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GMAT Training Manual Rule 1: Basic math computations must be perfect. Rule 2: You need to be able to TEACH this manual. Rule 3: Required homework is always the same: master relevant parts of manual (using tutored material as context). Re-do tricky math problems, several times if necessary. Rule 4: Writing must be clear & well-organized (not too large). Rule 5: Cancellations within 48 hours means I might do the same. Rule 6: Violation of any rule always results in a disappointing score. On test day: “Now remember, when things look bad and it looks like you're not gonna make it, then you gotta get mean. I mean plumb, mad-dog mean. 'Cause if you lose your head and you give up then you neither live nor win. That's just the way it is.” Josey Wales (Clint Eastwood), The Outlaw Josey Wales The GMAT might seem like Superman, but you’re Batman: “I want you to remember this, Clark…in all the years to come…in your most private moments…I want you to remember my hand…at your throat…I want you to remember…the one person who beat you.” 55-year-old Bruce Wayne (Batman) to Clark Kent (Superman) The Dark Knight Returns How the GMAT feels about you: “Business bad? F**k you, pay me. Oh, you had a fire? F**k you, pay me. Place got hit by lightning huh? F**k you, pay me." Henry Hill (Ray Liotta), Goodfellas Table of Contents Quantitative Problem Solving: the “trigger” method Basic GMAT math When to pick numbers: 3 triggers 1) Percents, fractions, or ratios in the question/answers AND an unknown whole value 2) Number properties/ must or cannot be true 3) Variables in answer choices When to plug in answers: 2 triggers 1) Tricky but standard algebra with basic numbers in the answers 2) Tricky word problem with basic numbers in the answers (esp. “least/greastest”) Arithmetic 1) Divisibility rules 2) Numbers to memorize 3) Lowest common denominator/ greatest multiple 4) Random counting formulas Data Sufficiency 1) Value 2) Yes/no 3) Algebraic rules of Data Sufficiency 4) 4-step method for Data Sufficiency 5) Picking numbers Algebra 1) Physical entities/ positive integers 2) A word problem with numbers in the answers 3) A word problem with variables in the answers 4) Number vs cost/value 5) Multiple equations and multiple variables 6) x2 with an equation 7) Compound expressions 8) Symbolism/number series Number Properties 1) Two-digit numbers/units digits 2) “1” 3) Primes 4) Even/odd 5) Positive/negative 6) Inequalities 7) Large numbers/ “product of” 8) Exponents 9) Decimal places/ 10x 10) Radicals 11) Absolute Value 12) Multiple/factor or “divisible by” 13) Consecutive numbers 14) Product of consecutive numbers (factorials) 15) Remainder 16) Minimize/maximize 17) Doubling Proportions 1) Fractions, rations, or percents AND an unknown whole value 2) “What fraction/percent of y is x?” 3) Fractions 4) “Largest/smallest value” 5) Ratios 6) Percents 7) Rates involving distance/work 8) Differing units of measurement 9) Rates involving working together with unknown work 10) Two (or more) percents, averages, costs, or wages * Mixure, dilution, weighted average, etc Overlapping sets 1) Two groups 2) Three groups Interest Rates 1) Simple interest 2) Compound interest Statistics Mean, median, mode, range, standard deviation, graphs & tables Counting Problems (“how many”) 1) General approach 2) Permutations 3) Distinct/same elements 4) Combinations Probability 1) Single event 2) Multiple events (“exactly/or”) 3) Multiple events (“at least”) Geometry 1) Eyeballing 2) Any figure 3) “The area…” 4) Triangles 5) Right triangles 6) Complex figures 7) Shaded region 8) Parallel lines 9) Circles 10) Angles inscribed in circles 11) Volume 12) Surface area 13) Maximum distance 14) Coordinate geometry (y = mx + b) Quantitative Problem Solving: the 2-step trigger method 1) Scan the question and the answers, looking for triggers (take your time). 2) Based on the trigger, DO THE RIGHT THING (get moving). Basic GMAT math * Use answer choices to help guide you to the solution or to eliminate wrong answers. * Eliminate answers repeated in question stem. * When stuck: Rephrase/simplify – write down anything you can figure out. * Fractions: reduce or eliminate. * Eliminate multiple fractions by multiplying each term by common denominator. * Addition or subtraction involving common elements: factor common elements. * Parentheses: multiply through & look to combine common elements. * Decimals: multiply terms by 10, 100, etc (or use 10-x) * Computation with several numbers: prime factor, reduce/combine exponents, use answers. * Look to write down what “remains” (try to define/specify). * The correct answer might take a different form from your solution. * Re-read the question before selecting an answer. When to pick numbers: 3 triggers Stay organized: define each element, starting in the upper-left hand corner 1) Percents, fractions, or ratios in the question/answers AND an unknown whole value: always pick a workable number for an unknown value. Percents: 10 or 100 for the whole element (immediately following “of”) Fractions: common denominator for the whole element (immediately following “of”) Ratios: common denominator for largest term OR pick “1” for one term. 2) Number properties/ must or cannot be true: pick acceptable numbers – only if necessary. Recall -1, -1/2, 0, 1/2, 1, 2 (only even prime) and the same numbers (for x and y). Look to pick more than one number for each variable 3) Variables in the answers: pick different numbers for each variable – if necessary. 