GRE Math 1 Flashcards
Prime Number
Only divisible by 1 and itself ; must be greater than zero
smallest prime number is 2
2, 3, 5, 7, 11, 13, 17, 19, 23,29
Zero
The only non-negative and non-positive integer
0 is an even number
is a multiple of every number
Integers
Whole numbers, negative numbers and Zero
Numbers we count with
Impossible to determine whether non-integers are even or odd
Digits
Math alphabet: they form numbers
2
the only even prime number
Factors
Small positive integer that divides evenly into a bigger number
GCF: The largest factor shared by a set of numbers
finite set of factors
Multiple
an integer that is the result of multiplying an integer by another integer …big numbers generated by multiplying smaller numbers
LCM: The smallest multiple shared by a set of numbers
infinite set of multiples
Remainder
always smaller than the number we are dividing by
Divisible by 2
if the number is even
if the units digits (the last number: 252…2) is even
Divisible by 3
if the sum of the digits is divisible by 3
Divisible by 4
if the 2-digit number formed by the last two digits is divisible by 4
Divisible by 5
if the last digit is 0 or 5
Divisible by 6
if the number is divisible by both 2 and 3
if and only if it is even and the sum of its digits is divisible by 3
Divisible by 9
if the sum of the digits is divisible by 9
Divisible by 10
if the last digit is 0
Average
Sum of values / number of values
Standard Deviation
A measure of how spread out a set of numbers is from the average. the larger the standard deviation, the more spread apart the numbers in a set
Prime Factorization
A number as expressed as the product of its prime factors
20 = 2 x 2 x 5
Converting a fraction into a decimal
divide the numerator by the denominator using long division
Converting a decimal to a fraction
numerator: move the decimal to the right until the number is an integer (whole)
denominator: count the number of places the decimal was moved, and then put the same number of 0s after a 1
ex) .45 = 45/100 = 9/20
ex) 6.75 = 6 and 3/4
Improper Fractions
Numerator is greater than the denominator
Mixed Fractions
Fractions that contain a whole number: 1 and 7/8
convert mixed fraction to manipulate them:
- multiply the whole number by the denominator
- add the numerator
- place total over denominator
ex) 1 and 7/8 = 15/8
Multiplying Fractions
Multiple numerators and denominators separately
Dividing Fractions
Multiply the first fraction by the reciprocal of the second fraction
the reciprocal of any number between 0 and 1, is greater than 1
Ratios
must be expressed in lowest, reduced terms
Relationships to each other ; values represent smallest possible quantity of each item… but you can have any multiple of each item
Real vs. Relationship Quantities - Relationship quantities do not dictate the actual quantity they are just a representation of the smallest possible quantity
Ration Formula:
(ratio)(x/total)
(ratio)(actual)
4/9 = x/36; x = 16
*if given only an actual number, the total of the ratio (a:b…a+b) must be a factor of the actual number ex) pg. 13
lab has 55 rabbits what could be the ratio of white rabbits to brown rabbits? 3:8 (8+3) bc 11 is a factor of 55
- when dealing with ratios, the total number needs to be divisible by the sum of the pieces of the ratio
when dealing with fractions, always try to clear the fractions:
- if you are given one fraction and a whole number multiply by the denominator to clear the fraction
- if you are given more than one fraction… do the division problem and then reduce … put colon between final fraction: 16/3 becomes 16:3
Percents
practice converting precent to fraction
type of ratio in which the total is always 100 … always try to get denominator of fraction to 100 o figure out what to multiple the numerator by
part/whole –> part/100
ex) 18 is what percent of 3?
