Math Section 1 - Overview Flashcards
If a quant comp question contains only numbers, …
the answer can’t be (D). Any quant comp problem that contains only numbers and no variables must have a single solution.
Digits
Digit refers to the numbers that make up other numbers. There are 10 digits: 0,1,2,3,4,5,6,7,8,9
Numbers
A number is made up of either a digit or collection of digits. There are an infinite number of numbers.
Integers
Integers are the numbers that have no fractional or decimal part, such as -3,-2,-1,0,1,2,3
Zero
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
Fractions are neither …..
even nor odd
Any integer is … if its units digit is …
Even, Odd
even + even = odd + odd = even + odd = even x even = odd x odd = even x odd =
= even = even = odd = even = odd = even
Consecutive Integers
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
Absolute Value
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.
Factors
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.
Multiples
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.
Prime Numbers
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
Prime Number Facts 0 1 2 +/-
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.
Divisibility
An integer is always divisible by its factors
Divisibility Rules: 2 3 4 5 6 8 9 10
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
Sum
the result of addition
Difference
the result of subtraction
Product
the result of multiplication
Quotient
the result of division
Divisor
the number you divide by
Numerator
the top number in a fraction
Denominator
the bottom number in a fraction
Consecutive
in order from least to greatest
Terms
the numbers and expressions used in an equation
Order of Operations
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)
Multiplication and Division + x + = - x - = + x - = + / + = - / - = + / - =
= positive = positive = negative = positive = positive = negative
Associative Laws
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)
Distributive Law
*Often tested on GRE a(b+c) = ab + ac a(b-c) = ab - ac
Five Rules for Exponents
1) a2 =
2) a2 • a3 =
3) (a2)3 =
4) a2 / a3 =
5) a12 - a11 =
1) a • a
2) (a • a)(a • a • a) = a2+3 = a5
3) (a • a)(a • a)(a • a) = a2•3 = a6
4) a • a / a • a • a = 1 / a = a 2-3 = a-1
5) a11(a - 1)
Multiplication with Exponents
As long as the numbers have the same base all you do is add the exponents.
22 x 24 =
22 + 4 = 26
This rule does not apply to addition. There is no quick and easy way to add numbers with exponents.
Addition with Exponents
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)
Division with Exponents
Dividing two or more numbers with teh same base that are rased to exponents is simple - just subtract the exponents.
29 / 23 = 29 - 3 = 26
Subtraction with Exponents
Like Addition, there is not cut and fast way to subtract numbers with exponents. You can factor though.
26 - 22 = 22 (24 - 1)
Dividing Exponents with a negative exponent
Take its reciprocal (put a 1 over the expression) and write the negative exponent as a positive.
3-2 = 1 / 32 = 1 / 9
Exponents to Exponents
When a number with an exponent is raised to another power, you simply mulitply the exponents.
(45)2 = 45 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)
Oddities of Exponents
Raising a fraction that’s between … and … to a power greater than … results in ….
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
Oddities of Exponents
A negative number raised to an even power results in…
A negative number raised to an even power results in a postitive number.
(-2)2 = 4
Oddities of Exponents
A negative number raised to an odd power results…
A negative number raised to an odd power results in a negative number. (-2)3 = -8
Oddities of Exponents
A number raised to a negative power is … the … of …
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
Oddities of Exponents
Any nonzero number raised to … is …
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.
Square Root Nuances
(1) result of solving for variable
eg x2 = 16
(2) ETS asking for value of a square root of any number
(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
Square Root Rules
Multiply Square Roots
√a x √b = ?
Divide Square Roots
√a/b = ?
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
Square Root Rules
Adding and Subtracting Square Roots
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.
Estimating and Simplifying Roots
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.
Some Perfect Squares
√4 √9 √16
√25 √36 √49
√64 √81 √100
√121 √144 √169
Some Perfect Squares
22 32 42
52 62 72
82 92 102
112 122 132
Perfect Squares that are Good to Know
√1
√2
√3
√4
√1 = 1
√2 = 1.4 (about)
√3 = 1.7 (about)
√4 = 2
The Golden Rule of Manupulating Equations
Whatever you do on one side of the equals sign you must also do on the other side.
Clearing a Fraction
5x + 3/2 = 7x
Multiply both sides by 2
10x + 3 = 14x
3 = 4x
3/4 = x
Inequalities
≠
>
<
≥
≤
≠ is not equal to
> is greater than
< is less than
≥ is greater than or equal to
≤ is less than or equal to
Inequality Rule for Manipulation
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
How to Crack Ranges for two Variables
If 0 ≤ x ≤ 10 and -10 ≤ y ≤ -1, then what is the range for x - y?
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
How to solve equations with two variables
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
Quadratic Equations
FOIL
(x+4)(x-7)
FOIL = First, outside, inside, last
(x+4)(x-7)
x2 -7x + 4x - 28
x2 - 3 - 28
Frequent Quadratics to Know
Factored form x2 - y2
Factored form (x + y)2
Factored form (x - y)2
Unfactored form (x+y)(x-y)
Unfactored form x2 + 2xy + y2
Unfactored form x2 - 2xy + y2
Quadratic Equations and Factoring
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
Simultaneous Equations
If 5x + 4y = 6 and 4x + 3y = 5, what does x+y equal?
Add or subtract equations from each other.
5x + 4y = 6
- 4x + 3y = 5 .
x + y = 1
Algebra questions and plugging in
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
What to Plug in?
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.
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
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*
Plugging in the Answers (PITA)
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.
PITA steps
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
Plugging in on Quant Comp Questions
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*