1) Pick basic and different numbers for each variable (don’t pick 0 or 1). 2) Determine the answer as a number value. 3) Plug the “picked numbers” into each choice, looking for the same answer. When to plug in answer choices: 2 triggers 1) Tricky but standard algebra with basic numbers in the answers: plug in answers * Consider plugging in answers after setting up the algebra (for word problems) 2) A tricky word problem with basic numbers in the answers (esp. “least/greatest”) Backsolving – 3 step method. 1) Set the question equal to answer (B) or (D) 2) Based on answer (B)/(D), write down everything you can figure out. 3) Determine whether (B)/(D) “fits”. Arithmetic Divisibility rules: x is divisible by when 0 1 2 3 4 5 6 7 8 9 10 11 12 never always x is even sum of digits = multiple of 3 last two digits divisible by 4 last digit is 0 or 5 x is divisible by 2 and 3 n/a n/a sum of digits divisible by 9 last digit = 0 n/a x is divisible by 3 and 4 Numbers to memorize 22 to 26, 32 to 34, 42 to 44, 52 to 54, 62 to 63, 72 to 152, “prime multiplication table” up to 19 x 19 1/3 = .33333… ¼ = .25 1/5 = .2 15% = 3/20 331/3% = 1/3 40% = 2/5 “4 choose 2” = 6; “5 choose 3 or 2” = 10 Calculate percentages by using multiples of 10%. LCD/LCM and GCF Lowest common denominator (LCD)/ multiple (LCM) = axbycz * a, b, c = all distinct prime factors of all numbers * x, y, z = largest exponent of each distinct prime factor. * Use answer choices before doing computation. Greatest common factor (GCF) = axbycz * a, b, c = all distinct prime factors common to all terms. * x, y, z = lowest exponent of each distinct prime factor (common to all terms). * Use answer choices before doing computation. Data Sufficiency (1 2 T E N) – rephrase and simplify Value questions: only a single value is sufficient. * If the 2 statements have single value in common, answer = C. * If the 2 statements have more than one in common, answer = E. * Consider picking numbers, trying to get more than one value. Yes/no questions: “Always yes” = sufficient; “Always no” = sufficient. * “Sometimes yes, sometimes no” = insufficient. * PLACE THE QUESTION MARK APPROPRIATELY * Consider picking numbers, trying to get a “yes” and a “no” 4-step method for data sufficiency 1. Scan the question stem and the statements, looking for triggers (take your time). * Question may be answerable by just thinking about it. * Consider picking numbers. 2. Rephrase and simplify the given information and the question (if possible) * Look to create a multi-variable equation or expression. * Reduce/eliminate fractions, factor/combine common elements * Equation in given information: look to plug into the question * Think about what would be sufficient to answer the question. 3. Rephrase and simplify the statements before deciding sufficiency (if possible) * Look to rephrase the statements into the same form as the question * Look to plug simplified statements into question/given information * Always use information provided in the question stem * A lack of information means “anything goes” (“x” could be anything) 4. Recall algebraic rules of data sufficiency (when appropriate) * “x” distinct linear equations are sufficient to solve for “x” variables. * Make sure that the equations are actually distinct and linear. * x2 or xy is not linear (x or y could have 2 values). * Non-linear could still be sufficient (it just doesn’t have to be). * Multiple variables can sometimes be treated as a single variable * Physical entities/geometry: non-linear is sufficient (only one value) * A compound expression [Ex: (x – y), (x+y)/z] in the question: look to rephrase the statements into the same form as that expression. * Linear equation when adding positive integer variables: could be sufficient. * See “Algebra” for solving Question/statements can’t be rephrased OR aren’t in form of equation/expression: pick #s! Picking numbers: try to get a “yes” and a “no” OR more than one value: 3 steps 1) Pick numbers consistent with the statement and the given information. * Recall -1, -1/2, 0, 1/2, 1, 2, and non-distinct numbers as possibilities. 2) Plug those numbers into the question and take note of the answer. 3) Repeat step 1 using different numbers, trying to get a different answer found in step 2. * Combining statements: use prior numbers and look for commonalities. * Pick numbers consistent with both statements and the given information. Algebra 1) Physical entities (word problems) are always positive integers. * Solving for two positive integers with only one equation: 1) Reduce/factor equation. 2) Identify the common factor for 2 of the terms. 3) Variable in 3rd term must be multiple of same “identified” factor as other two terms. 4) Pick numbers for the variable (multiples of the common factor); only one should fit. * DS: determine whether more than one number is possible for the variable. 2) A word problem with numbers in the answers: 3 options 1) Create two distinct equations. * Sometimes add/subtract both sides by a “new” value. 