ALL MATH in one GRE

consecutive #s

#s of a certain type, following one another without interruption # may be consecutive in ascending/descending order ex: series of... consecutive integers: (-2,-1,0,1,2...) consecutive even #s: (-4,-2,0,2,4,6...) consecutive multiples of 3: (-3,0,3,6,9..) consecutive prime #: (2,3,5,7,11...)

mult. and division of pos/neg #s

-Multiplying/dividing 2 #s with the same sign gives a pos. result - Multiplying/dividing 2 #s with dif. signs gives neg. result

Rules of divisibility - help assess whether one # is evenly divisible by another

1. An integer is divisible by 2 if its last light is divisible by 2 2. An integer is divisible by 3 if its digits add up to a multiple of 3 3. An integer is divisible by 4 if the last 2 digits are multiple of 4 4. An integer is divisible by 5 if its last digit is 0 or 5 5. An integer is divisible by 6 if it is divisible by both 2 and 3 An integer is divisible by 9 if its digits add up to a multiple of 9

Polynomials and quadratics

1. FOIL method 2. Factoring the product of binomials 3. factoring the dif. of 2 squares 4. Factoring polynomials of the form a^2 + 2ab + b^2 OR a^2 - 2ab + b^2

Rules of addition/subtraction operations with roots and radicals

1. Only "like" radicals can be added/subtracted from one another EX: 2√3 + 4√2 - √2 - 3√3 = (4√2 - √2) + (2√3 - 3√3) = 3√2 - √3 This expression can't be simplified any further

Rules of operations with exponents

1. to multiply 2 powers with the same base, keep the base and add the exponents together 2^2 x 2^3 = 2^2+3 = 2^5 2. To divide 2 powers with the same base, keep the base and subtract the exponent of the denominator from the exponent of the numerator 4^5 divided by 4^2 = 4^5-2 = 4^3 3. To raise a power to another power, multiply the exponents (3^2)^4 = (3x3)^4 = 3^2x4 = 3^8 4. To multiply 2 powers with dif. bases but the same power, multiply the bases together, then raise to the power (3^2)(5^2) = (3x3)(5x5) = (3x5)(3x5) = (3x5)^2 = 15^2 5. A base with a neg. exponent indicates the reciprocal of that base to the pos. value of the exponent 5^-3 = 1/5^3 = 1/125 6. Raising any non-zero # to an exponent of zero equals 1 5^0 = 1; 161^0 = 1; (-6)^0 = 1

Properties of fractions b/w -1 and +1

1.) Reciprocal of a fraction b/w 0 and 1 is greater than both the original fraction and 1. EX: reciprocal of 2/3 = 3/2, which is > than 2/3 and 1 2.)Reciprocal of fraction b/w -1 and 0 is less than both the original fraction and -1. EX: reciprocal of -2/3 = -3/2 or - 1 1/2, which is < -1 and -2/3 3.) The square of a fraction b/w 0 and 1 is less than the original fraction. EX: (1/2)^2 = 1/4 BUT square of any fraction b/w 0 and -1 is > than original fraction, bc. multiplying 2 neg. #s yields pos. product & pos. # is always > than neg. # 4.) Multiplying any pos. # by a fraction b/w 0 and 1 gives a product smaller than original #. EX: 6(1/4) = 6/4 = 3/2 5.) Multiplying any neg # by a fraction b/w 0 and 1 gives a product > than original # EX: (-3)(1/2) = -3/2 (it yields a smaller value of the original neg. #)

Converting percents - reference page 224

1.) To change a fraction to its % equivalent, multiply by 100%. EX: 5/8 percent equivalent = 5/8(100%) = 500% / 8 = 62 1/2 % 2.) To change decimal to %, use the rules for multiplying by a power of 10 - move the decimal point two places to the right and insert % sign. EX: 0.17 = 0.17(100%) = 17% 3.) To change % to its fractional equivalent, divide by 100%. EX: 32% / 100% = 8/25 4.) To change % to its decimal equivalent, use rules for dividing by power of 10 - move the decimal place to the left 2 places; EX: 32% = 0.32 or 32%/100% = 32/100 = 0.32 5.) when you divide a % by another %, the % sign cancels out - just as you would cancel out a common factor 6.) When you divide a % by a regular # (not by another %), the % sign remains EX: 100% / 5 = 20%

Multiple Figures - Area of shaded regions examples of 2 approaches pg 302!!!

A common Multiple-figures Qs involves a diagram of a geometrical figure that has been broken up into different, irregularly shaped areas, often w/ one region shaded. You will usually be asked to find the area of the shaded (or unshaded) portion of the diagram. Best bet is to take one of the following approaches: 1.) Break the area into smaller pieces whose separate areas you can find; add those area together 2.) Find the area of the whole figure; find the area of the region(s) that you're not looking for; subtract the latter from the former

Quadrilaterals

A quadrilateral is a 4-sided polygon Regardless of shape, the 4 interior angles sum to 360 deg.

Factoring polynomials of the form a² + 2ab + b² or a² - 2ab + b² Pg. 250 NEEDS PRACTICE

Any polynomial of this form is the square of a binomial expression, as you can see by using the FOIL method to multiply (a + b) (a +b) or (a - b)(a - b) To factor a polynomial in this form, check the sign in front of the 2ab term - 1.) if it is a plus sign (+), the polynomial is equal to (a + b)² 2.) if it is a minus symbol (-), polynomial is equal to (a - b)² EX: factor polymonial: x² + 6x + 9 x² and 9 are both perfect squares, and 6x is 2(3x), which is twice the product of x and 3 Thus, this polynomial is in the form a² + 2ab + b² with a=x and b= 3 Since there is a plus sign (+) in front of the 6x: x² + 6x + 9 = (a + 3)²

Frequency distribution Probability distribution or relative Frequency distribution Random variable X REVIEW EXAMPLES on pg. 265 - 266

Frequency distribution = a description of how often certain data values occur in a set and is typically shown in a table or histogram (265) Probability distribution/ Relative Frequency distribution = the frequencies with which given values occur is given in decimal form rather than as percentages Random variable X = The value of a randomly chosen value from a known distribution of data is called a random variable X NOTE: you can calculate the mean by using the weighted average approach (ex. pg 266 on probability distribution) Weighted mean = A weighted mean is a kind of average. Instead of each data point contributing equally to the final mean, some data points contribute more "weight" than others. To find the weighted mean: 1. Multiply the numbers in your data set by the weights. 2. Add the results up. EX: table on 266 0.05 (0) + 0.10(1) + 0.20(2) + 0.30(3) + 0.25(4) + 0.10(5) = 0 + 0.10 + 0.40 + 0.90 + 1.00 + 0.50 = 2.90 As mentioned above freq. distribution can be presented in table or histogram - if the sample set of an experiment is large enough, the histogram can begin to closely resemble a continuous curve

Set Operations Intersection Union Mutually exclusive Universal set NEEDS SERIOUS REVIEW page 270

Intersection: the intersection of 2 sets is a set that consists of all the elements that are contained in both sets (think of it as the overlaps b/w the sets) The intersections of sets A and B is written as A ∩ B Union: The union of 2 sets is the set of all the elements that are elements of either or both sets and is written as A ∪ B Mutually exclusive: If sets have no common elements, they are referred to as mutually exclusive, and their intersection is the empty set **Drawing a Venn diagram is a helpful way to analyze the relationship among sets Universal set: the set of all possible elements that have the characteristics of the sets represented by the circles in a Venn diagram & represented by U (For instance, U could be the set of all species in the world, A the set of species in Europe, B the set of species in Asia, and C those native to Australia) PG. 269

Least common multiple (LCM)

LCM of 2 or more integers is the smallest # that is a multiple of each of the integers To find the LCM: 1. determine the prime factorization of each integer 2. write out each prime # with the max # of times it appears in any one of the prime factorizations 3. Multiply those prime #s together to get the LCM of the original integers EX: LCM of 6 and 8 prime factors of 6 and 8: 6 = (2)(3) 8 = (2)(2)(2) LCM = (2)(2)(2)(3) = 24

Legs

Legs of a triangle: 1. The 2 equal sides of an isosceles triangle 2. The 2 shorter sides of a right triangle (the lines forming the right angle) Note: 3rd, unequal side of an isosceles triangle is called the base

sequence

Lists that have infinite # of terms, in order Terms often indicated by a letter with a subscript indicating the position of the # in the sequence ex: "a3" ( 3 would be subscript- small in bottom right of a) denotes the third # in sequence ex2: "an" ( n would be subscript- small in bottom right of a) indicates the nth term in sequence

Factoring out binomials Pg. 249

Many polynomials on the GRE can be factored into a product of 2 binomials using FOIL backwards EX: factor the polynomial: X^2 - 3x + 2 1. you can factor into 2 binomials, each containing an x term X^2 - 3x + 2 = (x )(x ) 2. need to find the missing term in each binomial factor - 3.the PRODUCT of the 2 missing terms will be the Last term in the original polynomial: 2. 4. the SUM of the 2 missing terms will be the coefficient of the second term of the polynomial: -3 5. Find the factors of 2 that add up to -3. Since (-1) + (-2) = -3, you can fill the empty spaces with -1 and -2 Thus: x^2 - 3x + 2 = (x - 1)(x - 2) WHENEVER YOU FACTOR A POLYNOMIAL, YOU CAN CHECK YOUR ANSWER BY USING FOIL TO MULTIPLY THE FACTORS AND OBTAIN THE ORIGINAL POLYNOMIAL

Simultaneous equations 2 methods to solve pg 245 - 247 has various examples of the 2 methods

One GRE, you will often have to solve 2 simultaneous equations - equations that give you dif. info about the same 2 variables - 2 methods to solve Make sure you have really have 2 distinct equations !! Method 1: Substitution Step 1: solve 1 equation for 1 variable in terms of the 2nd Step 2: substitute the result back into the other equation and solve EX: x - 15 = 2y and 6y + 2x = -10, what is the value of y? 1.) solve for x by adding 15 to both sides: x = 2y + 15 2.) substitute 2y + 15 for x into the second equation to find y Method 2: Adding to cancel 1.) Combine equations in such a way that one of the variables cancels out EX. on 245 4x + 3y = 8 and x + y = 3 Multiply both sides of 2nd eq. by -3 to get: -3x + -3y = -9 Add the 2 equations - the 3y and -3y cancel, leaving x = -1 Plug x in to 1 of the eq. to solve for y FOR EITHER METHOD, YOU CAN CHECK THE RESULT OF YOUR WORK BY PLUGGING BOTH VALUES BACK INTO BOTH EQUATIONS AND MAKING SURE THEY FIT

