Quantitative GRE Flashcards
factor / divisor
An integer that is multiplied together with another integer. For example: 2, 3, and 10 are factors or divisors of 60.
multiple
An integer that is produced when other integers, or factors, are multiplied together. For example: 60 is a multiple of the factors 2, 3, and 10.
divisible
We say that an integer is divisible by each of its divisors. For example: 60 is divisible by the divisors 2, 3, and 10.
Every nonzero integer has how many multiples?
Infinitely many
least common multiple
The least common multiple of two nonzero integers a and b is the least positive integer that is a multiple of a and b. For example: the least common multiple of 30 and 75 is 150.
greatest common divisor / greatest common factor
The greatest common divisor of two nonzero integers a and b is the greatest positive integer that is a divisor of both a and b. For example: the greatest common divisor of 30 and 75 is 15.
quotient and remainder
If a is divided by b, but b is not a divisor of a, then the result can be viewed as a fraction, decimal, or a remainder and quotient. Both the quotient and remainder are integers.
Find the greatest multiple of b that is less than or equal to a. That multiple can be represented as qb, where q is the quotient. Then the remainder r = a - qb.
For example: If 19 (a) is divided by 7 (b), the largest multiple of 7 that is less than 19 is 14. The quotient then is 2 and the remainder is 19 - 14 = 5. So we say the result is “2 remainder 5”.
the sum of two even integers is
an even integer
the sum of two odd integers is
an even integer
the sum of an odd and even integer is
an odd integer
the product of two even integers is
an even integer
the product of two odd integers is
an odd integer
the product of an even integer and an odd integer is
an even integer
prime number
an integer greater than 1 that has only two divisors: 1 and itself. Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29
What is the only even prime number?
2
Is 1 a prime number?
No
prime factorization
Every integer greater than 1 is either a prime number or can be uniquely expressed as a product of factors that are prime numbers, or prime divisors. This expression is called prime factorization.
For example: 81 = (3)(3)(3)(3) or 3^4
composite number
Any integer greater than 1 that is not a prime number
how to divide one fraction by another fraction
multiply one fraction by the reciprocal of the other
a negative number raised to an even power is always
positive
a negative number raised to an odd power is always
negative
For all nonzero numbers a, a^0 =
1
For all nonzero numbers a, a^(-x) =
1/(a^x)
0^0 =
undefined
All nonnegative, nonzero numbers have how many square roots?
2 - one positive and one negative
The square root of a negative number is
not defined in the real number system
The square root of zero is
0
√a√b =
√(ab)
(√a) / (√b) =
√(a/b)
A square root is a root of order
2
A cube root is a root of order
3
For odd-order roots, there are exactly how many roots for every number n?
1, regardless of whether n is positive or negative
For even-order roots, there are exactly how many roots for every number n?
2 if n is positive and 0 if n is negative. For example: -8 has no fourth root but 8 has two fourth roots (2 and -2)
Symbol for a repeating decimal
bar over the numbers that repeat
irrational number
a decimal that does not terminate and does not repeat
real numbers
the set of all rational numbers and all irrational numbers, including all integers, fractions, and decimals, which can be represented on the real number line
the triangle inequality
|a+b| is less than or equal to |a| + |b|
percent increase or decrease
amount of increase or decrease / base, where the base is the starting number
to find total percent change of two or more changes
(first percent change) x (second percent change) = total percent change
For example: first there is an 8% decrease, followed by a 6% increase. Total percent change = (.92) x (1.06) = .9752, or 97.52%, or a 2.48% decrease
like terms
terms that have the same variables. For example x and 3x, or y and 5y, or x^2 and 4x^2
x and x^2 are not like terms
when can you factor out
when a number or a variable is a factor of each term in an algebraic expression
x^(-a) =
1 / x^a
(x^a)(x^b) =
x^(a+b), if base is the same and you multiply, add the exponents
(x^a) / (x^b) =
x^(a-b) = 1 / x^(b-a), if base is the same and you divide, subtract the exponents
(x^a)(y^a) =
(xy)^a, if bases are different but exponent is the same, multiply the bases
(x/y)^a =
(x^a) / (y^a)
(x^a)^b =
x^(ab)
(x^a)(y^b) is not equal to
(xy)^(a+b) because bases are not the same
(x^a)^b is not equal to
(x^a)(x^b)
(x+y)^a is not equal to
x^a + y^a, can’t just distribute the exponent if the bases are different and added
(-x)^2 is not equal to
-(x^2), carefully note where the minus sign appears
√(x^2 + y^2) is not equal to
x+y
a/(x+y) is not equal to
a/x + a/y, can’t distribute the denominator
(x+y)/a =
x/a + y/a, can distribute the numerator
substitution method for solving a linear equation in two variables
manipulate one equation to express one variable in terms of the other, then substitute it into the other equation
elimination method for solving a linear equation in two variables
make the coefficients of one variable the same in both equations so that variable can be eliminated by adding or subtracting the equations to/from each other
quadratic equation
ax^2 + bx + c = 0
quadratic formula
to solve a quadratic equation:
x = (-b +/- √(b^2 - 4ac)) / 2a
another way to solve some quadratic equations
by factoring
when both sides of an inequality are divided by the same nonzero constant,
the direction of the inequality is preserved if the constant is positive but reversed if the constant is negative. The new inequality is equivalent to the original.
