Math Workout for the GRE, 3rd Edition (2013)

Chapter 2. The End of Mathophobia

WHEN THEY EXPECT YOU TO ZIG, ZAG

Before we get into the nuts and bolts of math review in Chapter 3, we should first discuss what kind of mindset you want to adopt toward the quantitative sections of the GRE. As you know, the GRE is a “standardized” test, in that it tests what ETS believes is a “standard” body of math skills, in what is a mostly standard, multiple-choice format. ETS creates test questions in a standard way, usually by predicting the same, standard errors that most test takers make. Therefore, much of your success on these tests will stem from training yourself to think in a way that ETS hasn’t predicted.

In other words, the GRE has set a bunch of traps along the path that most people follow, so you’re a lot less likely to fall into them if you take a detour.

STEADY AS SHE GOES

Developing these new ways of thinking—and feeling confident that you’ve invested so much time and effort into improving your score—can be the greatest asset toward building the most formidable skill possible: poise. Poise will help you react calmly when challenges present themselves so that you can search your brain for the right course of action. Poise will help you keep your head about you. Poise will make you realize that since people fear what they don’t understand, a heightened familiarity with math can cure even the most debilitating case of mathophobia.

THINK NUMERICALLY

First, there’s the simple matter of getting used to working with numbers. If you don’t come in contact with them very often or if you rely on calculators or spreadsheets to do the work for you, it’s time to ease your toes back into the numerical hot tub. Even though you’ll have a calculator, there will be plenty of times when calculating numbers on paper and in your head will save you valuable time. So get back into the numerical swing of things.

· Balance your checkbook without using a calculator.

· Figure out a 15 percent tip by calculating 10 percent and adding half again as much (because 10 percent plus 5 percent equals 15 percent).

· Take a few measurements and find out exactly how much storage space that old armoire has.

· Calculate the exact miles per gallon your car got on that last trip to your sister’s house.

· Figure out what fraction of your monthly budget is taken up by housing expenses, food, utilities or loan payments.

Even something as mundane as re-memorizing your times tables from zero to twelve can be useful because it reminds your brain that you were able to speak math fluently not too long ago and that you can do it again. It will also keep you from relying on the calculator for every little math calculation, thus freeing up time for the really important thinking that can be the difference between a good score and a great score.

THINK LIKE A TEST WRITER

Any test writer can write a math question and provide the correct answer. The true “skill” (if you can call it that) of writing questions is in choosing decoy answer choices that an unwitting test taker might choose if he or she made a predictable, careless error like forgetting to carry a 2 or omitting a minus sign.

As you work through all the questions in this book (and in any other resource you use), become acquainted with how GRE questions are written. Take them apart and look under the hood so you can see where you usually make errors and learn how not to make them.

DON’T THINK LIKE ETS THINKS YOU’LL THINK

Quick: Think of a number and write it in this space. _____

If you’re like most people, you chose a positive integer, probably between zero and ten. This is the kind of thinking that ETS can anticipate from its test takers, because humans usually think in integers, or “counting numbers.” Very few people will respond to the above question by writing –456.49 or 6 or π or even though it’s just this type of contrary, outside-the-box thinking that often provides good insights to solving math problems.

So when you think about numbers, be sure to consider all types of numbers—negatives, fractions, decimals, irrationals, the really huge and the really tiny—because they’re as much a part of the number family as boring old integers are.

THINK BACKWARD

Throughout our scholastic lives, we’ve all been told over and over again to look at a question, show all of our work, and provide our own answer in the blank. There’s a reason for this: It’s a lot harder to pull an answer out of thin air than to choose the right one from an array of five. A lot of your work on GRE prep will involve taking advantage of this multiple-choice format. So whenever a question with multiple answers makes you feel stuck, remember that the right answer is right there staring back at you. All you have to do is distinguish it from the four (or more!) decoys.

