Math Section 1 - Overview

Question 1

If a quant comp question contains only numbers, …

Answer 1

the answer can’t be (D). Any quant comp problem that contains only numbers and no variables must have a single solution.

Question 2

Digits

Answer 2

Digit refers to the numbers that make up other numbers. There are 10 digits: 0,1,2,3,4,5,6,7,8,9

Question 3

Numbers

Answer 3

A number is made up of either a digit or collection of digits. There are an infinite number of numbers.

Question 4

Integers

Answer 4

Integers are the numbers that have no fractional or decimal part, such as -3,-2,-1,0,1,2,3

Question 5

Zero

Answer 5

Zero itself is neither positive nor negative Zero is even Zero plus any other number is equal to that other umber Zero multiple by any other number is equal to that other number You cannot divide by zero

Question 6

Fractions are neither …..

Answer 6

even nor odd

Question 7

Any integer is … if its units digit is …

Answer 7

Even, Odd

Question 8

even + even = odd + odd = even + odd = even x even = odd x odd = even x odd =

Answer 8

= even = even = odd = even = odd = even

Question 9

Consecutive Integers

Answer 9

Are integers listed in order of increasing value without any integers missing in between them. 0,1,2,3,4,5 or -2,-1,0,1,2,3,4 for example

Question 10

Absolute Value

Answer 10

The absolute value of a number is equal to its distance from 0 on the number line. The absolute value of any number is always positive.

Question 11

Factors

Answer 11

A factor of a particular integer is a number that will divide evenly into the integer in question. To find all the factors of a particular integer, write down the factors systematically in pairs of integers starting with 1 and the integer itself.

Question 12

Multiples

Answer 12

Multiples of an integer are all the integers for with the original integer is a factor. There are an infinite number of multiples for any given number. Zero is a multiple of every number, though the concept is rarely tested on the GRE.

Question 13

Prime Numbers

Answer 13

A prime number is an integer that only has two factors: itself and one. List of primes less than 50: 2,3,5,7,11,13,17,19,23,29,31,37,41,43,47

Question 14

Prime Number Facts 0 1 2 +/-

Answer 14

0 is not a prime number 1 is not a prime number 2 is the only even prime number Prime numbers are positive integers. There is no such thing as a negative prime number or prime fraction.

Question 15

Divisibility

Answer 15

An integer is always divisible by its factors

Question 16

Divisibility Rules: 2 3 4 5 6 8 9 10

Answer 16

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 4 if its last two digits form a number that is divisible by 4 …. by 5 if its units digit is either 0 or 5 …. by 6 if it is divisible by both 2 and 3 …. by 8 if its last three digits form a number that is divisible by 8 …. by 9 if the sum of its digits is divisible by 9 …. by 10 if its units digit is 0

Question 17

Sum

Answer 17

the result of addition

Question 18

Difference

Answer 18

the result of subtraction

Question 19

Product

Answer 19

the result of multiplication

Question 20

Quotient

Answer 20

the result of division

Question 21

Divisor

Answer 21

the number you divide by

Question 22

Numerator

Answer 22

the top number in a fraction

Question 23

Denominator

Answer 23

the bottom number in a fraction

Question 24

Consecutive

Answer 24

in order from least to greatest

Question 25

Terms

Answer 25

the numbers and expressions used in an equation

Question 26

Order of Operations

Answer 26

PEMDAS (Please Excuse My Dear Aunt Sally) Parentheses, Exponents, Multiplication Division (do all M+D together in the same step going from left to right), Addition Subtraction (do all A+S together in one step from left to right)

Question 27

Multiplication and Division + x + = - x - = + x - = + / + = - / - = + / - =

Answer 27

= positive = positive = negative = positive = positive = negative

Question 28

Associative Laws

Answer 28

When you are adding or multiplying a series of numbers, you can regroup the numbers in any way you would like. (a + b) + (c + d) = a + (b + c + d) (ab)(cd) = a(bcd)

Question 29

Distributive Law

Answer 29

*Often tested on GRE a(b+c) = ab + ac a(b-c) = ab - ac

Question 30

Five Rules for Exponents

1) a² =
2) a² • a³ =
3) (a²)³ =
4) a² / a³ =
5) a¹² - a¹¹ =

Answer 30

1) a • a
2) (a • a)(a • a • a) = a²⁺³ = a⁵
3) (a • a)(a • a)(a • a) = a^2•3 = a⁶
4) a • a / a • a • a = 1 / a = a ^2-3 = a^-1
5) a¹¹(a - 1)

Question 31

Multiplication with Exponents

Answer 31

As long as the numbers have the same base all you do is add the exponents.

2² x 2⁴ =

2^{2 + 4} = 2⁶

This rule does not apply to addition. There is no quick and easy way to add numbers with exponents.