where is the function f(x) = 2x / (x-6) undefined?
Where x = 6, because the function would be equal to 12/0
simple interest formula
V = P(1 + rt/100), where P is the principal, r is the simple annual interest rate, t is the number of years, and V is the value after t years
compound interest formula
V = P(1 + r/(100n)) ^ (nt), where r is the annual interest rate, n is the number of times per year it is compounded, and t is the number of years
how to compute a fourth root
take the square root twice = √√x
equation for a straight line
y = mx + b, where m is the slope and b is the y intercept
slope of a line passing through (x1,y1) and (x2,y2) =
(y2 - y1) / (x2 - x1), also called rise over run
slope of a vertical line
not defined, since run is 0
slopes of two parallel lines are
equal
slopes of two perpendicular lines are
negative reciprocals of each other
ie y = 2x + 5 is perpendicular to y = -1/2x + 9
reflections across the line y = x
y = x passes through the origin and has a slope of 1. For any point with coordinates (a,b), the point (b,a) is a reflection about the line y = x, or we say these points are symmetric about the line y = x.
Similarly, interchanging the variables y and x in the equation of any graph yields another graph that is symmetric about y = x.
equation of a parabola
ax^2 + bx + c = 0, if a is positive, the parabola opens upward, and if a is negative, it opens downward
x-intercepts of a parabola
can be found by solving the quadratic equation ax^2 + bx + c = 0, using the quadratic formula if needed
equation of a circle
(x-a)^2 + (y-b)^2 = r^2, where the center is at point (a,b) and radius is r
piecewise defined function for h(x) = |x|
h(x) = {x, x>= 0 and -x, x
graph of √x is
half of a parabola lying on its side, where the other half would be the function -√x
the graph of h(x) + c is the graph of h(x) shifted
upward by c units
the graph of h(x) - c is the graph of h(x) shifted
downward by c units
the graph of h(x + c) is the graph of h(x) shifted
to the left by c units
the graph of h(x - c) is the graph of h(x) shifted
to the right by c units
the graph of ch(x) when c > 1 is the graph of h(x)
stretched vertically by a factor of c
the graph of ch(x) when 0
shrunk vertically by a factor of c
congruent line segments
line segments of equal length
opposite angles
also called vertical angles, the angles across from each other when two lines intersect. The angles have the same measure
acute angle
less than 90 degrees
obtuse angle
greater than 90 degrees but less than 180 degrees
convex polygon
a polygon in which the measure of each interior angle is less than 180 degrees
a polygon with n sides can be divided into how many triangles?
n - 2
the sum of the measures of the interior angles of an n-sided polygon is
(n-2)(180) degrees
regular polygon
a polygon in which all sides are congruent and all interior angles are congruent
equilateral triangle
a triangle with three congruent sides, where all interior angles are congruent and measure 60 degrees
isosceles triangle
a triangle with at least two congruent sides, where the angles opposite those sides are also congruent
hypotenuse
in a right triangle, the side opposite the right angle, which is also the longest side. The other two sides are the legs
Pythagorean theorem
if c is the length of the hypotenuse of a RIGHT triangle and a and b are the lengths of the legs, then a^2 + b^2 = c^2
45, 45, 90 triangle
In a 45, 45, 90 right triangle, the legs measure x and the hypotenuse measures x√2
30, 60, 90 triangle
In a 30, 60, 90 right triangle, the side across from the 30 degree angle measures x, the side across from the 60 degree angle measures x√3, and the side across from the hypotenuse measures 2x
area of a triangle
(base x height) / 2, where the height is the line segment from the opposite vertex to the base, or an extension of the base
two triangles are congruent if
they have the same size and shape. More specifically:
- If the three sides of one triangle are congruent to the three sides of another
- If two sides and an included angle of one triangle are congruent to two sides and the included angle of another triangle
- If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle
similar triangles
two triangles that have the same shape but not necessarily the same size. This is true if the corresponding angles are congruent or the corresponding sides all have the same ratio of lengths (scale factor of similarity)
parallelogram
quadrilateral where both pairs of opposite sides are parallel. In a parallelogram, opposite sides are congruent and opposite angles are congruent
trapezoid
a quadrilateral in which two opposite sides are parallel
area of a parallelogram
base x height
area of a trapezoid
(1/2) x (b1 + b2) x height, or the average of the lengths of the bases x the height
chord
any line segment joining two points on a circle. the diameter is any chord that passes through the center of the circle
circumference of a circle
C = πd, where d is the diameter
also C = 2πr, where r is the radius
measure of an arc of a circle
is equal to the measure of its central angle
length of an arc
the ratio of the length of an arc to the circumference of the circle is the same as the ratio of the arc’s central angle to 360 degrees
area of a circle
π(r^2)
area of a sector of a circle
a sector of a circle is a region bound by an arc of the circle and two radii.
