Crash Course for the New GRE, 4th Edition (2011)
Part II. Ten Steps to Scoring Higher on the GRE
Step 5. Know Your Math Vocabulary
IT’S A READING TEST
ETS says that the math section of the GRE tests the “ability to reason quantitatively and to solve problems in a quantitative setting.” Translation: It mostly tests how much you remember from the math courses you took in seventh, eighth, and ninth grades. That means good news for you: GRE math is easier than SAT math. As you might know, many people study little or no math in college. If the GRE tested “college-level” math, everyone but math majors would bomb. So, junior high it is. By brushing up on the modest amount of math you need to know for the test, you can significantly increase your GRE math score.
So, ETS is limited to the math that nearly everyone has studied: arithmetic, basic algebra, basic geometry, and basic statistics. There’s no calculus (or even precalculus), no trigonometry, and no major-league algebra or geometry. Because of these limitations, ETS has to resort to tricks and traps in order to create hard problems. Even the most difficult GRE math problems are typically based on pretty simple principles; what makes some difficult is that the simple principles are disguised with tricky wording. In a way, this is more of a reading test than a math test.
MATH VOCABULARY
Vocabulary in the math section? Well, if the math section is just a reading test, then in order to understand what you read, you have to know the language, right?
Quick—what’s an integer? Is 0 even or odd? How many even prime numbers are there? These terms look familiar, but it’s been a while, right? (We’ve sorted the terms in alphabetical order, but feel free to skip around.) Review the following:
1. Consecutive—Integers listed in order of increasing value without any integers missing in between. For example: −3, −2, −1, 0, 1, 2, 3.
2. Decimals—When you’re adding or subtracting decimals, just pretend you’re dealing with money. Simply line up the decimal points and proceed as you would if the decimal points weren’t there.
34.500
87.000
123.456
+ 0.980
245.936
Subtraction works the same way:
17.66
− 3.20
14.46
To multiply, just do it as if the decimal points weren’t there. Then put the point in afterward, counting the total number of digits to the right of the decimal points in the numbers you are multiplying. Then, place the decimal point in your solution so that you have the same number of digits to the right of it:
3.451
× 8.9
30.7139
Except for placing the decimal point, we did exactly what we would have done if we had been multiplying 3,451 and 89.
To divide, set up the problem as a fraction, then, move the decimal point in the divisor all the way to the right. You must then move the decimal point in the other number the same number of spaces to the right. For example:
3. Difference—the result of subtraction.
4. Digit—The numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Just think of them as the numbers on your phone dial. The number 189.75 has five digits: 1, 8, 9, 7, and 5. Five is the hundredths digit, 7 is the tenths digit, 9 is the units digit, 8 is the tens digit, and 1 is the hundreds digit.
5. Divisible—Capable of being divided with no remainder. An integer is divisible by 2 if its units digit is divisible by 2. An integer is divisible by 3 if the sum of its digits is divisible by 3. An integer is divisible by 5 if its units digit is either 0 or 5. An integer is divisible by 10 if its units digit is 0.
6. Even/odd—An even number is any integer that can be divided evenly by 2 (like 4, 8, and 22); any integer is even if its units digit is even. An odd number is any integer that can’t be divided evenly by 2 (like 3, 7, and 31); any integer is odd if its units digit is odd. Even + even = even; odd + odd = even; even + odd = odd; even × even = even; odd × odd = odd; even × odd = even. If you’re not sure, just put in your own numbers. Don’t confuse odd and even with positive and negative. Fractions are neither even nor odd.
7. Exponent—Exponents are a sort of mathematical shorthand. Instead of writing (2)(2)(2)(2), we can write 2^{4}. The little 4 is called an “exponent” and the big 2 is called a “base.”
HERE ARE SOME RULES ABOUT EXPONENTS
Raising a number greater than 1 to a power greater than 1 results in a bigger number. For example, 2^{2} = 4.
Raising a fraction between 0 and 1 to a power greater than 1 results in a smaller number. For example, .
A negative number raised to an even power becomes positive. For example, (−2)^{2} = 4.
A negative number raised to an odd power remains negative. For example, (−2)^{3} = −8.
When you see a number raised to an negative exponent, just put a 1 over it and get rid of the negative sign. For example, , which = .
