Arithmetic
Arithmetic is the fundamental building block of math. The other three
subject areas tested in GRE Math are all pretty much unthinkable without
arithmetic. You’ll certainly need to know your arithmetic to power through
algebra, geometry, and data analysis problems, but the Math section also
includes some pure arithmetic problems as well. So it makes sense to start Math
101 with a discussion of numbers and the typical things we do with them.
Common Math Symbols
You may remember these from way back when, but in case you need a
quick refresher, here’s a list of some of the most commonly used math
symbols you should know for the GRE. We’ll discuss some of them in this
arithmetic section and others later in the chapter.
Symbol

Name

Meaning

<

Less than

The quantity to the left of the symbol is less than
the quantity to the right.

>

Greater than

The quantity to the left of the symbol is greater
than the quantity to the right.

≤

Less than or equal to

The quantity to the left of the symbol is less than
or equal to the quantity to the right.

≥

Greater than or equal to

The quantity to the left of the symbol is greater
than or equal to the quantity to the right.


Square root

A number which when multiplied by itself equals the
value under the square root symbol.

 x 

Absolute value

The positive distance a number enclosed between two
vertical bars is from 0.

!

Factorial

The product of all the numbers up to and including a
given number.



Parallel

In geometry, two lines separated by this symbol have
the same slope (go in exactly the same
direction).


Perpendicular

In geometry, two lines separated by this symbol meet
at right angles.

°

Degrees

A measure of the size of an angle. There are 360
degrees in a circle.

π

Pi

The ratio of the circumference of any circle to its
diameter; approximately equal to 3.14.

Number Terms
The test makers assume that you know your numbers. Make sure you do by
comparing your knowledge to our definitions below.
Number

Definition

Example

Whole numbers

The set of counting numbers, including
zero

0, 1, 2, 3

Natural numbers

The set of whole positive numbers except
zero

1, 2, 3, 4

Integers

The set of all positive and negative whole numbers,
including zero, not including fractions and decimals.
Integers in a sequence, such as those in the example to the
right, are called consecutive integers.

–3, –2, –1, 0, 1, 2, 3

Rational numbers

The set of all numbers that can be expressed as
integers in fractions—that is, any number that can be
expressed in the form , where m
and n are integers


Irrational numbers

The set of all numbers that cannot be expressed as
integers in a fraction

π, ,
1.010100001000110000

Real numbers

Every number on the number line, including all
rational and irrational numbers

Every number you can think of

Even and Odd Numbers
An even number is an integer that is divisible by 2 with no remainder,
including zero.
Even numbers: –10, –4, 0, 4, 10
An odd number is an integer that leaves a remainder of 1 when divided
by 2.
Odd numbers: –9, –3, –1, 1, 3, 9
Even and odd numbers act differently when they are added, subtracted,
multiplied, and divided. The following chart shows the rules for addition,
subtraction, and multiplication (multiplication and division are the same in
terms of even and odd).
Addition

Subtraction

Multiplication/Division

even + even = even

even – even = even

even × even = even

even + odd = odd

even – odd = odd

even × odd = even

odd + odd = even

odd – odd = even

odd × odd = odd

Zero, as we’ve mentioned, is even, but it has its own special
properties when used in calculations. Anything multiplied by 0 is 0, and 0
divided by anything is 0. However, anything divided by 0 is undefined, so
you won’t see that on the GRE.
Positive and Negative Numbers
A positive number is greater than 0. Examples include
, 15, and 83.4. A negative number is
less than 0. Examples include –0.2, –1, and –100. One tipoff is the
negative sign (–) that precedes negative numbers. Zero is neither positive
nor negative. On a number line, positive numbers appear to the right of
zero, and negative numbers appear to the left:
–5, –4, –3, –2, –1, 0, 1, 2, 3, 4, 5
Positive and negative numbers act differently when you add, subtract,
multiply, or divide them. Adding a negative number is the same as
subtracting a positive number:
5 + (–3) = 2, just as 5 – 3 = 2
Subtracting a negative number is the same as adding a
positive number:
7 – (–2) = 9, just as 7 + 2 = 9
To determine the sign of a number that results from multiplication or
division of positive and negative numbers, memorize the following rules.
Multiplication

Division

positive × positive = positive

positive ÷ positive = positive

positive × negative = negative

positive ÷ negative = negative

negative × negative = positive

negative ÷ negative = positive

Here’s a helpful trick when dealing with a series of multiplied or
divided positive and negative numbers: If there’s an even number of negative
numbers in the series, the outcome will be positive. If there’s an odd
number, the outcome will be negative.
When negative signs and parentheses collide, it can get pretty ugly.
However, the principle is simple: A negative sign outside parentheses is
distributed across the parentheses. Take this question:
3 + 4 – (3 + 1 – 8) = ?
You’ll see a little later on when we discuss order of operations that
in complex equations we first work out the parentheses,
which gives us:
3 + 4 – (4 – 8)
This can be simplified to:
3 + 4 – (– 4)
As discussed earlier, subtracting a negative number is the same as
adding a positive number, so our equation further simplifies to:
3 + 4 + 4 = 11
An awareness of the properties of positive and negative numbers is
particularly helpful when comparing values in Quantitative Comparison
questions, as you’ll see later in chapter 4.
Remainders
A remainder is the integer left over after one number has been divided
by another. Take, for example, 92 ÷ 6. Performing the division we see that 6
goes into 92 a total of 15 times, but 6 × 15 = 90, so there’s 2 left over.
In other words, the remainder is 2.
Divisibility
Integer x is said to be divisible by integer
y when x divided by y
yields a remainder of zero. The GRE sometimes tests whether you can
determine if one number is divisible by another. You could take the time to
do the division by hand to see if the result is a whole number, or you could
simply memorize the shortcuts in the table below. Your choice. We recommend
the table.
Divisibility Rules
1

All whole numbers are divisible by 1.

2

A number is divisible by 2 if it’s even.

3

A number is divisible by 3 if the sum of its
digits is divisible by 3. This means you add up all the
digits of the original number. If that total is
divisible by 3, then so is the number. For example, to
see whether 83,503 is divisible by 3, we calculate 8 + 3
+ 5 + 0 + 3 = 19. 19 is not divisible by 3, so neither
is 83,503.