the ratio of the area of a sector of a circle to the entire circle is equal to the ratio of the degree measure of its arc to 360 degrees
if a line is tangent to a circle, then
a radius drawn to the point of tangency is perpendicular to the tangent line
a polygon is inscribed in a circle if
all its vertices lie on the circle, you can also say the circle is circumscribed about the polygon
if one side of an inscribed triangle is a diameter of a circle, then
the triangle is a right triangle, with the diameter being the hypotenuse
a polygon is circumscribed about a circle if
each side of the polygon is tangent to the circle, you can also say the circle is inscribed in the polygon
concentric circles
two or more circles with the same center
volume of a rectangular solid
length x width x height
surface area of a rectangular solid
sum of the areas of its six faces
2(lw + lh + wh)
right circular cylinder
a circular cylinder whose axis is perpendicular to its bases
volume of a right circular cylinder
π(r^2) x height
surface area of a right circular cylinder
2(πr^2) + 2πrh
frequency distribution
a table or graph that presents the categories or numerical values along with their associated frequencies
relative frequency
the frequency of a category or numerical value divided by the total number of data
the sum of the relative frequencies in a relative frequency distribution is
always 1
in a circle graph, the measure of the central angle of a sector is
the proportional to the percent of the 360 degrees that the sector represents
in a histogram, there are no
regular spaces between the bars. If there are spaces, this represents no data in those intervals
statistical measures are often grouped into these three categories:
measures of central tendency, measures of position, measures of dispersion
measures of central tendency
mean, medium, mode
median
if n is odd, the medium is the middle number in the ordered list of numbers. If n is even, the median is the average of the two middle numbers
quartiles
there are three quartiles numbers (Q1, Q2, Q3) that divide a set of data into four roughly equal groups
In all cases, Q2 = M, the median
These are L(lowest value) to Q1, Q1 to M, M to Q3, and Q3 to G(greatest value)
Q1 is the median of the values less than M, and Q3 is the median of the values greater than M
percentiles
there are 99 percentile numbers that divide a set of data into 100 roughly equal groups
range
the difference between the greatest number and the least number in a group (G-L)
interquartile range
Q3 - Q1
box and whisker plot
marked with lines from L to Q1, then a box from Q1 to Q3 with a line at M, then lines from Q3 to G
standard deviation takes into account
how much each value differs from its mean and then takes an average of these differences. The more the data are spread away from the mean, the greater the standard deviation, and the more clustered around the mean, the lesser the standard deviation
the standard deviation for n values is calculated by
(1) calculating the mean of the n values, (2) finding the difference between the mean and each of the n values, (3) squaring each of the differences, (4) finding the average of the n squared differences, and (5) taking the nonnegative square root of the average squared difference
sample standard deviation
slightly different from the standard deviation in that it is computed by dividing the sum of the squared differences by (n-1) instead of n
how to find how many standard deviations a number x is above the mean
take the absolute value of x - mean, then divide by the standard deviation
standardization
the process of subtracting the mean from each value and then dividing the result by the standard deviation. This is useful because it provides for each data value a measure of its position relative to the rest of the data, independently of the variable for which the data were collected and the units of the variable.
In any group of data, most of the data are within about how many standard deviations from the mean?