what percent of means (x/100)
18 = (x/100)(3)
x=600
Percents to Fractions & Vice Versa:
*to convert a percent to a fraction divide the percent by 100% ex) 1/4% = (1/4)/100 = (1/4)(1/100) = 1/400 *
- to convert a fraction to a percent multiply it by 100% ex)
(12/5)(100%)=(12)(20%)=240%
*review photo in phone of common percent equivalents
Decimals to Percents and Vice Versa:
- to convert decimal to percent multiply it by 100% ex) (.002)(100%)= 0.2%*
- use the rules for multiplying by a power of 10… move the decimal point two places to the right and insert a percent sign
- to convert a percent to its decimal equivalent use the rules for dividing by powers of 10… move the decimal point two places to the left… what is the decimal equivalent of 32%? .32
if the denominator is 25, when trying to convert a fraction to a percent, you only have the multiply both the numerator and denominator by 4 in order to get the fraction to be over 100, which will give you the percent ex)
(7/25)(100%)= (7)(4%)= 28%
Percent Formula:
- part/whole(100%)
Percent Increase Formula:
- increase(100%)/original
Percent Decrease Formula:
- decrease(100%)/original
- the original is the base from which the change occurs…it may or may not be the first number mentioned in the problem*
ex) Two years ago, 450 seniors graduated from Inman High School. Last year 600 seniors graduated. By what percentage did the number of graduating seniors increase? - the original is the figure from the earlier time: 450
- the increase is 600 - 450, or 150. So the percent increase is: 150(100%)/450 = 33 and 1/3 %
Average and Statistics
AVG = Sum of Values / Number of Values
or…
AVG as the balance: difference between each term and the avg to determine what the unknown value needs to be in order to balance the other side of the avg
ex) avg of 3,4,5 and x is 5 … what is x?
diff bw 3&5 is 2; diff bw 4 and 5 is 1; diff bw 5 and 5 is 0… thus x must be 5+2+1+0 to balance the avg… x = 8
test question
the avg of six numbers is 6. If 3 is subtracted from each of four of the numbers, what is the new avg?
x = sum of the 6 numbers
x/6=6 (multiply both sides by 6 to solve for x) x = 36
but…
now we know that 3 is subtracted from 4 of those numbers so we multiply 3(4) and subtract that from 36… 24 becomes the new sum
but…
now we have to find the new avg. so we do 24/6 and find that the new average is 4.
Avg of consecutive, EVENLY spaced numbers:
when consecutive numbers are evenly spaced, the avg will be the middle value: ex) 6,7,8…7 is the avg
if there is an even number, there won’t be on middle value. so to find the avg add the values and then divide by 2
Combining averages: iff there is an equal number of terms in each set, you can combine
ex) suppose there are two bowlers and you must find their average score per game.. one has an avg of 100 and the other 200. Iff they bowled the same number of games, you can find their combined average…they both bowled 4 games so plug that into the formula and get:
4(100) + 4(200)/8 = 150
Exponents
count the number of times something is multiplied by itself
Raising a positive fraction less than 1 to a positive exponent greater than 1 results in a smaller value… The higher the exponent the smaller the result.
When a negative number is raised to an even = the result is positive
When a negative number is raised to an odd exponent, the result is negative
Any number raised to the 0 power = 1
- An important rule to keep in mind is that we can only multiply and divide powers if the base numbers are the same. If our base numbers are different, then we cannot add, subtract, multiply, or divide exponents.*
- You cannot combine exponents when adding or subtracting the bases!*
To add or subtract with powers, both the variables and the exponents of the variables must be the same. You perform the required operations on the coefficients, leaving the variable and exponent as they are:
2x^2 + 3x^2 = 5x^2
8y^3 - 5y^3 = 3y^3
In the testmaker’s favorite exponent trap you’ll see a variable squared. The square of a number is positive, so the student assumes that the number is positive, when it could be positive or negative. Keep in mind the rules from the previous slide.
Don’t forget: If x^2 = 4, that means that x could be 2 or - 2.
Radicals
follow the same rules as exponents
Note: The square root symbol refers to the positive root only. In other words:
If x2 = 25, then x equals 5 or - 5. But root 25 equals 5 only.
In order to multiply or divide numbers with exponents, all exponents needed to have the same base. The same is true for operations involving roots, as evidenced by the following rule and examples.
