GRE Quantitative
Geometry Problem Solving
1. Draw or redraw figures and fill in all given information. 2. Identify relationships and create equations. 3. Solve the equations for the missing value(s). 4. Make inferences from the figures
Non-Terminating Decimal Examples
4/9=0.4444..., 23/99=0.2323..., 1/11=9/99=0.0909..., 3/11=27/99=0.2727...
What number is 50% greater than 60?
50% of 60= 30, 60+30=90, 90 is 50% greater than 60
Equilateral triangle
A triangle that has 3 equal angles (all 60 degrees) and three equal sides
Isosceles triangle
A triangle that has two equal angles and two equal sides (opposite the equal angles)
Area of a Trapezoid
A= 1/2 (b1+b2) x h ; the height refers to a line perpendicular to the 2 bases which are parallel
Area of a triangle
A= 1/2 bh ; the base and the height must be perpendicular to each other, in a triangle one side of the triangle is the base and the height is formed by dropping a line from the opposite point of the triangle straight down towards the base so that it forms a 90 degree angle with the base
Area of a parallelogram
A=bh ; the height refers to a line perpendicular to the base
Area of a Circle
A=πr²
Area of a circle
A=πr²
Right triangle
Any triangle in which one of the angles is aright angle
Circumference of a circle
C= 2πr ; C=πd ; think of it as the perimeter of a circle
Circumference of a Circle
C=2πr ; r=radius
Using Extreme Values in Inequalities
Can use this technique for problems with multiple inequalities where the question involves the potential range of values for variables in the problem and problems involving both equations and inequalities; use the highest possible values and lowest possible values to solve the problem
Fraction Bars as Grouping Symbols
Even though fraction bars do not fit into the PEMDAS hierarchy, they do take precedence. In any expression with a fraction bar, you should pretend that there are parentheses around the numerator and denominator of the fraction. This may be obvious as long as the fraction bar remains in the expression, but it is easy to forget if you eliminate the fraction bar or add or subtract fractions, so put parenthesis in to remind yourself. SEE PAGE 43 in book.
How do you use m and b to sketch a line?
Example: plot the line y= 1/2 x -2, start with the y-intercept, you know that the line crosses the y-axis at y=-2 so begin by plotting that point on the coordinate plane, every slope whether an integer or a fraction should be thought of as a fraction, in this equation m is 1/2, 1/2 = Rise/Run = Change in y/Change is x (the numerator of your fraction tells you how many units you want to move in the y-direction and the denominator tells you how many units you want to move in the x-direction, for this equation the slope is 1/2 which means you want to move up 1 unit and right 2 units
FOIL
F-multiply the first term in each of the parentheses, O- multiply the outer term in each, I-multiply the inner term in each, L-multiply the last term in each
Percent Change vs. Percent of Original
If a quantity is increased by x percent, then the new quantity is (100+x)% of the original. Thus, a 15% increase produces a quantity that is 115% of the original. If a quantity is decreased by x percent, then the new quantity is (100-x)% of the original. Thus, a 15% decrease produces a quantity that is 85% of the original.
Successive Percents Problems
If a ticket increased in price by 20%, and then increased again by 5%, by what percent did the ticket price increase in total? The answer is not 25%; Auccessive percents can't simply be added together; This applies for both successive increases and decreases and for combinations of the 2; A great way to solve successive percent problems is to choose real numbers and see what happens; Usually, 100 will be the easiest real number to choose for percent problems
Inequality Multiplication and Division
If you multiply or divide by a negative number, you must switch the direction of the inequality sign; if you are multiplying or dividing by a positive number than the sign stays the same; try not to divide by a variable when solving an inequality because you don't know if that value is negative or not
What does it mean when two lines intersect in the coordinate plane?