3
when data are standardized, the mean is transformed to
0
members or elements
the objects in a set
finite set
a set whose members can be completely counted
infinite sets
sets that are not finite
empty set
set with no members, denoted by ∅
subset
if A and B are sets and all members of A are also members of B, then A is a subset of B
∅ is a subset of
every set, by convention
list
like a finite set, but the members are ordered, so rearranging the list makes it a different list. Furthermore, members can be repeated and repetitions matter.
For example, {1,2,3} and {2,3,1} are different lists, as are {3,2,3} and {2,3}
in a set, are repetitions counted?
No. The sets {1,2,3,2} and {3,1,2} are the same
For any finite set S, the number of elements of S is denoted by
|S|
intersection of S and T is denoted by
S ∩ T
union of S and T is denoted by
S ∪ T
disjoint or mutually exclusive
when two sets have no elements in common
universal set
a rectangle sometimes drawn outside of Venn diagrams to denote a set of which all other sets shown are subsets
inclusion-exclusion principle
the number of elements in the union of two sets equals the sum of their individual numbers of elements minus the number of elements in their intersection
|A ∪ B| = |A| + |B| - |A∩B|
multiplication principle
if there are two choices to be made sequentially and the second choice is independent of the first choice, and there are k possibilities for the first choice and m possibilities for the second choice, then there are km possibilities for the pair of choices
number of permutations of n objects
found by n factorial, or n!
For example, if there are 4 different objects, then there are 4! = 4(3)(2)(1) = 24 permutations
0!
= 1
permutation vs. combination
With permutations, the order DOES matter, for example ABD is different from ADB.
With combinations, the order DOES NOT matter, for example ABD is the same as ADB
suppose that k objects will be selected from n objects and ordered, where k is less than or equal to n, how many ways are there to select and order k objects out of n objects?
this is the permutation of n objects taken k at a time
n! / (n-k)!
suppose k objects will be chose from a set of n objects, where k is less than or equal to n, but the objects will NOT be put in order, how many ways are there to do this?
this is the combination of n objects taken k at a time
n! / (k!(n-k)!)
also called n choose k, or nCk, or
(n)
(k)
n choose k is always equal to
n choose n-k
sample space
the set of all possible outcomes of a random experiment
event
any particular set of outcomes of a random experiment
If an event E is certain to occur, then P(E) =
1
If an event E is certain not to occur, then P(E) =
0
If an event E is possible but not certain to occur, then
0
If E is an event, then the probability of E is
the sum of the probabilities of the outcomes in E
The sum of the probabilities of all possible outcomes of an experiment is
1
The event that both E and F occur is written as
E ∩ F
The event that E or F, or both, occur is written as
E ∪ F
mutually exclusive events
are those that cannot occur at the same time
P(E or F or both) =
P(E) + P(F) - P(both E and F, or E ∩ F)
If E and F are mutually exclusive, then P(E or F or both) =
P(E) + P(F) since P(both E and F) = 0
independent events
when the occurrence of one event does not affect the occurrence of the other
P(both E and F) when E and F are independent =
P(E) x P(F)
If P(E) does not equal 0 and P(F) does not equal 0, then can E and F be both independent and mutually exclusive?
No, since if E and F are independent, P(both E and F) = P(E)P(F) which does not equal 0, and if mutually exclusive, then P(both E and F) = 0
If two events that happen sequentially are not independent, then
the probability that both events happened is the probability that the first event happened multiplied by the probability that given the first event has already happened, the second event happens as well
mean and median as halving vs balance point
median is the halving point, mean is the balance point
For a random variable that represents a randomly chosen value from a distribution of data, the probability distribution of the random variable is the same as
the relative frequency distribution of the data
expected value
another name for the mean of a random variable X, which is the sum of each value of X multiplied by its corresponding probability P(X), or the sum of XP(X)
In a histogram for a random variable, the area of each bar is proportional to
the probability represented by the bar
In approximately normally distributed data, the following properties are true
- The mean, median, and mode are all nearly equal
- The data are grouped fairly symmetrically about the mean
- About 2/3 of the data are within 1 stdev of the mean
- Almost all of the data are within 2 stdev of the mean
In a normal distribution
the mean, median, and mode are exactly the same and the distribution is perfectly symmetric about the mean
for a continuous probability distribution of a continuous random variable,
the total area of the region under the curve is 1, and the areas of the vertical slices of the region are equal to the probabilities of a random variable associated with the distribution
standard normal distribution
is a normal distribution with a mean of 0 and standard deviation of 1. To transform a normal distribution to a standard normal distribution, you standardize the values, that is you subtract the mean m from each value and divide the result by the stdev.
How do you factor something in the form x^2 - 1?