Roots Tip #3: Square roots with the same base can be added or subtracted: 4 root 3 + 7 root 3 = 11 root 3
Roots Tip #6: When taking the square root of a fraction between 0 and 1, the result will always be larger than the fraction itself: the square root of 1/4 is 1/2 which is greater than 1/4… just as a fraction is always larger than its square.
square root of 169 is 13
when dealing with radicals in QC, use both the positive & negative:
ex) radical (x^sqaured+39) = 8
so we need to figure out what plus 39 will give 64. 25 will.
so x = pos 5 & neg 5…test both when deciding the QC answer bc they could result in D being the answer
*refer to picture in phone
Linear Equations - Isolating a Variable
use PEMDAS to isolate variables
- Eliminate any fractions by multiplying both sides: (multiply by the LCM to eliminate fractions)
(you can also cross multiply if the fractions are equal ex -
12+b/3 = b+10/3… 6(12+b) = 3(b+10) - Put all terms with the variable you are solving for on one side by adding or subtracting on both sides
- Combine like terms
- Factor out the desired variable
- Divide to leave the desired variable by itself
Systems of Linear Equations
To solve for all of the variables in a system of equations, we must have at least as many distinct linear equations n as we have distinct variables n
There are two ways to solve systems:
- Substitution - solve one equation for one of the variables, and substitute that variable into the other equation (easy when one of the equations is solved for)
- Combination - add or subtract one equation from another equation to cancel out one of the variables
(create equivalent fractions with LCM so that one of the variables cancels)
(use when one of the variables will easily drop out)
Quadratic Equations
refer to picture in phone
polynomials & quadratics:
- the FOIL method
- difference of two squares:
a^squared - b^sqaured can be factored into (a+b)(a-b)
9x^squared - 1 = (3x+1)(3x-1)
- factoring polynomials of the form a^sqaured+2ab+b^squared:
a & b are perfect squares and the middle is the product of 2 times the perfect squares
the sign in front of the 2ab term will determine whether the sign is positive or negative… (a+b)^squared or (a-b)^squared:
ex) x^squared + 6x + 9 = (x+3)^squared
Solving quadratic equations:
- factor then set each binomial equal to zero & solve for each variable :
(x-1)(x-2) = 0
x-1=0 or x-2=0… x = 1 or 2 … plug both values back into the unfactored equation and see if the equation works
Inequalities
Greater than: >
Less than: <
dash under implies or equal to
Should be treated like equations with two exceptions:
1. when we multiply or divide an inequality by a negative number, we must reverse the direction of the inequality sign
- Single-variable equations are usually solved for a specific value, whereas inequalities can only be solved for a range of values
Pythagorean Triplets (side ratios)
3: 4:5
5: 12:13
7: 24:25
8: 15:17
9: 40:41
also appear in multiples
Isosceles Right: (45 degrees, 45 degrees, 90 degrees)
x:x:x(root 2)
30, 60, 90 right triangle:
x:x(root 3):2x
a^2+b^2=c^2
ex) what is the length of the hypotenuse of an isosceles right triangle of area 32?
area of an isosceles right triangle is v^2/2
v^2/2 = 32
v^2= 64
v=8…the hypotenuse is 8(root)2
Word Problems
In order to minimize the chances that you’ll overlook something, you should follow these three steps every time you encounter a word problem:
Determine the math concept being tested
Determine what specific piece of information the question is asking you to find & label the variables or unknowns in a way that is easy to remember what they stand for
Look for equations and/or other relationships in the language of the text…translate the problem into one or more equations sentence by sentence
refer to screenshot on desktop
Associative Laws of Addition and Multiplication
Addition & multiplication are associative: regrouping the numbers does not affect the result:
(3+5)+8 = 3+(5+8) 8+8 = 3+13 16 = 16
(a+b)+c = a(b+c)
(ab)c = a(bc)
Lines and Angles
The sum of angles around a point is 360 degrees; straight line is equivalent to an angle of 180 degrees
Parallel lines never intersect, when they are cut by a transversal, corresponding angles are equal
complementary angles: their angles sum to 90
acute angles: measure less than 90
obtuse angles: measure bw 90 and 180
supplementary: their measure sums to 180. adjacent angles are supplementary bc they lie along a straight line
angle bisectors: a line or line segment bisects an angle if it splits the angle into two smaller, equal angles
Triangles
Sum of interior angles is 180 degrees
p = sum of all sides; a = 1/2(b)(h), where b & h must form a right angle
Triangle inequality theorem: third side must be greater than the difference but smaller than the sum of the other two sides… *gives the relationship that sides of a triangle have to each other
isosceles: two sides and two angles are equal
equilateral: three sides are equal and all angles equal 60 degrees
right: have one 90 degree angle
interior and exterior angles:
the sum of interior angles is 180
an exterior angle is equal to the sum of the remote interior angles…three exterior angles of any triangle add up to 360
QC and triangles:
- if you know two angles, you know the third