Other rates on GRE (L per min or proportions ...) *ALL RATE PROBLEMS CAN BE SOLVED USING THE SPEED FORMULA AND ITS VARIANTS - by conceiving "speed" as "rate and "distance" as "quantity" Time = Quantity/Rate T = Q/R; Q = (T)(R); R= Q/T

Other rates on GRE besides speed - Liters per min, cost per unit *ALL RATE PROBLEMS CAN BE SOLVED USING THE SPEED FORMULA AND ITS VARIANTS - by conceiving "speed" as "rate and "distance" as "quantity" Time = Quantity/Rate T = Q/R; Q = (T)(R); R= Q/T EX: how many hours will it take to fill a 500-Liter tank at a rate of 2L per min? T = Q/R 1.) T = 500L/ 2L per min = 250 minutes 2.) convert min to hours 250 min / 60 min per hour = 4 and 1/6 or 4.167 hours GRE tests ability to convert hours to min and min to hours - PAY ATTENTION TO WHAT UNITS THE Q IS ASKING FOR In some cases, you should use proportions to answer rate Qs EX of proportion: if 350 widgets cost 20$, how much will 1400 widgets cost at the same rate? # widgets/ cost = 350 W/ 20$ = 1400/ x$ A: cross multiply: (20$ x 1400)/350 = 80 So 1400 widgets will cost 80$ at rate of 350 widgets for 20$

Order of operations: PEMDAS or Please Excuse My Dear Aunt Sally

Parenthesis Exponents Multiplication - from L to R Division - from L to R Addition - from L to R Subtraction - from L to R

circumference

Perimeter = the distance around a polygon Circumference = the distance around a circle Ratio of the circumference of any circle to its diameter is a constant - pi π = 3.14 π = the ratio of the circumference, C, to the diameter,d π = C/d formula for the circumference of a circle: C = πd Circumference of a circle can also be stated in terms of the radius, r. Since diameter is twice the length of the radius, that is, d = 2r & C = 2πr

Pie charts EX. on pg 260

Pie charts show how things are distributed - the fraction of circle occupied by each piece of the "pie" indicates what fraction of the whole that piece represents - top. the percentage of the pie occupied by each "slice" will be shown on the slice itself or outside the circle with an arrow/line pointing to the appropriate slice (for narrow slices) The total size of the whole pie is usually given at the top/bottom of the graph, either as "TOTAL = xxx" or as "100% = xxx" To find the approximate amt. represented by a particular piece of the pie, multiply the whole by the appropriate percent IMPORTANT NOTE about pie charts: if you aren't given the whole and you do not know both the percentage and the actual # that at least one slice represents, you won't be able to find the whole. Pie charts are ideal for presenting the kind of info that ratio problems present in words

Polygons

Polygon = a closed figure whose sides are straight line segments Families/classes of polygons are named according to their # of sides: Triangle = 3 sides Quadrilateral = 4 sides Pentagon = 5 sides Hexagon = 6 sides LOOK AT PIC FOR MORE NAMES *** Triangles and Quadrilaterals are by far most important polygons on GRE The sum of exterior angles in a polygon is always equal to 360 degrees.

Square

Product of a # multiplied by itself A squared # has been raised to the 2nd power ex: 4^2 = (4)(4) = 16 4 = square root of 16

REFERENCE pages 229 - 230 Terms you may encounter in more complicated percent word problems: Profit Discount or Percent discount Sale Price Interest - principle, simple interest and compound interest

Profit made on an item = seller's price minus the cost to the seller Discount on an item = the original price minus the reduced price - a discount is often represented as a % of he original price Sale price = final price after discount or decrease Interest - given as a % per unit of time, such as 5% per month Principle - the sum of money invested is the principle Simple interest = the interest payments received are kept separate from the principle; most common type of interest Compound interest = the money earned as interest is reinvested - the principle grows after every interest payment received

Diagrams of polygons

Slash marks can provide useful info on diagram of polygon Sides with same # of slash marks are equal in length Angles with same # of slash marks thru circular arcs have same measure

Speed - most commonly tested rate on GRE Other rates on GRE

Speed usually expressed in miles/mph or km per hour/kmph Relationship b/w speed (S) , distance(D) & time(T) is formula: S = D/T T = D/S D = (S)(T) Other rates on GRE besides speed - Liters per min, cost per unit *ALL RATE PROBLEMS CAN BE SOLVED USING THE SPEED FORMULA AND ITS VARIANTS - by conceiving "speed" as "rate and "distance" as "quantity"

Standard deviation pg 264 ** Prob won't have to calculate SD on test day, but need to know how it behaves - SO it's worthwhile to calculate it for a couple lists of #s to be familiar

Standard deviation is a way to measure how spread out the values in a given data set are (like range and interquartile range) How to calculate Standard Deviation: 1.) find average of data points 2.) Find difference b/w average and each data point 3.) square each of the differences 4.) Find the average of the squared differences 5.) Take the square root of that average NOTE: 1. The farther the data points are away from the mean, the greater the standard deviation will be 2. two sets whose data points are the same distance from the mean will have the same standard deviation EX: sets {2, 4, 6} and {8, 10, 12} will have the same standard deviation

Coordinates

The #s that designate distance from an axis in coordinate geometry. The first # is the x-coordinate; 2nd # is the y-coordinate. In ordered pair (7,8), 7 is the x-coordinate and 8 is the y-coordinate

Percents

The word percent means "hundredths" and the percent sign % means 1/100. Ex: 25% = 25(1/100) = 25/100 Percents measure a part-to-whole relationship b/w a specified part and a whole - a whole that is 100 part/whole (100%) = %

set

Well-defined collection of items, typically collection of #s, objects, events The bracket symbols {} are normally used to define sets of #s ex: {2,4,6,8} is a set of #s

Multiplication Principle

When Choices/events occur one after the other and the choices or events are independent of one another, the total # od possibilities is the product of the # of options for each EX: If a ballot offers 3 candidates choices for office: A = 4, B = 3, C= 2 The total # of ways a voter could fill out the ballot is 4 x 3 x 2 = 24 Qs like this may require analysis of the # of options for each choice. EX: if a website calls for a 3-letter password bu no two letter can be the same, the total possibilities would be 26 x 25 x 24 = 15600 because stipulation that no 2 letters can be the same reduces the # of choices for the second and third letters In instances where choices are "or" instead of "and" as long as the 2 groups are mutually exclusive, ADD instead of multiplying. EX: menu has 3 choices for soup and 4 for salad, dinners are permitted to select a soup or salad w/ their dinners - the total # of choices available is 3 + 4 = 7

Fractional Exponents

When the exponent is a fraction, you're looking for a root of the base. The root corresponds to the denominator of the fraction. EX: For example, take "125 raised to the 1/3 power," or 125^1/3. The denominator of the fraction is 3, so you're looking for the 3rd root (or cube root) of 125. Because 5 x 5 x 5 = 125, the 3rd root of 125 is 5. Thus, 125^1/3 = 5. EX: Now try 256^1/4. You're looking for the 4th root of 256. Since 4 x 4 x 4 x 4 = 256, the answer is 4.

part: part ratios and converting to part:whole ratios

You can convert a part:part ratio to a part:whole ratio (or vise versa) only if there are no missing parts and no overlap among the parts - that is, if the whole is equal to the sum of the parts EX: the ratio of domestic sales rev to foreign rev of a certain product is 3:5. What fraction of the total sales rev comes from domestic sales? A: you aren't given the actual $ figures for domestic sales, but since all sales are either domestic or foreign, "total sales revenue" must be the sum of the revenues from domestic and foreign sales ***ie: you can convert the part:part ratio to part:whole because the su of the parts equals the whole

absolute value

absolute value of a # is value of a # w/o its sign - can be thought of as #s distance from zero on # line It is written as 2 vertical lines, one on either side of the # and its sign Ex: |-3| = |+3| = 3 Ex2: if you are not told that |x| = 5, then x could = -5 or 5

Binomial NEEDS REVIEW confused

an algebraic expression of the sum or the difference of two terms. (a +b) (a - b) 5x - 1 Can use foil to multiply out 2 binomials to get a polynomial

Divisibility

concepts of multiples and factors tied together by the idea of divisibility # is said to be evenly divisible by another # if the result of the division is an integer w/ no remainder A # that is evenly divisible by 2nd # is also a multiple of 2nd # EX: 52/ 4 = 13, which is an integer - so 52 is evenly divisible by 4 and also a multiple of 4

Polynomial & polynomial forms

https://www.mathplanet.com/education/algebra-1/factoring-and-polynomials/monomials-and-polynomials A polynomial as oppose to the monomial is a sum of monomials where each monomial is called a term. The degree of the polynomial is the greatest degree of its terms. A polynomial is usually written with the term with the highest exponent of the variable first and then decreasing from left to right. The first term of a polynomial is called the leading coefficient. 4x5+2x2−14x+12 other forms: x² - 3x + 2 = (x - 1 )(x - 2) a² - b² = (a + b) (a - b) a² + 2ab + b² = (a + b)² a² - 2ab + b² = (a - b)²

Quadratic equations pg. 250 NEEDS REVIEW

quadratic equation is an eq. in the form: ax² + bx + c = 0 Many quadratic equations have 2 solutions, in other words, the eq. will be true for 2 dif. values of x When you see a quadratic equation on the GRE, you'll generally be able to solve it by factoring the algebraic expression, setting each of the factors equal to 0, and solving the resulting equations EX: x² - 3x + 2 = 0. solve for x 1. to find the solutions, or roots, start by factoring x² - 3x + 2 = 0 into (x - 2) (x - 1) = 0 2. the product of the 2 quantities equals 0 only if one (or both) of the quantities equals 0. So if you set each go the factors equal to 0, you will be able yo solve the resulting equations for the solutions of the original quadratic eq. 3. Setting the 2 binomials equal to 0 gives you: x - 2 = 0 OR x - 1 = 0 4. This means that x can equal 1 or 2. To check, you can plug each of those values into x² - 3x + 2 = 0 and you will see that either value makes the eq. work.

# Line

straight line (extending infinitely in either direction) on which numbers are represented as points # to the right of 0 = pos & to the left of 0 = neg. - 0 is neither neg/pos Irrational numbers (ex sq. root of 2), decimals and fractions can also be depicted on a # line pi can be on # line **pg. 198 for picture**

Discount on an item

the original price minus the reduced price - a discount is often represented as a % of he original price

The difference

the result of subtraction

Add/subtract polynomials

to add/subtract polynomials, combine like terms

Interest

- given as a % per unit of time, such as 5% per month

Sale price

= final price after discount or decrease

Calculating probability of multiple events for Dependent probability

To calculate probability of 2 or more dependent events occurring, multiply their individual probabilities, but you must calculate each dependent event as if the preceding event had resulted in the desired outcome or outcomes EXAMPLE 275

Mean, median, mode (covered in statistics section)

To find the mean, add up the values in the data set and then divide by the number of values that you added. - find the average To find the median, list the values of the data set in numerical order and identify which value appears in the middle of the list. To find the mode, identify which value in the data set occurs most often - which value is repeated the most

Shortest distance from a point to a line Perpendicularity and Right Angles

To find the shortest distance form a point to a line: 1. draw a line segment from the point to the line such that the line segment is ⟂ to the line 2. Measure the length of that segment The distance (or perpendicular distance) from a point to a line is the shortest distance from a fixed point to any point on a fixed infinite line in Euclidean geometry. It is the length of the line segment which joins the point to the line and is perpendicular to the line.

