To see this week’s answer choices and to submit your pick, visit our Challenge Problem page.**

### The Math Beast Challenge Problem of the Week – December 2, 2013

What is the greatest prime factor of 399?

** To see this week’s answer choices and to submit your pick, visit our Challenge Problem page.**

### GRE Quantitative Comparison: Don’t Be a Zero, Be a Hero

When it comes to quantitative comparison questions, zero is a pretty important number, because it’s a weird number. It reacts differently from other numbers when placed in some of the situations. And zero isn’t the only weirdo out there.

Most of us equate “number” with “positive integer”, and for good reason. Most of the numbers we think about and use in daily life are positive integers. Most of our math rules were learned, at least at first, with positive integers.

The GRE knows this, and takes advantage of our assumption. That’s why it’s important to remember all the “other” numbers out there. In particular, when testing numbers to determine the possible values of a variable, there are a few categories of numbers you want to keep in mind.

If I’m going to think about picking numbers, I want to pick numbers that are as different as possible. I try to choose my numbers from a mixture of seven categories, which can be remembered with the word FROZEN:

FR: fractions (both positive and negative)

O: one and negative none

ZE: zero

N: negatives

So we’ve got positive and negative integers (the bigger the absolute value, the better), positive and negative one, positive and negative fractions, and zero. Don’t forget, zero is an integer too!

There are other categories of numbers to think about, particularly if they are mentioned in the problem: odd versus even, prime versus non-prime, etc. But the seven groups listed above account for most of the different ways that numbers behave when you “do math” to them. Because of that fact, picking numbers from different categories can be a fast way to understand the limits of a problem.

To illustrate my point, let’s think about the value of x raised to the power of y. What happens to the value of that expression as y gets bigger? Let’s simplify our lives even further by stipulating that y is a positive integer.

What first comes to mind is the idea that as we increase the value of the exponent, we increase the value of the expression. Well, if x is a positive integer, that’s true: the expression gets exponentially bigger as y increases. Unless x is the positive integer 1, in which case the expression stays the same size, regardless of the value of y. The same is true if x is equal to 0. If x is a positive proper fraction, the expression gets smaller as the value of y increases.

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### The Math Beast Challenge Problem of the Week – September 30, 2013

A retailer previously bought an item from a wholesaler for $20 and sold it to consumers for a retail price of $35. After a wholesale cost reduction, the retailer reduces the retail price by 10%, yet the retailer’s profit on the item still increases by 20%. By what percent did the wholesale cost decrease?

See the answer choices and submit your pick over on our Challenge Problem page.

### The Math Beast Challenge Problem of the Week – September 9, 2013

If

cis a positive integer, which of the following could be the remainder after (c+ 1)3 –c3 is divided by 6?

See the answer choices and submit your pick over on our Challenge Problem page.

### The Math Beast’s Challenge Problem of the Week – October 24th

Each week, we post a new Challenge Problem for you to attempt. If you submit the correct answer, you will be entered into that week’s drawing for two free Manhattan Prep GRE Strategy Guides.

This week’s question is below. Get out their scrap paper and start solving!

{8, 10, 11, 16, 20, 22, 25,

x}In the set above,

xis an odd integer between 13 and 21, inclusive. Each possiblexvalue is equally probable.Which of the following statements has the highest probability of being true?