2) Create two expressions and set them equal to one another. 3) Backsolve. 3) A word problem with variables in the answers: pick numbers when necessary. 4) Number vs cost/value * Number: add individual variables. * Cost/value: multiply each variable by its relevant cost/value before adding. EXCEPTION: when only cost/value is involved (no “number”). 5) Multiple equations and multiple variables: try to combine equations (adding/subtracting). * Inequalities: Line up inequalities in same direction then add or subtract them. * Multiple variables (compound expression) can sometimes be used as a single variable. 6) x2 with an equation: use reverse FOIL to factor, often setting the equation equal to zero and solving for x. * Memorize the three classic quadratics: 1) (x + y)2 = (x +y)(x + y) = x2 + 2xy + y2 2) (x – y)2 = (x – y)(x – y) = x2 – 2xy + y2 3) x2 – y2 = (x + y)(x – y) * x2 – 1 = x2 – 12 = (x – 1)(x + 1) 7) A compound expression in the question: look to solve for that expression as a whole, (most often using combination method) rather than solving for each variable. 8) Symbolism/functions: follow the directions of the question. Pick numbers? * Na, Na-1 , (number series): use substitution method for solving. 9) Rates, percents, averages, costs, or wages: Use 3-part grid (see Proportions) Number Properties: Prime factorization! 1) 1: implied in and factor of all numbers; product of any numbers includes 1. 2) Two-digit number: rephrase as 10t + u * Units digits: multiplying units digits is easy… 3) Prime numbers (a number divisible by only 2 distinct integers, itself & 1). * “2” is the lowest and only even prime number (pick 2 when possible) 4) Even/odd: think in terms of the rules of even and odd (try not to pick numbers). * Use “1” and “2” to help recall rules. i) “0” is even, but neither positive nor negative. * Even x (even or odd) = even. * Positive odd + positive odd = prime ONLY for 1 + 1 5) Positive/negative or x</>0: think in terms of rules of positive/negative * Inequalities: recall positive/negative issues. * a – b = x implies b – a = -x and vice-versa. * xeven > 0, although x may be positive or negative. * x(even)(y) > 0 * (-x)(even)(y) = x(even)(y) * xodd = same “sign” as x. * Product of terms = 0: at least one term = 0 6) Inequalities: multiplying or dividing by a negative number: switch the direction of the sign. * D.S. (yes/no): look to multiply/divide by a variable whose sign is unknown. * D.S. (value): look to pick numbers. * x < y implies x – y < 0 (negative) and vice-versa. 7) Large numbers/ “product of”: prime factor * Prime factor: fractions, exponents, radicals, multiple/factor, factorials * Prime factorization of a large number: break up the number into 2 (or more) non-prime factors, and work from there. 8) Exponents * Prime factor to get a “common base” and apply rules (one-step down). * Adding/subtracting common bases: no one-step down BUT factor out terms. * Adding/subtracting exponents: look to “break-out” into common bases. * No common bases: break out terms to create common exponents, re-combine terms. * Exponents can be distributed only among multiplied or divided terms * Variables as exponents: set common bases equal, solve for the variable. 9) Decimal places/large numbers: use 10-x or 10x to rephrase or eliminate. Prime factor? * 10x: decimal moves x places to the right; 10-x: x places to the left. 10) Doubling “x” y times: 2yx * Look to use 10x or 10-x; prime factor. 11) Radicals (square roots): same rules as exponents. * Large number under radical: prime factor to find perfect squares. * Radical in equation: isolate the radical and square both sides. * Eliminate radicals in denominator by multiplying by “1” * (square root of x)/(square root of x) = 1 * Look to rephrase radicals into exponents. 12) Absolute value: the distance from zero. * Rephrase absolute value as two equations (positive & negative) * Consider picking numbers or backsolving. * Absolute value with an inequality: rephrase as a normal absolute value, but switch the direction of the sign for the negative side. 13) Multiple/factor or “divisible by”: rephrase as M/F = I (multiple/factor = integer): 5 steps 1. Substitute numbers or variables for the multiple & factor, always keeping “I”. 2. Prime factor all numbers. 3. Reduce the fraction, if possible (but don’t eliminate it). 4. Given a variable only in numerator, factor out all numbers in numerator. * More than one variable or variable w/ exponent: pick numbers 5. Recall basic rules of integers: I +/- I = I; I x I = I; I/I = fraction or I. If x/a = I & x/b = I then x/(ab) = I; a & b = prime (ab)/x = I implies a/x = I or b/x = I (and vice-versa); a & b = prime (multiple of x) +/-/x (multiple of x) = multiple of x * Look to pick numbers. 