Reciprocals of fractions

To get reciprocal of common fraction, flip fraction upside down so that the numerator becomes denominator & vise versa If fraction has a numerator of 1, the fraction's reciprocal will be equivalent to an integer Ex: reciprocal of 1/25 = 25

Multiplying fractions

To multiply fractions, multiply the numerators and multiply the denominators EX: 5/7(3/4) = 15/28 Reducing fractions before you multiply can help minimize errors in your calculations ^EX. on pg 217: (10/9)(3/4)(8/15) Cancel 5 out of the 10 & 15, 3 out of the 3 & 9, and 4 out of the 4 & 8 (2/3)(1/1)(2/3) = 4/9

Multiplication and division of radicals

To multiply/divide one radical by another, multiply/divide the # outside the radical signs, then the #s inside the radical signs EX of multiplication: (6√3)2√5 = (6)(2)(√3)(√5) = 12√15 EX of division: 12√15 divided by 2√5 = (12/2)(√15/√5) = 6√15/5 = 6√3

Total surface area of a cylinder

Total surface area of a cylinder = Areas of circular ends + Lateral surface area = 2πr^2 + 2πr(h)

Isosceles triangle

Triangle w/ 2 equal sides, which are opposite 2 equal angles 3rd, unequal side is called the base

Right triangle

Triangle with one interior angle of 90 deg, or right angle

Permutation

W/in any group of items or people, there are multiple arrangements or Permutations possible Permutations differ from combinations in that permutations are ordered. By definition, each combination larger than 1 has multiple permutations (arrangements) On GRE, a question asking: "How many ways/arrangements/schedules are possible? generally indicates a permutation problem To find permutations, think of each place that needs to be filled in a particular arrangement as a blank space. The first place can be filled w/ any item in the larger group. Second place can be filled with any item EXCEPT the item being used to fill the first place and so on for third, fourth... places EX: W/in a group of 3 items: A B C, there are 6 permutations (ABC, ACB, BAC, BCA, CAB, CBA)

Powers of 10

When 10 is raised to an exponent that is a pos. integer, the exponent tells how many zeros the number would contain if it were written out To multiply a # by a power of 10, move the decimal point the same # of places to the right as the value of the exponent (or as the # of zeros in that power of 10). To divide by a power of 10, move decimal point the corresponding # of places to the left

Avg. of consecutive, evenly spaced #s

When consecutive #s are evenly spaced, the average is the middle value If there is an even # of evenly spaced #s, there is no middle value - then the avg is midway (avg of) b/w the 2 middle values Can use this technique whenever you have consecutive integers, consecutive odd/even #s, consecutive multiples of an integer, or any other consecutive #s evenly spaced

Sum of interior angles of a polygon

You can find the Sum of interior angles of a polygon by Dividing the polygon into Triangles ! 1.) Draw diagonals from any vertex to all of the nonadjacent vertices 2.) Multiply the # of triangles by 180 deg. to get the sum of the interior angles of polygon *** The sum of the interior angles of any Triangle is always 180 deg.

Constant (term)

a # not multiplied by any variable(s)

Commutative Law of addition and multiplication

a +b = b +a ab = ba Addition and multiplication are both commutative - meaning that switching the order of any 2 #s being added/multiplied together does no affect the result * Subtraction and division are not commutative

variable

a letter/symbol representing an unknown quantity

Averages

average of a group of # is the sum of the terms divided by the # of terms Avg = sum of terms/# of terms # of terms = sum of terms/ Avg Sum of terms = (# of terms)(Avg) Arithmetic mean is just another term for average

Basic algebraic operations

combining like terms adding/subtracting polynomials factoring algebraic expressions

Sum

result of addition

Profit made on an item

seller's price minus the cost to the seller

Algebraic equation

two algebraic expressions separated by an equal sign or one algebraic expression separated from a # by an equal sign

Remainder

what is left over in a division problem remainder is always smaller than the # you are dividing by Ex: 17/3 = 5 with a remainder of 2 12/6 = 2 with a remainder of 0 GRE writers often disguise remainder problems *** see pg. 209 in GRE book !!!***

Plane

Plane is a flat surface that extends indefinitely in any direction

Tables, Graphs, & Charts

Qs, esp. in data interpretation, combine #s and texts with visual formats Formats that appear most freq. on GRE math Qs are tables, bar graphs, line graphs, & pie charts

Rectangle

Rectangle is a Parallelogram with 4 right angles Opposite sides are equal; diagonals are equal Square = is a rectangle with equal sides

Right triangle

Right angle (90 deg) is always the largest angle in right triangle - therefore the hypotenuse, which lies opposite right angle, is the longest side

Sets elements/members Finite sets Infinite sets Empty sets Nonempty sets

Sets are groups of values that have some common property, such as the neg. odd integers greater than -10 or all pos. integers that are evenly divisible by 3 Items in sets are called elements or members Finite set: If all the elements in a set can be counted (ex: # of species of birds in north america) Infinite set: If the elements in a set are limitless (ex: all pos. #s that are evenly divisible by 3) Empty set: Set with no elements - represented by symbol Ø. By definition, the empty set is a subset of all sets Nonempty set: logically, a set with any members is nonempty Subset: If all elements of set A are among the elements of set B, then A is a Subset of B. By definition, the empty set is a subset of all sets ****Important characteristic of sets is that elements are unique - that is, they are not repeated (for instance, the set of #s 1, 1, 2, 2, 3 is {1, 2, 3} *** Order does NOT matter in sets - {1, 2, 3 } is the same as {3, 2, 1 } Sets are usually enclosed in curly brackets { }

Uniform solid

Solid that could be cut into congruent cross sections (parallel "slices" of equal size and shape) along a given axis Solids on GRE will almost certainly be uniform solids

Rectangular solid

Solid with 6 rectangular faces, all edges meet at right angles EX: cereal boxes, bricks

Perimeter

The distance around a polygon The sum of the lengths of all its sides

Median ex. pg 261

The middle term in a group of terms that are arranged in numerical order. To find the median of a group of terms, first arrange the terms in numerical order: If there is an odd # of terms in the group, the median is the middle term If there is an even # of terms in the group, the median is the average of the 2 middle terms The median of a group of #s is often different from its average!

Circle

The set of all points in a plane at the same distance from a certain point This point is called the center of the circle Circle is labeled by its center point; circle O, means the circle with center point O

Sides and angles of triangles

The sum of the lengths of any 2 sides of a triangle os greater than the length of the 3rd side If the lengths of 2 sides of a triangle are unequal, the greater angle lies opposite the longer side & lesser angle lies opposite of the shorter side Since the 2 legs of an isosceles triangle have same length, the 2 angles opposite the legs must have same measure

Triangle inequality Theorem

The triangle inequality theorem states that: 1.) the third side of a triangle will always be greater than the positive difference between the triangle's other two sides 2.) third side of triangle it will be less than the sum of these two sides.

Acute angle

An acute angle measure less than 90 deg.

Area of a triangle

Area of triangle formula = A = (1/2) (base)(height) A = (1/2) (B)(H) Area of right triangle is simple to find - think of one leg as the base and the other as the height & area of half of the product of the legs (2 legs making the right angle)

Quizbank 5 : Q17

TAKEAWAY: When an equation involves exponents, the first step should almost always be to rewrite it so that as many of the powers as possible have the same base.

integer

a whole number, including pos/neg whole #s and zero # w/o fractional or decimal parts # that completes itself *** All integers are multiples of 1 ex: (...-5,-4,-3,-2,-1,0,1,2,3,4,5...)

Distributive Law of multiplication

a(b + c) = ab + ac Allows you to "distribute" a factor over #s that are added or subtracted You do this by multiplying that factor by each # in the group ex: 4(3+7) = (4)(3) + (4)(7) = 40 ** Law works for numerator in division also** does NOT work in denominator 9/4+5 does not equal 9/4 + 9/5

properties of odd/even #s

odd + odd = even even + even = even odd + even = odd odd x odd = odd even x even = even odd x even = even ** multiplying an even # by any integer always produces another even # ** review pg. 206 example on picking #s

Square root

square root of any non-negative # x is a # that when multiplied by itself, yields x Every positive # has 2 square roots - one pos. and one neg. EX: square root of 25 = 5 bc. 5^2 = 25. The neg. square root of 25 = -5 bc. (-5)^2 = 25 By convention, the radical symbol √ stands for the pos. square root only - therefore √9 = 3 only, even tho both 3^2 and (-3)^2 = 9

coefficient

the numerical constant by which one or more variables are multiplied. 3x^2 - coefficient is 3 A variable/product of variables without a numerical coefficient (z or xy^3) is understood to have coefficient of 1

Product

the result of multiplication

Combined rate problems

Rates can be added!! Ex. pg 235 Add rates together Divide total area/quantity by combined rate to get the time

cube

# raised to the 3rd power ex: 4^3 = 64

Exponent

# that denotes the power to which another #/variable is raised to Typ. written as a superscript to a # Can be referred to as a "power" (5^3 = 5 to the 3rd power) Exponents can be pos/neg integers or fractions and they may include variables

Lateral surface area of cylinder

(Circumference of base/circle) (height) = 2πr(h)

Associative laws of addition and multiplication

(a + b) + c = a + (b +c) (ab)c = a(bc) Add. and Multiplication are commutative and associative Regrouping the numbers does not affect the result

Volume of a rectangular solid

(area of base)(height) = (Length x Width)(height) = lwh

alternative strategies for mult. choice algebra problems if don't know how to solve Review Examples on pg. 251 - 253

1. Back solving - sub answer choices into eq. until one works - can waste time - think about which choices would work best before going thru all of them 2. Picking numbers - When answer choices consist of variables or algebraic expressions, picking numbers can make the problem seems less abstract Evaluate the answer choices and he info in the Q stem by picking a # and substituting it for the variable wherever variable appears in expression *** When using this method of picking #s, ALWAYS check every answer choice to make sure you haven't chosen a # that works for more than one answer choice - if thats the case, you'll need to pick a new number to sub for the variable 3. Using picking numbers method to solve for one unknown in terms of another - If the first # you pick doesn't lead to a single correct answer, be prepared to pick a new # (and spend more time on the prob) or settle for guessing strategically among answers you have not eliminated

