GRE formulas Flashcards
1
Q
What is the formula for the volume of cylinder?
A
Pi r^2 h
2
Q
How are roots represented as fractional exponents?
A
The square root of an exponent can also be represented as being raised to the 1/2 power; the cube root of an exponent can also be written as being raised to the 1/3 power; the fourth root of an exponent can be written as being raised to the 1/4 power and so on.
3
Q
What is the formula for finding percents?
A
Is/of = x/100
4
Q
Inequality Rule of a Triangle:
A
The length of a given side of a triangle is less than the sum of the lengths of the other two sides and greater than the difference between those two sides.
5
Q
Number of integers from x to y inclusive:
A
Y-x+1, where y represents the larger and x the smaller of two given integers.
6
Q
How to find the SUM of all integers from x to y inclusive:
A
X+y(y-x+1)/2
Step 1: Integer 1 + integer 2
Step 2: multiply the answer in step 1 by the larger integer minus the smaller integer plus one
Step 3: divide the answer in step 2 by two
Tada!
7
Q
Product of the greatest common denominator (GCD) and the least common multiple (LCM):
A
If x = GCD of a+b and y=LCM of a+b, then ab = xy
8
Q
How to solve quadratic equations with a coefficient greater than one in front of x^2:
A
Look at the equation and see if the coefficient can be factored out -
For example, 2x^2 -20x -48 = 0 can be factored as follows: 2 (x^2 - 10x-24)
9
Q
Integers:
A
Any counting number including negative numbers (i.e. -3, -1, 2, 7..but NOT 2.5)
10
Q
Real Numbers
A
Numbers that appear on the number line (i.e. one that is not imaginary) - including pi, the square root of 2, etc.
11
Q
Order of Operations
A
PEMDAS (please excuse my dear aunt sally)
Complete any arithmetical operation in the following order:
- Parentheses
- Exponents
- Multiplication/Division
- Addition/subtraction
12
Q
Prime Numbers
A
A prime number is one that is divisible only by itself and one. In other words, a positive integer with exactly 2 positive divisors. This includes 2, 3, 5, 7, 11, 13…not 9, because 9 is divisible by 3.
1 is NOT a prime. 2 is the smallest prime number and the only prime number that is even.
13
Q
Prime Numbers below 60
A
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59
14
Q
Prime Factorization
A
The prime factorization of a number is dividing a number into its constituent primes - i.e. 21 can be divided into 7 and 3, both of which are prime numbers. When two primes do not multiply to equal a given number, its prime factorization can be found by factoring the number until all factors are prime.
For instance, to find the prime factors of 24, first factor 24 - 12 and 2
2 is already prime
12 can be factored as 6 and 2
6 can be factored as 2 and 3
now we have all prime numbers one 3 and 3 2s - so the prime factorization is 2^3 x 3
15
Q
How to find the number of factors a number has:
A
Step 1: Find the prime factorization of the given number
Step 2: Add 1 to any exponents in the answer (including an exponent of 1) and multiply each answer together.
For example, the prime factors of 21 are 7 and 3; the exponents of both 7 and 3, are 1, so add 1 to each and you get 2X2 = 4, so 21 has 4 factors in total (check: 1, 3, 7, 21).
16
Q
Factor/Divisor
A
Any number that divides evenly into a given number.
17
Q
Greatest Common Factor/Greatest Common Divisor
A
The biggest factor shared by two numbers. The easiest way to find the GCF is to take the prime factorization of both numbers and multiply all of the primes in each number. For example, the GCF of 12 and 30 can be calculated as follows:
12 prime factors: 2, 3
30 prime factors: 2, 3, 5
2 and 3 appear in both - so 2 x 3 = 6 prime factors
If two numbers share no primes, the GCF is 1.
18
Q
Least Common Multiple
A
The smallest positive integer that two numbers share as a factor. So, the LCM of 4 and 6 is 12 - it is the smallest number that they BOTH divide evenly into. To find LCM, multiply each of the shared primes with each other, then multiply that answer by each number’s unshared primes.