4

A number is divisible by 4 if its last two
digits, taken as a single number, are divisible by 4.
For example, 179,316 is divisible by 4 because 16 is
divisible by 4.

5

A number is divisible by 5 if its last digit is 0
or 5. Examples include 0, 430, and –20.

6

A number is divisible by 6 if it’s divisible by
both 2 and 3. For example, 663 is not divisible by 6
because it’s not divisible by 2. But 570 is divisible by
6 because it’s divisible by both 2 and 3 (5 + 7 + 0 =
12, and 12 is divisible by 3).

7

7 may be a lucky number in general, but it’s
unlucky when it comes to divisibility. Although a
divisibility rule for 7 does exist, it’s much harder
than dividing the original number by 7 and seeing if the
result is an integer. So if the GRE happens to throw a
“divisible by 7” question at you, you’ll just have to
suck it up and do the math.

8

A number is divisible by 8 if its last three
digits, taken as a single number, are divisible by 8.
For example, 179,128 is divisible by 8 because 128 is
divisible by 8.

9

A number is divisible by 9 if the sum of its
digits is divisible by 9. This means you add up all the
digits of the original number. If that total is
divisible by 9, then so is the number. For example, to
see whether 531 is divisible by 9, we calculate 5 + 3 +
1 = 9. Since 9 is divisible by 9, 531 is as
well.

10

A number is divisible by 10 if the units digit is
a 0. For example, 0, 490, and –20 are all divisible by
10.

11

This one’s a bit involved but worth knowing.
(Even if it doesn’t come up on the test, you can still
impress your friends at parties.) Here’s how to tell if
a number is divisible by 11: Add every other digit
starting with the leftmost digit and write their sum.
Then add all the numbers that you
didn’t add in the first step and write
their sum. If the difference between the two sums is
divisible by 11, then so is the original number. For
example, to test whether 803,715 is divisible by 11, we
first add 8 + 3 + 1 = 12. To do this, we just started
with the leftmost digit and added alternating digits.
Now we add the numbers that we didn’t add in the first
step: 0 + 7 + 5 = 12. Finally, we take the difference
between these two sums: 12 – 12 = 0. Zero is divisible
by all numbers, including 11, so 803,715 is divisible by
11.

12

A number is divisible by 12 if it’s divisible by
both 3 and 4. For example, 663 is not divisible by 12
because it’s not divisible by 4. 162,480 is divisible by
12 because it’s divisible by both 4 (the last two
digits, 80, are divisible by 4) and 3 (1 + 6 + 2 + 4 + 8
+ 0 = 21, and 21 is divisible by 3).

Factors
A factor is an integer that divides into another integer evenly, with
no remainder. In other words, if
is an integer, then
b is a factor of
a.
For example, 1, 2, 4, 7, 14, and 28 are all factors of 28, because they go
into 28 without having anything left over. Likewise, 3 is
not a factor of 28 since dividing 28 by 3 yields a
remainder of 1. The number 1 is a factor of every number.
Some GRE problems may require you to determine the factors of a
number. To do this, write down all the factors of the given number in pairs,
beginning with 1 and the number you’re factoring. For example, to factor 24:
 1 and 24 (1 × 24 = 24)
 2 and 12 (2 × 12 = 24)
 3 and 8 (3 × 8 = 24)
 4 and 6 (4 × 6 = 24)
Five doesn’t go into 24, so you’d move on to 6. But we’ve already
included 6 as part of the 4 × 6 equation, and there’s no need to repeat. If
you find yourself beginning to repeat numbers, then the factorization’s
complete. The factors of 24 are therefore 1, 2, 3, 4, 6, 8, 12, and 24.
Prime Numbers
Everyone’s always insisting on how unique they are. Punks wear
leather. Goths wear black. But prime numbers actually
are unique. They are the only numbers whose sole
factors are 1 and themselves. More precisely, a prime number is a number
that has exactly two positive factors, 1 and itself. For example, 3, 5,
and 13 are all prime, because each is only divisible by 1 and itself. In
contrast, 6 is not prime, because, in addition to being
divisible by 1 and itself, 6 is also divisible by 2 and 3. Here are a
couple of points about primes that are worth memorizing:
 All prime numbers are positive. This is because every negative
number has –1 as a factor in addition to 1 and itself.
 The number 1 is not prime. Prime numbers must
have two positive factors, and 1 has only one positive factor,
itself.
 The number 2 is prime. It is the only even prime number. All
prime numbers besides 2 are odd.
Here’s a list of the prime numbers less than 100:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59,
61, 67, 71, 73, 79, 83, 89, and 97
It wouldn’t hurt to memorize this list. In addition, you can
determine whether a number is prime by using the divisibility rules
listed earlier. If the number is divisible by anything other than 1 and
itself, it’s not prime.
If a number under consideration is larger than the ones in the
list above, or if you’ve gone and ignored our advice to memorize that
list, here’s a quick way to figure out whether a number is prime:

Estimate the square root of the number.

Check all the prime numbers that fall below your
estimate to see if they are factors of the number. If no prime
below your estimate is a factor of the number, then the number
is prime.
Let’s see how this works using the number 97.

Estimate the square root of the number:

Check all the prime numbers that fall below 10 to see
if they are factors of 97:
Is 97 divisible by 2? No, it does not end with an even
number.
Is 97 divisible by 2? No, it does not end with an even
number.
Is 97 divisible by 3? No, 9 + 7 = 16, and 16 is not
divisible by 3.
Is 97 divisible by 5? No, 97 does not end with 0 or 5.
Is 97 divisible by 7? No, 97 ÷ 7 = 13, with a
remainder of 6.
Therefore, 97 is prime. (Of course, you knew that already from
familiarizing yourself with the prime numbers less than
100. . . .)
Prime Factorization
Come on, say it aloud with us: “prime factorization.” Now imagine
Arnold Schwarzenegger saying it. Then imagine if he knew how to do it.
Holy Moly. He would probably be governor of the entire United States!
A math problem may ask you to directly calculate the prime
factorization of a number. Other problems, such as those involving
greatest common factors or least common multiples (which we’ll discuss
soon), are easier to solve if you know how to calculate the prime
factorization. Either way, it’s good to know how to do it.
To find the prime factorization of a number, divide it and all its
factors until every remaining integer is prime. The resulting group of
prime numbers is the prime factorization of the original integer. Want
to find the prime factorization of 36? We thought so:
36 = 2 × 18 = 2 × 2 × 9 = 2 × 2 × 3 × 3
That’s two prime 2s, and two prime 3s, for those of you keeping
track at home.
It can be helpful to think of prime factorization in the form of a
tree:
As you may already have noticed, there’s more than one way to find
the prime factorization of a number. Instead of cutting 36 into 2 and
18, you could have factored it into 6 × 6, and then continued from
there. As long as you don’t screw up the math, there’s no wrong
path—you’ll always get the same result.
Let’s try one more example. The prime factorization of 220 could
be found like so:
220 = 10 × 22
10 is not prime, so we replace it with 5 × 2:
10 × 22 = 2 × 5 × 22
22 is not prime, so we replace it with 2 × 11:
2 × 5 × 22 = 2 × 2 × 5 × 11
2, 5, and 11 are all prime, so we’re done. The prime factorization
of 220 is thus 2 × 2 × 5 × 11.
Greatest Common Factor
The greatest common factor (GCF) of two numbers is the largest
number that is a factor of both numbers—that is, the GCF is the largest
factor that both numbers have in common. For example, the GCF of 12 and
18 is 6, because 6 is the largest number that divides evenly into 12 and
18. Put another way, 6 is the largest number that is a factor of both 12
and 18.
To find the GCF of two numbers, you can use their prime
factorizations. The GCF is the product of all the numbers that appear in
both prime factorizations. In other words, the GCF is the overlap of the
two factorizations.
For example, let’s calculate the GCF of 24 and 150. First, we
figure out their prime factorizations:
24 = 2 × 2 × 2 × 3
150 = 2 × 3 × 5 × 5
Both factorizations contain 2 × 3. The overlap of the two
factorizations is 2 and 3. The product of the overlap is the GCF.
Therefore, the GCF of 24 and 150 is 2 × 3 = 6.
Multiples
A multiple can be thought of as the opposite of a factor: If
is an integer, then
x
is a multiple of
y. Less formally, a multiple is
what you get when you multiply an integer by another integer. For example,
7, 14, 21, 28, 70, and 700 are all multiples of 7, because they each result
from multiplying 7 by an integer. Similarly, the numbers 12, 20, and 96 are
all multiples of 4 because 12 = 4 × 3, 20 = 4 × 5, and 96 = 4 × 24. Keep in
mind that zero is a multiple of every number. Also, note that any integer,
n, is a multiple of 1 and
n, because 1
×
n =
n.
Least Common Multiple
The least common multiple (LCM) of two integers is the smallest
number that is divisible by the two original integers. As with the GCF,
you can use prime factorization as a shortcut to find the LCM. For
example, to find the least common multiple of 10 and 15, we begin with
their prime factorizations:
10 = 5 × 2
15 = 5 × 3
The LCM is equal to the product of each factor by the maximum
number of times it appears in either number. Since 5 appears once in
both factorizations, we need to include it once in our final product.
The same goes for the 2 and the 3, since each of these numbers appears
one time in each factorization. The LCM of 10 and 15, then, is 5 × 3 × 2
= 30. In other words, 30 is the smallest number that is divisible by
both 10 and 15. Remember that the LCM is the least
common multiple—you have to choose the smallest number that is a
multiple of each original number. So, even though 60 is a multiple of
both 10 and 15, 60 is not the LCM, because it’s not the smallest
multiple of those two numbers.
This is a bit tricky, so let’s try it again with two more numbers.
What’s the LCM of 60 and 100?
First, find the prime factorizations:
60 = 2 × 2 × 3 × 5
100 = 2 × 2 × 5 × 5
So, 2 occurs twice in each of these factorizations, so we’ll need
to include two 2s in our final product. We have one 5 in our
factorization of 60, but two 5s in our factorization of
100. Since we’re looking to include the maximum number
of appearances of each factor, we’ll include two 5s in our product.
There’s also one 3 in the first factorization, and no 3s in the second,
so we have to add one 3 to the mix. This results in an LCM of 2 × 2 × 3
× 5 × 5 = 300.
Order of Operations
What if you see something like this on the test:
You basically have two choices. You can (a) run screaming from the
testing site yelling “I’ll never, ever, EVER get into graduate school!!!” or
(b) use PEMDAS.
PEMDAS is an acronym for the order in which mathematical operations
should be performed as you move from left to right through an expression or
equation. It stands for:
 Parentheses
 Exponents
 Multiplication
 Division
 Addition
 Subtraction
You may have had PEMDAS introduced to you as “Please Excuse My Dear
Aunt Sally.” Excuse us, but that’s a supremely lame 1950sstyle acronym. We
prefer, Picking Eminem Made Dre A Star. Whatever. Come up with one of your
own if you want. Just remember PEMDAS.
If an equation contains any or all of these PEMDAS elements, first
carry out the math within the parentheses, then work out the exponents, then
the multiplication, and the division. Addition and subtraction are actually
a bit more complicated. When you have an equation to the point that it only
contains addition and subtraction, perform each operation moving from left
to right across the equation. Let’s see how this all plays out in the
context of the example above:
First work out the math in the parentheses, following PEMDAS even
within the parentheses. So here we focus on the second
parentheses and do the multiplication before the subtraction:
Now taking care of the subtraction in both sets of parentheses:
Now work out the exponent (more on those later):
Then do the multiplication:
Then the division:
12 – 7 + 17
We’re left with just addition and subtraction, so we simply work from
left to right:
5 + 17
And finally:
22
Piece of cake! Well, not exactly, but it beats fleeing the room in
hysterics. PEMDAS is the way to crunch down the most difficultlooking
equations or expressions. Take it one step at a time, and you’ll do just
fine.
Fractions
Much of what we’ve covered so far concerns whole numbers. Now we enter
the vast universe that exists between those nice round
numbers: the world of fractions. The GRE loves fractions.
The number of questions on the Math section that involve fractions in some
way or another is nothing short of stupefying. This means you must know
fractions inside and out. Know how to compare them, reduce them, add them,
and multiply them. Know how to divide them, subtract them, and convert them
to mixed numbers. Know them. Love them like the GRE does. Make them your
friend on the test, not your enemy.
To begin, here are the basics: A fraction is a part of a whole. It’s
composed of two expressions, a numerator and a denominator. The numerator of
a fraction is the quantity above the fraction bar, and the denominator is
the quantity below the fraction bar. For example, in the fraction
, 1 is the numerator and
2 is the denominator. The denominator tells us how many units there
are in all, while the numerator tells us how many units out of that total
are specified in a given instance. For example, if your friend has five
cookies and offers you two of them, you’d be entitled to eat of her cookies.
How many you sneak when she’s not looking is up to you.
The general concept of fractions isn’t difficult, but things can get
dicey when you have to do things with them. Hence, the following subtopics
that you need to have under your belt.
Fraction Equivalencies
Fractions represent a part of a whole, so if you increase both the
part and whole by the same multiple, you will not change the
relationship between the part and the whole.
To determine if two fractions are equivalent, multiply the
denominator and numerator of one fraction so that the denominators of
the two fractions are equal (this is one place where knowing how to
calculate LCM and GCF comes in handy). For example, because if you
multiply the numerator and denominator of
by 3, you get:
. As long as you multiply or
divide both the numerator and denominator of a fraction by the same
nonzero number, you will not change the overall value of the
fraction.
Reducing Fractions
Reducing fractions makes life simpler, and we all know life is
complicated enough without crazy fractions weighing us down. Reducing
takes unwieldy monsters like and makes them into smaller, friendlier
critters. To reduce a fraction to its lowest terms, divide the numerator
and denominator by their GCF. For example, for
, the GCF of 450 and 600 is 150.
So the fraction reduces down to
, since
and
.
A fraction is in its simplest, totally reduced form when the GCF
of its numerator and denominator is 1. There is no number but 1, for
instance, that can divide into both 3 and 4, so
is a fraction in its lowest form,
reduced as far as it can go. The same goes for the fraction
, but
is a different story because 3
is a common factor of both the numerator and denominator. Dividing each
by this common factor yields
, the fraction in its most reduced form.
Adding, Subtracting, and Comparing Fractions
To add fractions with the same denominators, all you have to do is
add up the numerators and keep the denominator the same:
Subtraction works similarly. If the denominators of the fractions
are equal, just subtract one numerator from the other and keep the
denominator the same:
Remember that fractions can be negative too:
Some questions require you to compare fractions. Again, this is
relatively straightforward when the denominators are the same. The
fraction with the greater numerator will be the larger fraction. For
example,
is greater than
, while
is greater than
. (Be careful of those negative
numbers! Since –5 is less
negative than –13, –5 is
greater than –13.)
Working with fractions with the same denominators is one thing,
but working with fractions with different denominators is quite another.
So we came up with an easy alternative: the Magic X. For adding,
subtracting, and comparing fractions with different denominators, the
Magic X is a lifesaver. Sure, you can go ahead and find the least common
denominator, a typical way of tackling such problems, but we don’t call
our trick the “Magic X” for nothing. Here’s how it works in each
situation.
Adding.
Consider the following equation:
You could try to find the common denominator by multiplying
by 9 and
by 7, but then you’d be working
with some pretty big numbers. Keep things simple, and use the Magic X.
The key is to multiply
diagonally and up, which in this
case means from the 9 to the 3 and also from the 7 to the 2:
In an addition problem, we add the products to get our numerator:
27 + 14 = 41. For the denominator, we simply multiply the two
denominators to get:
Believe it or not, we’re already done! The numerator is 41, and
the denominator is 63, which results in a final answer of
.
Subtracting.
Same basic deal, except this time we subtract the products that
we get when we multiply diagonally and up. See if you can feel the magic
in this one:
Multiplying diagonally and up gives:
The problem asks us to subtract fractions, so this means we need
to subtract these numbers to get our numerator: 24 – 25
= –1. Just like in the case of addition, we multiply across the
denominators to get the denominator of our answer:
That’s it! The numerator is –1 and the denominator is 30, giving
us an answer of
. Not the
prettiest number you’ll ever see, but it’ll do.
Comparing.
The Magic X is so magical that it can also be used to compare
two fractions, with just a slight modification: omitting the step where
we multiply the denominators. Say you’re given the following
Quantitative Comparison problem. We’ll explain much more about QCs in
chapter 4, but for now remember that the basic idea is to compare the
quantity in Column A with the quantity in Column B to see which, if
either, is bigger. (In some cases, the answer will be that you
can’t determine which is bigger, but as you’ll
learn, when the two quantities are pure numbers with no variables, that
option is impossible.) See what you can make of this sample QC:
Column A