- to find the area, you need the base and the height
- in a right triangle, if you have two sides, you can find the third. and if you have two sides you can find the area
- in isosceles right triangles and 30/60/90 triangles, if you know one side you can find everything
when a triangle is inscribed in a circle in such a way that one side of the triangle coincides with the diameter of the semicircle, the triangle is a right triangle …may be the hypotenuse of the right triangle
Circles
area = pie(radius squared)
diameter = 2r
circumference = 2(pie)r or pie(d)
pie = the ratio of the circumference of any circle to its diameter…c/d
area of a sector: as a fraction of the circles area is equal to the degree measure of the corresponding central angle as a fraction of 360…
area of sector/area = x(is the central angle)/360
arc length: as a fraction of circle’s circumference is equal to the degree measure of the corresponding central angle as a fraction of 360…
length of arc/circumference = x(is the central angle)/360
A line that has exactly one point in common with a circle is tangent to the circle, and it is perpendicular to the radius
If a triangle is inscribed in a circle so that one of its sides is a diameter of the circle, then the triangle is a right triangle
central angle: an angle formed by two radii
chord: a line segment that joins two points on a circle. the diameter is the longest chord of a circle; half of the area of the circle is behind the diameter
tangent: a line that touches only one point on the circumference of a circle…a line drawn tangent to a circle is perpendicular to the radius at the point of tangency
Polygons
closed plane figure formed by 3 or more line segments
Sum of interior angles of polygon with n sides: (n-2)180
- you can also figure out the interior angles by dividing the polygon into triangles…draw a diagonal from any vertex to all nonadjacent vertices…then multiply the number of triangles by 180
quadrilaterals: any 4 sided figure … two triangles put together (special right triangles, pythagorean triplets are important)
parallelogram: quadrilateral whose opposite sides are parallel - including rectangles, squares
- opposite sides and angles are equal
- area = (b)(h)
- the diagonals bisect each other
- rectangle: diagonals are equal, but are not perpendicular; perimeter = 2(l+w)
- square: diagonals are equal & perpendicular so you can use the 45,45,90 rule to find the length of the diagonal
QC w parallelograms:
- in a parallelogram, if you know two adjacent sides, you know all of them; and if you know two adjacent angles you know all of them
- in a rectangle if you know two adjacent sides, you know the area
- in a square if you are given virtually any measurement (area, length of a side, length of a diagonal) you can figure out the other measurements
a polygon is inscribed in a circle if all the vertices of the polygon lie on the circle.
a polygon is circumscribed about (drawn around) a circle if all of the sides of the polygon are tangent to the circle
Solids
Uniform solids: solids in which the measure of each dimension is constant through the entire object
volume of uniform solid: (area of base)(height)
volume of cylinder - (pie)(r squared)(height)
volume of a rectangular solid - (l)(w)(h)
volume of a cube - edge cubed ; SA of cube - 6(edge squared)…there are 12 edges in a cube
SA - sum of the area of all of the faces of the solid (top,ends, sides)
SA rectangular solid - 2lw+2lh+2hw
Lateral SA of a cylinder - (circumference of height)(base) = (2)(pie)(r)(h)
Total SA of a cylinder - areas of circular ends + lateral surface area = (2)(pie)(r-squared) + (2)(pie)(r)(h)
lateral (pipe) surface of a cylinder: unrolls into a rectangle with a length equal to the circumference of the circular base and the height of equal to that of the cylinder
Coordinate Geometry
Slope of line - change in y / change in x
equation of a straight line in the two dimensional coordinate plane is typically expressed as y = mx+b ; b = y-intercept ; m is the slop
to find the x-int, set y=0 & solve for x
to find the y-int, set x=0 & solve for y
if line is parallel to y-axis slope is undefined … the change in x = 0… therefore has the equation x=a, where a is the x-intercept
lines that are parallel to the x-axis have a slope of 0 and therefore have the equation y=b (y-intercept)
distance in the coordinate plane - pythagorean theorem : in order to find the distance between two points on the coordinate plane you have to draw right triangle…if the line segment is not parallel to either of the axes, it becomes the hypotenuse of a right triangle
to find the length of a line segment parallel to one of the axes, all you have to do is find the difference bw the end point coordinates that do change:
- parallel to x use the x coordinates
- parallel to y use the y coordinates
Comparing Positive Fractions
If neither the numerator nor the denominator are the same, you have three options:
- turn both fractions into their decimal equivalents
- express both fractions in terms of a common denominator and then see which new equivalent fraction has the largest numerator
- cross multiply the numerator of each fraction by the denominator of the other… the greater result will wind up next to the greater fraction
- ex) which is greater 5/6 or 7/9?
