Manhattan Math GRE Prep
Opposite angles and sides of a Triangle
In a triangle, the ides correspond to their opposite angles. The longest side is opposite the largest angle. Smallest side is opposite the smallest angle.
How can you determine the distance between two points on a plane?
Using the pythagorean theorem. Draw a triangle connecting the two point. Find the lengths of the two legs by finding the rise and run. Use the pythagorean theorem.
Rule for exponents
When you see a base without an exponent, write in the exponent one.
Adding and Subtracting Remainders
You can add and subtract remainders directly as long as you correct excess or negative numbers. For example, if x leaves a remainder of 4 after division by 7 and y leaves a remainder of 2, then x+y leaves a remainder of 6. If x leaves a remainder of 4 and 2 leaves a remainder of 5, then the two remainders together yield 9 so subtract 7 until the number is under 7.
Multiplying Remainders
You can multiply remainders as long as you correct excess remainders at the end. If x has a remainder of 4 upon division by 7 and 2 has a remainder of 5, 4 by 5 is 20. Two 7's can be taken out so that x by z will have a remainder of 6.
slope- rise/run
change in y over change in x
Equation for percent Increase/Decrease
change over original= percent over 100 and Original multiplied by 1+ Percent Increase or multiplied by 1- Percent Decrease equals the new number.
Central Angle
degree measure between the two radii
Inscribed Angle=
half of the arc it intercepts
The Factor Foundation Rule,
if a is divisible by b and b is divisible by c, a is also divisible by c. For example 150 is divisible by 5 and 3 because one if its factors , 15, is divisible by 5 and 3.
Type of Rectangle
is a square. All 4 sides are equal and area is side squared. Perimiter is 4 times side.
Volume of 3 dimensional shape
length times width times height
Area of a rectangle
length times with
Central Angle Vertex
lies at center point of circle
Formula for the sum of interior angles of a polygon
n minus 2 in parentheses multiplied by 180
Maximum area of a quadrilateral
of all quadrilaterals with a given perimeter , the square has the largest area. Of all quadrilaterals with a given area, the square has a minimum perimeter.
Volume of a cylinder
pie r squared times height
y intercept
point on line where x=0
x intercept
point on line where y equals 0
Diagonal of a square
side radical 2
Diagonal of a cube
side radical 3
Number of faces in a rectangular solid and cube
six and all faces in the cube are the same.
Surface Area
sum of the areas of all of the faces
What is an equalateral triangle made up of?
two 30, 60 , 90 triangles.
Rules of Exponents
x to the a times x b equals x to the a+b a to the x times b to the x equals (ab) to the x (2 to the 2nd times 3 to the 2nd is six to the second) x to the a over x to the b is equal to x to the a minus b a to the x plus a to the x plus a to the x equals 3 a to the x.
What are the three x and y rules
x²-y²=(x+y) (x-y) x²+2xy+y²= (x+y) (x+y)= (x+y)² x²-2xy+y²= (x-y) (x-y) = (x-y)²
equation of a line
y= mx+b
What must a product of a number and it's reciprocal equal?
1
What is a good real number when working with percents?
100
Area of a Trapezoid
Base 1 plus base 2 in parentheses divided by 2 times the height
Area of a parallelogram
Base times height
How do you know a number is divisible by another number?
It needs to contain all the same prime factors that the other # contains
Circumference
Measure of the distance around the circle. Diameter times pie is circumference.
Trapezoid Properties
One pair of opposite sides is parallel
Rhombus
Opposite sides and opposite angles are equal. All sides are equal.
Counting Integers: Consecutive Integers
Take the difference of the first and last numbers and add one before you are done. ex: integers from 14 to 765, you do 751 plus 1 = 752
What to do if something is term is squared or c ubed
Take the root of the equation
Inscribed angle
Vertex is on the circle itself
Radical x times radical x
equals x
Evenly Spaced Sets
sequences of numbers whose values go up or down by the same amount (4,7, 10, 13, 16)
Properties of evenly spaced sets
1. The arithmetic mean and median are equal to each other. 2.Mean and median are equal to the average of the first and last terms. 3. The sum of the elements in the set is equal to the mean times the number of items in the set.
Consecutive Primes
11,13,17,19
What is the only even prime number?
2
What is the perimeter of a sector
2 Radii plus arc length
Surface area of a cylinder.
2 circles + rectangle= 2 (pie r squared) +2 pie r times height
Prime Numbers up to 20
2,3, 5,7,11,13,17,19
What does each angle in a 30 60 90 triangle correspond to?
30 is opposite the short leg, which is x. 60 is opposite the long leg which is x radical 3 and 90 is opposite the hypotenuse or 2x.
Parallelogram
4 sided figure in which opposite sides are parallel and equal and opposite angles are equal. Diagonal divides parallelogram into two equal triangles.
Isosceles Right Triangle is also known as a :
45, 45 90 triangle. It is exactly half of a square.
When should I convert to decimals or percents?
Addition, Subtraction, Estimating Numbers or Comparing Numbers
Rectangle
All angles are 90 degrees and opposite sides are equal
Square
All angles are 90 degrees. All sides are equal.
Rectangle
All same properties of a parallelogram and all angles are right angles. Also, one pair of sides is length and the other is width.
Base
Always perpendicular to the height
Base raised to an even power
Any base raised to an even power will always result in a positive answer. This means that x squared equals nine would make x either negative 3 or 3. Odd powers, however, are harless. They always keep the original sign of the base.