Column B



Now, if you were a mere mortal with no magic at your fingertips,
this would be quite a drag. But the Magic X makes it a pleasure. Again,
begin by multiplying diagonally and up:
Now compare the numbers you get: 161 is larger than 150, so
is greater than
. Done.
Why does this work? Who knows? Who cares? It just does. (Actually,
the rationale isn’t too complex, but it doesn’t add anything to your GRE
repertoire, so let’s skip it.) Learn how to employ the Magic X in these
three circumstances, and you’re likely to save yourself some time and
effort.
Multiplying Fractions
Multiplying fractions is a breeze, whether the denominators are
equal or not. The product of two fractions is merely the product of
their numerators over the product of their denominators:
Want an example with numbers? You got one:
Canceling Out.
You can make multiplying fractions even easier by canceling out.
If the numerator and denominator of any of the fractions you need to
multiply share a common factor, you can divide by the common factor to
reduce both numerator and denominator before multiplying. For example,
consider this fraction multiplication problem:
You could simply multiply the numerators and denominators and then
reduce, but that would take some time. Canceling out provides a
shortcut. We can cancel out the numerator 4 with the denominator 8 and
the numerator 10 with the denominator 5, like this:
Then, canceling the 2s, you get:
Canceling out can dramatically cut the amount of time you need to
spend working with big numbers. When dealing with fractions, whether
they’re filled with numbers or variables, always be on the lookout for
chances to cancel out.
Dividing Fractions
Multiplication and division are inverse operations. It makes
sense, then, that to perform division with fractions, all you have to do
is flip the second fraction and then multiply. Check it out:
Here’s a numerical example:
Compound Fractions.
Compound fractions are nothing more than division problems in
disguise. Here’s an example of a compound fraction:
It looks intimidating, sure, but it’s really only another way of
writing
, which now looks
just like the previous example. Again, the rule is to invert and
multiply. Take whichever fraction appears on the bottom of the compound
fraction, or whichever fraction appears second if they’re written in a
single line, and flip it over. Then multiply by the other fraction. In
this case, we get
. Now we
can use our trusty canceling technique to reduce this to
, or plain old 6. A far cry from
the original!
Mixed Numbers
Sick of fractions yet? We don’t blame you. But there’s one topic
left to cover, and it concerns fractions mixed with integers.
Specifically, a mixed number is an integer followed by a fraction, like
. But operations such as
addition, subtraction, multiplication, and division can’t be performed
on mixed numbers, so you have to know how to convert them into standard
fraction form.
Since we already mentioned , it seems only right to convert it.
The method is easy: Multiply the integer (the 1) of the mixed
number by the denominator of the fraction part, and add that product to
the numerator: 1 × 3 + 2 = 5. This will be the numerator. Now, put that
over the original denominator, 3, to finalize the converted fraction:
.
Let’s try a more complicated example:
Pretty ugly as far as fractions go, but definitely something we
can work with.
Decimals
A decimal is any number with a nonzero digit to the right of the
decimal point. Like fractions, decimals are a way of writing parts of
wholes. Some GRE questions ask you to identify specific digits in a decimal,
so you need to know the names of these different digits. In this case, a
picture is worth a thousand (that is, 1000.00) words:
Notice that all of the digits to the right of the decimal point have a
th in their names.
In the number 839.401, for example, here are the values of the
different digits.
Left of the decimal point