Question 32

Addition with Exponents

Answer 32

There is no quick and easy way to add numbers with exponents.

If they have the same bases, however, you can factor.

So, 2² + 2⁴ = 2² (1 + 2²)

Question 33

Division with Exponents

Answer 33

Dividing two or more numbers with teh same base that are rased to exponents is simple - just subtract the exponents.

2⁹ / 2³ = 2^{9 - 3} = 2⁶

Question 34

Subtraction with Exponents

Answer 34

Like Addition, there is not cut and fast way to subtract numbers with exponents. You can factor though.

2⁶ - 2² = 2² (2⁴ - 1)

Question 35

Dividing Exponents with a negative exponent

Answer 35

Take its reciprocal (put a 1 over the expression) and write the negative exponent as a positive.

3^-2 = 1 / 3² = 1 / 9

Question 36

Exponents to Exponents

Answer 36

When a number with an exponent is raised to another power, you simply mulitply the exponents.

(4⁵)² = 4^{5 x 2} = 4¹⁰

Remember the exponent applies to everything inside the parentheses. (3x)² = (3x)(3x) = 9x² not 3x² and not 9x.

The same is true of fractions raised to exponents.

(3/2)² = (3/2)(3/2) = (9/4)

Question 37

Oddities of Exponents

Raising a fraction that’s between … and … to a power greater than … results in ….

Answer 37

Raising a fraction that’s between 0 and 1 to a power greater than 1 results in a smaller number

(1/2)² = 1/4

Question 38

Oddities of Exponents

A negative number raised to an even power results in…

Answer 38

A negative number raised to an even power results in a postitive number.

(-2)² = 4

Question 39

Oddities of Exponents

A negative number raised to an odd power results…

Answer 39

A negative number raised to an odd power results in a negative number. (-2)³ = -8

Question 40

Oddities of Exponents

A number raised to a negative power is … the … of …

Answer 40

A number raised to the negative power is equal to the reciprocal of the number raised to the positive power.

2^-2 = 1 / 2² = 1 / 4

Question 41

Oddities of Exponents

Any nonzero number raised to … is …

Answer 41

Any nonzero number raised to the 0 power is 1, no matter what the number is.

1,000⁰ = 1

Note, however, that 0 to the 0 power is undefined.

Question 42

Square Root Nuances

(1) result of solving for variable

eg x² = 16

(2) ETS asking for value of a square root of any number

Answer 42

(1) if x² = 16, then x = +/- 4 be extra careful about this
(2) If ETS asks you for the value of √16, or the square root of any number, they are askign you for the positive root only

Question 43

Square Root Rules

Multiply Square Roots

√a x √b = ?

Divide Square Roots

√a/b = ?

Answer 43

To multiply: √a x √b = √ab

√3 x √12 = √36 = 6

To divide take the square root of a fraction:

√a/b = √a / √b

√16/4 = √16 / √4 = 4 / 2 = 2

Question 44

Square Root Rules

Adding and Subtracting Square Roots

Answer 44

You cannot add or subtract square roots unless the roots are the same.

So, √2 + √2 = 2√2, but √2 + √3 does not equal √5

In order to add different roots, you need to estimate their values first and then add them.

Question 45

Estimating and Simplifying Roots

Answer 45

You can guesstimate a range for a root if it falls between two known square roots.

You could also try to simplify it using the rule for multiplying roots. Find a factor that is a perfect square. √32 - 32 can be split into 16x2, which means √32 is the same things as √16x2

We can get the square root of 16 and move that outside the square root symbol, giving us 4√2.

Question 46

Some Perfect Squares

√4 √9 √16

√25 √36 √49

√64 √81 √100

√121 √144 √169

Answer 46

Some Perfect Squares

2² 3² 4²

5² 6² 7²

8² 9² 10²

11² 12² 13²

Question 47

Perfect Squares that are Good to Know

√1

√2

√3

√4

Answer 47

√1 = 1

√2 = 1.4 (about)

√3 = 1.7 (about)

√4 = 2

Question 48

The Golden Rule of Manupulating Equations

Answer 48

Whatever you do on one side of the equals sign you must also do on the other side.

Question 49

Clearing a Fraction

5x + 3/2 = 7x

Answer 49

Multiply both sides by 2

10x + 3 = 14x

3 = 4x

3/4 = x

Question 50

Inequalities

≠

>

<

≥

≤

Answer 50

≠ is not equal to

> is greater than

< is less than

≥ is greater than or equal to

≤ is less than or equal to

Question 51

Inequality Rule for Manipulation

Answer 51

When you multiply or divide both sides of an inequaity by a negative number, the direction of the inequality symbol must change. That is if x>y then -x<-y

12 - 6x > 0 -6x > -12

12 > 6x OR -6x/-6 < -12/-6

2 > x x < 2

Question 52

How to Crack Ranges for two Variables

If 0 ≤ x ≤ 10 and -10 ≤ y ≤ -1, then what is the range for x - y?