Primes of 4: 2
Primes of 6: 2 and 3
shared primes - 2 x 2 = 4
unshared primes = 3 X 4 (shared) = 12
19
Q
A number is divisible by 3 if:
A
The sum of the digits in the number is divisible by 3 (i.e. 33 - 3 +3 = 6 so it is divisible by 3)
20
Q
A number is divisible by 4 if:
A
The number formed by the last two digits of a number are divisible by 4 (for instance, 144 - the last 2 digits form 44, so 144 is divisible by 4).
21
Q
A number is divisible by 5 if:
A
The last digit of the number is a 5 or a zero
22
Q
A number is divisible by 6 if:
A
It is an even number AND the sum of the digits is divisible by 3 I.e. 66 is even and 6+6 =12, which is divisible by 3, so it is divisible by 6.
23
Q
A number is divisible by 8 if:
A
The last three digits form a number that is divisible by 8.
24
Q
A number is divisible by 9 if:
A
The sum of digits is divisible by 9.
25
Absolute value
A number's distance from 0 on the number line - absolute value is always positive, since distance can never be negative! (i.e. the absolute value of -8 is 8)
26
Percent change
To find the percentage change, calculate the value of the change or difference, and divide that by the original value. For instance, if the price of something goes from $40 to $52, then you do 52-40/40, which is 12/40 = 30%
27
Calculating value from percentage change
If a price decreases by a given percentage, find the new value by multiplying the original value by the difference between 100% and the percent change. FOR INSTANCE, if a price falls by 15%, multiply the original value by 1-.15 = .85) to find the new value. If a price increases by 15%, simply multiply the original value by (1+.15 = 1.15).
28
Calculating the original value of an item when given a new value and the percentage that the original changed:
let x = original value
x + percentage change * (x) = new value
x - percentage change * (x) = new value
example: if the new price of an item is $50, and it was discounted by 10%, then to find the original price you do the following equation:
x - .10x = $50
.9x = 50
x = 55.5
29
Dividing numbers with like bases that have exponents:
x^a/x^b = x^(a-b)
30
When is a fraction undefined?
When the denominator = 0.
31
Power law of exponents (how to calculate a base raised to more than one power)
When something raised to a power, is in turn raised to another power, multiply the powers to combine them.
(x^a)^b = x^(a*b)
32
Fractions as exponents
x^(1/2)= square root of x, x^(2/3) = cube root of x squared
The denominator is the ROOT outside the radical and the numerator is the power that the value in the radical is being raised to.
33
Negative exponents
a^ (-m) = 1/a^m
If the base is raised to a negative exponent, take the reciprocal of the base to rewrite the negative exponent as a positive exponent.
34
Multiplying numbers with like bases that have exponents:
x^a * x^b = x^(a+b)
35
Negative Bases
When a negative number is raised to an even number, the resulting value is positive. When a negative number is raised to an odd number, the resulting value is negative.
36
How find the value of 10 raised to ANY power:
Add that many zeros after the 1 (i,e. 10^5 is 100,000 - a one plus 5 zeros).
37
Properties of numbers inside a radical
Positive numbers inside a radical always result in a positive root, so if you are asked for the square root of 49, -7 is an extraneous root, since it is negative. The answer is 7.
If a negative number is inside a radical, then whether the root is real or imaginary depends on if the power is even or odd. For example, the square root of -49 is imaginary because square=2=even (-7 * -7 can never be -49). BUT, the cube root of -8 is -2.
38
Simplifying Roots
To simplify roots inside a radical, separate the number into its prime factors and simplifying.
For instance the square root of 48 can be simplified by putting 48's prime factors under the radical sign - 3 * 2 *2 * 2 * 2 - essentially, rewritten you see that the question is now looking for the square root of 3* 2^4, which is the square root of 3 * 16 - since the square root of 16 is 4, the 4 can be moved outside of the radical sign, and the final answer is radical 3 with a coefficient of 3.
39
Adding roots
Roots can be added like variables - i.e. the numbers inside each of the radicals must be the same, otherwise they cannot be added.
40
FOIL
Pneumonic for the order in which polynomials should be multipled- first, outer, inner last.
41
Difference of Squares
Pattern that should be recognizable to help with factoring! Is used all the time in GRE questions!