It means that at the point of intersection, BOTH equations representing the lines are true; the pair of numbers (x,y) that represents the point of intersection solves BOTH equations, finding this point of intersection is equivalent to solving a system of two linear equations; if two lines in a plane do not intersect then the lines are parallel, if this is the case there is no pair of numbers (x,y) that satisfies both equations at the same time
Compound Interest Formula
P( 1 + r/n) ^nt ; P=principal, r= rate (decimal), n=number of times per year, t=number of years
PEMDAS
P= parentheses, E= exponents, M= multiplication, D= division, A= addition, S= subtraction; if you have two operations of equal importance you should do them in left to right order
Percents as Fractions
Part/Whole = Percent/100
Simple Interest Formula
Principle (P) x Rate (r) x Time (t)
Estimate to the closest integer: What is 11/40 of 5/16 of 120?
Read this problem as what is 5/16 of 120, multiply the two values and once you have that number multiply that by 11/40 and that will give you the final answer
Multiplying Square Roots
Simply multiply the numbers within the square by each other. Example in picture
Percent increase and decrease
The price of a cup of coffee increased from 80 cents to 84 cents. By what percent did the price change? Price change=84-80=4 cents which is the "part" and the original price=80 cents is the "whole". 4/80= 1/20= x/100, 20x=100, x=5
How to see if fractions are reciprocals or not
The product of a number and it's reciprocal must equal 1; To test whether two numbers are reciprocals, multiply them; If the product is not 1, they are not reciprocals
Basic Properties of Triangles
The sum of any two sides of a triangle will always be greater than the third side length; the internal angles of a triangle must add up to 180 degrees; sides correspond to their opposite angles meaning that the longest side is opposite the largest angle, and the smallest side is opposite the smallest angle
The Three Special Products
Three quadratic expressions called special products come up so frequently on the GRE
Benchmark Values: 10% and 5%
To find 10% of any number, just move the decimal point to the left one place, example: 10% of 500 is 50. Once you know 10% of a number it is easy to find 5% of that number because 5% is half of 10%, example: 10% of 500 is 50 and 5% of 500 is 50/2=25
Comparing Fractions: Cross Multiply
To find which fraction is larger when comparing two fractions, instead of finding the common denominator, you can cross multiply and that will show you which fraction is larger and which one is smaller
Volume of a Cube
V=s³
Volume of a Cylinder
V=πr²h
Pythagorean triples
a certain subset of right triangles in which all three sides have lengths that are integer values, MEMORIZE these to save time on the GRE
polygons
a closed shape formed entirely by line segments; the polygons tested on the GRE include three-sided shapes (triangles), four-sided shapes (quadrilaterals), and other polygons with n sides (where n is 5 or more)
Function Graphs
a function can be visualized by graphing it in the coordinate plane; the input variable is considered the domain of the function, or the x coordinate; the corresponding output is considered the range of the function or the y coordinate
The Intercepts of a Line
a point where a line intersects a coordinate axis is called an intercept; there are 2 types of intercepts: the x-intercept, where the line intersects the x-axis, and the y-intercepts, where the line intersects the y-axis; the x-intercept is the point on the line at which y=0, the y-intercept is the point on the line at which x=0
Sequence Formulas
a sequence is a collection of numbers in a set order; the order of a given sequence is determined by a rule; Example of a sequence rule:
Lines and Angles
a straight line is 180 degrees; Parallel lines are lines that lie in a plane and never intersect; Perpendicular lines are lines that intersect at a 90-degree angle
Inscribed triangles
a triangle is said to be inscribed in a circle if all of the vertices of the triangle are points on the circle; if one of the sides of an inscribed triangle is a diameter of the circle then the triangle must be a right triangle
Exterior Angles of a Triangle
an exterior angle of a triangle is equal in measure to the sum of the two non-adjacent (opposite) interior angles of the triangle
Quadratic equation
any equation for which the highest power on a variable is the second power (eg. x^2)
Sector
any time you have a fractional portion of a circle (example: half a circle, pizza slice of a circle); every question about sectors involves determining what fraction of the circle the sector is
Pythagorean Theorem
a²+b²=c² ; for any right triangle the relationship is a²+b²=c² where a and b are the lengths of the sides forming the right angle, also known as the legs, and c is the length of the side opposite the right angle also known as the hypotenuse
Y intercept
b= y-intercept ; this tells you where the line crosses the y-axis, any line or curve crosses the y-axis when x=0
Diameter of a circle
d= 2r
Diagonal of a Square
d=s√2 ; s is the side of square, this is also the diagonal of a face of a cube
Main Diagonal of a Cube
d=s√3
Recursive Sequence Formula
each item of a sequence is defined in terms of the value of previous items in the sequence