Right of the decimal point

Units: 9

Tenths: 4

Tens: 3

Hundredths: 0

Hundreds: 8

Thousandths: 1

Converting Fractions to Decimals
So, what if a problem contains fractions, but the answer choices
are all decimals? In that case, you’ll have to convert whatever
fractional answer you get to a decimal. A fraction is really just
shorthand for division. For example,
is exactly the same as 6 ÷ 15.
Dividing this out on your scratch paper results in its decimal
equivalent, .4.
Converting Decimals to Fractions
What comes around goes around. If we can convert fractions to
decimals, it stands to reason that we can also convert decimals to
fractions. Here’s how:

Remove the decimal point and make the decimal number
the numerator.

Let the denominator be the number 1 followed by as
many zeros as there are decimal places in the original decimal
number.

Reduce this fraction if possible.
Let’s see this in action. To convert .3875 into a fraction, first
eliminate the decimal point and place 3875 as the numerator:
Since .3875 has four digits after the decimal point, put four
zeros in the denominator following the number 1:
We can reduce this fraction by dividing the numerator and
denominator by the GCF, which is 125, or, if it’s too difficult to find
the GCF right off the bat, we can divide the numerator and denominator
by common factors such as 5 until no more reduction is possible. Either
way, our final answer in reduced form comes out to
.
Ratios
Ratios look like fractions and are related to fractions, but they
don’t quack like fractions. Whereas a fraction describes a part of a whole,
a ratio compares one part to another part.
A ratio can be written in a variety of ways. Mathematically, it can
appear as
or as 3:1. In words, it
would be written out as “the ratio of 3 to 1.” Each of these three forms of
the ratio 3:1 means the same thing: that there are three of one thing for
every one of another. For example, if you have three red alligators and one
blue alligator, then your ratio of red alligators to blue alligators would
be 3:1. For the GRE, you must remember that ratios compare parts to parts
rather than parts to a whole. Why do you have to remember that? Because of
questions like this:



For every 40 games a baseball team
plays, it loses 12 games. What is the ratio of the
team’s losses to wins? 
 (A) 
3:10 
 (B) 
7:10 
 (C) 
3:7 
 (D) 
7:3 
 (E) 
10:3 

