GRE Quantitative Reasoning
Calculate percent increase
(New Value - Original Value) / Original Value * 100
Calculate percent decrease
(Original Value - New Value) / Original Value * 100
Arc length =
(angle / 360) x (circumference) or (angle/360) x ( (2)(pi)(r) )
Area of a sector of a circle =
(angle/360) x ((pi)(r^2))
Circumference =
(pi)(diameter) or (pi)(2r)
The mean of a data set is 40 and the SD is 10. What value is 1.5 SDs below the mean?
- (10 / 1 SD) x (1.5 SD) = 15 terms - 40 - 15 = 25!
Special properties of a standard normal distribution curve
- centered around 0 - SD is calculated to 1
Since arcs are associated with ______, they can be measured in _____
- central angles - degrees
A bag contains 10 marbles, 4 of which are blue and 6 of which are red. If 2 marbles are removed without replacement, what is the probability that both marbles removed are red?
- dependent events - First draw: 6/10 - Second draw with now 9 marbles total: 5/9 - Total probability: (6/10) x (5/9) = (3/5) x (5/9) = 1/3
what is the multiplication principle of sets?
- is the product of the number of options for each choice - used then choices or events occur one after the other and are independent of one another - i.e. a ballot that offers 3 candidate choices for office A, 4 for office B, and 2 for office C ---> the total number of different wats to fill out the ballot is 3 x 4 x 2 = 24 possibilities
Find the SD of 1,3,8,11,12
- mean: (1+3+8+11+12)/5 = 7 - difference between each term and the mean: (1-7), (3-7), (8-7), (11-7), (12-7) - square the differences: (-6)^2 , (-4)^2 , (1)^2 , (4)^2 , (5)^2 - mean of the squares: (36 + 16 + 1 + 16 +25)/5 = 18.8 - positive square root of the mean: (18.8)^(1/2) = 4.34
Convert 5/8 to a percent
- multiply by 100% = 500%/8 - 500/8 = 62 1/2 %
Convert 0.125 to a fraction
- multiply by 1000/1000 - simplify to 1/8
Properties of a normal distribution curve
- peak is the mean - data are symmetrically distributed around the mean - is a continuous probability distribution - mean = median = mode - the probability of selecting a value under the curve is 100% or 1 - 68% of the area is within +/- 1 SD of the mean - 95% of the area is within +/- 2 SD of the mean
In a spelling contest, the winner will receive a gold medal, the second-place finisher will receive a silver medal, the third-place finisher will receive a bronze medal, and the fourth- place finisher will receive a blue ribbon. If there are 7 entrants in the contest, how many different arrangements of award winners are there?
- recognize: "arrangements" indicates permutation problem (7) * (6) * (5) * (4) = 840 arrangements
If a ballot offers 3 candidate choices for office A, 5 for office B, and 10 for office C, what is the total number of different ways to fill out the ballot?
- recognize: choices are independent of one another soooo multiplication principle - 3 x 5 x 10 = 150 different possibilities
the 4 finalists in a spelling contest win commemorative plaques. If there are 7 entrants in the spelling contest, how many possible groups of finalists are there?
- recognize: creating subgroups from an existing group means combination formula - (7!)/((4!)(7-4)!) (7*6*5*4*3*2*1)/((4*3*2*1)(3*2*1)) = (7*6*5) / (3*2*1) = (210) / (6) = 35 35 potential groups of spelling contest finalists
What is the probability of tossing a fair coin four consecutive times and having it land heads up exactly once?
- recognize: each toss has 2 possible outcomes Total number of outcomes = 2 x 2 x 2 x 2 = 16 Number of desired outcomes = 4 (HTTT, THTT, TTHT, TTTH) Probability of one heads = 4/16 = 1/4 = 0.25
If a website calls for a 3-letter password but no 2 letters can be the same, what is the total number of possible passwords?
- recognize: letters are independent of one another - the second and third letter would have one less option than its prior because one letter would have already been used - 26 x 25 x 24 = 15,600
A menu has 3 choices for soup and 4 for salad options; diners are permitted to select a soup or salad with their dinners. How many options are there?
- recognize: that "or" and mutually exclusive options means we will add - 4 + 3 = 7 total different choices
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?