14) Consecutive numbers * Total number of terms= last – first + 1; Average of terms = (first + last)/2 * Sum of consecutive numbers= (average)(number of terms) * “The sum of 4 consecutive even integers”: x + (x+2) + (x+4) + (x+6) 15) Product of consecutive numbers (factorials): rephrase as prime factors, re-combine terms. * If Y has any different distinct prime factors from X, then: X/Y does NOT = I * If any of Y’s distinct prime factors outnumber X’s, then X/Y does NOT = I * Product of consecutive integers+ 1: Prime factors of the product are not prime factors of (product +1). a) Product of even consecutive numbers: take out (divide) the 2! * X! + Y: If all of Y’s prime factors can be found within X!, then (X! + Y)/Y = I 16) Remainder: Pick numbers or plug in answer choices whenever possible. Otherwise: * Rephrase as remainder/divisor = decimal; use 10-x and prime factor. * All terms are integers * Recall: 2/3 = 0 remainder 2 (ie a remainder can follow 0) 17) “Minimize/maximize”: look to maximize/minimize other terms. Backsolve? Proportions (fractions, ratios, percents, rates, averages) 1) Fractions, ratios, or percents, AND an unknown whole value: pick a workable number for the unknown value. 2) “What fraction/percent of y is x?” = x/y “X is what fraction/percent of y?” = x/y * The words immediately following “of” = denominator (other term = numerator). 3) Fractions: look to reduce or eliminate. Pick numbers? * Master the arithmetic rules of fractions (adding, dividing, reducing, cross-multiplication, and reverse cross-multiplication). Data sufficiency: look to use “reverse cross” when: x – y = 0 or x = y * 1/3 of x = 1/3; 1/3 greater than x = 4/3x; 1/3 less than x = 2/3x * Look to write down the “remainder” of the whole. * Sometimes simplify a fraction by multiplying by a version of “1” * (-1)/(-1), (square root/square root) = 1 * Decimals can be converted to fractions (and vice-versa) * Given a complex fraction, consider “breaking it out” into multiple fractions. * Terminating decimal: denominator has only 5 and/or 2 as factors. * Repeating decimal: denominator has 3 as a factor. *(xy)/99 = .xyxyxyxyxy… 4) “Largest/smallest value” with fractions/decimals. * Look to factor out common terms. * Convert denominator of each fraction into (approximately) 100 * Divide numerator by denominator to generate decimals (if necessary). * Larger positive denominators/decimal places = smaller numbers * Larger negative denominators/decimal places = larger numbers. 5) Ratios: rephrase as fractions or add the elements. Pick numbers? * Ratio with a (possible) whole value: ax + bx + cx = total * Two-part ratio with numbers in answers: rephrase as fractions. * Three part: rephrase as fractions or use colons. * Ratios/fractions in answers: rephrase as fractions, eliminate the fractions, pick a simple number (1,2, or a common denominator) for one variable, work from there. * Conversion (old ratio into new ratio): “old” +/- x = “new” (x in numerator/denom) * Multiplying/dividing elements of the ratio: creates new known ratio * Adding to/subtracting from elements of the ratio: creates new unknown ratio 6) Percents: rephrase as decimals or fractions. Pick numbers? * Look to write down the remainder of the whole. * Percent change = actual change/original. *Percent increase = (new – old)/old * Percent decrease = (old – new)/old * “x is increased by 15%”: 1.15(x); “x is decreased by 75%”: 1/4(x) * “x is y percent of z”: x = (y/100)z * “x is increased/decreased by y percent: (1 + y/100)x / (1 – y/100)x * Convert fractions into percents/decimals by creating (approx.) 100 in denominator 7) Rates involving distance/work use 3-part grid (distance/work = rate x time): 4 initial steps 1. Fill in the given (easy) information (often the rates) . 2. Assign variables and relative values to time, rate, and/or distance/work. * Time, rate, or distance is often the same for both entities. * One entity often has greater/lesser distance/time than the other. 3. Look for total distance/work or total time (may not apply). 4. Multiply/divide across, add/subtract down (don’t add rates) * Total distance/work or total time is provided: add the equations, solve for unknown. * R1T1 + R2T2 = total D/W * D1/R1 + D2/R2 = total T * Catching up/overtaking (no totals): subtract the equations, solve for unknown. * Consider setting equations equal to one another, solve for unknown. * Average rate = (total D)/(total T) (not (R1 + R2)/2) * Average rate question when given rates and ratio of times or distances: assign a workable number for the time or distance. 8) Differing units of measurement (feet vs inches, miles vs rotations): * Multiply or divide appropriately – use the answer choices to help. 9) Rates involving working together with unknown work: 2 options 1. Amount of job completed in a certain time: 1/A + 1/B (+ 1/C + 1/D) = 1/T * A, B, (C, D) = time (typically in hours) * 1/T = amount of job completed in one hour. 2. Time taken to complete entire job: (AB)/(A + B) = T * T = total time taken to complete a job. * Optional: 1/A + 1/B (+ 1/C + 1/D) = 1/T; solve for T. * Multiple entities doing a job in multiple days: apply formula * Work = rate x time x number; W = (R)(T)(#) (W often = 1) 10) Two (or more) percents, averages, costs, or wages: apply the 3-part grid, looking to assign variables and their relative values (when necessary). * Averages: sum = (average) x (number) [S = (A)(#)] * Percents (mixture): part = (percent) x (whole) [P = (%)(W)] * Label the “part” and “percent”. * Dilution: the part and percent of one element is zero. * Costs: total cost = (unit cost) x (number) [TC = (UC)(#)] * Wages: total wage = (unit wage) x (number) [TW = (UW)(#)] * Values for any 2 cells in middle