It will always be (x + 1)(x - 1)
moving the decimal place of a number by x places to the right is equivalent to multiplying by…
10^x
moving the decimal place of a number by x places to the left is equivalent to multiplying by…
10^(-x)
In order to add inequalities together, the inequality signs…
must face in the same direction
To find the number of terms in a sequence of constant step size…
1) figure out what the largest and smallest terms are in the sequence that are WITHIN the specified range, 2) subtract the smallest term from the largest term, 3) divide by the step size, 4) add 1: this gives you the number of terms
If a sequence of numbers are equally spaced, the mean equals
the median
If a sequence of numbers are equally spaced, the overall average is equal to
the average of the first and the last terms
(x+y)^2 =
x^2 + 2xy + y^2
(x-y)^2 =
x^2 - 2xy + y^2
(x+y)(x-y) =
x^2 - y^2
For similar figures, the ratio of the areas is
the square of the ratio of the sides
An increase of x percent is the same as multiplying the original number by
1 + (x/100)
With a series of nested square roots, you can…
interchange the order of operations
When predicting what the units digit of a product will be, we only need to look at
the units digits of each factor
A triangle inscribed in a circle where the diameter is one of the sides is…
always a right triangle
Any integer raised to consecutive powers will…
manifest a clear units digit pattern of one, two, or four terms
For a given perimeter, the rectangle that has the largest area will always be…
a square
A fraction is only defined when…
the denominator is not equal to zero (if x is in the denominator, the fraction is NOT defined for any values of x where the denominator is zero!)
For an equilateral triangle with sides of length x, the height of the triangle is
((√3)/2)r
Factoring out factorials
To simplify factorials in a fraction, be sure to represent in terms of any factorials that could be canceled out. For example: 18!/(216!) can be (181716!)/(216!), which can then be simplified to just (18*17)/2
(xy)^2 =
x^2 * y^2
What are % of data distributed 1, 2, and 3 std deviations from the mean?
68% within 1 std dev, 95% within 2 std dev, 99.7% within 3 std. dev (this is total below AND above the mean)
How to find the least common multiple of x and y using prime factorization
Break x and y into the prime factorizations. Find all the prime factors that are repeated in both x and y. Multiply these together (DON’T double count). Then also multiply any of the non-repeated prime factors. The final product gives you the LCM.
For example: LCM of 24 and 108
24 = 2223
108 = 2233*3
Two 2’s are repeated, and one 3. Count those once, don’t double count.
Then you are left with the non-repeated factors: one 2 and two 3’s. So the final product is 22233*3 = 2^3 * 3^3
The prime factorization of a square number must contain
only even exponents
Method for solving comparisons by simplification
Set up a comparison of the two quantities with a ? in the middle as the placeholder. Then simplify both sides using the normal rules of algebra. Use any information given to decide whether >,
For the line segment PQ connecting points P and Q, all of the points that are equidistant from P and Q are
all the points on the line that is the perpendicular bisector of PQ
Work/rates problems
Remember that work done/distance traveled = rate*time.
One way to approach if you are given how much something can do in x hours is to divide and rearrange to find out how much something can do in 1 hour. If you have multiple things working, you can add them together to see what they can do together in 1 hour.
Probabilities with multiple combinations
Suppose you want to find the probability of a certain combination or combinations occurring (those that fit certain criteria)
Sometimes one combination becomes the denominator. This is the total number of possible combinations.
Then you need to figure out how many combinations match a particular description.
If you are calculating separate combinations or probabilities for different possibilities, then the overall number of combinations is the product of them.
If you know the probabilities of two events but nothing about their relationship, and you want to know the probability of one, the other, or both occurring (A U B), then…
You can only compute the minimum and maximum probabilities.
The minimum probability is if one event is a subset of another event, in which case the probability of either or both occurring is the higher probability of the two (think of the venn diagram where one circle is entirely within the other, then the union is just the larger circle). The maximum probability is that they both occur but are mutually exclusive and don’t intersect, which is the sum of the two probabilities.
Think of the two venn diagrams moving closer together. The highest total union area (probability of A U B) is if the two circles are not overlapping at all. That total area decreases as the intersection increases, until the point when one is entirely inside the other, which is the minimum area.
Break into cases approach to probabilities/combinations
Try to divide into different cases if possible and then find how many combinations/permutations are possible for each case. Then add the number of cases together.
the nth root of a can also be written
a^(1/n)
If asked to find the sum of a sequence of terms…
try writing out the terms added together - sometimes things cancel out and make it very easy!
