GRE Notes
area of an equilateral triangle using side length
((s^2)3^(1/2))/4
Perfect Squares 1-15
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225
Identity Strategy
If you can find two things that equal x then you can set them equal to each other to solve
Isosceles Triangle Special Properties
Line from the vertex to the base has 4 properties angle bisector perpendicular bisector altitude median
rectangle
all 90 degree angles plus 4 parallelogram props
Average Speed Formula
total distance/total time
Solving obtuse triangles
you can expand lines to create altitudes to solve using the area formula or pythagorean
Growth/Decay Problems
you will always be given the rate and should solve one step at a time
sum of an evenly spaced list
(N/2)(a_1+a_n)
Trapezoid Area Formula
(base 1+base 2/2)h average of the bases times the height
sum of angles in an n sided polygon
(n-2)180
hours in seconds
1 hr = 3600 s
Prime Numbers under 60
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59
Pythagorean triples
3,4,5 5,12,13 8,15,17 7,24,25
rhombus
4 equal sides, perpendicular bisecting diagonals and 4 parallelogram props
Work Questions
A=RT (amount produced = production rate * time) Two types of questions: 1) proportion problems ->ensure units are the same and be careful of simplification across an equals 2)shared work -> the sum of individual work rates equals to the shared work rate. mk/(m+k) is time tom complete one unit
area of a semicircle
A=½πr²
Inscribed Chords
An inscribed chord with the same length must have the same angle
inscribed angle theorem
If an angle is inscribed in a circle, then the measure of the angle equals one half the measure of the central angle of its intercepted arc.
Trapezoid Angle Theorem
In a trapezoid, consecutive angles between a pair of parallel sides are supplementary.
Number Sense on the GRE
Integers include negatives 0 is an integer 0 is an even
Gap Questions
Involve two objects going in the same or opposite directions and hence have a gap between them Strategy: Use the Gap to your advantage If in same direction then gap changes by the difference (subtraction) of the two speeds If in opposite directions then the gap changes by the sum (addition) of the two speeds Hence a D=RT equation can be used to model the gap distance, and gap speed (So D is the GAP traversed in a given time)
Sum of a Series when average is known
Is the average times the quantity
LCM formula
LCM = (P*Q)/GCF OR P*Q = LCM*GCF
Counting Identical Items
MISSISPPI Rule: first consider the number of ways you can order the items as if they were the same then divide by the way you can sort each item in each category
Exactly definintion
Must be completely equal, not more or less
Similar Triangles
Must have 2 angles of the same degree (do not need to know angles if they share a parallel side it is also true) if so, then the triangles are proportional by some scale factor k additionally, the area is proportional by k^2 (.5kbkh=k^2(.5bh))
See x-a in a denominator
NOTE that x=a cannot be an answer
How to find an adjacent square
N^2 +n +n +1 = (N+1)^2
Divisibility Rule for 4
The last 2 digits are divisible by 4
Triangle Inequality Theorem
The sum of the lengths of any two sides of a triangle is greater than the length of the third side. a-b<c<a+b
space diagonals
Use 3D pythagorean cube: SD= s*root(3) rectangle SD^2=l^2+w^2+h^2
Combinations (definition)
Used when order matters ie ABBA = ABAB
Permutations (definition)
When order does not matter
Squaring multiples of 10
X0^2 is the same as X^2*100 ie (X^2)(10^2), works the same for any amount of zeros. On the exam look for a perfect square followed by a couple zeros, if 1 then it is 100, 2 then it is 10,000, 3 is 1,000,000 etc
Did you read the entire question
YES!!!!
age questions
make variables start today, then use addition and subtraction to note before and after today be careful and make sure you monitor the difference
Inscribed Angle Meeting a Semi Circle on the diameter
the inscribed angle must be 90 degrees if the vertex is on the semicircle and rays terminate on the diameter
The Largest Possible Triangle
the largest triangle is the one with two perpendicular bisectors
Divisibility rule for 9
the sum of digits is divisible by 9.