Direct Sequence Formula
each item of the sequence is defined as a function of n, the place in which the term occurs in the sequence
Digits
every number is composed of digits; there are only 10 digits in our number system: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9; the term digit refers to one building block of a number, it does not refer to a number itself; for example: 356 is a number composed of 3 digits 3, 5, and 6
Inequalities
expressions that use <,>,< or >(with a line underneath) to describe a relationship between two values
GRE Sequence Problems
for simple linear sequences, in which the same number is added to any term to yield the next term, you can use the following alternative method- rather than find the rule or definition for the sequence, you can sometimes logically derive one item in the sequence based on the information given
Maximum Area of a Parallelogram or Triangle
general rule is if you are give two sides of a triangle or parallelogram, you can maximize the area by placing those two sides perpendicular to each other
Inscribed angle
has its vertex on the circle itself (rather than on the center of the circle); the central angle defines the arc and the inscribed angle is equal to half of the arc it intercepts in degrees
Decimal Division
if there is a decimal point in the dividend (the inner number) only, you can simply bring the decimal point straight up to the answer and divide normally (written out); if there is a decimal point in the divisor (the outer number), you should shift the decimal point to the right in both the divisor and the dividend to make the divisor a whole number, then bring the decimal point up and divide and make sure to shift the decimal in both numbers before dividing; ALWAYS shift the decimals on top and bottom so you are dividing by whole numbers; if you shift the decimal points you have to do it to both the divisor and the dividend (you move the decimals points in the same direction)
Knowing just one coordinate
if you know the x coordinate but not the y coordinate or vice versa, then you know the point lies somewhere on that vertical or horizontal line
Decimal Multiplication
ignore the decimal point until the very end; multiply the numbers as you would if they were whole numbers; count the total number of digits to the right of the decimal point in the factors; the product should have the same number of digits to the right of the right of the decimal point
Knowing ranges
instead of knowing the actual x coordinate, see what happens if all you know is a range of possible values for x. For example, what if all you know is x > 0? you can shade in the part of the coordinate plane: the part to the right of 0 to represent the knowledge you have of x
Rules for Integers with Basic Operations
integer + integer= always an integer. integer - integer= always an integer. integer x integer= always an integer. integer (divided by) integer= not always an integer.
Slope
m= slope ; tells you how steep the line is and whether the line is rising or falling; m>0 is a positive slope, m<0 is a negative slope, m>1 is a steep slope, 0<m<1 is a gentle slope
Optimization Problems
minimization and maximization problems; focus on the largest and smallest possible values for each of the variables, as SOME combination of them will usually lead to the largest or smallest possible results (do not have to only multiply the lowest value by the lowest value or the highest value by the highest value), Notice that in the example after subtracting the values the GTs change to LTs and vice versa
Heavy Division Shortcut
more the decimals in the same direction and round to whole numbers; the goal is to get a single digit to the left of the decimal in the denominator
If you are multiplying a very large number and a very small number
move the decimals in the opposite direction the same number of places; this works because you are multiplying and then dividing by the same power of 10, you are trading decimal places in one number for decimal places in another number
Trapezoid
one pair of opposite sides are parallel
Perfect Square Quadratics
one solution quadratics, both roots are the same
Rhombus
opposite sides and opposite angles are equal; all sides are equal
Parallelogram
opposite sides and opposite angles are equal;a diagonal will divide the parallelogram into 2 equal triangles
Powers of 10: Shifting the Decimal
place values continually decrease from left to right by powers of 10; when you multiply any number by a positive power of 10, move the decimal forward (right) the specified number of places, this makes positive numbers larger; when you divide any number by a positive power of 10, move the decimal backward (left) the specified number of placed, this makes the numbers smaller; a negative power of 10 is the opposite process for both multiplication and division
3 Dimensions: Surface Area
surface area= the SUM of the areas of ALL of the faces (both a rectangular solid and a cube have six faces)
Reading a graph
the GRE will mathematically define each line or curve so you will never be forced to guess visually where a point falls. In fact, if more specific information is not given for a coordinate problem on the GRE, you cannot infer the location of a point based solely on visual cues.