Commonly tested properties of powers -many quantitative comparison problems test your understanding of what happens when neg. #s and fractions are raised to a power

1. Raising a fraction b/w 0 and 1 to a power produces a smaller result EX: (1/2)^2 = (1/2)(1/2) = 1/4 2. Raising a neg. # to an even power produces a positive result EX: (-2)^2 = 4 3. Raising a neg. # to an odd power gives a neg. result EX: (-2)^3 = -8 4. Raising an even # to any exponent gives an even # EX: 8^5 = 32,768 = even 5. Raising an odd # to an exponent gives an odd # EX: 5^8 = 390,625 = odd

Hypotenuse

Longest side of a right triangle Hypotenuse is always opposite of the right angle

Special word problem tips

1.) Do not try to combine several sentences into one equation; each sentence usually translates into separate equation 2.) Pay attention to what the question asks for and make a note to yourself if it is not one of the unknowns in the equation(s). Otherwise, you may stop working on the problem too early

Comparing positive fractions

1.) Given 2 positive fractions w/ same denominator, the fraction with the larger numerator will have larger value (3/8 < 5/8) 2.) BUT if given 2 positive fractions with the same numerator but different denominators, the fraction with the smaller denominator will have larger value (3/4 > 3/8) 3.) If neither the numerator/denominator are the same, you have 3 options: A.) Turn both fractions into their decimal equivalents B.) Express both fractions in terms of some common denominators and then see which new equivalent fraction has the largest numerator C.) 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? cross multiply 9x5 = 45 and 6x7 = 42 THUS 5/6 is greater pg. 219

Isosceles right triangle 2 types of Special Right triangles

1.) Isosceles right triangle: 45°/45°/90° R. triangles * Sides of Isosceles right triangle are in ratio: x: x: x√2 x√2 representing the hypotenuse ***Remember the longest side has to be opposite the greatest angle

scientific notation

123,000,000,000 = 1.23 x 10^11 0.000000003 = 3.0 x 10^-9 **Multiplying by 10^-9 is equivalent to dividing by 10^9

complementary angle

2 angles are complementary to each other is their measures sum up to 90 deg.

Supplementary angle

2 angles are supplementary if their measures sum up to 180 deg.

Adjacent angles & Opposite/ Vertical angles

2 intersecting lines form 4 angles Adjacent Angles = The angles that are adjacent to each other are Supplementary (add up to 180 deg.) because they lie along a straight line Opposite/ Vertical angles = The 2 angles that are not adjacent, are vertical or opposite angles. Opposite angles are equal in measure because each of them is supplementary to the same adjacent angle (ie they are on either side of the same angle and thus supplementary) 2 intersecting lines form 4 angles & each angle is supplementary to each of its 2 adjacent angles

Solid

3D figure Dimensions usually called length, width, and height (l,w,h) or height, width, depth (h, w, d) Only 2 types of solids that appear w/ any frequency on the GRE: rectangular solids (including cubes) and cylinders

Angles around parallel lines intersected by a transversal refer to Diagram on pg. 279

A Transversal is: A line that intersects 2 parallel lines Each of the parallel lines intersects the third line (the transversal) at the same angle *** When 2 or more parallel lines are cut by a transversal: 1. All acute angles formed are equal 2. All obtuse angles formed are equal 3. Any acute angles formed are supplementary to any obtuse angle formed (sum to 180 deg.) If a transversal intersects 2 parallel lines, then the pairs of corresponding angles are congruent Congruent Angles have the same angle (in degrees or radians). That is all. They don't have to point in the same direction. They don't have to be on similar sized lines.

Angle bisectors

A line or line segment bisects an angle if it splits the angle into 2 smaller, equal angles The 2 smaller angles are each half the size of the original angle

Diagonal of a polygon

A line segment connecting any 2 nonadjacent vertices

Diameter

A line segment that connects 2 points on the circle and passes thru the center of the circle. AB is a diameter of circle O above (pg. 292)

Radius

A line segment that connects the center of the circle with any point on the circle Radius is 1/2 of the diameter

Chord

A line segment that joins 2 points on the circle. The longest chord of a circle is its diameter

Transversal

A line that intersects 2 parallel lines If a transversal intersects 2 parallel lines, then the pairs of corresponding angles are congruent

Tangent

A line that touches only 1 point on the circumference of a circle. A line drawn tangent to a circle is perpendicular to the radius at the point of tangency

Lists

A list is like a finite set, except 1.) Order does matter 2.) Duplicate members can be included SO, 1, 2, 3 and 3, 2, 1 are different lists and 1, 2, 3, 2 is a valid list Bec. order DOES matter, elements can be uniquely identified by their position, such as "1st element" or "5th element" Lists are not enclosed in { } brackets

Vertex of a polygon

A point where 2 sides of a polygon intersect Polygons are named by assigning each vertex a letter and listing them in order Pic is named: AEDCB

Inscribed/Circumscribed figures

A polygon is inscribed in a circle if all the vertices of the polygon lie on the circle. A polygon is circumscribed about a circle if all the sides of the polygon are tangent to the circle Pg. 302: Square ABCD is inscribed in circle O. We can also say that circle O is circumscribed about square ABCD Pg. 303: Square PQRS is circumscribed about circle O. We can also say that circle O is inscribed in square PQRS Pg. 304: When a triangle is inscribed in a semicircle in such a way that one side of the triangle coincides w/ the diameter of the semicircle, the triangle is a right triangle REVIEW EXAMPLE OF CHORD PG 303-304

Triangles

A polygon with 3 straight sides and 3 interior angles that add up to 180 deg *** The sum of the interior angles of any Triangle is always 180 deg.

Cylinder

A uniform solid whose horizontal cross section is a circle Cylinders measurements generally given interns of its radius, r, and its height, h EX: soup can or a pipe that is closed at both ends

Comparing decimals

Add zeros after last digit to the right of the decimal point in each decimal fraction until all the decimals you're comparing have the same # of digits - essentially giving all the fractions the same denominator so that you can just compare numerators EX on page 222 - arrange decimals in order from smallest to largest

Triangles and quantitative comparison

All quantitative comparison Qs require you to judge whether enough info has been given to make a comparison For triangles, keep in mind: 1.) If you know 2 angles, you know the 3rd 2.) To find the area, you need base and height 3.) In a right triangle: if you have 2 sides, you can find the third if you have 2 sides, you can find the area 4.) in isosceles right triangles and 30°/60°/90° Right triangles, if you know one side, you can find everything

Altitude or height of a triangle

Altitude or height of a triangle is the perpendicular distance from a vertex to the side opposite the vertex. The altitude may fall inside or outside the triangle, or it may coincide with one of the sides

Monomial

An algebraic expression with only 1 term To multiply monomials, multiply the coefficients and the variables separately: 2a x 3a = (2 x 3) (a x a) = 6a^2 A monomial is a number, a variable or a product of a number and a variable where all exponents are whole numbers. That means that 42,5x,14x12,2pq all are examples of monomials whereas

Central Angle

An angle formed by 2 radii. Pg. 292: In circle O, AOC is central angle, COB anf BOA are also central angles (BOA happens to be 180 deg) Total degree measure of a circle is 360 degrees

Semicircle

An arc that is exactly half of the circumference of its circle is semicircle

algebraic expression

An expression containing 1 or more variables, 1 or more constants, and possibly one or more operation symbols. In the case of the expression x, there is an implied coefficient of 1. AN EXPRESSION DOES NOT CONTAIN AN = SIGN 3x^2 + 2x 7x + 1 / 3x^2 - 14

Obtuse Angle

An obtuse angle measures b/w 90 - 180 deg.

Angles

Angles are measured in degrees Angles may be named according to their vertices - When 2 or more angles share a common vertex, an angle is named according to 3 points: 1. a point along one of the lines or line segments that form the angle 2. the vertex point 3. Another point along the other line or line segment Diagrams sometimes show a letter inside the angle, this can also be used to name the angle

Converting decimals

Any decimal fraction is equivalent to some common fraction with a power of 10 in the denominator To convert a decimal b/w 0 and 1 to a fraction, determine the place value of the last non-zero digit and set that value as the denominator. Then use all the digits of the decimal # as the numerator (w/o decimal pt.), if nec. reduce fraction to lowest terms To convert fraction to decimal, simply divide numerator by denominator EX: 0.875 convert to fraction in lowest terms Denominator will be 1,000 bc. 5 is last non-zero number and is in the thousandths place, numerator will be 875. 875/1000 divide top and bottom by 25 to get 35/40 and divide top and bottom by 5 to get the lowest terms: 7/8

Plotting points on coordinate plane

Any point on coordinate plane can be identified by an ordered pair consisting of x-coordinate and y-coordinate. Every point that lies on the x-axis has a y-coordinate of 0, and every point that lies on the y-axis, has an x-coordinate of 0

Quantitative Comparison: pg. 309 - 319

Asked to compare 2 math expressions, Quantity A and Quantity B. Some questions include additional centered info - this centered info applies to both quantities and is essential in the comparison. This type of Q is about the relationship b/w the 2 quantities, you usually won't need to calculate a specific value for either quantity - so don't rely on screen calculator

Histograms

Bar graphs that show relative frequencies or #s of occurrences are called histograms - these graphs can be useful in visualizing patterns and trends in the data The y-axis (up and down) on histograms shows the frequencies The x-axis (horizontal) might show category definitions, values, or ranged depending on what is being graphed One drawback of histograms is that estimating the mean of the data can be very difficult

Boxplot/box-and-whisker plot

Boxplot OR box-and-whiskerplot: a straightforward way to visually display data dispersion by quartiles Boxplot visual depiction uses 5 values: L: the least # in the data G: the greatest # in the data M: the median (middle #) Q1: 1st quartile Q3: 3rd quartile The interquartile range (which includes M) is drawn as a rectangular box, and straight lines extend from the sides of the box to the least and greatest values (L and G). A # line is drawn below the box plot to show the numerical values of these points EXAMPLE pg. 264

adding/subtracting fractions

Cannot add/subtract fractions unless they have the same denominator If same denominator, then add/subtract the numerators (NOT denominators) & if necessary simplify/reduce resulting fraction to lowest terms If denominator is not the same, the least common denominator of the 2 fractions will be the least common multiple of the 2 denominator #s

Combination

Combination Q asks you how many unordered subgroups can be formed from a larger group Some Qs on GRE can be solved w/o any computation just be counting/ listing possible combos