column and ratio of any 2 cells in another column: pick a workable number for “ratio” column. * Ratio = fraction, percent, etc. * Using the grid: multiply/divide across, add/subtract down (don’t add middle) 11) Data sufficiency: look to use grids to rephrase question/given information. * Values for any 2 cells in middle column and ratio of any 2 cells in another column: sufficient for value of third middle cell. 12) Grids: MULTIPLY/DIVIDE ACROSS; ADD/SUBTRACT DOWN!! Overlapping sets: look to use answer choices 1) Two groups where membership overlaps * Use Venn diagram when necessary. * Whole groups: group 1 + group 2 – both + neither = total * Only groups: group 1 only + group 2 only + both + neither = total * Group 1 only = group 1 – both; both < group 1; both < group 2; both < total * “Of group x, 30% are part of group y”: .3x = both 2) Three groups * (sum of each group) – (sum of each double overlap) – (2 x 3rd overlap) + neither = total * Recall: group 1 only = group 1 – (sum of each relevant overlap) * P.S.: Look to create equation with a single variable. * Single variable may represent every group NOT provided in given information. * D.S.: Look to create equations with multiple variables. * Any overlap < Any group or total Interest rates 1) Interest based only on the principal (simple interest) * Apply formula: total interest = (interest rate)(time) * Express “interest rate” as a decimal; “time” in terms of years (often as a fraction). 2) Interest based on principal + previous interest (compound interest) * Apply formula: final = (principal)[1 + (interest rate/C)] (time)(C) * C = number of times compounded annually (usually = 1) Statistics (number sets) 1) Mean = average; rephrase as sum/number = average * Outliers can skew averages 2) Median = middle term * Median can differ significantly from average * Even number of terms: median = average of middle terms 3) Mode = number that shows up the most * More than one mode is possible; no mode is possible 4) Range: difference between highest and lowest term. 5) Standard deviation = the average distance of each term from the average of the set. * “2 SDs” = twice the average distance. * Larger spread = larger standard deviation (and vice-versa). * Sufficient to find SD: variance, all the numbers in the set, average spread of set. 6) Graphs & tables: Read all information from graph before reading the question. Counting problems (“how many”): multiply elements (sketch out?) 1) How many arrangements/specific assignments: permutation logic (5 x 4 x 3) or formula * (n!)/(n – k)! (n = # of entities; k = # of available places) * # of orderings for x elements in x places: x! (0! = 1). * # of re-arrangements of x letters: x!/(y!)(z!) y & z = # of repeated letters * Circular sequencing for x elements: (x – 1)! * # of orderings for x elements where one element must be behind/ahead of another: (x – 1)[(x – 2)!] * # of arrangements of x entities where 2 paired: 2(x – 1)[(x – 2)!] * NOT paired: All arrangements – [2(x – 1)][(x – 2)!] 2) How many small groups from a larger group: use combination formula. * (n!)/[k!(n – k)!] (n = large; k = small) (“n choose k”) * (n choose k) = (n choose n – k) *(n choose n) and (n choose 0) = 1 * (n choose 1) and (n choose n – 1) = n * Largest # of small groups = (n choose .5n) 3) How many distinct/same elements (usually digits): 4 steps 1) Lay out the number of digits using slots (_ _ _ _) 2) Underneath slots: label first w/ “S” (same), repeat, remainder w/ “D” (distinct) 3) (a) Label the first slot with the number of elements (digits) that could go first. (b) Label each additional “S” slot with “1” (only 1 digit could be the same as previous) (c) Label each “D” slot with 9, 8, 7, etc. (number of digits distinct from previous). (d) Multiply across. 4) Determine number of arrangements for the letters (see permutations) 5) Multiply result of step 3 by result of step 4. Probability (# of desired)/(# of total possibilities): use answer choices 1) “and”: multiply; “or”: add; “exactly”: 5-step method; “at least”: subtract from 1. * Absence of above terms: look to use counting methods (combinations/permutations). * Probability involving geometry: (desired area)/(total area) 2) Probability of multiple events (“exactly/or”): 5 step method 1) Lay out the number of events using slots (_ _ _ _) 2) Write down one specific example of the desired outcome (underneath each slot). 3) Label each specific example with its relevant probability & multiply across. * Selection of items: items might not be replaced. 4) Determine the number of possibilities for the desired outcome. * Sketch out possibilities or use counting methods. * Combination: n = # of events; k = # of desired * Permutation: how many ways to re-arrange the “word”/letters. 5) Multiply result of step 3 by result of step 4 (given same specific probabilities) * Different specific probabilities (result of step 3): add them together. 3) Probability of multiple events (“at least”): P of non-occurrence; subtract from 1. 4) k elements selected from n elements; probability that x specific elements will be selected: (# of desired)/(# of total possibilities) = [(n – x) choose (k – x)]/(n choose k) Geometry 1) Problem-solving figures are drawn to scale (so you can eyeball) * Data sufficiency figures are NOT drawn to scale. 