POE SHALL SET YOU FREE

The Process of Elimination (POE) is a beautiful thing, because in many circumstances it will allow you to choose the correct answer without necessarily knowing why it’s correct. For many multiple choice questions, all you have to know is that four answer choices are definitely wrong in order to know that the answer choice that’s left is correct by default.

It is very empowering when a student in a Princeton Review classroom raises her hand and says, “I got this question correct, but I don’t know why.” This often means that the student has subverted the system by using POE to choose the correct answer instead of giving up when she didn’t know how to approach the problem.

USE THE ANSWER CHOICES AGAINST EACH OTHER

On some questions you’ll be able to approximate, or “ballpark,” what the correct answer should be and then eliminate the answer choices that are obviously too big or too small. Take this problem, for example:

Question 1 of 20

If the price of a dress with an initial price of $100 is reduced by 20% and the reduced price is then reduced by an additional 25%, what is the final price of the dress?

$45

$55

$60

$120

$150

Before you attempt anything algebraic—which is an old instinct that you should work hard to eradicate—take a look at the answer choices. If the dress used to sell for $100 and its price was lowered twice, the new price wouldn’t be greater than $100, would it? So right away you can eliminate (D) and (E).

TURN ALGEBRA INTO ARITHMETIC

As you’ll see in Chapter 4, in certain situations you’ll be able to plug numbers into the answer choices in order to make them a little easier to analyze. Sometimes you’ll be able to avoid algebra altogether and plug the answer choices themselves back into the question. Most questions will be quite vulnerable to these techniques, and it will be in your best interest to exploit this vulnerability.

THINK CONFIDENTLY

Since you’re just getting started with this book, you’re at a crossroads. At this point, you can become one of the following two types of people:

· those who prepare as much as possible and arrive at the test center knowing they’ve done all they can, or

· those who arrive at the test center doubting themselves because they know they could have done more to prepare.

Resolve right now to end up in the former category.

But Give Yourself a Break

Students taking the GRE also commonly think too much about their score and become paralyzed with the fear of getting a bad one. This is misdirected energy. Instead, tell yourself that you are going to do the absolute best you can do, and that’s all anyone can ask of you—especially yourself.

Once you face down this fear of failure and realize that the world won’t end if you don’t do well, the fear won’t gnaw at you as much and you’ll be able to concentrate on the task at hand.

BITE-SIZED PIECES

Whether you like it or not, the quantitative portion of the GRE reads a lot like a verbal section when it comes to word problems. If you’re a little intimidated by word problems because of all the information they give you to process, don’t consider all of the information at once. Instead, break the question down into bite-size pieces and consider each one separately. Take a look at an example.

Question 4 of 20

Point B is 18 miles east of point A, and point C is 6 miles west of point B. If point D is halfway between points B and C, and point E is halfway between points D and B, how far, in miles, is point E from point D ?

At first glance, this questions looks like an indigestible mouthful of alphabet soup. But you can make sense of it if you’re patient and consider it in little sips:

· Point B is 18 miles east of point A: Draw a line segment with points A and B on either end.

· Point C is 6 miles west of point B: Add point C to the diagram, 6 miles from B.

· Point D is halfway between points B and C: Add point D, which is 3 miles from both points B and C.

· Point E is halfway between points D and B: Add point E, which is 1½ miles from both points B and D.

· And now the question is a piece of cake. How far, in miles, is point E from point D? The answer is 1.5 miles.

See how this kind of step-by-step analysis can help make what looks like an impossible problem relatively simple?

THINK PATIENTLY…

The first time you work on some of these problem types, it may go slowly. That’s fine and perfectly normal. When you take the test you’ll answer these questions more quickly, but don’t worry about that yet. For now, worry about getting the techniques down and doing each question flawlessly. Once you’ve done that type of question slowly and carefully several times, you will naturally get a little faster at it. Keep striving to answer each question correctly, and speed will come naturally.