You probably won’t have to worry about adding or subtracting exponents, but you might be asked to multiply or divide. Just remember this phrase: When in doubt, expand it out. In other words:
2^{2} × 2^{4} = (2 × 2) (2 × 2 × 2 × 2) = 2 × 2 × 2 × 2 × 2 × 2 = 2^{6}
Same thing with division
2^{6} ÷ 2^{2} = (2 × 2 × 2 × 2 × 2 × 2) ÷ (2 × 2) = 2 × 2 × 2 × 2 = 2^{4}
And don’t forget PEMDAS (if you don’t remember what PEMDAS is, see number 14):
(4^{5})^{2} = (4 × 4 × 4 × 4 × 4)(4 × 4 × 4 × 4 × 4) = 4 × 4 × 4 × 4 × 4 × 4 × 4 × 4 × 4 × 4 = 4^{10}
8. Factor—a is a factor of b if b can be divided by a without leaving a remainder. For example, 1, 2, 3, 4, 6, and 12 are all factors of 12. All numbers, even prime numbers, have at least two factors. They are one and the number itself. How many times does 12 go into 12? It goes in once with nothing left over.
9. Fractions—A fraction is just shorthand for division. On the GRE, you’ll probably be asked to compare, add, subtract, multiply, and divide them. In multiplication, you just go straight across:
In division, you multiply by the second fraction’s reciprocal; in other words, turn the second fraction upside down. In other words, put its denominator (the bottom number) over its numerator (the top number), then multiply:
If you were asked to compare and , all you have to do is multiply diagonally up from each denominator, as shown:
Now, just compare 42 to 49. Because 49 is bigger, that means is the bigger fraction. This technique is called the Bowtie. You can also use the Bowtie to add or subtract fractions with different denominators (because to add or subtract, the fractions need the same denominator). Just multiply the denominators of the two fractions, and then multiply diagonally up from each denominator, as shown:
If the denominators are the same, you don’t need the Bowtie. You just keep the same denominator and add or subtract the numerators:
10. Integer—The integers are the “big places” on the number line: −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, 6. Note that fractions, such as , are not integers. Neither are decimals.
11. Multiple—A multiple of a number is that number multiplied by an integer other than 0. 10, 20, 30, 40, 50, and 60 are all multiples of 10.
12. Order of operations—Also known as PEMDAS, or Please Excuse My Dear Aunt Sally. Parentheses > Exponents > Multiplication = Division > Addition = Subtraction. This is the order in which the operations are to be performed. For example:
10 − (6 − 5) − (3 + 3) − 3 =
Start with the parentheses. The expression inside the first pair of parentheses, 6 − 5, equals 1. The expression inside the second pair equals 6. Now rewrite the problem as follows:
10 − 1 − 6 − 3 =
9 − 6 − 3 =
3 − 3 =
= 0
Here’s another example:
Say you were asked to compare (3 × 2)^{2} and (3) (2^{2}). (3 × 2)^{2} = 6^{2}, or 36, and (3) (2^{2}) = 3 × 4, or 12.
Note that with multiplication and division, you just go left to right (hence the “=” sign in the description of PEMDAS above). Same with addition and subtraction. In other words, if the only operations you have to perform are multiplication and division, you don’t have to do all multiplication first, because they are equivalent operations. Just go left to right.
13. Positive/negative—Positive integers get bigger as they move away from 0 (6 is bigger than 5); negative integers get smaller as they move away from zero (–6 is smaller than −5). Positive × positive = positive; negative × negative = positive; positive × negative = negative. Be careful not to confuse positive and negative with odd and even.
14. Prime—A prime number is a number that is evenly divisible only by itself and by 1. Zero and 1 are not prime numbers, and 2 is the only even prime number. Other prime numbers include 3, 5, 7, 11, and 13 (but there are many more).
15. Probability—Probability is equal to the outcome you’re looking for divided by the total outcomes. If it is impossible for something to happen, the probability of it happening is equal to 0. If something is certain to happen, the probability is equal to 1. If it is possible for something to happen, but not necessary, the probability is between 0 and 1, otherwise known as a fraction. For example, if you flip a coin, what’s the probability that it will land on “heads”? One out of two, or . What is the probability that it won’t land on “heads”? One out of two, or . If you flip a coin nine times, what’s the probability that the coin will land on “heads” on the tenth flip? One out of two, or . Previous flips do not affect anything.