2) Data Sufficiency with triangles: create equations based on a2 + b2 = c2 OR angles = 180. 2) Any figure: write down anything you can figure out. 3) “The area of the figure is x”: rephrase as (formula of area) = x * Quadrilateral: area = base x height; perimeter = sum of sides i) Perimeter of rectangle = 2b + 2h ii) Perimeter of square = 4x iii) Interior angle = 360 degrees *Triangle: area = ½(base x height); perimeter = sum of sides i) Base and height are always perpendicular ii) Interior angles = 180 degrees * Circle: area = pi(r)squared; circumference = 2pi(r) i) Interior angles = 360 degrees * Angles creating a straight line: sum of angles = 180 4) A triangle: classify it. * Isosceles: 2 opposite sides (& angles) are equal. * Equilateral: sides and angles are equal (angles = 60 degrees) * Right triangles: one side = 90 degrees * Similar triangles (2 triangles with the same angles): lengths of sides are all same ratio * Scalene = no equal angles * Interior angles = 180 degrees i) Sum of two sides > third side ii) Difference of two sides < third side * Exterior angle of triangle = sum of the two remote angles. * The larger the angle, the longer the opposite side (and vice-versa). * Length of two sides and value of enclosed angle: sufficient for length of third side. 5) Right triangles: look for the four classic right triangles * 45-45-90: sides = x, x, x(square root of 2) * 30-60-90: sides = x, x(square root of 3), 2x * 3x-4x-5x & 5x-12x-13x triangles * Two sides/angles needed for special right triangles * No special right triangle: use a2+ b2= c2 6) A complex figure: consider creating an equation out of a “hidden” polygon. * Interior angles of a polygon: 180(# of sides – 2) * All angles must be labeled with a value or single variable * Area: divide figure into triangles/quadrilaterals 7) Shaded region: rephrase as subtraction. 8) Parallel lines: look for a line that goes through both (transversal) * Figure out what you can about the angles. 9) A circle (or part of a circle): identify the radius (as a value or unknown) * Diameter: rephrase as 2r * A sector: (pie slice – related to area): (x/360)(pi(r)2) * Angle on edge of circle formed from 2 outer points of sector = ½(sector) * An arc: (pie crust – related to circumference): (x/360)(2pi(r)) 10) Angles (or triangle) inscribed in a circle * Intercepted arc = ½ (sector). * A triangle with diameter as hypotenuse creates a right angle. i) A right angle creates a diameter. 11) Volume of a solid = (area of base)(H) * Volume of triangle = (1/2bh)(H) * Volume of rectangle = bh(H) * Volume of cube = s3 * Volume of cylinder = pi(r2)(H) * Volume of sphere = 4/3(pi)(r3) * Hemisphere: half of a sphere. 12) Surface area = sum of the areas of each side. 13) Maximum distance within a solid quadrilateral: square root of a2 + b2 + c2 14) Points on a graph (Equation of a line): y = mx + b * (x,y) = coordinates on a graph. * Given any values for x & y, plug them in. * m = slope = rise/run = (y2 – y1)/(x2 – x1) * b = y-intercept (when x = 0) * x-intercept: (set y = 0) * You’re stuck: 2 options a) Calculate slope (recall (0,0) as a possibility) b) Isolate “y” in an equation. * Perpendicular lines: the slopes are m = (1/-m) * Positive slope: line goes up from left to right; Negative slope: down from left to right. * Zero slope: horizontal line; Undefined slope: vertical line. * Distance between 2 points: square root of: (x2 – x1)2 + (y2 – y1)2 * Equation of line given two (x,y) points: 3 steps 1) Calculate slope 2) Plug one set of (x,y) numbers in y = mx + b 3) Solve for b Reading Comprehension – Think Basic! 1) Read the passage quickly, (3 – 4 minutes) taking basic notes on each paragraph. a) Focus notes on all opinions (especially the author’s), voices (critics, historians), and/or theories. b) Write down only what you understand. c) Consider writing down “crystal clear” details. NOTE: The more abstract the passage, the more basic the notes should be! 2) Review notes and summarize all opinions, focusing on the author. a) Author’s opinion might be neutral (to explain, to discuss, etc) 3) Predict a basic answer to each question before evaluating the answers. a) If possible, determine which paragraph the question refers to. b) Use notes to predict a basic answer c) Refer back to the passage only when necessary. d) Main point questions: select answer that discusses all opinions. 4) Be attracted to “nice and general” answers. a) The correct answer will often act as an “umbrella” over other answers. b) Extreme language usually indicates a wrong answer. c) Extreme language is OK if it reflects an opinion. 5) When stuck: select the answer that comes the closest to the relevant opinion or the author’s main idea. 6) Reading Comp homework a) Try a timed passage on your own. b) Review all correct answers, thinking about how you could have taken them down in your notes, especially as opinions. c) Identify those questions where referring back to passage was necessary Sentence Correction: Answer must be parallel to non-underlined!! 1. Look for “triggers”. 2. Recall the relevant rule (students must learn specific rules through self-study). 