Sum of Interior Angles of polygons
the SUM of interior angles of a polygon= (n-2) x 180
Remainder
the amount left over when a number cannot be divided equally; is always a whole number;
Maximum Area of a Quadrilateral
the best known maximum area problem is one that asks you to maximize the area of a quadrilateral with a fixed perimeter; of all quadrilaterals with a given perimeter the square has the largest area; also works when the principle is turned around: of all quadrilaters with a given area the square has the minimum perimeter
Central Angle
the central angle of a sector is the degree measure between the two radii; For example: take a look at a quarter of a circle, there are 360 degrees in a full circle to find the degree measure of the angle between the two radii you do the same thing that you do to the area and circumference, 360/4= 90 degrees is the central angle for a quarter circle
The Distance Between Two Points
the distance between any two points in the coordinate plane can be calculated by using the Pythagorean theorem;
Quadrants
the four quarters of the coordinate plane are called quadrants; each quadrant corresponds to a different combination of signs of x and y
Intersecting Lines
the interior angles formed by intersecting lines form a circle so the sum of these angles is 360 degree; interior angles that combine to form a line sum to 180 degrees (supplementary angles); angles found opposite each other where these two lines intersect are equal (vertical angles); these rules apply to more than two lines that intersect at a point it can be more than two lines
45-45-90 degree triangle
the lengths of the legs of every 45-45-90 triangle have a specific ratio which you must memorize
A square root being multiplied by a number
the square root that contains the x is being multiplied by 4 so to solve for x your next step would be to divide both sides by 4
Parallel Lines Cut by a Transversal
there are 8 angles that are formed by the transversal but there are only 2 different angle values; all the acute angles are equal and all the obtuse angles are equal; the GRE will disguise a transversal by presenting it in a "Z" shape
Variable Inside an Absolute Value
there are two numbers that could be the same distance away from 0; if a variable is in an absolute value then it has two answers, a positive and a negative answer; when there is more than just the variable inside the absolute value (such as a number with the variable or a number being added to the variable) then you have to solve for both solutions by having the whole absolute value positive and the whole absolute value negative
Formulas with Unspecified Amounts
these questions typically focus on the increase or decrease in the value of a certain formula given a change in the value of the variables, solve these problems by picking numbers to plug in
time= distance/rate
time=distance/rate
Unit Digit Problems
to find the units digit of a product or a sum of integers, only pay attention to the units digits of the numbers you are working with. drop any other digits
Surface Area of a Cylinder
two circles and a rectangle combine to form a 3-D shape called a right circular cylinder; to find the surface area, sum the areas of the 3 surfaces (A=πr² and A=bh); the length of the rectangle is equal to the circumference of the circle and the width of the rectangle is equal to the height of the cylinder so A= 2πr x h
Compound inequalities
two inequalities in one statement
Volume of 3 dimensional shapes
volume= length x width x height
Solving quadratic fractions with 0 in the denominator
when 0 appears in the denominator of an expression, then that expression is undefined; see example in the picture
Squared Variable
when a variable is squared it becomes possible that there will be two solutions to the equation; whenever you see a squared variable you need to recognize that the equation may have two solutions and know how to find both solutions
Arc length
when dealing with sectors the portion of the circumference that remains is called the arc length
Integers
whole numbers, that means that they are numbers that do not have any decimals or fractions attached; can be positive or negative; 0 is an integer also
Lines in a plane
you can generalize this relationship, any relationship of the following form represents a line: y=mx+b
Fraction Rule
you can only cross out values that are both in the numerator and the denominator if they are the exact same; if the variable in the numerator has a value in front of it and the same variable in the denominator does not than you can not cancel out that value; same goes for what is is being added to;
Manipulating Compound Inequalities
you can perform operations on a compound inequality as long as you remember to perform those operations on every term in the inequality, not just the outside terms