59= 45 ; 67=42… since 45 is bigger 5/6 is the larger of the two
Properties of Fractions bw -1 and 1
- the reciprocal of a fraction bw 0 and 1 is greater than both the original fraction and 1 :
ex) the reciprocal of 2/3 is 3/2, which is greater than both 1 and 2/3 - the reciprocal of a fraction bw -1 and 0 is less than both the original fraction and -1
ex) the reciprocal of -2/3 is -3/2, which is less than both -1 and -2/3
Speed
relationship bw speed, distance & time:
speed = quantity / time
this can also be written as ….
time = distance/speed distance = (speed)(time)
in general for rate problems speed equations works with rate & quantity as substitutes:
rate = quantity/time
ex) The shortest distance between the starting point and the ending point of a trip if a vehicle is driven 60 miles north and 80 miles east … this means draw a right triangle with 60 as the height since it is mentioned first and 80 as the base… the hypotenuse is the actual answer
Combined Work Problems (given hours per unit of work)
question type: how long it takes a number of people working together to complete a task
two people or two machines:
T= ab/a+b
a = amount of time it takes person a to complete the task
b = amount of time it takes person b to complete the task
more than two people:
1/a + 1/b + 1/c = 1/T
Solving for one unknown in terms of another
in order to solve for the NUMERICAL value of an unknown, you need as many distinct equations as you have variables … if there are two variables you need two distinct equations
Implications for QC: to have enough information to compare the two quantities, you must have at least as many equations as you have variables
- make sure you REALLY have two distinct equations:
ex) 2x+3y = 8 and 4x+6y=16 are the same equations in different forms*
when asked to solve for one variable in terms of the other, isolate the desired variable on one side of the equation and move all of the constants and other variables to the other side
Sequences
list of numbers. the value of a number in a sequence is related to its position in the list…
the subscript part of each number will give the position of the each element in the series… most questions give a formula that defines each element
Combination
question asks how many unordered subgroups can be formed from a larger group
n!/k!(n-k)!
n = number of items in the group as a whole
k = the number of items in each subgroup formed
! = factorial … the product of an integer & all of the integers below it… descending order down to 1… 4! = 24…(4)(3)(2)(1)
when the same numbers are in the numerator and denominator you can cancel them in order to reduce the terms
when asked to find the potential combinations from
multiple groups:
1. find the # of possible combinations in each group and then multiply them to get the actual answer (pg.213, premier)
ex) how many groups can be formed consisting of 2 people from room A and 3 people from room B if there are 5 people in room A and 6 people in room B
- use the formula for room A: the number of combinations of 2 in a set of 5
- use the formula for room B: the number of combinations of 3 in a set of 6
- multiply the results
when asked to find the number of possible subgroups when choosing from one item from a set … the number of possible subgroups will always equal the number of items on the set:
ex) Restaurant A has 5 appetizers, 20 main courses, and 4 desserts. If a meal consists of 1 appetizer, 1 main course, and 1 dessert, how many different meals can be ordered at Restaurant A?
- the number of possible outcomes from each set is the number of items in the set: there are 5 appetizers, 20 main courses, 4 possible desserts …(5)(4)(20)= 400 different meals
Permutation
permutations differ from combinations in that permutations are ordered… by definition each combination larger than 1 has multiple permutations.