Radius
Any line segment connecting the center of the circle and any point on the circle is a radius.
Rule for exponents
Any nonzero base raised to the power zero equals 1.
What do exponents and roots do?
Cancel each other out. x to the cubed equals 8 can be radicaled to be 2.
Circle Formula for non-full circle
Central angle over 360 is equal to sector area over circle area which is equal to arc length over circumference
Divident Divisor Formula
Dividend Over divisor is equal to quotient plus remainder over the Number. For example, 13 divided by 6 is equal to two and one sixth.
Inclusive
Do include extremes.
Exclusive
Don't include extremes
Sum of Two Primes
Equals even unless 1 of them is the number 2.
Bases 0 and 1
Exponential expression with a base of 0 will always yield 0 regardless of the exponent. Exponential expression with a base of 1 always yields 1 , regardless of the exponent. Exponential expression with a base of negative one yields 1 when exponent is even and negative one when exponent is odd.
To find all of the factors of a small # use a :
Factor pairs table. Example: For the number 60, Use a small number 1, Large number 60, Small number 2, large number 30 and so on
Sector for a circle
Fractional portion of a circle. Area and Circumference will be cut in half.
Multiples
If 2 numbers are multiples of the same # , the sum i also a multiple of that # (14 and 21 are multiples of 7 and add up to a multiple of 7)
Multiplication of Odds and Evens
If any integers are even, the result will be even. If no integers are even, the result will be odd.
Inscribed Triangles
If one of the sides of an inscribed triangle is a diameter of the circle, then it is a right triangle. Conversely, any right triangle inscribed in a circle must have the diameter as one of its sides.
How do you define a terminating fraction?
If the denominator has only factors of 2 and or 5
Adding and Sutracting Multiples
If you add or subtract multiples of a number, the result is a multiple of the number. For example, 2 multiples of 7, 35 and 21 make 56, another multiple of 7.
Maximum area of a parallelogram or triangle
If you are given two sides of a triangle or parallelogram, you can maximize the area by placing those two sides perpendicular to each other.
To find sector area
If you have the central angle, put it over 360 and then multiply by the area of the circle
Remainders
If you need a number that leaves a remainder of 4 after division by 7, first pick a multiple of 7. Then add 4 to get 18, which satisfies the rqurement.
Consecutive Integers are...
Integers that follow one after another from a given starting point, without skipping any integers. Consecutive Even integers are 8, 10, 12, 14
When should I convert to fractions?
Multiplication or Division
Can you ever split the terms of the denominator in a fraction?
No
Division of Odds and Evens
No guarantees. Odd divided by Odd or Even does not equal even. Odd divided by even will not be an integer. Even divided by even equals odd or even or non integer. Even divided by odd, equals even or non integer.
Prime Numbers
Numbers that have only 2 factors. 1 and Itself.
Addition and Subtraction of Odds and Evens
Odd and Odd equals even. Even and Even equals even. Odd and Even equals odd. Two different signs will equal an odd number. Two same signed numbers equal an even number.
Number Lines Strategy
On any number line, numbers get bigger as they move from left to right. You know the relative position of each point but you don't know the relative distance between points.
Units Digits Problem
Only pay attention to the units digit of the numbers you are working with. drop any other digits. Ex: 8²*9²* 3³ > 64, 81, 27 so you do 4*2*7= 28
Parallelogram
Opposite sides and angles are equal
Compound Interest
P (1 plus r over n) to the exponent nt where p = principal , r = rate in the forms of a decimal, n = number of times per year and t equals number of years.
How to find the y intercept
Plug in zero for x
Arc Length
Portion of circumference that remains
What can you use to simplify a root?
Prime factorization method. Find the prime factors that you can pair off (for example, two number 2's) and these can be brought out of the square root. For example, 75 is three times 25 and 25 is 5 times 5. The prime factors are 3 , 5 and 5, which becomes 5 radical 3.
Simple Interest
Principal *Rate*Time
Negative Exponents
Put the term containing the exponent into the denominator of a fraction and make the exponent positive. For example, 5 to the negative 1 is equal to one over 5.
What should be my first strategy when working with successive percents?
Real Numbers
Types of parallelograms
Rectangle
With an Unknown Exponent...
Rewrite the equation so you have the same base on both sides of the equal sign. For instance, 2 to the x power equals 8 is the same as 2 to the x = 2 to the 3rd. X is then known to be 3.
Area of a Circle
Space inside the circle. Pie r squared.
Consecutive Multiples
Special cases of evenly spaced sets: all of the values in the set are multiples of the increment. (ex: 12, 16, 20, 24). The values increase from one to the next by 4 and each element is a multiple of 4.
Consecutive integers
Special cases of multiples. All integers are multiples of 1 and all values increase by 1.
Exterior angle of a triangle equals
Sum of the two interior angles of a triangle
Volume
The amount of "stuff" a figure can hold
Triangle Side Formula
The sum of any 2 sides of a triangle are greater than the third side. The third side will always be greater than the difference between the two side lengths. Third side will be less than the sum, greater than the difference.
Isosceles right triangle corresponding angles and formula
The two legs are opposite the 45 degree angles each and the hypotenuse is opposite the 90 degree angle. Lengths of the legs are x and x. Hypotenuse length is x radical 2.
Diameter
Two radii laid end to end