(a+b) (a-b) = a^2-b^2
42
Factoring polynomials using Greatest Common Factor
The greatest common number that goes into several given terms. Very helpful to use when factorng/simplifying polynomials!!
i.e. - 6x^3+12x^3+33x = 3x (2x^2+4x+11)
43
Factoring using quadratic polynomials
ax^2+bx+ c= (x+m) (x+n), where a is the sum of m and n and b is their product.
I.e. - the factored numbers must add up to a and multiply to b.
44
Factoring a quadratic equation in which the leading coefficient is GREATER THAN ONE
To factor a quadratic equation in which the leading coefficient is 2 or greater, first multiply that coefficient by the last term in the equation to determine the product of the factored values. Once a product is established, the potential values of m and n are limited only to ones that form the product, AND they also must add up to the coefficient in front of the second term in the equation. Once those terms are worked out, it's time to use them to factor the equation - the easiest way is to draw a box, dividing it into 4 parts, and write the ax^2 in one box, which is diagonal to c. In the other two boxes, write the terms for bx in each. Once the boxes are drawn and filled, factor out the GCF in each row/column. Then combine the answers going horizontally and vertically, respectively, into separate parentheses to get the answer.
45
How to eliminate fractions from equations:
To eliminate a fraction as a coefficient, multiply the reciprocal of the fraction on both sides.
In an equation with multiple fractions, the fractions can be multiplied by the lowest common denominator that goes into each numerator. For instance, each term in the equation 3x/4 +1.2 = x/3 can be multiplied by 12/1, yielding 9x + 6 = 4x.
Fractional equations that are set equal to each other can be cross-multiplied to factor/simplify/find x.
46
Quadratic formula:
x= -b + or - the square root of b^2-4ac/2a
47
Substitution method for solving systems of equations:
In systems of equations with two variables, isolate one variable and simplify it, setting it equal to the side of the equation with the other variable. Then plug that side of the equation into the other equation to solve for a given variable.
For instance, given 17 -3x = y and another equation with the same variables, replace y in the other equation with (17-3x) in order to solve for x by making all of the variables uniform. Then plug x in to solve for y.
This method also works for systems with 3 variables - it just takes longer.
48
Elimination method for solving systems of equations:
Examine the system of equations carefully to see if a variable can be cancelled out.
FOR EXAMPLE, for the equations 6x + y = 34 and 2x - 2y =6, the -2y and 2y cancel out because they equal zero when combined.
49
What is important to look for when solving equations on the GRE?
One of the GRE's favorite tricks is to disguise two equations that are exactly the same as being different, thus forcing you to do a lot more work than you have to find an answer! If you have a system of equations that are the same, you cannot find an answer/there is no answer.
50
Function Notation
If given f(x) = and asked for the value of f (something else), simply replace every instance of "x" in something else expression with whatever is now in the parentheses.
51
Strange operators
Sometimes, to torture and confuse students for fun, the GRE throws into equatons bizarre symbols, like triangles, or # signs, for the sake of this example - if given, x # y = 3x + y^2, for instance, and asked to find 5 # 2, you would plug the 5 in as x and the 2 in as y.
52
Multiplying or dividing a negative # by an inequality:
Inequalities can be treated, for the most part, as regular equations EXCEPT that when multiplying or dividing a negative number by an inequality, one MUST switch the sign of the inequality around!!!
53
Solving quadratic inequalities:
First solve the inequality by setting the equation equal to zero, then use the number line to determine which answers fit the inequality and are solutions.
54
Inequalities with absolute values:
Because absolute value indicates distance from zero on the number line, inequalities with absolute values indicate that a given number x, has two options to satisfy the inequality.
For instance, if the absolute value of x is less than 3, that means that -3<3, because x can be 3 away from zero on the number line in either the positive or negative direction.
55
Manipulating numbers inside radicals to find out their order:
If given a question in which radicals with coefficients need to be ordered, check the coefficients to determine if they are also square roots - if so, rewrite them in that form and find their product with the number already inside the radical. This is helpful for comparative purposes, because when radicals are simplified, it can be difficult to determine their value differences.
56
Identifying percentage change correctly
Be careful not to look at the value of the change itself if percent change is asked for!! This means, if unemployment went from 3% in one year to 7% in another year, even though that is a difference of 4%, the *percentage change* is greater than 50%, because 7% is over twice as great as 3%.