3. Re-read the answer back into the sentence (as necessary). * When stuck: check parallel structure. * Answer must be parallel to non-underlined. * The correct answer will never change the meaning of the sentence. * Eliminate wordiness. Rules and triggers 1. Rule: subject-verb agreement (singular subjects with singular verbs, plural subjects with plural verbs). Triggers: prepositional phrase, comma phrase (both will separate the subject from the verb). 2. Rule: verb tense (past, present, future, etc). Trigger: a sequence of events (past tense or future tense). * Think about the logical sequence of events and use your ear to evaluate answer choices. 3. Rule: pronouns (must be unambiguous and in the correct form). Trigger: pronouns (“it”, “this”, “there”, “they”, etc). * Determine who/what the pronoun refers to. 4. Rule: modification (a descriptive phrase (modifier) must be placed right next to who/what is being described OR right next to another modifier that describes the same thing). Trigger: a comma phrase that can be removed from the sentence. * Determine who/what is being described. 5. Rule: parallel structure (elements must be in similar form). Triggers: commas indicating a list, conjunctions (and, but), prepositions * Think about how connected elements should be parallel. * Answer must be parallel to non-underlined. 6. Rule: comparisons (compared elements must be logically comparable). Trigger: comparison words (than, like, unlike, as, with). * Determine who/what is being compared. 7. Rule: Style and usage (4 sub-rules: few triggers). 1) Idioms (socially accepted words with no real logic). 2) Eliminate passive voice (subject is after the verb) * Triggers: “by”, “there is/are/were”, “it is/was” * If non-underlined is passive, then make underlined passive. 3) Stay away from gerunds (“-ing” words: verbs that act as nouns). * Eliminate “having” and “being” * “That”: words following are necessary; “which”: words following not necessary. Critical Reasoning How to identify conclusion (main point) & evidence (assumed to be true). 1) Look for keywords a. Conclusion: thus, therefore, hence, so, clearly, suggests, “as a result”, “for this reason”, “this is why” These words do not necessarily indicate the conclusion. b. Evidence: because, since, for, to, due to, in order to, to, after all These words always indicate evidence; always circle them. c. Contrast: but, yet, however, although, while These words indicate either a shift in evidence or the author’s conclusion; always circle them. 2) Use the why test Identify what you believe to be the conclusion. Ask “why?” Information in the stimulus must provide an answer to this “why”. “The conclusion because the evidence.” * 3 types of conclusion: the author’s, someone else’s, intermediate * 3 types of evidence: supporting (“why”) evidence, counter-evidence, background. * Given a dialogue, identify at least one conclusion. If only one conclusion is apparent, the other speaker will most often disagree with that conclusion. Tone (always exaggerate extreme or mild language) Extreme: each, any, all, always, will, must, most important, only, exactly, same, independent, primary, never, no, cannot Strong: most, tend, probably, generally, small, few, “only a few” Neutral: often, many Mild: some, possible, may, “no guarantee”, “not all”, “at least a few” *some = magic GMAT word = “at least one…” * some/sometimes in an answer = rarely correct for weaken and strengthen; very often correct for inference and necessary assumption. Classic answer type: A comparison. A comparison is deceptive and often irrelevant. Ex: “Alex is smarter than Greg.” This does not mean that Alex is smart because Greg might be a dumb-ass. In addition, we do not know how much smarter is Alex than Greg; by .001%? 200%? Critical Reasoning: 4-step method 1) Identify the question type * EXCEPT questions: focus on eliminating wrong answers. 2) Read the stimulus appropriately, always taking note of the tone and any keywords. * Identify conclusion & why evidence for all questions, except for inference & resolve. 3) Use the why test for all questions except for inference & resolve (predict an answer). 4) Evaluate the answer choices appropriately by asking the right questions. General rules for evaluating answer choices 1. If the correctness of an answer is not readily apparent, give it a “maybe” and move on. 2. Never eliminate those answers you do not understand. 3. Only eliminate those answers that you could easily explain as to why they are wrong. 4. 1 answer is definitely correct and 4 are definitely incorrect. 5. Never eliminate answers that address the evidence. 6. The answer need not discuss the conclusion directly, but it will always affect it. 5. Evidence is always true; it can’t be flawed, weakened, or disagreed with. Evidence can, however, be shown to be irrelevant. 6. Tone tone tone tone tone Weaken: the correct answer will weaken the conclusion (often by making evidence irrelevant), most often by addressing the evidence and using strong language. 1) “weaken” “undermines” “casts the most doubt” 2) Identify conclusion & why evidence 3) Use the why test to summarize the argument. 