- recognize: we are creating subgroups of existing groups; there are 2 separate subgroups being made Group A: (5!) / ( (2!)(5-2)! ) = (5*4*3*2*1) / (2*1)(3*2*1) = (5*4) / (2*1) = 20/2 = 10 Group B: (6!) / ( (3!)(6-3)! ) = (6*5*4*3*2*1) / (3*2*1) (3*2*1) = (5*4) = 20 Group A and B with these specific parameters are independent groups soooo multiplication = (10)(20) = 200 200 possible groups
What is the sum of the interior angles of a pentagon?
- remember that the sum of the interior angles of any triangle equals 180 degrees - draw diagonals from any vertex to all nonadjacent vertices THEN multiply the number of triangles by 180 degrees in this case 180 x 3 = 540 degrees
What is the probability of tossing a fair coin four consecutive times and having the coin land heads up 0,2,3, or 4 times?
- remember: the probability of any event happening is 1/1 - a coin landing on heads 1 time at all is (4 / (2x2x2x2) ) = 1/4 - SO any other number of times a coin landing on heads is 1 - 1/4 = 3/4
what is a percentile?
- you would multiply p% by the total number of terms and the resulting value is the pth place of the term you are being asked to find (i.e. 75th percentile of a set with 60 values. 60 x 0.75 = 45th value of the set in ascending order) -the value such that p% of the observations fall at or below it - used when we are most interested in seeing where an individual observation stands relative to the other individuals in the distribution
Combined work formula
1/(person 1) + 1/ (person 2) = 1/T
the exterior angles of any triangle add up to...
360 degrees
a quartile is...
4 equal sections or intervals of the data created after ordering the terms numerically L, Q1, Q2, Q3, G
what is a chord on a circle?
A chord is a line segment joining two points on a circle.
quadratic function
A function in which the greatest power of the variable is 2 y=x^2 .....
what is a tangent on a circle?
A line that touches only one point on the circle forming a right angle that intercepts the arc.
a rhombus
A parallelogram with four equal sides
a trapezoid
A quadrilateral with exactly one pair of parallel sides
area of a triangle
A=1/2 bh -- Area = 1/2 base X height
Calculate compound interest
A=P(1+r / n)^nt A= investment's future value with interest P= Principle R= Annual Interest Rate N= How many times interest is compounded annually T= how long the money has been invested in years
What is a permutation?
An arrangement of things in a particular order Think of each place that needs to be filled in a particular order as a blank space. The first place can be filled with any of the items in the larger group. The second place can be filled with any of the items in the larger group except for the one used to fill the first place. The third place can be filled with any of the items in the group except for the two used to fill the first two places, etc. ie: how many different ways can you arrange 5 statues on a shelf? 5x4x3x2x1 = 120 different ways
Calculate simple interest
Interest = Principle x Rate x Time To calculate interest for months, multiply the principle by the rate by the number of months
Probability equation when all possible outcomes of the experiment are equally likely to occur
Probability = (Number of Desired Outcomes) / (Number of Possible Outcomes)
interquartile range
Q3-Q1 - because outliers fall into the bottom and top quartiles, they do not affect this value
Special Right Triangle 45-45-90
Right Isosceles triangle. Both legs are equal. Hypotenuse= Leg * √ 2
Special Right Triangle 30-60-90
Short Leg is opposite the smallest angle (30°). Long Leg is opposite the 60° angle. Long Leg = (short leg) *√3. Hypotenuse = 2 * Short Leg
what is the triangle inequality theorem
The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
Standard deviation
a computed measure that determines how far away, on average, the values in a set are from the mean (1) find the mean (2) find the difference between the mean and each value (3) find the square of each of those differences (4) find the mean of the squared differences (5) take the non-negative square root of the mean
Define pi
a constant that defines the relationship between the circumference of a circle to the diameter
What is a set in math?
a group of values that have some common property, such as the negative integers greater than -10 - are defined by unique, nonrepeatable elements
a factor is...
a number that is multiplied by another number to find a product
a quadratic function will produce
a parabola
the altitude of a triangle is...