But Don’t Over-think It

If you’re going to score well on the GRE, naturally you’ll have to prepare and practice. But you should also train yourself to manage stress by not over-thinking the process.

Taking the GRE is hard enough; why complicate the process by thinking too much about your score and its consequences? It’s too easy to plague yourself with doubts.

Even if you are worried about competitive graduate programs, your GPA, your GRE scores, or intimidating admissions committees, they don’t mean anything while you’re taking the test. When you’re in the middle of a section, your entire universe should be that section only. The past is gone, the future is unknowable. So don’t think about either.

THE CALCULATOR

There is a simple on-screen calculator when you take the Math section of the GRE. It doesn’t have many functions: plus, minus, multiply, divide, and square root. That’s about it. Seems like they’re not giving you much, but in reality you don’t need that much from a calculator for the GRE. Most of the questions aren’t going to require too many calculations, and most of the calculations are straightforward.

Use the calculator only when necessary.

In fact, before the GRE implemented the on-screen calculator, they tested out how people did with and without the calculator, using unscored Research sections at the end of the test. They found that for most questions, about the same number of people got that question correct with the calculator as did without the calculator. Why? The GRE is not a test of simple calculation. Instead, it tests how well you know how to set up and solve problems.

Feel free to use the calculator for any simple arithmetic you do. Of course, as you do, make sure you write each result on your scratch paper. You will quickly find that you generally don’t need to use the calculator more than once or twice per problem, if at all. If you find yourself using the calculator a lot on a single problem, you may either be misreading the problem or doing more work than necessary.

The calculator is simply a tool. It will not solve the questions for you, and it will not replace a good knowledge of basic mathematics. It is better for you to know that = 12 than it is to pull up the calculator, type in 144, and then hit square root. In fact, the calculator can easily lead you in the wrong direction if you don’t know some math rules. For instance, if you type in 5 + 3 × 2 into the calculator, you will get an answer of 16, even though the answer is actually 11, due to the order of operations. 5 + 3 × 2 is not the sort of math for which you should use the calculator.

When to Use the Calculator

Only use the calculator after you have written down the setup for the problem on your scratch paper. Generally, you’ll only use it for multiplication and division, and for addition and subtraction of large numbers.

Question 7 of 20

A customer receives a 15% discount off the $300 regular price of an appliance. A sales tax of 10% is applied to the sales price at the time of purchase.

Here’s How to Crack It

Since this is a Quant Comp question, write down A B C D vertically on the left side of your paper. We need to take 15% percent of 300, which is the same as 0.15 × 300. Write down 0.15 × 300 on your scratch paper. 10% of $300 is $30 so 5% of $300 is $15 and thus 15% of $300 is $45, so the sale price is $45 less than the original. Write down 300 – 45. Since 300 – 45 is 255, $255 is the sale price. Now we need to calculate the sales tax. 10% of $255 is 0.10 × 255 = 25.50. So the final price, including sales tax, is the sale price plus the sales tax, which is $255 + $25.50 = $280.50. Since Quantity A is larger than Quantity B, the answer is (A).

Notice that before using the calculator, we wrote down the exact formula we were going to enter. Don’t keep any information in your head or on the calculator, because in either case there is the risk of losing that information. Write everything down on your scratch paper.

This was a fine question to use the calculator on. The math was fairly straightforward, and we knew how to set up each equation. The calculator simply assisted us with a couple small portions. (If you’re not comfortable with how we set up the equations to find each percentage, we’ll discuss percentage problems in detail in the next chapter, Nuts and Bolts.)

When NOT to Use the Calculator

There are some questions on the GRE that are created specifically to get you to use the calculator and waste time. If your first impulse is to plug numbers into the calculator, stop. Write down the setup for the problem as usual, and see if there’s something else you can do instead. Can you estimate, plug in the answers, or cancel out? Some questions, such as questions involving quadratics, fractions, or exponents, are solvable using a calculator, but take a while unless you look for a different method.