The combination formula

Combination Q asks you how many unordered subgroups can be formed from a larger group Some Qs on GRE can be solved w/o any computation just be counting/ listing possible combos Combination Qs can use #s that make quick, non computational solving difficult, so we use the combination formula: See pic ! symbol means factorial *When you are asked to find the potential combinations from multiple groups, multiply the potential nominations from each group - EX on pg 272: use the formula for the 2 sep groups in the Q and then multiply them **Sometimes, GRE will ask you to find the # of possible subgroups when choosing one item from a set. In this case, the # of possible subgroups will always = # of items in the set (Ex 272)

Congruent Angles ≅ congruent to equivalence of geometric shapes and size

Congruent Angles have the same angle (in degrees or radians). That is all. They don't have to point in the same direction. They don't have to be on similar sized lines. Symbol for congruent angles: ≅ congruent to equivalence of geometric shapes and size

Data Interpretation

Data Interpretation Questions are based on info located in tables or graphs, and they are often statistics oriented Data may be located in one table or graph, but you also might need to extract data from 2 or more tables or graphs

Symbolism Ex. on 247- 248

Don't panic if you see strange symbols like stars or diamonds on GRE Problems like this typ. don't require more than substitution - read Q stem carefully for definition of the symbols and for any examples of how to use them. Follow given model, substituting the #s that are in the Q stem

Equilateral triangle

Equilateral Triangle = 1.) 3 sides are all equal in length and 2.) 3 interior angles each measure 60 deg

Estimation/rounding on GRE

Ex: to round off 235 to the tens place, look at the units place - to the right of the tens place there is a 5 - so round the 3 up to get the # 240. If it were a 4 instead of a 5, the # would have been rounded down to 230 Rounding off #s before calculations will allow you to quickly estimate the correct answer - estimating can save valuable time on GRE but look at answer choices - if choice values are relatively close together, you'll have to be more accurate

Percentiles

For large groups of #s, position of given data points is sometimes stated in % rather than quartiles. The principle is the same as for quartiles, but there are 100 subdivisions instead of 4 Converting quartiles to % is easy: Q1 = 25th percentile Q2 = 50th percentile Q3 = 75th percentile Q4 = 100th percentile

For word problems

For word problems ADD when.. 1. you are given amounts of individual quantities and asked to find the total 2. you are given an original amount and an increase and are then used to find new amount SUBTRACT when... 1. ) you are given the total and one part of the total and you want to find the remaining part/parts 2.) you are given 2 #s and asked how much more or how much less one # is than the other # - the amount is called the difference MULTIPLY when.... 1.) you are given an amount for one item and asked for the total amount of many of these DIVIDE when... 1.) you are given a total amount for many items and asked for the amount for 1 item 2.) you are given the size of 1 group and the total size for many such identical groups and are asked how many of the small groups fit into the larger one

Greatest common factor

GCF, or greatest common divisor, of a pair of integers is the largest factor that they share Ex: GCF of 36 and 48? to find the GCF breakdown both integers into their prime factorizations and multiply all the prime factors they have in common: 36 = (2)(2)(3)(3), and 48 = (2)(2)(2)(2)(3) ^ they have two 2s and one 3 in common, so the GCF is (2)(2)(3) = 12 pg. 207 in GRE book

Polynomial

General name for an algebraic expression with more than 1 term An algebraic expression with 2 terms is called binomial expression

Solving for one unknown in terms of another

Generally, in order to solve for the value of an unknown, you need as many distinct equations as you have variables (ie you have 2 variables, you need 2 distinct equations) Some GRE Qs don't require solving for numerical value of unknown. Instead, you are asked to solve for 1 variable in terms of the other(s) - to do so, isolate the desired variable on one side of the equation and move all the constants/ other variables to the other side EX. on 244 - 245

Simplifying radicals

If # inside radical is a multiple of a perfect square, the expression can be simplified by factoring out the perfect square EX: √72 = (√36)√2 = 6√2

Scatterplots Refer to example on pg 259

If 2 measured variables are related to each other, the data are called BIVARIATE data A scatterplot is often best way to graphically display such data - one variable is plotted on the x-axis and the other plotted on y-axis Thus, each ordered pair of measured values represents one data point that is plotted on the graph Scatterplots are useful for visualizing the relationships b/w 2 variables - a trend line shows the nature of that relationship and clearly highlights data points that deviate significantly from the general trend - trend line can either be straight or curved, and will often be drawn on the scatterplot in the Q Trend lines can be used to make predictions by interpolating along the trend line or extrapolating beyond the trend line - Interpolation is guessing data points that fall within the range of the data you have, i.e. between your existing data points. Extrapolation is guessing data points from beyond the range of your data set. Scatterplots are also useful to spot outliers - individual data points that deviate from the trend

Coordinate Geometry

In Coordinate Geometry, the locations of points in a plane are indicated by ordered pairs of real #s

Logic problems

In logic problems, issue is not translating english to math, but simply using your head/common sense Problem may call for non math skills: such as the ability to organize and keep track of different possibilities, ability to visualize something (ex: reverse side of a symmetrical shape), the ability to think of the exception that changes the answer to a problem, or the ability to deal w/ overlapping groups

Probability of Multiple events

In probability Qs involving more than 1 event, the events are independent - meaning one event doesn't effect the other. There are also other cases where the results are NOT independent. EX: ^ If there are 4 red disks and 4 green disks in a bag and 2 disks w/drawn at random w/o replacement, the probability for the result of the second draw is Dependent on the result of the 1st draw. What about the probability of one or another event occurring? On the GRE, you can interpret "the probability of A or B" to mean "the probability of A or B or both" and the formula for calculating this is similar to the inclusion-exclusion principle for sets: P(A or B) = P(A) + P(B) - P(A and B) EX 275 - 276

Picking #s with percents

In problems in which no actual values are mentioned, just percents - assign values to the percents you are working with and you'll find the problem less abstract You should almost always pick 100 in % problems, because its relatively easy to find %s of 100.

Prime #

Integer greater than 1 that only has 2 factors: itself and 1 # 1 is not considered prime because it is divisible on by itself #2 is smallest prime # and the only even prime # - every other even # must have 2 as a factor and thus it can't be prime 1st ten prime #s: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29

Interquartile range & Boxplot/box-and-whisker plot pg. 263 Not many GRE Qs address quartiles or interquartile range, but can be helpful to achieve high score

Interquartile range is the dif. b/w the values of the 3rd and 1st quartile values: Q3 - Q1 ***Interquartile range is a way to measure how spread out the values in a given data set are Bec. outliers fall into the bottom and top quartiles, they do not affect Interquartile range Boxplot OR box-and-whiskerplot: a straightforward way to visually display data dispersion by quartiles Boxplot visual depiction uses 5 values: L: the least # in the data G: the greatest # in the data M: the median (middle #) Q1: 1st quartile Q3: 3rd quartile The interquartile range (which includes M) is drawn as a rectangular box, and straight lines extend from the sides of the box to the least and greatest values (L and G). A # line is drawn below the box plot to show the numerical values of these points EXAMPLE pg. 264

Ratios of more than 2 terms

It is possible to set up ratios with more than 2 terms - they convey more info/express more relationships but the principle of two-term ratios are just as applicable to ratios of more than 2 terms EX on pg. 233

Kaplan's additional tips for data interpretation questions

Kaplan's additional tips for data interpretation questions 1.) Slow down - Lots going on in data interpretation Qs! If you slow down the first time thru, you can avoid calculation errors and having to reread Qs and charts 2.) Pace yourself wisely - Remember each Q type has the same value of points!! - ^^ If you must miss a few Qs in a section, make them the ones that would take you the longest to answer, not the ones at the end of the section that you could have answered correctly but simply didnt get to - Data interpretation Qs tend to be some of the more time-consuming Qs to answer, if they are not your strong suit, move on and save them for the end

Line Graph

Line graphs follow same general principles as bar graphs, except instead of using the lengths of bars to represent #s, they use points connected by lines - the lines further emphasize the relative values of the #s

Line Straight Line

Line is a 1Dimensional (1D) geometrical abstraction that is infinitely long, w/o a width Straight line = shortest distance b/w any 2 points; There is exactly 1 straight line that tased thru any 2 points

Line segment

Line segment is a section of a straight line of finite length, with 2 end points Line segment is named for its end points, EX: segment AB

Translating english to math in part-whole problems on the GRE

Many fractions/percents appear in word problems - solve the problems by plugging #s you're given into some variation of one of the three basic formulas: Part/whole = fraction Part/whole = decimal Part/whole (100) = percent *** Avoid careless errors and look for key words IS and OF IS (or ARE) often introduces the part OF almost invariably introduces the whole

Mixed #s and Improper fractions

Mixed # consists of an integer and a fraction Improp fraction is a fraction whose numerator is > than its denominator To convert improper fraction to Mixed #, divide the numerator by the denominator. The # of "whole" times the denominator goes into the numerator will be the integer portion of the improper fraction. The remainder will be the numerator of the fraction portion (over the original denominator) EX: 23/4 = 5 3/4 To convert mixed # to an improper fraction: multiply the integer portion by the denominator and add the numerator - the result of this is your new numerator and the denominator will not change. EX: 2 3/7 = 17/7

decimal system - is the only numbering system used on GRE

Numbering system based on the powers of 10 Each figure or digit in a decimal # occupies a particular position - from which it derives its place value ** diagram on pg 198 of GRE book** Integer part on left of decimal point- from R. to L. of decimal: units, tens, hundreds, thousands, 10 thousands, 100 thousands, millions Decimal part on right of decimal point - from L. to R: tenths, hundredths, thousandths, 10 thousandths, 100 thousandths, millionths

Digit

One of the numerals: 0,1,2,3,4,5,6,7,8,9 A # ca have several digits ex: 542 has 3 digits (5, 4, and a 2) ex2: 321,321,000 has 9 digits, but only 4 distinct/dif digits (3, 2, 1, and 0)

Common percentage equivalents - be familiar with to save time on GRE

Page 226 in book - make paper flashcards

Parallelogram

Parallelogram is a quadrilateral with 2 paid of parallel sides Opposite sides are equal in length; Opposite angles are equal in measure Angles that are not opposite are supplementary to each other

Part:whole ratios

Part:whole ratios - whole is the entire set, while part is a certain subset of the whole on GRE, the word fraction generally indicates a part:whole relationship

Pythagorean theorem

Pythagorean theorem holds true for all right triangles & NO other triangle Pythagorean theorem states that the square of the hypotenuse is equal to the sum of the squares of the legs (Leg 1)² + (Leg 2)² = (Hypotenuse)² C² = A² + B² C = √A² + B²