57
Comparing percentages
If a question asks you to compare percentage values of different items, the easiest way is to choose a variable for one item and determine the percentages as functions of that item. For example, if one was comparing weights of different people and was given information about percentages, set the person's weight you are trying to find equal to a certain variable, let's say T. Then, let's say someone else is 60% heavier - that would be 1.60T, because that person is T (100%)+(60%). Someone who is 25% lighter would be .75T.
58
Slope Formula
y = mx + b, where coefficient m = a slope
59
What happens to decimals being raised to a positive integer?
When decimals are raised to powers of positive integer, they will DECREASE after being raised to the first power, and continually approach zero.
60
Integers:
Any counting number including negative numbers (i.e. -3, -1, 2, 7..but NOT 2.5)
61
Real Numbers
Numbers that appear on the number line (i.e. one that is not imaginary) - including pi, the square root of 2, etc.
62
Order of Operations
PEMDAS (please excuse my dear aunt sally)
Complete any arithmetical operation in the following order:
1. Parentheses
2. Exponents
3. Multiplication/Division
4. Addition/subtraction
63
Prime Numbers
A prime number is one that is divisible only by itself and one. In other words, a positive integer with exactly 2 positive divisors. This includes 2, 3, 5, 7, 11, 13...not 9, because 9 is divisible by 3.
1 is NOT a prime. 2 is the smallest prime number and the only prime number that is even.
64
Prime Numbers below 60
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59
65
Prime Factorization
The prime factorization of a number is dividing a number into its constituent primes - i.e. 21 can be divided into 7 and 3, both of which are prime numbers. When two primes do not multiply to equal a given number, its prime factorization can be found by factoring the number until all factors are prime.
For instance, to find the prime factors of 24, first factor 24 - 12 and 2
2 is already prime
12 can be factored as 6 and 2
6 can be factored as 2 and 3
now we have all prime numbers one 3 and 3 2s - so the prime factorization is 2^3 x 3
66
How to find the number of factors a number has:
Step 1: Find the prime factorization of the given number
Step 2: Add 1 to any exponents in the answer (including an exponent of 1) and multiply each answer together.
For example, the prime factors of 21 are 7 and 3; the exponents of both 7 and 3, are 1, so add 1 to each and you get 2X2 = 4, so 21 has 4 factors in total (check: 1, 3, 7, 21).
67
Factor/Divisor
Any number that divides evenly into a given number.
68
Greatest Common Factor/Greatest Common Divisor
The biggest factor shared by two numbers. The easiest way to find the GCF is to take the prime factorization of both numbers and multiply all of the primes in each number. For example, the GCF of 12 and 30 can be calculated as follows:
12 prime factors: 2, 3
30 prime factors: 2, 3, 5
2 and 3 appear in both - so 2 x 3 = 6 prime factors
If two numbers share no primes, the GCF is 1.
69
Least Common Multiple
The smallest positive integer that two numbers share as a factor. So, the LCM of 4 and 6 is 12 - it is the smallest number that they BOTH divide evenly into. To find LCM, multiply each of the shared primes with each other, then multiply that answer by each number's unshared primes.
Primes of 4: 2
Primes of 6: 2 and 3
shared primes - 2 x 2 = 4
unshared primes = 3 X 4 (shared) = 12
70
A number is divisible by 3 if:
The sum of the digits in the number is divisible by 3 (i.e. 33 - 3 +3 = 6 so it is divisible by 3)
71
A number is divisible by 4 if:
The number formed by the last two digits of a number are divisible by 4 (for instance, 144 - the last 2 digits form 44, so 144 is divisible by 4).
72
A number is divisible by 5 if:
The last digit of the number is a 5 or a zero
73
A number is divisible by 6 if:
It is an even number AND the sum of the digits is divisible by 3 I.e. 66 is even and 6+6 =12, which is divisible by 3, so it is divisible by 6.
74
A number is divisible by 8 if:
The last three digits form a number that is divisible by 8.
75
A number is divisible by 9 if:
The sum of digits is divisible by 9.