“how many ways/arrangements/orders/schedules are possible? indicates a permutation question
Probability
the numerical representation of the likelihood of an event or combination of events…expressed as a ratio of the number of desired outcomes to the total number of possible outcomes
probability of any event occurring cannot exceed 1 (probability of 1 represents a 100% chance of an event occurring), and it cannot be less than 0 (a probability of 0 represents a 0% chance of an event occurring)
see examples on phone of pg.215
probability of dependent events: when the probability of a later event occurring varies according to the results of an earlier event…see example on phone of page 216
11^2
121
13^2
169
14^2
196
15^2
225
2^3
2^4
2^5
2^6
8
16
32
64
3^3
3^4
3^5
27
81
243
4^3
4^4
64
256
5^3
5^4
125
625
Divisible by 12
Sum of digits is a multiple of 3 and last two digits is a multiple of 4
Number prop
pos, neg, zero, one , fraction
Quant comparison
- add/subtract same value
- multiply/divide same POS value
- square both columns to remove radical
- cross multiply to compare fractions
Ex Quadratic factors
x^2 - 9 = (x+3)(x-3)
x^2+6x+9 = (x+3)(x+3) = (x+3)^2
x^2-6x+9= (x-3)(x-3) = (x-3)^2
x^2+2x(radical 7)+7 = (x+radical 7)(x+radical 7)
x^2+x+1/4 = (x+1/2)(x+1/2)
Parallelogram
parallelogram: quadrilateral whose opposite sides are parallel - including rectangles, squares, rhombus
- opposite sides and angles are equal
- area = (b)(h)
- the diagonals divide into two identical triangles
QC w parallelograms:
- in a parallelogram, if you know two adjacent sides, you know all of them; and if you know two adjacent angles you know all of them
Rectangle
- all angles are 90 degrees
- opposite sides are equal
- diagonals are equal in length and bisect each other , but are not perpendicular;
- perimeter = 2(l+w)
- type of parallelogram
QC
- in a rectangle if you know two adjacent sides, you know the area
Square
- all angles are 90 degrees
- all sides are equal
- diagonals are equal in length, bisect each other and meet at right angles; diagonals divide square into two right angles of equal size (they are perpendicular so you can use the 45,45,90 rule to find the length of the diagonal)
- area = side squared or diagonal squared/2
- type of rectangle
QC
- in a square if you are given virtually any measurement (area, length of a side, length of a diagonal) you can figure out the other measurements
Trapezoid
- One pair of opposite sides is parallel.
- Isosceles trapezoid has equal leg lengths
- Area = base 1 + base 2 / 2 * height
(average of the bases * height)
Rhombus
- opposite sides are parallel
- opposite angles are equal
- all sides are equal
- diagonals bisect one another and meet at right angles
- Area= diagonal 1 * diagonal 2 / 2
A number to the zero power
equals 1
a^0 = 1
Isosceles Triangle
two equal sides and two equal angles
A square can turn into two isosceles right triangles
45-45-90
Equilateral Triangle
can be split into 30-60-90 triangle
Area = root 3 / 4 * side squared
Height = 1/2 (side) * root 3
45 degrees from the X- axis
positive slope line equation is y = x and the slope is 1 so a point on the line would be (1,1)
negative slope line equation is y = -x and the slope is -1 so a point on the line would be (-1,1)
Regular Shape
means all of the angles are equal and all of the sides are equal
Kilo
Thousand
Deca
10
Deci
Tenth
Centi
Hundredth
Milli
Thousandth
Feet/Yard
3 Feet/Yard
Factors
Can also be called divisors
Composite Number
A positive integer that is not prime
“Non-negative and Non-positive”
include zero because it means positive or zero
Magnitude
measure of how far from zero
Counting Integers
(Last - First) + 1
Ex) How many integers are there from 14 to 765, inclusive?
765 - 14 + 1 = 752
Counting Consecutive Multiples
(Last - First) / Increment + 1
“How many multiples of 4?”
“How many even numbers?”
Ex) How many multiples of 7 are there between 100 and 150?
105 is the first number and 147 is the last number (not 100 and 150 because those are not multiples of 7)
147 - 105 = 42
42 / 7 = 6
6 + 1 = 7