Rates

Rate relates one kind of quantity to a completely dif. kind (instead of relating a part to the whole or to another part) When we talk about rates, we usually use the word per, as in miles per hour, cost per item... Since per means "for one" or "for each" we express the rates as ratios reduced to a denominator of 1 Speed is most commonly tested rate on GRE - but others exist such as liters per min, or cost per unit *ALL RATE PROBLEMS CAN BE SOLVED USING THE SPEED FORMULA AND ITS VARIANTS - by conceiving "speed" as "rate and "distance" as "quantity"

Ratios

Ratio = the proportional relationship b/w 2 quantities Ratio/relationship b/w 2 #s can be expressed with a colon b/w #s, words, or as a fraction (2:3 or ratio of 2 to 3 or 2/3) A ratio is a constant proportion, it can be multiplied/divided by any # without losing its meaning, as long as the mult/division are applied to all components of the ratio To translate ratio in words to fractional ratio, use whatever follows the word OF as numerator and whatever follows TO as denominator

Ratio vs Actual #

Ratios usually reduced to their simplest form - if the ratio of men to women in a room is 5:3, you cannot necessarily infer that there are exactly 5 men and 3 women If you knew the total # of people in the room, in addition to the male-to-female ratio, you could determine the # of men and # of women in the room

Sector

Sector is a portion of a circle's area that is bounded by 2 radii and an arc Like arcs, sectors are associated with central angles. To find the area of a sector: 1.) find the degree measure of the sector's central angle and figure out what fraction that degree measure is of 360 deg 2.) Multiply the area of the whole circle by that fraction Ex: in a sector whose central angle measures n degrees: Area of sector = n/360 (area of a circle) = n/360 πr²

Volume of a cube

Since a cube is a rectangular solid for which l = w = h, the formula for its volume can be stated in terms of any edge Volume of a cube = kWh = (edge)(edge)(edge) = e^3

Normal Distribution REVIEW EXAMPLE ON 268 AND REVIEW GRAPH

Special kind of frequency distribution = normal distribution - closely tied to concept of standard deviation Normal distribution is often referred to as a Bell Curve bc of its shape Only 2 parameters are needed to define normal distribution: 1.) the mean and 2.) the standard deviation In a normal distribution, the mean equals the median, and the data are symmetrically distributed around the mean, so the curve to the left of the mean is a mirror image of the curve to the right Graph on pg. 266 show some important probability values that hold true for ALL normal distributions - The percentage of the area under any portion of a distribution curve equals the probability that a randomly selected event will fall within that area's range Graph is on desktop in math symbols doc

Quartiles pg. 263 Not many GRE Qs address quartiles or interquartile range, but can be helpful to achieve high score

Steps in determining quartiles: 1.) Arrange/re-arrange terms in data set in numerical order 2.) Subdivide the set of terms into 4 quarters, each containing an equal # of terms 3.) The largest # in the first (lowest) group will represent the first quartile - often written as Q1 4.) Largest in the second group is Q2. Q3 is top in the third quartile, & Q4 is the max value of the set NOTE: the rules for determining quartiles if the # of terms is not evenly divisible by 4 can vary, so it's unlikely a GRE Q would require you to determine quartiles of a set where the # of terms is not a multiple of 4

Surface area of a rectangular solid

Sum of areas of faces = 2lw + 2lh + 2hw

Range EX. pg. 262

The distance b/w the greatest and least values in a group of data points Find the range of a set of #s by subtracting the smallest # in the set from the largest *** range is a way to measure how spread out the values in a given data set are NOTE: sets with the same mean or median may have very different ranges

Factors

The factors, or divisors of an integer are pos. integers by which it is evenly divisible (leaving no remainder) Ex: factors of 36? 36 has 9 factors: 1,2,3,4,6,9,12,18,36 Can group these factors into pairs: (1)(36) = (2)(18) = (3)(12) = (4)(9) = (6)(6)

x-axis and y-axis

The horizontal (x) and vertical (y) lines that intersect perpendicularly to indicate location on a coordinate plane. Each axis is a number line

Arc length

The length of an arc is the same fraction of a circle's circumference as its degree measure is of 360 deg.(deg measure of a whole circle) For an arc with a central angle measuring n°: arc length = n/360 (circumference) arc length = n/360 (2πr)

Origin

The point where the x and y axs intersect - its coordinates are (0,0)

PRABAILITY

The probability that a number greater than 4 will first appear on the third or fourth roll is equal to the probability that a number greater than 4 will first appear on the third roll plus the probability that a number greater than 4 will first appear on the fourth roll. The probability formula is Probability=Number of desired outcomesNumber of possible outcomes. When a number greater than 4 is rolled, the number must be a 5 or a 6. When a fair die is rolled, and the event is rolling a number greater than 4, there are 2 desired outcomes, which are 5 or 6, and the number of possible outcomes is 6, since one of the numbers 1, 2, 3, 4, 5, or 6 will be rolled. The probability that a number greater than 4 is rolled is 26=13. The probability that an event does not occur is equal to 1 minus the probability that the event occurs. So the probability that a number that is not greater than 4 shows when the fair die is rolled is 1−1/3=3/3−1/3=3−1/3=2/3. To have a number greater than 4 first appear on the third roll, there must be a number less than or equal to 4 on each of the first 2 rolls, and a number greater than 4 on the third roll. The rolls of the die are independent of one another. The probability that a number greater than 4 first appears on the third roll is 2/3×2/3×1/3=2×2×1/3×3×3=4/27. To have a number greater than 4 first appear on the fourth roll, there must be a number less than or equal to 4 on each of the first 3 rolls, and a number greater than 4 on the fourth roll. Again, the rolls of the die are independent of one another. The probability that the first number greater than 4 appears on the fourth roll is 2/3×2/3×2/3×1/3=2×2×2×1/3×3×3×3=8/81. Thus, the probability that a number greater than 4 first appears on the third roll is 427, and the probability that a number greater than 4 first appears on the fourth roll is 881. Then the probability that a number greater than 4 first appears on either the third roll or the fourth roll is 4/27+8/81=4/27×3/3+8/81=12/81+8/81=12+8/81 =20/81.

Sum of angles around a point

The sum of the measures of the angles around a point is 360 degrees

Mode EX. pg 262 - worded example and table example REVIEW

The term that appears most frequently in a set A set may have more than one mode if 2 or more terms appear an equal # of times within the set and each appears more times than any other term If every element in the set occurs an equal # of times, then the set has no mode

Segmented bar graphs OR stacked bar graphs

This type of bar graph display multiple quantities on each bar - these quantities represent different subgroups that sum to the amount at the top of each bar The first segment value can be read directly - values for other categories must be calculated by subtracting the value at the bottom of the portion of the bar for that category from the value at the top of that portion on the bar

Combining averages

When there is an equal # of terms in each set, and ONLY when there is an equal # of terms in each set you can average averages Can combine averages by concept of balanced average or by using average formula Can also solve a weighted average by concept of balanced average or by using average formula EX. pg 240-241

Solving Equations EX pg. 243

When you manipulate any equation, always do the same thing on both sides of the = sign TO solve algebraic equation w/o exponents for a particular variable, you have to manipulate the equation until that variable is on one side of the equal sign with all the numbers/other variables on the other side - can perform add, sub, mult, and division ( as long as quantity by which you are dividing does not = 0)

Quadrants of coordinate plane

When you start at origin (0,0) and move: 1. To the right ..... X is positive 2. To the left......... X is negative 3. Up ..... y is positive 4. Down ..... Y s negative Coordinate plane is divided up into 4 quadrants PIC

decimal

a fraction written in decimal system format (ex: 0.6) to convert fraction to decimal, divide the numerator by denominator ex: 5/8 = 5 divided by 8 = 0.625

Multiples

a multiple is the product of a specified # and an integer (3,12, 90 are all multiples of 3) Multiples don't have to be of integers, but all multiples must be product of a specific # and integer ex: 2.4, 12, 132 all multiples of 1.2 2.4 = (1.2)(2), 12 = (1.2)(10), 132 = (1.2)(110)

Term

a numerical constant; also, the product of a numerical constant and one or more variables

part

a specified # of the equal sections that compose a whole

distinct

different from eachother Ex: 12 has three prime factors (2, 2, 3) but only 2 distinct facors (2 and 3) * factors of a # are number or quantity that when multiplied with another produces a given number or expression * factors of a # can be found by making a factor tree and simplifying

fraction

division of a part by a whole part/whole = fraction ex: 3/5 **The only way a fraction can equal 0 is if the numerator is equal to 0. As for the denominator, recall that the denominator of a fraction can never equal 0.

Combining like terms

process of simplifying an expression by adding together or subtracting terms that have the same variable factors *The commutative, associative & distributive laws that govern arithmetic operations with ordinary #s also apply to algebraic terms and polynomials

Whole

quantity that is regarded as a complete unit

pi

ratio of the circumference of a circle to the diameter

Mean

the average of a set of #s

prime factorization of a #

the expression of the # as a product of its prime factors - factors that are prime #s Write out prime factorization by putting the prime #s in increasing order ex: (2)(3)(5)(5)(7) Prime factorization can also be expressed in exponential form: (2)(3)(5^2)(7) 2 methods to determine a #s prime factorization - pg. 207 GRE book METHOD 1: work your way up through the prime #s, starting with 2 - knowing the first 10 prime #s will be useful and save time METHOD 2: figure out 1 pair of factors, then determine their factors, continuing the process until you're left with only prime #s pg. 207 GRE book!!!!!

numerator

top # of a fraction, representing the part being divided by the whole

Principle

- the sum of money invested is the principle

FOIL method

-When 2 binomials are multiplies, each term is multiplied by each term in the other binomial - Called the FOIL method because: adding the products of the First, Outer, Inner, and Last terms EX: to multiply out (x+ 5)(x - 2), the product of the first terms is: X^2, the product of the Outer terms is: -2x, the product of the inner terms is 5x, and the product of the last terms is -10. Adding, the answer is x^2 + 3x - 10. **It's crucial that you not only know the mechanics of FOIL and reverse FOIL for the GRE, but that you also understand the implications of setting a quadratic equal to 0. When two factors multiplied give a product of 0, one or both of those factors must be 0.