Again, the calculator is not a substitute for knowing the math. If you realize you’re writing down really long, ugly decimals, or doing many more steps than usual on the calculator, then you may be missing something in the question.

Question 11 of 20

Amy’s bucket contains 4 gallons of water.

She pours out and of the remaining water evaporates. How many gallons of water were removed from the bucket?

Give your answer as a fraction.

gallons

Here’s How to Crack It

Your impulse may be to convert all those fractions into decimals. Don’t. First off, they’re fairly messy as decimals ( ≈ 0.42857 …), and secondly, there are some nice simplifications we can do with the fractions. Start with that 4gallons. Convert it into an improper fraction, which means it is 4 = = . Now we want to find of , and then of that. So we want to find × × . Notice that we can simplify the 3 in the denominator of the first fraction with the 3 in the numerator of the last fraction, leaving us × × . Simplify a 7 from the top of the first fraction and the bottom of the last, leaving × × . Now simplify out a 2 from the first and middle fractions, leaving × × = . Enter in 1 on the top of the fraction box, and 2 on the bottom.

When You Can’t Use the Calculator

The calculator is pretty limited. It can only hold numbers with up to 8 digits, and it can’t do exponents, cube roots, or multi-step equations. There will be many problems that simply can’t be solved using the calculator. If you know the limits of the on-screen calculator, you’ll be less likely to waste time with the calculator on problems you should be solving without it.

Question 15 of 20

If a = (−5)²³, which of the following is greater than a ?

Indicate all such values.

2⁴

(−3)⁴⁰

5²²

1

(−6)²⁷

(−5)²⁵

(−4)¹⁶

(−5)²⁴

Here’s How to Crack It

Since this is an All That Apply question with 8 answer choices, write down A B C D E F G H. First off, the calculator cannot calculate (−5)²³. Sorry. It has more digits than the calculator can display. Even if it could display it, then you’d still be stuck multiplying (−5) × (−5) × (−5) and on and on. There’s got to be a shortcut here. As we’ll discuss more in detail in Chapter 3: The Nuts and Bolts, any negative number raised to an odd exponent is always negative, e.g. (−2)³ = (−2)(−2)(−2) = −8. Any negative number raised to an even exponent is always positive: (−2)⁴ = (−2)(−2)(−2)(−2) = 16. The exponent on (−5)²³ is odd, which means that (−5)²³ is going to be some massively negative number. Since we want to know all the numbers greater than (−5)²³, all the positive numbers will automatically be greater than our massively negative number. Put check marks next to (A), (B), (C), (D), (G), and (H). All we have left now are (E) and (F). At this point, remember that when dealing with negative numbers, the normal greater/lesser rules are reversed: −3 is larger than −40. Think of it this way: Which is colder, −10 degrees out or −100 degrees? −100 is much colder, because it’s a much smaller number than −10. (−6)²⁷ is going to be less than (−5)²³, because 6²⁷ is greater than 5²³. Cross off (E). The same is true of (F): (−5)²⁵ is less than (−5)²³. Cross off (F). The answers are therefore (A), (B), (C), (D), (G), and (H).

Since the calculator is limited to an 8-digit screen, you may occasionally come up against rounding errors for any number which can’t fit within the screen. For instance, entering in 0.00001 × 0.00001 on the calculator will give the result as 0, which is definitely wrong. The actual answer is 0.000000001, too small for the calculator to show. Any numbers too large for the calculator to show will result in an Error display.

The calculator is nice to have, but think of it as a backup tool. Make sure you know, without using the calculator, the multiplication table up to 12 × 12, the perfect squares up to 15², the cubes up to 5³, and the decimal and percentage equivalents of most basic fractions. Know that to multiply large numbers ending in zeros, you only have to multiply the non-zero digits then tack on the zeros at the end: 200 × 30 = 6,000. These and other bits of simple math are things you could do with the calculator, but will be much quicker and easier without it.

So let’s get to it.