4) Ask: Does this answer weaken the conclusion (or make the evidence irrelevant)? Does it address the evidence? Does it use strong language? * Mild language is OK if the conclusion is extreme. * “some” or “sometimes” in an answer is almost always wrong. * The correct answer can discuss “outside” evidence (still needs “inside” evidence). * An answer that definitely weakens the conclusion is definitely correct. Strengthen: the correct answer will strengthen the conclusion, most often by addressing the unique evidence and using strong language. 1) “strengthens” “answer supports stimulus” “justifies” 2) Identify conclusion & why evidence 3) Use the why test to summarize the argument. 4) Ask: Does this answer strengthen the conclusion? Does it discuss the evidence? Does it use strong language? * “some” or “sometimes” in an answer is almost always wrong. * The correct answer can discuss “outside” evidence (still needs “inside” evidence). * An answer that definitely strengthens the conclusion is definitely correct. Necessary assumption: the correct answer will be necessary in order for the conclusion to be true (as shown by the negation test), most often by addressing the evidence and using negative or mild language. 1) “relies” “depends” “required” “necessary” “assumption” 2) Identify conclusion & why evidence 3) Use the why test to summarize the argument. 4) Ask: When I negate this answer, does the conclusion fall apart? Does it address the evidence? Does it have negative or mild language? (BIG DEAL!!) * Use the negation test where it is easy to do so. * A lack of mild or negative language or an “un-negatable” answer is most often wrong. * An answer that is definitely required for the conclusion to be true is definitely correct. The negation test: if an answer is negated and the conclusion falls apart as a result, then that answer is the necessary assumption (“some” negates into “none” and vice-versa). The two basic necessary assumptions to every argument: * The evidence is in some way relevant to the conclusion. * The evidence does not lead to a different conclusion. Sufficient assumption: the correct answer will prove the conclusion to be true most often by addressing the evidence and using strong language. 1) “conclusion follows logically” “enables the conclusion to be properly drawn/inferred” 2) Identify conclusion & why evidence. 3) Use the why test to summarize the argument. 4) Ask: Does this answer prove the conclusion to be true? Does it address the evidence? Does it use strong language? * An answer that definitely proves the conclusion to be true is definitely correct. Inference: the correct answer will be inferable from the stimulus and most often use mild language. 1) “infers” “(the answer) follows logically” “the above supports the answer” “the answer is supported by the above” “must be true” 2) Paraphrase the stimulus, treating everything as evidence (think basic). * Consider using notes to deal with the stimulus. * Take note of any conclusions (“claims, scientists, critics”). * The “claim” was made, but its truth is unknown. 3) Try to infer anything from the stimulus (think basic). 4) Ask: Is this answer inferable from the stimulus? Does it use mild language? * Wrong answers typically use extreme language. * Extreme language is acceptable if the stimulus is extreme. Resolve/explain: the correct answer will resolve a discrepancy or explain a situation, most often by addressing elements of the discrepancy. 1) “discrepancy” “resolve” “explain” 2) Paraphrase the discrepancy/situation. 3) Try to resolve/explain discrepancy/situation (think basic) 4) Ask: Does this answer resolve/explain the stimulus? Does it discuss elements of the discrepancy? * Correct answer will explain how both contradictory elements could be true. Bolded statement(s): the correct answer will describe the function of the bolded statement(s) and use the same tone as the stimulus. 1) “role” “function” 2) Identify conclusion & evidence. * Look for other conclusions (someone else, intermediate) 3) Determine how the statement(s) relate to the argument (or each other). 4) Ask: Does the statement(s) function in this way? Does the tone match? Strange question: Answer the question directly, after identifying conclusion and evidence Method of Argument (rare): the correct answer will describe how the author uses evidence to affect a conclusion and use the same tone as the stimulus. 1) “does which of the following” “argumentative strategy” “responds by” 2) Identify conclusion & evidence 3) Use the why test to think about how the author uses evidence to affect a conclusion 4) Ask: Does the author actually do this? Does the tone match that of the stimulus? Flaw: the correct answer will be a flaw committed by the author, most often by addressing the evidence (rare). 1) “flaw” “calls into question” “most vulnerable to criticism” no question mark 2) Identify conclusion & evidence 3) Use the why test to summarize the argument. * Extreme conclusions are often flawed due to their tone. 4) Ask: Does the author actually do this? Is it a flaw? Does this answer address the evidence? * Flaw = the conclusion is incorrect or unknowable. * Wrong answers may be too extreme or too mild * Given no “unique evidence”, focus on killing the conclusion. * Flaw translations: “ignores the fact”, “fails to consider”: Usually committed by author. “takes for granted that” “assumes without warrant”: author believes...