Answer 52

Set up a table

X Y X - Y

Big 10 Big -1 11 .

Big 10 Small -10 20 .

Small 0 Big -1 1 .

Small 0 Small -10 10 .

The range for x-y is 1≤ x-y ≤ 20

Question 53

How to solve equations with two variables

Answer 53

Stack them and then combin them by adding or subtracting to cancel out one of the variables. Plug the solution for the other variable back in to find the other.

4x + 7y = 41

2x + 3y = 19

(multiply 2nd by 2)

y = 3

2x + 9 = 19 x = 5

Question 54

Quadratic Equations

FOIL

(x+4)(x-7)

Answer 54

FOIL = First, outside, inside, last

(x+4)(x-7)

x² -7x + 4x - 28

x² - 3 - 28

Question 55

Frequent Quadratics to Know

Factored form x² - y²

Factored form (x + y)²

Factored form (x - y)²

Answer 55

Unfactored form (x+y)(x-y)

Unfactored form x² + 2xy + y²

Unfactored form x² - 2xy + y²

Question 56

Quadratic Equations and Factoring

Answer 56

Factoring quadratics “undoes” the FOIL process

When you solve a quadratic equation, you usually get two answers.

x² + 2x - 15 = 0

Quantity A = 2 Quantity B = x

(x+5)(x-3) - set each to zero and solve

x = -5 , x = 3

Question 57

Simultaneous Equations

If 5x + 4y = 6 and 4x + 3y = 5, what does x+y equal?

Answer 57

Add or subtract equations from each other.

5x + 4y = 6

• 4x + 3y = 5 .

x + y = 1

Question 58

Algebra questions and plugging in

Answer 58

1) you can plug in on any proble that has variables in the answer choices.
2) You should write out your answer choices and solve on paper.
3) Plug in - Come up with numbers but avoid 1 or 0.
4) Solve for the target.
5) Always check all answer choices

Question 59

What to Plug in?

Answer 59

Youcan plug in any numbers you like as long as they are consistent with any restrictions stated in the problem. Smaller numebrs are easier to work with than larger. Good to start with 2, but avoid 1 or 0.

In a problem solving percentages, 10 and 100 are ood to use.

In a problem involving minutes or seconds 30 or 120 are often good choices.

Question 60

Rules for Plugging In

1) Don’t plug in … or …
2) Dont plug in … that are ….
3) Don’t plug in the … for … variables
4) Avoid … numbers

Answer 60

1) Don’t plug in 1 or 0. These numbers have special properties
2) Don’t plug in numbers that are already in the problem
3) Don’t plug in the same number for multiple variables
4) Avoid conversion numbers. eg dont use 60 for a problem involving minutes and hours

*Remember to check all five answer chocies when plugging in*

Question 61

Plugging in the Answers (PITA)

Answer 61

The answers may not have variables, but you may be able to solve by plugging in the answers. Signs for PITA may include if you are thinking about using variables, and that the question asks for a specific amount and that there are numbers in the answer choices.

Question 62

PITA steps

Answer 62

1) Recognise the opportuntity - trigger phrases include ‘how much…’, ‘how many…’, ‘what is the value of…’. Second tip off is specific numbers in the answer in ascending or descending number. Third is your own inclination to write your own algebraic formulas and invent your own variables.
2) Set up scratch paper - list numbers in answer choices in upper left hand corner of your scratch paper
3) label the first column - what do these numbers represent? Write down what these numbers represent.
4) Start with Choice (C). It will always be the number in the middle. This is the most efficient place to start.
5) Creat your spreadsheet. Work through the problem in bite size pieces
6) Rinse and repeat

Question 63

Plugging in on Quant Comp Questions

Answer 63

1) Recognize opportuntity. When a quant comp questions pops up and you see variables, you know you can plug in.
2) Set up scratch paper

A abcd B

y=

y=

y=

3) Plug in and eliminate. Start wiht a normal number such as 2 or 5, but follow any condidtions in problem. Calculate value in quantity A and then quantity B. If quantity A is bigger, eliminate choices B and C. If quantity B is bigger, eliminate A and C. If both are the same, eliminate choices A and B
4) Rinse and repeat. Try to get a different result by messing with problem. Use FROZEN

Fractions, Repeats (numbers from problem), One, Zero, Extremes (big numbers like 100), and Negatives. Goal is to eliminate choices A B C. If everying on the checklist and A B or C is stills tanding, that is your answer.

*Always plug in at least twice on quant comp questions*