a perpendicular segment from a vertex to the line containing the opposite side; it may fall inside or outside the triangle, or it may coincide with one of the sides
a parallelogram
a quadrilateral whose opposite sides are both parallel and equal in length opposite angles are equal in measure angles that are adjacent are supplementary to each other (180 - x)
the union of two sets is...
a set that contains all the elements of both sets
what does the value of the interquartile range communicate?
a smaller IQR means that the data set is more closely spaced
Prime number
an integer (or whole number) that has exactly 2 factors, 1 and itself
Composite number
an integer (or whole number) that has more than 2 factors
area of a trapezoid
area = average of parallel sides X Height OR area = (1/2)(b1 + b2) x (h)
Area of a parallelogram
area = bh
A combination question is defined as a question that...
asks how many unordered subgroups can be formed from a larger group
To the 0 power
equals 1 - because if you have a term over itself in a fraction, you would subtract the exponents to simplify and obtain a 0 exponent
even + even =
even
even x even =
even
odd + odd =
even
odd x even =
even
an integer is divisible by 6 if...
it is divisible by both 2 and 3
A line segment bisects an angle if...
it splits a preexisting angle into two smaller, equal angles
an integer is divisible by 3 if...
its digits add up to a multiple of 3
an integer is divisible by 9 if...
its digits add up to a multiple of 9
an integer is divisible by 5 if...
its last digit is 0 or 5
an integer is divisible by 2 if...
its last digit is divisible by 2
an integer is divisible by 4 if...
its last two digits are a multiple of 4
what does the value of SD communicate?
larger SD = more spread from the mean smaller SD = terms are relatively close to the mean
what is a list in math?
like a finite set whose order of elements DOES matter and DOES include duplicate elements - elements are then defined by their position, such as "first element" or "fifth element"
The hypotenuse is always the ___ side of a right triangle
longest
In y=mx+b, what letter represents the slope?
m
Convert degrees to radians
multiply by π/180
do outliers affect the interquartile range?
no, because outliers fall into the bottom and top quartiles (outside of the Q1 and Q3 range)
even + odd =
odd
odd x odd =
odd
perimeter and area of a rectangle (and square)
perimeter = 2l + 2w - for square, perimeter = 4s area = lw - for square, area = (side)^2
perimeter and area of a circle
perimeter = circumference = (pi)(diamter) = (pi)(2r) area = (pi)(r^2)
Associative laws of addition and multiplication
regrouping the numbers does not affect the results. (a+b)+c=a+(b+c) (ab)c=a(bc)
slope (m) =
rise / run = (change in y) / (change in x) = (y2-y1) / (x2-x1)
Commutative laws of addition and multiplication
switching the order of any two numbers being added or multiplied together does not affect the result
to find the value of a quartile...
take the median of the respective section (usually found from Q2 to Q1 and Q3)
the mean is...
the average of a distribution
Regardless of a quadrilateral's shape...
the four interior angles always sum to 360 degrees
GCF (greatest common factor)
the greatest number that is a factor of two or more numbers
To find the length of a line segment that is not parallel to the axes of a plane, treat the line segment as _____
the hypotenuse of a triangle, and use Pythagorean theorem to find the length
the median is...
the middle term after all terms are in numerical order (or the 2 middle terms divided by 2)
a multiple is...
the product of a specific number and an integer
the intersection between two sets is...
the set that consists of all the elements contained in or shared by both sets
what property of a normal distribution curve determines if the SD is high or low?
the spread of the curve - wider curve = higher SD - narrower curve = smaller SD
the sum of an exterior angle of a triangle is equal to..
the sum of the remote interior angles
the mode is...
the term that appears the most (or no mode if more than 1 term appears for the same number of times)
two angles are supplementary to each other if...
their measures sum to 180 degrees
two angles are complementary to each other when...
their measures sum to 90 degrees
Dividing by 0
undefined; cannot be done
Inclusion-Exclusion Principle equation
used to find the number of elements in the union of two or more events or groups
words that indicate a permutation problem?
ways arrangements orders schedules
In y=mx+b what does "b" represent?
y-intercept
what is the set of the numbers below? 1,1,2,2,3
{1,2,3} (the unique elements) (does not have to be written in this order either)