The question says that the team loses 12 of every 40 games, but it
asks you for the ratio of losses to wins, not losses to
games. So the first thing you have to do is find out
how many games the team wins per 40 games played: 40 – 12 = 28. So for every
12 losses, the team wins 28 games, for a ratio of 12:28. You can reduce this
ratio by dividing both sides by 4 to get 3 losses for every 7 wins, or 3:7.
Choice C is therefore correct. If you instead calculated
the ratio of losses to games played (part to whole), you might have just
reduced the ratio 12:40 to 3:10, and then selected choice A.
For good measure, the test makers include 10:3 to entice anyone who went
with 40:12 before reducing. There’s little doubt that on ratio problems,
you’ll see an incorrect part : whole choice and possibly
these other kinds of traps that try to trip you up.
Proportions
Just because you have a ratio of three red alligators to one blue
alligator doesn’t mean that you can only have three red alligators and
one blue one. It could also mean that you have six red and two blue
alligators or that you have 240 red and 80 blue alligators. (Not that we
have any idea where you’d keep all those beasts, but you get the point.)
Ratios compare only relative magnitude. To know how many of each color
alligator you actually have, in addition to knowing the ratio, you also
need to know how many total alligators there are. This
concept forms the basis of another kind of ratio problem you may see on
the GRE, a problem that provides you with the ratio among items and the
total number of items, and then asks you to determine the number of one
particular item in the group. Sounds confusing, but as always, an
example should clear things up:
Egbert has red, blue, and green marbles in the
ratio of 5:4:3, and he has a total of 36 marbles. How many blue
marbles does Egbert have?
First let’s clarify what this means. For each
group of 5 red marbles, Egbert (who does sound like a
marble collector, doesn’t he?) has a group of 4 blue
marbles and a group of 3 green marbles. If he has one
group of each, then he’d simply have 5 red, 4 blue, and 3 green marbles
for a total of 12. But he doesn’t have 12—we’re told he has 36. The key
to this kind of problem is determining how many groups of each item must
be included to reach the total. We have to multiply the total we’d get
from having one group of each item by a certain factor that would give
us the total given in the problem. Here, as we just saw, having one
group of each color marble would give Egbert 12 marbles total, but since
he has 36 marbles, we have to multiply by a factor of 3 (since 36 ÷ 12 =
3). That means Egbert has 3 groups of red marbles with 5 marbles in each
group, for a total of 3 × 5 = 15 red marbles. Multiplying the other
marbles by our factor of 3 gives us 3 × 4 = 12 blue marbles, and 3 × 3 =
9 green marbles. Notice that the numbers work out, because 15 + 12 + 9
does add up to 36 marbles total. The answer to the question is therefore
12 blue marbles.
So here’s the general approach: Add up the numbers given in the
ratio. Divide the total items given by this number to get the factor by
which you need to multiply each group. Then find the item type you’re
looking for and multiply its ratio number by the factor you determined.
In the example above, that would look like this:
5 (red) + 4 (blue) + 3 (green) = 12
36 ÷ 12 = 3 (factor)
4 (blue ratio #) × 3 (factor) = 12 (answer)
For the algebraicminded among you, you can also let x
equal the factor, and work the problem out this way:
5x + 4x + 3x
= 36
12x = 36
x = 3
blue = (4)(3) = 12
Percents
Percents occur frequently in Data Interpretation questions but are
also known to appear in Problem Solving and Quantitative Comparison
questions as well. The basic concept behind percents is pretty simple:
Percent means divide by 100. This is true whether you see the word percent
or you see the percentage symbol, %. For example, 45% is the same as
or .45.
Here’s one way percent may be tested:
4 is what percent of 20?
The first thing you have to know how to do is translate the question
into an equation. It’s actually pretty straightforward as long as you see
that “is” is the same as “equals,” and “what” is the same as
“x.” So we can rewrite the problem as 4 equals x
percent of 20, or:
Since a percent is actually a number out of 100, this means:
Now just work out the math:
Therefore, 4 is 20% of 20.
Percent problems can get tricky, because some seem to be phrased as if
the person who wrote them doesn’t speak English. The GRE test makers do this
purposefully because they think that verbal tricks are a good way to test
your math skills. And who knows—they may even be right. Here’s an example of
the kind of linguistic trickery we’re talking about:
What percent of 2 is 5?
Because the 2 is the smaller number and because it appears first in
the question, your first instinct may be to calculate what percent 2 is of
5. But as long as you remember that “is” means “equals” and “what” means
“x” you’ll be able to correctly translate the word
problem into math:
So 5 is 250% of 2.
You may also be asked to figure out a percentage based on a specific
occurrence. For example, if there are 200 cars at a car dealership, and 40
of those are used cars, then we can divide 40 by 200 to find the percentage
of used cars at the dealership:
.
The general formula for this kind of calculation is:
Percent of a specific occurrence =
Converting Percents into Fractions or Decimals
Converting percents into fractions or decimals is an important GRE
skill that may come into play in a variety of situations.
 To convert from a percent to a fraction, take the
percentage number and place it as a numerator over the denominator
100. If you have 88 percent of something, then you can quickly
convert it into the fraction .
 To convert from a percent to a decimal, you
must take a decimal point and insert it into the percent number two
spaces from the right: 79% equals .79, while 350% equals
3.5.
Percent Increase and Decrease
Percent Increase and Decrease
One of the most common ways the GRE tests percent is through the
concept of percent increase and decrease. There are two main varieties:
problems that give you one value and ask you to calculate another, and
problems that give you two values and ask you to calculate the percent
increase or decrease between them. Let’s have a look at both.
One Value Given
In this kind of problem, they give you a single number to
start, throw some percentage increases or decreases at you, and then
ask you to come up with a new number that reflects these changes.
For example, if the price of a $10 shirt increases 10%, the new
price is the original $10 plus 10% of the $10 original. If the price
of a $10 shirt decreases 10%, the new price is the original $10
minus 10% of the $10 original.
One of the classic blunders test takers make on this type of
question is to forget to carry out the necessary addition or
subtraction after figuring out the percent increase or decrease.
Perhaps their joy or relief at accomplishing the first part
distracts them from finishing the problem. In the problem above,
since 10% of $10 is $1, some might be tempted to choose $1 as the
final answer, when in fact the answer to the percent increase
question is $11, and the answer to the percent decrease question is
$9.
Try the following example on your own. Beware of the kind of
distractor we’ve just discussed.



A vintage bowling league shirt
that cost $20 in 1990 cost 15% less in 1970.
What was the price of the shirt in 1970? 
 (A) 
$3 
 (B) 
$17 
 (C) 
$23 
 (D) 
$35 
 (E) 
$280 

First find the price decrease (remember that 15% = .15):
$20 × .15 = $3
Now, since the price of the shirt was less back in 1970,
subtract $3 from the $20 1990 price to get the actual amount this
classic would have set you back way back in 1970 (presumably before
it achieved “vintage” status):
$20 – $3 = $17
Seventeen bucks for a bowling shirt!? We can see that . . . If
you finished only the first part of the question and looked at the
choices, you might have seen $3 in choice A and
forgotten to finish the problem. B is the choice that
gets the point.
Want a harder example? Sure you do! This one involves a
doublepercent maneuver, which should be handled by only the most
experienced of percent mavens. Do not attempt this at home! Oh,
wait . . . Do attempt this at home, or wherever
you’re reading this book.
The original price of a banana in a store
is $2.00. During a sale, the store reduces the price by 25% and
Joe buys the banana. Joe then raises the price of the banana 10%
from the price at which he bought it and sells it to Sam. How
much does Sam pay for the banana?
This question asks you to determine the cumulative effect of
two successive percent changes. The key to solving it is realizing
that each percentage change is dependent on the last. You have to
work out the effect of the first percentage change, come up with a
value, and then use that value to determine the effect of the second
percentage change.
We begin by finding 25% of the original price:
Now subtract that $.50 from the original price:
$2 – $.50 = $1.50
That’s Joe’s cost. Then increase $1.50 by 10%:
Sam buys the banana for $1.50 + $.15 = $1.65. A total ripoff,
but still 35 cents less than the original price.
Some test takers, sensing a shortcut, are tempted to just
combine the two percentage changes on doublepercent problems. This
is not a real shortcut. It’s more like a dark alley filled with
cruel and nasty people who want you to do badly on the GRE. Here, if
we reasoned that the first percentage change lowered the price 25%,
and the second raised the price 10%, meaning that the total change
was a reduction of 15%, then we’d get:
Subtract that $.30 from the original price:
$2 – $.30 = $1.70 = WRONG!
We promise you that if you see a doublepercent problem on the
GRE, it will include this sort of wrong answer as a trap.
Two Values Given
In the other kind of percent increase/decrease problem, they
give you both a first value and a second value, and then ask for the
percent by which the value changed from one to the other. If the
value goes up, that’s a percent increase problem. If it goes down,
then it’s a percent decrease problem. Luckily, we have a handy
formula for both:
percent increase =
percent decrease =
To borrow some numbers from the banana example, Sam pays $1.65
for a banana that was originally priced at $2.00. The percent
decrease in the banana’s price would look like this:
percent decrease =
So Sam comes out with a 17.5% discount from the original
price, despite lining Joe’s pockets in the process.
A basic question of this type would simply provide the two
numbers for you to plug into the percent decrease formula. A more
difficult question might start with the original banana question
above, first requiring you to calculate Sam’s price of $1.65 and
then asking you to calculate the percent decrease from the original
price on top of that. If you find yourself in the deep end of the
GRE’s question pool, that’s what a complicated question might look
like.
Common Fractions, Decimals, and Percents
Some fractions, decimals, and percents appear frequently on the
GRE. Being able to quickly convert these into each other will save time
on the exam, so it pays to memorize the following table.
Fraction