76
Absolute value
A number's distance from 0 on the number line - absolute value is always positive, since distance can never be negative! (i.e. the absolute value of -8 is 8)
77
Percent change
To find the percentage change, calculate the value of the change or difference, and divide that by the original value. For instance, if the price of something goes from $40 to $52, then you do 52-40/40, which is 12/40 = 30%
78
Calculating value from percentage change
If a price decreases by a given percentage, find the new value by multiplying the original value by the difference between 100% and the percent change. FOR INSTANCE, if a price falls by 15%, multiply the original value by 1-.15 = .85) to find the new value. If a price increases by 15%, simply multiply the original value by (1+.15 = 1.15).
79
Calculating the original value of an item when given a new value and the percentage that the original changed:
let x = original value
x + percentage change * (x) = new value
x - percentage change * (x) = new value
example: if the new price of an item is $50, and it was discounted by 10%, then to find the original price you do the following equation:
x - .10x = $50
.9x = 50
x = 55.5
80
Dividing numbers with like bases that have exponents:
x^a/x^b = x^(a-b)
81
When is a fraction undefined?
When the denominator = 0.
82
Power law of exponents (how to calculate a base raised to more than one power)
When something raised to a power, is in turn raised to another power, multiply the powers to combine them.
(x^a)^b = x^(a*b)
83
Fractions as exponents
x^(1/2)= square root of x, x^(2/3) = cube root of x squared
The denominator is the ROOT outside the radical and the numerator is the power that the value in the radical is being raised to.
84
Negative exponents
a^ (-m) = 1/a^m
If the base is raised to a negative exponent, take the reciprocal of the base to rewrite the negative exponent as a positive exponent.
85
Multiplying numbers with like bases that have exponents:
x^a * x^b = x^(a+b)
86
Negative Bases
When a negative number is raised to an even number, the resulting value is positive. When a negative number is raised to an odd number, the resulting value is negative.
87
How find the value of 10 raised to ANY power:
Add that many zeros after the 1 (i,e. 10^5 is 100,000 - a one plus 5 zeros).
88
Properties of numbers inside a radical
Positive numbers inside a radical always result in a positive root, so if you are asked for the square root of 49, -7 is an extraneous root, since it is negative. The answer is 7.
If a negative number is inside a radical, then whether the root is real or imaginary depends on if the power is even or odd. For example, the square root of -49 is imaginary because square=2=even (-7 * -7 can never be -49). BUT, the cube root of -8 is -2.
89
Simplifying Roots
To simplify roots inside a radical, separate the number into its prime factors and simplifying.
For instance the square root of 48 can be simplified by putting 48's prime factors under the radical sign - 3 * 2 *2 * 2 * 2 - essentially, rewritten you see that the question is now looking for the square root of 3* 2^4, which is the square root of 3 * 16 - since the square root of 16 is 4, the 4 can be moved outside of the radical sign, and the final answer is radical 3 with a coefficient of 3.
90
Adding roots
Roots can be added like variables - i.e. the numbers inside each of the radicals must be the same, otherwise they cannot be added.
91
FOIL
Pneumonic for the order in which polynomials should be multipled- first, outer, inner last.
92
Difference of Squares
Pattern that should be recognizable to help with factoring! Is used all the time in GRE questions!
(a+b) (a-b) = a^2-b^2
93
Factoring polynomials using Greatest Common Factor
The greatest common number that goes into several given terms. Very helpful to use when factorng/simplifying polynomials!!
i.e. - 6x^3+12x^3+33x = 3x (2x^2+4x+11)
94
Factoring using quadratic polynomials
ax^2+bx+ c= (x+m) (x+n), where a is the sum of m and n and b is their product.
I.e. - the factored numbers must add up to a and multiply to b.
95
Factoring a quadratic equation in which the leading coefficient is GREATER THAN ONE
To factor a quadratic equation in which the leading coefficient is 2 or greater, first multiply that coefficient by the last term in the equation to determine the product of the factored values. Once a product is established, the potential values of m and n are limited only to ones that form the product, AND they also must add up to the coefficient in front of the second term in the equation. Once those terms are worked out, it's time to use them to factor the equation - the easiest way is to draw a box, dividing it into 4 parts, and write the ax^2 in one box, which is diagonal to c. In the other two boxes, write the terms for bx in each. Once the boxes are drawn and filled, factor out the GCF in each row/column. Then combine the answers going horizontally and vertically, respectively, into separate parentheses to get the answer.