17. Product—the result of multiplication.
18. Quotient—the result of division.
19. Reducing fractions—To reduce a fraction, “cancel” or cross out factors that are common to both the numerator and the denominator. For example, to reduce , just divide both 18 and 24 by the biggest common factor, 6. That leaves you with . If you couldn’t think of 6, both 18 and 24 are even, so just start cutting them in half (or by thirds) till you can’t go any further. And remember—you cannot reduce numbers across an equal sign (=), a plus sign (+), or a minus sign (–).
20. Remainder—The remainder is the number left over when one integer cannot be divided evenly by another. The remainder is always an integer. Remember grade school math class? It’s the number that came after the big “R.” For example, the remainder when 7 is divided by 4 is 3 because 4 goes into 7 one time with 3 left over. ETS likes remainder questions because you can’t do them on your calculator.
21. Square root—The sign indicates the square root of a number. For example, means that something squared equals 2. You can’t add or subtract square roots unless they have the same number under the root sign (does not equal , but ). You can multiply and divide them just like regular integers:
You will have a square root symbol on your calculator, but it is often easier if you recognize the common ones on site. Here are a few square roots to remember that might come in handy:
Note: If you’re told that x^{2} = 16, then x = ± 4. You must be especially careful to remember this on quantitative comparison questions. But if you’re asked for the value , you are being asked for the positive root only, so the answer is 4. A square root is always positive.
22. Standard deviation—The standard deviation of a set is a measure of the set’s variation from its mean. You’ll rarely, if ever, have to actually calculate it, so just remember this: The bigger the standard deviation, the more widely dispersed the values are. The smaller the standard deviation, the more closely grouped the values in a set are around the mean. For example, the standard deviation of the numbers 6, 0, and 6 is bigger than the standard deviation of the numbers 4, 4, and 4, because 6, 0, and 6 are more widely dispersed than 4, 4, and 4.
23. Sum—The result of addition.
24. Zero—An integer that’s neither positive nor negative, but is even. The sum of 0 and any other number is that other number; the product of 0 and any other number is 0.
QUANTITATIVE COMPARISON
There are four question formats on the math section: five-choice problem-solving questions, four-choice quantitative comparisons (or quant comps), All that Apply consisting of three up to a possible eith answer choices, and Numeric Entry in which there are no answer choices and you have to supply your own answer. A quant comp is a math question that consists of two quantities, one in Quantity A and one in Quantity B. You are to compare the two quantities and choose:
(A) Quantity A is greater.
(B) Quantity B is greater.
(C) The quantities are equal.
(D) The relationship cannot be determined from the information given.
In this book, we’re going to phrase the answer choices exactly that way, although on your test it will be slightly different (but it will mean the same thing).
Quant comps have only four answer choices. That’s great: A blind guess has one chance in four of being correct. Always write A, B, C, D (but no E) on your scratch paper so you can cross off wrong answer choices as you go. The content of quant comp problems is drawn from the same basic arithmetic, algebra, and geometry concepts that are used on GRE math problems in other formats. In general, then, you’ll apply the same techniques that you use on other types of math questions. Still, quant comps do require a few special techniques of their own.
The Peculiar Behavior of Choice (D)
Any problem containing only numbers must have a single solution. Therefore, the fourth bubble, or choice (D), can be eliminated immediately on all such problems. For example:
Quantity A |
Quantity B |
Quantity A is greater.
Quantity B is greater.
The two quantities are equal.
The relationship cannot be determined from the information given.
You know the answer can be determined, so the answer could never be choice (D). So right off the bat, as soon as you see a quant comp that involves only numbers, you can eliminate (D) on your scratch paper. The answer to this one is (B), by the way. Use the Bowtie, so you end up with 8 versus 9.
Compare Before You Calculate
You don’t always have to figure out what the exact values would be in both columns before you compare them. The prime directive is to compare the two columns. Finding ETS’s answer frequently is merely a matter of simplifying, reducing, factoring, or unfactoring. For example:
Quantity A |
Quantity B |
Quantity A is greater.
Quantity B is greater.
The two quantities are equal.
The relationship cannot be determined from the information given.
The first thing to do is eliminate choice (D), because there are only numbers here. Then, notice that there are fractions in common to both columns; both contain and . If the same numbers are in both columns, they can’t make a difference to the total quantity. So just cross them off (after copying down the problem on your scratch paper, of course). Now, what’s left? In Quantity A we have , and in Quantity B we have . All we have to do now is compare to . Use the Bowtie and we get choice (B).