Decimal

Percent


0.125

12.5%


(the little line above the 6 means that the 6 repeats
indefinitely, so
0.166 = .1666666666 . . .)



0.2

20%


0.25

25%





0.375

37.5%


0.4

40%


0.5

50%


0.625

62.5%





0.75

75%


0.8

80%


0.875

87.5%

Exponents
An exponent is a shorthand way of saying, “Multiply this number by
itself this number of times.” In ab, a is
multiplied by itself b times. Here’s a numerical example:
2^{5} = 2 × 2 × 2 × 2 × 2. An exponent can also
be referred to as a power: 25 is “two to the fifth power.” Before jumping
into the exponent nittygritty, learn these five terms:
 Base. The base refers to the 3 in 35. In other words,
the base is the number multiplied by itself however many times specified
by the exponent.
 Exponent. The exponent is the 5 in 35. The exponent
tells how many times the base is to be multiplied by itself.
 Squared. Saying that a number is squared is a common
code word to indicate that it has an exponent of 2. In the expression
62, 6 has been squared.
 Cubed. Saying that a number is cubed means it has an
exponent of 3. In the expression 43, 4 has been cubed.
 Power. The term power is another way to talk about a
number being raised to an exponent. A number raised to the third power
has an exponent of 3. So 6 raised to the third power is 63.
Common Exponents
It can be very helpful and a real time saver on the GRE if you can
easily translate back and forth between a number and its exponential
form. For instance, if you can easily see that 36 = 62, it can really
come in handy when you’re dealing with binomials, quadratic equations,
and a number of other algebraic topics we’ll cover later in this
chapter. Below are some lists of common exponents.
Squares

Cubes

Powers of 2

We’ll start with the squares of the first ten
integers:

Here are the first five cubes:

Finally, the powers of 2 up to 210 are useful to
know for various applications:

1^{2} = 1

1^{3} = 1

2^{0} = 1

2^{2} = 4

2^{3} = 8

2^{1} = 2

3^{2} = 9

3^{3} =
27

2^{2} = 4

4^{2} =
16

4^{3} =
64

2^{3} = 8

5^{2} =
25

5^{3} =
125

2^{4} =
16

6^{2} =
36


2^{5} =
32

7^{2} =
49


2^{6} =
64

8^{2} =
64


2^{7} =
128

9^{2} =
81


2^{8} =
256

10^{2} =
100


2^{9} =
512



2^{10} =
1,024

Adding and Subtracting Exponents
The rule for adding and subtracting values with exponents is
pretty simple, and you can remember it as the inverse of the Nike
slogan:
Just Don’t Do It.
This doesn’t mean that you won’t see such addition and subtraction
problems; it just means that you can’t simplify them. For example, the
expression 2^{15} +
2^{7} does not equal
2^{22}. The expression
2^{15} + 2^{7} is
written as simply as possible, so don’t make the mistake of trying to
simplify it further. If the problem is simple enough, then work out each
exponent to find its value, then add the two numbers. For example, to
add 33 + 42, work out the exponents to get (3 × 3 × 3) + (4 × 4) = 27 +
16 = 43.
However, if you’re dealing with algebraic expressions that have
the same base variable and exponents, then you can add or subtract them.
For example, 3x4 + 5x4 =
8x4. The base variables are both x,
and the exponents are both 4, so we can add them. Just remember that
expressions that have different bases or exponents cannot be added or
subtracted.
Multiplying and Dividing Exponents with Equal Bases
Multiplying or dividing exponential numbers or terms that have the
same base is so quick and easy it’s like a little math oasis. When
multiplying, just add the exponents together. This is known as the
Product Rule:
To divide two samebase exponential numbers or terms, subtract the
exponents. This is known as the Quotient Rule:
Quick and easy, right?
Multiplying and Dividing Exponents with Unequal Bases
You want the bad news or the bad news? The same isn’t true if you
need to multiply or divide two exponential numbers that
don’t have the same base, such as, say,
. When two exponents have
different bases, you just have to do your work the oldfashioned way:
Multiply the numbers out and multiply or divide the result accordingly:
.
There is, however, one trick you should know. Sometimes when the
bases aren’t the same, it’s still possible to simplify an expression or
equation if one base can be expressed in terms of the other. For
example:
25 × 89
Even though 2 and 8 are different bases, 8 can be rewritten as a
power of 2; namely, 8 = 23. This means that we can replace 8 with 23 in
the original expression:
25 × (23)9
Since the base is the same for both values, we can simplify this
further, but first we’re going to need another rule to deal with the
(2^{3})^{9} term.
This is called . . .
Raising an Exponent to an Exponent
This one may sound like it comes from the Office of Redundancy
Office, but it doesn’t. To raise one exponent to another exponent (also
called taking the power of a power), simply multiply the exponents. This
is known as the Power Rule:
Let’s use the Power Rule to simplify the expression that we were
just working on:
2^{5} ×
(2^{3})^{9} =
2^{5} × 2^{3×9} =
2^{5} × 2^{27}
Our “multiplication with equal bases rule” tells us to now add the
exponents, which yields:
2^{5} × 2^{27} =
2^{32}
2^{32} is a pretty huge number, and the
GRE would never have you calculate out something this
large. This means that you can leave it as
2^{32}, because that’s how it would appear in
the answer choices.
To Recap:
Multiply the exponents when raising one exponent to another, and
add the exponents when multiplying two identical bases with exponents.
The test makers expect lots of people to mix these operations up, and
they’re usually not disappointed.
Fractions Raised to an Exponent
To raise a fraction to an exponent, raise both the numerator and
denominator to that exponent:
That’s it; nothing fancy.
Negative Numbers Raised to an Exponent
When you multiply a negative number by another negative number,
you get a positive number, and when you multiply a negative number by a
positive number, you get a negative number. Since exponents result in
multiplication, a negative number raised to an exponent follows these
rules:
 A negative number raised to an even exponent will be positive.
For example, (–2)^{4} = 16. Why? Because
(–2)^{4} means –2 × –2 × –2 × –2. When
you multiply the first two –2s together, you get positive 4 because
you’re multiplying two negative numbers. When you multiply the +4 by
the next –2, you get –8, since you’re multiplying a positive number
by a negative number. Finally, you multiply the –8 by the last –2
and get +16, since you’re once again multiplying two negative
numbers. The negatives cancel themselves out and vanish.
 A negative number raised to an odd exponent will be negative.
To see why, just look at the example above, but stop the process at
–2^{3}, which equals –8.
Special Exponents
It’s helpful to know a few special types of exponents for the GRE.
Zero
Any base raised to the power of zero is equal to 1. Strange,
but true:
123^{0} = 1
0.8775^{0} = 1
a million trillion
gazillion^{0} = 1
Like we said: strange, but true. You should also know that 0
raised to any positive power is 0. For example:
0^{1} = 0
0^{73} = 0
One
Any base raised to the power of 1 is equal to itself: 21 = 2,
–671 = –67, and x1 = x. This fact
is important to know when you have to multiply or divide exponential
terms with the same base:
The number 1 raised to any power is 1:
1^{2} = 1
1^{4,000} = 1
Negative Exponents
Any number or term raised to a negative power is equal to the
reciprocal of that base raised to the opposite power. Got that?
Didn’t think so. An example will make it clearer:
Here’s a more complicated example:
Here’s an English translation of the rule: If you see a base
raised to a negative exponent, put the base as the denominator under
a numerator of 1 and then drop the negative from the exponent. From
there, just simplify.
Fractional Exponents
Exponents can be fractions too. When a number or term is
raised to a fractional power, it is called taking the root of that
number or term. This expression can be converted into a more
convenient form:
The
symbol is
known as the
radical sign, and anything under the
radical is called the radicand. We’ve got a whole section devoted to
roots and radicals coming right up. But first let’s look at an
example with real numbers:
, because 4 × 4 × 4
= 64. Here we treated the 2 as an ordinary exponent and wrote the 3
outside the radical.
Roots and Radicals
The only roots that appear with any regularity on the GRE are square
roots, designated by a fancierlooking long division symbol, like
this:
. Usually the test
makers will ask you to simplify roots and radicals.
As with exponents, though, you’ll also need to know when such
expressions can’t be simplified.
Square roots require you to find the number that, when multiplied by
itself, equals the number under the radical sign. A few examples:
= 5, because 5 × 5 = 25
= 10, because 10 × 10 = 100
= 1, because 1 × 1 = 1
Here’s another way to think about square roots:
if
x^{n} =
y, then
=
x
When the GRE gives you a number under a square root sign, that number
is always going to be positive. For example,
is just 5, even though in real life
it could be –5. If you take the square root of a variable, however, the
answer could be positive or negative. For example, if you solve
x^{2} = 100 by taking the square root
of both sides,
x could be 10 or –10. Both values work
because 10 × 10 = 100 and –10 × –10 = 100 (recall that a negative times a
negative is a positive).
Very rarely, you may see cube and higher roots on the GRE. These are
similar to square roots, but the number of times the final answer must be
multiplied by itself will be three or more. You’ll always be able to
determine the number of multiplications required from the little number
outside the radical, as in this example:
= 2, because 2 × 2 × 2 = 8
Here the little 3 indicates that the correct answer must be multiplied
by itself a total of three times to equal 8.
A few more examples:
= 3, because 3 × 3 × 3 = 27
= 5, because 5 × 5 × 5 × 5 = 625
= 1, because 1 × 1 × 1 × 1 = 1
Simplifying Roots
Roots can only be simplified when you’re multiplying or dividing
them. Equations that add or subtract roots cannot be simplified. That
is, you can’t add or subtract roots. You have to work out each root
separately and then perform the operation. For example, to solve
, do not add the 9 and 4 together
to get
. Instead,
.
You can multiply or divide the numbers under the radical sign as
long as the roots are of the same degree—that is, both square roots,
both cube roots, etc. You cannot multiply, for example, a square root by
a cube root. Here’s the rule in general form:
Here are some examples with actual numbers. We can simplify the
expressions below because every term in them is a square root. To
simplify multiplication or division of square roots, combine everything
under a single radical sign.
You can also use this rule in reverse. That is, a single number
under a radical sign can be split into two numbers whose product is the
original number. For example:
The reason we chose to split 200 into 100 × 2 is because it’s easy
to take the square root of 100, since the result is an integer, 10. The
goal in simplifying radicals is to get as much as possible out from
under the radical sign. When splitting up square roots this way, try to
think of the largest perfect square that divides evenly into the
original number. Here’s another example:
It’s important to remember that as you’ve seen earlier, you can’t
add or subtract roots. You have to work out each root separately and
then add (or subtract). For example, to solve
, you cannot add 25 + 9 and put
34 under a radical sign. Instead,
= 5 + 3 = 8.
Absolute Value
The absolute value of a number is the distance that number is from
zero, and it’s indicated with vertical bars, like this: 8. Absolute values
are always positive or zero—never negative. So, the absolute value of a
positive number is that number: 8 = 8. The absolute value of a negative
number is the number without the negative sign: –12 = 12. Here are some
other examples:
5 = 5
 –4.234  = 4.234
=
=
0 = 0
It is also possible to have expressions within absolute value bars:
3 – 2 + 3 – 7
Think of absolute value bars as parentheses. Do what’s inside them
first, then tackle the rest of the problem. You can’t just make that –7
positive because it’s sitting between absolute value bars. You have to work
out the math first:
3 – 2 +  –4 
Now you can get rid of the bars and the negative sign from that 4.
3 – 2 + 4 = 5
You’ll see more of absolute value in the algebra section of this Math
101 chapter. And speaking of which, it’s time to head there now.