Double Matrix Method
two populations with 2 characteristics. Ex. cars & trucks and red and green. take the given no. of items and fill in spots based on info. This method is very quick and runs faster than probability tree intuition
counting diagonals of a polygon
use fundamental counting principle so pick number of vertiices, by number of non adjacent points divided by the way to sort those two points
slanty shapes
use rectangles and right triangles
Nested List Strategy
usually you can place lists on top of each other to see the difference
units of measure
will be given the conversion ratio if an area is needed you square the ratio if a volume is needed you cube the ratio
Number of integers from x to y inclusive
y-x+1
Comparing Inequalities
you can do lots of things and maintain inequalities cross multiply square add or subtract multiply (negative swaps inequality)
Symmetric List Properties
Have the same mean and median
Combinatorics case strategy
If a combinatorics question can be separated into cases, it will be easier to identify the combinations in each case and together
coordinate planes
Are drawn to scale
Sufficient definition
Can either tell us the answer is correct or incorrect
Distance and Speed Problem equations
D=R*T Avg Speed = total D/Total T calculate a D=R*T for each person/leg of journey if T becomes a function of D, then D will cancel when solving for R given D is a part of whole distance (also means we can guess the distance and calculate the rate)
When dealing with negative number algebra
DO NOT FORGET YOUR NEGATIVE SIGNS
Multiples of X between m and n
Determine the first factor is x *y Determine final factor is x*z number of factors is z-y+1
Calculator
Do Not Use It
counting factors
Do a prime factorization check if they are asking for just odd or even If so, drop off the odd or even then take remaining factors and add one to each exponent and multiply together
Venn Diagrams
Draw them
Proportionally Solving Right Triangles
Find the GCF of the sides of a right triangle and factor out, then solve for smaller values, and then scale back up never do large number pythagorean-+
Number Sense Testing Strategy
Positive integer Negative integer 0 positive/negative decimal
Simplification
REDUCE FIRST before operations recommend handle each side of an equals and operation then move around if you start simple then you can never get bogged down
Mixture Problems
a concentration is the amount of solute/total volume If given two unknown mixtures combined to create a final known mixture, you should create two equations, one for the final amount of volume and one for the final amount of solute
consecutive integer property
a list of n consecutive integers must have a multiple of n and all positive integers less than n
arithmetic sequence
a_n=a_1+(n-1)d illustrates a sequence that is evenly spaced starting at a specific numbers, all even numbers or odd numbers can be an arithmetic sequence, very powerful
backsolving
an option when there are 5 numeric options start with B, decide higher or lower, then D, higher or lower
Types of Bisectors
angle bisector, creates two equal angles perpendicular bisector, creates two equal lengths and an meets at right angle
arclength and sector area
arclength/2rpi = arcmeasure/360=sectorarea/2r^2pi
Equilateral Triangle decomposition
can be turned into 6 30-60-90 triangles
circumscribed right triangle
center is also the midpoint of the hypotenuse radius is the altitude from the 90 degree vertice to the mid point
comparing std deviations
compare average distance between points and the MEAN largest is greatest smallest is least
tangent lines and circles
creates a 90 degree angle with the radii that meets at the tangent point
prime factorization
divide by 2 until you cannot, then do 3 etc until you have a prime times a prime
Divide by 5 shortcut
double N and divide by 10 OR divide by 10 and double
testing odd and even terms
easiest way is 0 is an even and 1 is an odd and test quickly
isosceles trapezoid
equal legs, equal angles on each parallel side, equal diagonals
regular polygons
equals sides and equal angles, additionally the line that bisects (splits in half) a regular polygon must bisect the angle as well
Numbers ending in 5 must have a multiple of 5
ex 35 remove the 5 take what is left, 3, and multiple by n+1, 4, -> 3*4 make that number the left side of the number and add two spaces 12_ _ add 25 to open spaces 1225
trapezoid
exactly one set of parallel sides
altitude
find height of a triangle and creates two right triangles If looking for a side, will allow you to use area formula or Pythagorean to solve for a side length
geometry strategy
identify the BIG and the SMALL shapes
rewriting radical
if given an impossible square root to rewrite, it may be best to prime factorize and take out a perfect squares to determine the possibilities
picking a number
if there are variables in the answer and the question we can pick smart numbers and then solve for a target solution, the answer with that hits the target is correct 0 and 1 cannot be smart numbers
translating words into math
is/are means = *with percents 50% greater is 1.5
when you see the term AT LEAST or AT MOST
it can be the number the number or below or above BEWARE DURING COMBINATORICS
3 way venn diagram
key is to use the central value and work your way out
comparison strategy
make like terms
slope of 1
must create a 45-45-90 slope triangle
two radii in a circle
must form an isosceles triangle between them
sum of a series of consecutive integers starting at 1
n(n+1)/2
Combinations Formula
nCr = n!/r!(n-r)!
parallelograms
parrallel opposite sides equal opposite angles diagonals bisect each other equal opposite sides
divisible by something
prime factorize and then determine the units of the prime factorization, typically it simpifies to something like divisible by 2,3,4,9 where we have rules to make it quick
Finding the LCM
prime factorize the 2 numbers, then take the value of the prime factors multiplied together
what to do when you see the word divisor
prime factorize!!
profit formula
profit = reve-cost reve= profit+cost cost= reve-profit
Quadratic Identities
r^2-s^2 = (r+s)(r-s) (a+b)^2 = a^2 + b^2 + 2ab (a-b)^2 = a^2 + b^2 - 2ab On the exam if you see any element of this formula it may be leading you to use this identity
square
rectangle and a rhombus
Special Right Triangles
refers to the 45-45-90 and 30-60-90 right triangles
number of people in a group when given a ratio
remember the true breakdown of the group is just the ratio times some x/x value. May be easier to solve this way.
circle strategy
solve for the radius and use it to solve for everything else