Inequalities

2 difference b/w solving an inequality (2x < 5) and solving an equation (2x - 5 = 0) 1.) First dif: the solution to an inequality is almost always a range of possible values, rather than a single value - the range is more clearly seen when expressing it visually on # line EX. on pg 243 ^ for a range between -4 and 0 - the dots on -4 and 0 on a number line would NOT be shaded/filled in range expressed algebraically by the inequality -4 < x < 0 EX: range is greater than -1, up to and including 3 Dot on -1 is not filled in but the dot/point on 3 is filled in bc. it includes 3 algebriacally expressed by the inequality: -1 < x ≤ 3 (less than or equal to 3 ) 2.) 2nd Dif: if you multiply/divide the inequality by a negative #, you have to reverse the direction of the inequality (EX. on 244) -x ≥ -4 Mult. both sides by -1 and change sign and direction on # line x ≤ 4 and point on 4 of the #line is shaded in bc x is less than OR = to 4

Perpendicularity and Right Angles

2 lines are perpendicular (⟂) if they intersect at a 90 degree angle (ie Right Angle) If Line L 1 is ⟂ to line L2, we write L1 ⟂ L2 If lines L1, L2, L3 all lie in the same plane, and if (L1 ⟂ L2) & (L2 ⟂ L3), then (L1 || L3) (^ If lines L1, L2, L3 all lie in the same plane, and if Line 1 is perpendicular to Line2 and Line 2 is perpendicular to Line 3, then Line 1 and Line 3 are parallel

Properties of zero

Adding/subtracting zero to/from a # does not change the # Subtracting a # from zero, changes the numbers sign 0 - 3 = -3 The product of zero and any # is always zero Division by zero can't be done - so any fraction with 0 in denominator can't be done - when given a fraction with an algebraic expression in denom. be sure expression cannot equal zero

Concept of "balanced value" Another way to find a missing #

Another way to find a missing # is to understand the sum of the difference b/w each term and the mean of the set must equal zero EX: Avg rainfall for 1976 - 1979 was 26 inches per year. There were 24 in of rain in 1976, 30 in 1977, 19 in 1978, how many inches were there in 1979? (24 - 26) + (30 - 26) + (19 - 26) + (? - 26) = 0 -2 + 4 + -7 + ? = 0 -5 + (? - 26) = 0 ? = 31

Arc Arc length

Arc = the section of the circumference of a circle Any arc can be thought of as the portion of a circle cut off by a particular central angle Since arcs are associated with central angles, they can be measured in degrees. The degrees measure of an arc is equal to that of the central angle that cuts it off

Factoring the difference b/w 2 squares Pg. 250

Common factorable expression on GRE is dif. of 2 squares (ie. a² - b ²) - once you recognize polynomial as the dif. of 2 squares, you'll be able to factor it. Since any polynomial of the form a² - b² can be factored into a product to form: (a + b)(a - b) EX: factor expression 9x² - 1 9x² = (3x)² and 1 = 1², SO 9x² - 1 is the dif of 2 squares Therefore: 9x²- 1 = (3x + 1)(3x - 1) Must be a perfect square?? unless it is variables

properties of 1 and -1

Mult/dividing # by 1 does no change the # Mult/dividing a non-zero # by -1 changes the sign of the #

Percent increase & decrease

Percent increase = increase (100%) / Original Percent decrease = decrease (100%) / Original To find increase/decrease, take the difference b/w the original and new - note that the "original" is the base from which change occurs - it may or may not be the first # mentioned in problem BE WISE

Probability

Probability measures the likelihood that an event will occur. Can be expressed as a fraction, decimal, or percent. Every probability expressed a # b/w 0 and 1 inclusive, w/ a probability of 0 meaning "no chance" and a probability of 1, meaning "garunteed to happen" Higher the probability, greater the chance an event will occur An event may include more than one outcome. Many GRE probability Qs are based on random experiments w/ well-defined # of possible outcomes, such as drawing a single card form a full deck. If all the possible outcomes of the experiment are equally likely to occur, you can use this formula to calculate probability: Probability = # of desire outcomes / # of possible outcomes ***The total of the probabilities of all possible outcomes in an experiment must = 1. By this same logic, if P(E) is the probability that an event Will occur, then 1 - P(E) is the probability that an event will NOT occur

Substitution

Process of plugging values into equations, is used to evaluate an algebraic expression or to express it in terms of other variables Replace every variable in the expression with the # or quantity you are told is its equivalent, then carry out designated operations, remembering to follow the order of operations (PEMDAS) Example on pg. 242 - simply the expression as much as you can

Bar graphs

Somewhat less accurate than tables, but not a bad thing - can be helpful on GRE esp. bc estimating often saves time on calculations Handy bc. you can see which values are larger/smaller w/o reading the actual #s - #s are represented on a bar graph by the heights or lengths of the bars If height/length of the bar falls b/w 2 #s on the axis, you will have to estimate Display 1 value for each bar

Advanced algebraic operations

Substitution Solving equations Inequalities Solving for 1 unknown in terms of another Simultaneous equations Method 1 - substitution Method 2 - adding to cancel Symbolism Sequences

Dividing common fractions

To divide fractions, multiply the reciprocal of the # or fraction that follows the division sign Operation of division produces the same result as multiplication by the inverse EX: 1/2 divided by 3/5 = 1/2(5/3) = 5/6

Perimeter of quadrilaterals

To find perimeter of quadrilateral, add up all of its sides Perimeter of rectangle: 2x sum of the length and width Perimeter = 2(length + width) Perimeter of square: all 4 sides same length Perimeter = 4(side)

Ordered pair

Two #s or quantities separated by a comma and enclosed in parentheses (Ex: (7,8)). All ordered pairs on GRE will be in form (x,y), where first quantity x tells you how far the point is to the left or right of the y-axis, and 2nd quantity, y, tells you how far the point is below/above x-axis

Vertex

Vertex is the point at which 2 lines or line segments intersect to form and angle

Work formula for 2

When only 2 people/machines in combined work problem, we can use simplified work formula: T = ab/ a + b 1/a + 1/b = 1/T (ab)(1/a + 1/b) = (1/T)(ab) ab/a + ab/b = ab/T b + a = ab/T T(b+ a) = (ab/T)(T) T(b+ a) = ab T = ab/ a + b a = the amt of time it takes for person a to complete job b = the amt of time it takes for person b to complete job Example on pg. 237 - john and Mark gardening

Work problems (given hours per unit of work) Page 236

Work formula can be used to find out how long is takes a # of people working together to complete a task EX: say we have 3 people, the 1st takes "a" units of time to complete job, 2nd takes "b" units of time to complete job, 3rd takes "c" units of time to complete job If the time it takes for all 3 working together to finish the job is T, then 1/a + 1/b + 1/c = 1/T

Multi-step percent problems

You cannot add percents of different wholes On some problems, you'll be asked to find more than one percent or to find a percent of a percent EXAMPLE ON 229 IS EXTREMELY HELPFUL

Factoring algebraic expressions Page 242 REVIEW

factoring a polynomial means expressing it as a product of 2 or more simpler expressions. Common factors can be factored out by using distributive law EX: 3x^3 + 12x^2 - 6x all 3 terms contain factor of 3x, pulling out the common factor yields 3x(x^2 + 4x - 2)

operation

function/process performed on one or more #s 4 basic arithmetic operations: add, subtract, multiply, divide

Properties of 100%

since % means 1/100, 100% means 100/100, or 1 whole - the key to solving some GRE % problems is to recognize that all parts add up to one whole: 100% PAGE 224 example - look at!!

Compound interest =

the money earned as interest is reinvested - the principle grows after every interest payment received

30°/60°/90° Right triangles 2 types of Special Right triangles

2.) Right triangles w/ acute angles of 30° & 60°: 30°/60°/90° Right triangles ** sides of 30°/60°/90° Right triangles are in ratio: x: x√3: 2x 2x represents the hypotenuse x represents the side opposite of the 30 deg angle ***Remember the longest side has to be opposite the greatest angle

Sequences Ex. pg 249

Lists of #s Value of a # in a sequence is related to its position in the list You will be given a formula that defines each element

Parallel lines (// lines)

2 lines are parallel (//) if they lie in the same plane and never intersect Regardless of how far they are extended If line 1 (ℓ1) is // to ℓ2, we write: ℓ1 || ℓ2 If 2 lines are both // to a 3rd line, then they are // to each other also

Area of a circle

Area of a circle = πr²

Pythagorean Triples

Certain ratios of integers always satisfy the Pythagorean theorem = Pythagorean Triples 3, 4, 5 is most common on GRE - 2 legs are 3 and 4 and hypotenuse is 5 **Any multiple of these lengths makes another Pythagorean triple - thus 6, 8, 10 is a triple Common Pythagorean Triples - PIC 3, 4, 5 5, 12, 13 6, 8, 10

Exterior angles of triangle

Exterior angles of triangle is equal to the sum of the remote interior angles- in pic remote int angles are green and purple The 3 exterior angles of any triangle add up to 360 deg.

Use average to find a missing #

Given avg, total # of terms, and all but 1 of the actuals # you can find the missing # EX: Avg rainfall for 1976 - 1979 was 26 inches per year. There were 24 in of rain in 1976, 30 in 1977, 19 in 1978, how many inches were there in 1979? avg = 26 inches # of terms = 4 Sum of terms = 24 + 30 + 19 + ? 24 + 30 + 19 + ? = (26 inches)(4) inches of rain in 1979 = 31 inches

Tables

Most basic way to organize info, but cannot spot trends or extremes very readily Can ask for difference b/w 2 rows - subtract smaller # from larger Can ask to find average - Sum of all terms divided by the # of terms

Regular polygon

Polygon with sides of equal length and interior angles of equal measure

denominator

Quantity in the bottom of a fraction, representing the whole #

Sum of angles along a straight line

The sum of the measures of angles on ONE side of a straight line is 180 degrees Two angles are Supplementary to each other if their measures sum to 180 degrees

Volume of a cylinder

To find volume or surface area of a cylinder, you will need 2 pieces of info: the height of the cylinder and the radius of the base Vol of cylinder = (area of base(ie area of circle))(height) = πr2(h) V of cylinder = πr^2(h)

Supplementary Angles

Two angles are Supplementary to each other if their measures sum to 180 degrees

Calculating probability of multiple events for independent probability

to calculate the probability of 2 or more independent events occurring, multiply the probabilities of the individual events EX: prob of rolling a 3 four consecutive times on a 6-sided die would be (1/6)(1/6)(1/6)(1/6) = 1/1296

Surface area of a cube

Sum of areas of faces = 6e^2

Interior angles of a triangle

Sum of the interior angles of ant triangle is 180 deg

Kaplan's additional tips for quantitative comparison Questions

1.) Memorize the answer choices - memorize what the answer choices mean, the choices are always the same, the wording/order of answer choices never change/vary - this will save time 2.) When there is at least 1 variable in a problem, try to demonstrate different relationships b/w the quantities - **If you can demonstrate 2 different relationships, then choice D (not enough info given) is correct & there is no need to examine the question any further - Look at the expression(s) containing a variable and notice the possible values of the variable given the mathematic operation involved - Base your choices for # values of variables on mathematical properties you already know (EX: a positive fraction less than 1 becomes smaller than squared, but a # greater than 1 grows when squared) - Saves time and checks work 3.) Compare quantities piece by piece - Compare the value of each "piece" in each quantity - If every "piece" in one quantity is greater than a corresponding "piece" in other quantity, and the operation involved is either multiplication or addition, then the quantity with the greater individual values will have greater total value 4.) Make one quantity look like the other - When quantities A and b are expressed differently, you can often make the comparison easier by changing format of one of the quantity to look like other quantity. - Good approach to take when quantity look so dif that you cannot compare them directly 5.) do the same thing to both quantities - If quantities seem too complex to compare immediately, look closely to see if there is any addition, subtraction, division, or multiplication operation you can perform on both quantities to make them simpler (simplify quantities by doing same thing to both sides) 6.) Don't be tricked by misleading information - DONT assume anything - use only the info given or info that you know must be true based on properties or theorems (EX: don't assume angles are equal or lines parallel unless it is stated or can be deduced from other given info) - Common mistake is to assume that variables represent only positive integers. As we know, when using the Picking Numbers strategy, fractions or negative #s often show a different relationship b/w quantities 7.) Don't forget to consider other possibilities - If answer looks very obvious, it could be trap! EX: a question requires you to think of 2 integers whose product is 6. If you jump to conclusion that 2 and 3 are the integers, you will miss several other possibilities. There is 1 and 6 and pairs of negative integers: -2 and -3, -1 and -6 8.) Don't fall for look-alikes - 2 expressions maybe mathematically different even tho they look similar *** Be especially careful w/ expressions involving parenthesis or radicals!