96
How to eliminate fractions from equations:
To eliminate a fraction as a coefficient, multiply the reciprocal of the fraction on both sides.
In an equation with multiple fractions, the fractions can be multiplied by the lowest common denominator that goes into each numerator. For instance, each term in the equation 3x/4 +1.2 = x/3 can be multiplied by 12/1, yielding 9x + 6 = 4x.
Fractional equations that are set equal to each other can be cross-multiplied to factor/simplify/find x.
97
Quadratic formula:
x= -b + or - the square root of b^2-4ac/2a
98
Substitution method for solving systems of equations:
In systems of equations with two variables, isolate one variable and simplify it, setting it equal to the side of the equation with the other variable. Then plug that side of the equation into the other equation to solve for a given variable.
For instance, given 17 -3x = y and another equation with the same variables, replace y in the other equation with (17-3x) in order to solve for x by making all of the variables uniform. Then plug x in to solve for y.
This method also works for systems with 3 variables - it just takes longer.
99
Elimination method for solving systems of equations:
Examine the system of equations carefully to see if a variable can be cancelled out.
FOR EXAMPLE, for the equations 6x + y = 34 and 2x - 2y =6, the -2y and 2y cancel out because they equal zero when combined.
100
What is important to look for when solving equations on the GRE?
One of the GRE's favorite tricks is to disguise two equations that are exactly the same as being different, thus forcing you to do a lot more work than you have to find an answer! If you have a system of equations that are the same, you cannot find an answer/there is no answer.
101
Function Notation
If given f(x) = and asked for the value of f (something else), simply replace every instance of "x" in something else expression with whatever is now in the parentheses.
102
Strange operators
Sometimes, to torture and confuse students for fun, the GRE throws into equatons bizarre symbols, like triangles, or # signs, for the sake of this example - if given, x # y = 3x + y^2, for instance, and asked to find 5 # 2, you would plug the 5 in as x and the 2 in as y.
103
Multiplying or dividing a negative # by an inequality:
Inequalities can be treated, for the most part, as regular equations EXCEPT that when multiplying or dividing a negative number by an inequality, one MUST switch the sign of the inequality around!!!
104
Solving quadratic inequalities:
First solve the inequality by setting the equation equal to zero, then use the number line to determine which answers fit the inequality and are solutions.
105
Inequalities with absolute values:
Because absolute value indicates distance from zero on the number line, inequalities with absolute values indicate that a given number x, has two options to satisfy the inequality.
For instance, if the absolute value of x is less than 3, that means that -3<3, because x can be 3 away from zero on the number line in either the positive or negative direction.
106
Manipulating numbers inside radicals to find out their order:
If given a question in which radicals with coefficients need to be ordered, check the coefficients to determine if they are also square roots - if so, rewrite them in that form and find their product with the number already inside the radical. This is helpful for comparative purposes, because when radicals are simplified, it can be difficult to determine their value differences.
107
Identifying percentage change correctly
Be careful not to look at the value of the change itself if percent change is asked for!! This means, if unemployment went from 3% in one year to 7% in another year, even though that is a difference of 4%, the *percentage change* is greater than 50%, because 7% is over twice as great as 3%.
108
Comparing percentages
If a question asks you to compare percentage values of different items, the easiest way is to choose a variable for one item and determine the percentages as functions of that item. For example, if one was comparing weights of different people and was given information about percentages, set the person's weight you are trying to find equal to a certain variable, let's say T. Then, let's say someone else is 60% heavier - that would be 1.60T, because that person is T (100%)+(60%). Someone who is 25% lighter would be .75T.
109
Slope Formula
y = mx + b, where coefficient m = a slope
110
What happens to decimals being raised to a positive integer?
When decimals are raised to powers of positive integer, they will DECREASE after being raised to the first power, and continually approach zero.
111
What are the five most common Pythagorean triples?
```
3-4-5
5-12-13
8-15-17
7-24-25
9-90-41
```
112
Right triangle combinations
30-60-90
| 45-45-90
113
Formula for group problems with two groups:
group 1 + group 2 – both + neither = total.