Cube

A special rectangular solid in which all edges are of equal length, e, and therefore all faces are squares. EX: sugar cubes and dice w/o rounded corners = cubes

Distances on the coordinate plane

Distance b/w endpoints is equal to the length of the straight-line segment that has those 2 points as endpoints. If line seg. is parallel to x-axis, the y-coordinate of every point on the line segment will be the same and if line segment is parallel to the y-axis, the x-coordinate of every point on the line segment will be the same THEREFORE, to find the length of a line segment parallel to one of the axes, you just have to find the dif. b/w the endpoint coordinates that do change You can find the length of a line segment that is not parallel to one of the axes by treating the line segment as the hypotenuse of a right triangle. Siply draw the legs of the triangle parallel to the 2 axes - the length of each leg will be the dif b/w the x or y coordinates of its endpoints. Once you know the lengths of the legs, you can use the pythagorean theorem to find length of hypotenuse

Kaplan's additional tips for problem solving

Kaplan's additional tips for problem solving: 1.) Choose an efficient strategy -On GRE, The best way to each solution is often the quickest, and the quickest way is often not straightforward/textbook math - Thru practice you will become familiar with approaching each Q in more strategic way 2.) Rely on Kaplan math strategies - Use reasoning in conjunction w/ mathematics to answer Q quickly as possible - Combine approaches! For ex, using textbook math to simplify an equation, then picking manageable #s for the variables to solve that equation 3.) Picking Numbers -Picking numbers strategy can be good for problems that seem difficult - problems where either the answer choices have variables, the problem tests a # property you don't recall, or the problem and the answer choices deal w/ percents or fractions w/o using actual values *** If it is not explicitly states #s are integers, pick fractions and negative #s to rule out Answer choice D!!!! 4.) Backsolving - Similar strategy as picking numbers, but you will use one of the 5 answer choices as the number to pick - **Numerical answer choices are always in ascending or descending order! Use that to your advantage when backsolving - ^^Start w/ either B or D first bc you'll have a 40% chance of finding the correct answer based on your first round of calculations - ** If you don't pick the correct answer the first time, reason whether the # you started w/ was too large/small. If you test choice B. when the answers are in ascending order and choice B turns out to be too large, choice A is the correct answer. If choice B is too small, test answer choice D. If choice D is too large, choice C is the correct answer. If choice D is too small, then choice E is the correct answer. -^^ The opposite would be true if answer choices were listed in descending order ***Backsolving allows you to find the correct answer w/o ever needing to test more than 2 answer choices! 5.) Use Strategic Guessing - Good strategy is you can eliminate choices by applying number property rules or estimating bc gaps b/w answer choices are wide - If some of the choices are out of the rhelm of possibility, eliminate them and move on

Edge of a solid

Line segment that connects adjacent faces of a solid

Area of quadrilaterals

Multiply length of horizontal side by length of vertical side Area of rectangle = Length x width Area of square = (side)² Area of Parallelogram = (base)(height) ** designate 1 side as the base, then draw a line down from one of the vertices opposite the base so that it intersects the base at a right angle - this line segment will be called the height REMEMBER: 1.) In parallelogram, if you know 2 adjacent sides, you know all of them & if you know 2 adjacent angles, you know all of them 2.) In rectangle: if you know 2 adjacent sides, you know the area 3.) In square: if you are given virtually any measurement (area, length of side, length of diagonal), you can figure out the other measurements *** A rhombus is a parallelogram with four sides of equal length. If a rhombus has one angle measuring 120° and a side of length 2, we can infer several things about the figure in question. If one angle is 120°, then its opposite angle in the rhombus must also be 120° and their adjacent angles must be 60°. To find the area of a rhombus, we cannot multiply side by side, as if it were a square. ^^^In a rhombus we must find the product of base and height. We must draw in an altitude, perpendicular to the base

Problem solving Review test format for prob solving Q on pg 321 - 323

Problem solving can be broken up into several general mathematic categories: algebra, arithmetic, number properties, geometry You may be asked to solve a pure math problem or a word problem involving a real-world situation. You will be asked to enter your answer into an onscreen box, select one answer, or select one or more options that correctly answer the Q

Kaplan Method for Problem Solving

STEP 1: ANALYZE THE QUESTION - Look at what the Q is asking and what area of math is being tested. - Note what information is being given and note any particular trends in the answer choices (numbers/variables, integers/non-integers) - Unpack as much info as possible STEP 2: IDENTIFY THE TASK - Determine what Q is being asked before solving the problem, ask yourself "what does the correct answer represent?" - GRE intentionally provides correct answers for test takes who get the right answer to the wrong Q STEP 3: APPROACH STRATEGICALLY - Depending on the type of problem, you may use the textbook approach to solve OR other strategies: Picking Numbers, Backsolving, or Strategic Guessing -When picking numbers to substitute for variables, choose #s that are manageable and fit description given in problem -Backsolving in another form of picking numbers - you will start w/ one of the answer choices and plug that choice back into the Q -Strategic guessing can be a great time-saver on GRE: being able to make a smart guess on a Q is preferable to taking too much time and compromising your ability to answer other questions correctly STEP 4: CONFIRM YOUR ANSWER - Check that your answer makes sense and that you answered the question that was being asked TIPS/NOTES of how to apply: 1.) EX 2 pg. 326: Any # divided by itself gives remainder of 0. So if we need a remainder of 10, we want a # that is 10 more than the # we are dividing by. To check work, pick another # that yields a remainder of 10 when divided by same #

Kaplan method for data interpretation

STEP 1: ANALYZE THE TABLES AND GRAPHS - Tables, graphs, and charts often come in pairs that are linked in some way. Familiarize yourself w/ the info in both graphs/tables/charts and how the 2 are related before attacking the questions - Scan the figures for these components: 1.) Title: read the charts' titles to ensure you can get to the right chart/graph quickly 2.) Scale: Check the units of measurement. Does the graph measure miles per minute or hour? Missing units can dramatically change your answer 3.) Notes: Read any accompanying notes - GRE will typically give you info only if it is helpful or even critical to getting correct answer 4.) Key: if there are multiple bars or lines on a graph, make sure you understand the key so you can match up the correct quantities with the correct items STEP 2: APPROACH STRATEGY - Data interpretation Qs are designed to test understanding of fractions and percents and your attention to detail - make sure you are answering the correct question being asked!! - Qs tend to become more complex as you move thru a set. For instance, if Q contains 2 graphs, the first Q will likely refer to just 1 graph. A later Q will often combine data from both graphs - if you don't use both graphs to answer Q you're probably missing something - You can usually simplify single-answer multiple choice Qs by taking advantage of their answer choice format. - ^ By approximating the right answer rather than calculating it whenever possible, you can quickly identify the right answer. - Estimation can be one of the fastest ways to identify the correct answer in math problems ^^^Data interpretation problems benefit from this strategy as they tend to be the most time-consuming Qs

How the Kaplan method for quantitative comparison works

Step 1: Analyze the centered info and quantities - Notice whether the quantities contain #s, variables, or both - If there is centered information, notice how it affects the info given in the quantities - Note that a variable has the same value each time it appears w/in a question Step 2: Approach strategically - Think about strategy you can use to compare the quantities now that you have determined the info you have and the info you need - Variety of approaches (look at practice examples pg 310 - 313) Notes/Tips: 1.) When both quantities contain only #s, there is a definite value for each quantity and a relationship CAN be determined - SO answer choice D: "relationship cannot be determined from info given" is NEVER correct when the quantities contain only numbers 2.) reciprocal of any # b/w 0 and 1 is always greater than 1 3.) 1/4 + 1/4 + 1/4+ 1/4 = 1; 1/5, 1/6, 1/7, 1/8... < 1/4 4.) If every "piece" in one quantity is greater than a corresponding "piece" in the other quantity, and if the only operation is addition, then the quantity w/ greater individual values will have greater total values 5.) If possible, rewrite one of the quantities (A or B) so that relationship b/w quantities can be looked at in universal/like terms for a quicker/simpler comparison b/w quantities ^^ie: Make one quantity look like the other 6.) If possible, Simplify the quantities for easier/clearer comparison

Equations of lines

Straight lines can be described by linear equations Commonly: y = mx + b m = slope = rise/run (change in y/change in x): (y2 -- y1) / (x2 -- x1) b = point where line intercepts y-axis (value of y where x = 0) Lines that are parallel to the x-axis have a slope of 0, and therefore have the equation y = b Lines that are parallel to the y-axis have the equation x = a, where a is the x-intercept of that line

Perimeter of triangle

Sum of the lengths of the sides of triangle

Face of a solid

Surface of a solid that lies on a particular plane

Lateral surface of a cylinder

The "pipe" surface, as opposed to the circular "ends" The lateral surface of a cylinder is unlike most there surfaces of solids that you'll seen GRE bc: 1.) It does not lie in a plane 2.) It forms closed loop Think of it as the label around a soup can - if you could remove it from the can in one piece, you would have an open tube. If you then cut the label and unrolled it, it would form a rectangle w/ a length equal to the circumference of the circular base of the can and a height equal to that of the can

Base of a solid

the "bottom" face of a solid as oriented in any given diagram

Simple interest =

the interest payments received are kept separate from the principle; most common type of interest