KNEWTON KNOTES
GMAT Quantitative STRATEGIES AND APPROACHES
First steps
Timing strategies
Always be doing doing something! something! Once you’ve read read the problem problem (20-25 (20-25 seconds), seconds), choose an approach and dive in! Some first steps that open up many questions are: - Set up an equation equation
- Facto Factor/FOIL r/FOIL
- Set up a system system and solve with substitution
- Apply expone exponent nt rules
- Test Cases Cases
• Aim to spend 2 minutes per question—less on easier questions, a little more on harder ones. • If you’ve spent 3 minutes on a question, re-evaluate: Are you within 30 seconds of a solution? - If you are, continue and and solve quickly. quickly. - If you aren’t, use the work you’ve done to make an educated educated guess and move on. • NEVER leave questions unanswered at the end of the exam.
- Apply angle properties and and look for right triangles
Data Sufficiency
Strategies
Problem Solving
Approaches Approac hes
• Testing Cases – This strategy is especially useful on number properties/ inequalities/absolute value questions. - When Testing Testing Cases on Data Suffi Suf ficiency, use the constraints to try to find confl conflicting answers to the question in the prompt. - Only test numbers numbers that satisfy satisfy the constraints in the prompt. • Don’t Solve – Don’t solve equations or do calculations if you don’t have to! - A single-variable linear equation is enough to find the value of the variable. - A system of independent independent linear linear equations with as many equations as variables is enough to find the value of any variable.
• Always know what you need to determine before turning to the statements. • Always consider the statements separately before combining. • Remember that “No” can be sufficient: If you have enough information to answer “No” to the question in the prompt, that is sufficient information, and the answer cannot be E. • Memorize the answer choices! - State Statement ment 1 alone alone - State Statement ment 2 alone alone - Both statements statements combined combined - Each statement statement alone alone - Not even both both combined combined
Strategies
Testing Cases – On any Number
PIN – If you see variables in the answer
Properties question, you can Test Cases. • Test any number that fits the conditions in the prompt! • Look for clue phrases: - “coul “could d be true” true” - “must be true” true” - “coul “could d be false” false” - “must be false” false” Choice-driven questions – If you must look at the answer choices to solve (“Which of the following”-type questions), start with E and work upwards.
choices and “in terms of” in the prompt, you can use PIN. • Procedure: 1. Choose number(s). 2. Solve problem with chosen number(s). Find numerical answer. 3. Test all five answer choices. If multiple answer choices return same value as prompt, plug in new number(s) and repeat. Plugging in the Answer Choices – If the answer choices are numbers that can be substituted for a value in the question, and are steadily increasing or decreasing round numbers, you can plug them into the prompt. • Start with B or D, t hen try the other— this method can allow you to solve by testing only two answer choices instead of all five.
Approaches Approac hes • Always be sure you know what you are solving for (don’t solve for x when the answer is 3x). • Note that there is no extra information in Problem Solving questions—you will need to use every piece of information provided!
ALGEBRA
Inequalities • If you multiply or divide an inequality by a negative number, you must flip the sign. • NEVER multiply or divide an inequality by a variable if you don’t know its sign.
Operation words
Exponents and roots
< 1, x is is farther from 0 Roots: If -1 < x <
Exponent rules:
than x 2 is, and closer to 0 than x 2 is.
1
• ab ( ) n = ( a n )( b n ) m n • a ( )( a ) = a ( m + n ) • a ( m ) n = a ( mn ) • a ( b ) = ( ac ) b = c a b • ac = a ( b – c )
Even powers have both positive and
negative solutions ( x 2 = 4 x = = ±2), but √ x is is defi defined to be positive ( √4 = 2).
-plane xy -plane Distance formula: d = √ x ( 2 − x 1 )2 + y ( 2 − y 1 )2 Lines
• “sum” is result of adding numbers
1 Negative exponents: a = n a Roots:
• Slope formula:
• “difference” is result of subtracting numbers
Negative bases: Negative numbers raised
• “product” is result of multiplying numbers
to odd powers stay negative ((-3)3 = -27), but negative numbers raised to even powers are positive ((-3)4 = 81).
• Perpendicular lines have negative reciprocal slopes. • Parallel lines have the same slope. • Equation of a line: y = mx + b, where m is slope and b is y -intercept. -intercept.
–n
• “quotient” is result of dividing numbers
1
Fractional exponents: a n =
n
STATISTICS AND FORMULAS COMMON HIGHER POWERS Common Squares
Powers of 2
12 = 1
21 = 2
22 = 4
22 = 4
32 = 9
23 = 8
42 = 16 52 = 25
24 = 16 25 = 32
62 = 36
26 = 64
72 = 49
27 = 128
2
2 = 256
92 = 81 102 = 100
29 = 512 210 = 1,024
11 = 121
Powers of 3
122 = 144
31 = 3
132 = 169 142 = 196
32 = 9 33 = 27
152 = 225
34 = 81
162 = 256
Common Cubes
172 = 289 2
18 = 324 192 = 361
13 = 1
202 = 400 252 = 625
33 = 27 43 = 64
302 = 900
53 = 125
23 = 8
• Weighted averages: - using # of terms terms in each set:
= 0.2 = 20%
≈ 0.33 ≈ 3 33 3.3%
= 0.4 = 40%
≈ 0.66 ≈ 6 66 6.6%
= 0.6 = 60%
≈ 0.16 ≈ 1 16 6.6%
= 0.8 = 80%
MODE: The number that appears most often in a set.
≈ 0.33 ≈ 3 33 3.3%
= 0.125 = 12.5%
RANGE: The difference between the largest and smallest numbers in a set.
= 0.5 = 50%
= 0.25 = 25%
≈ 0.66 ≈ 6 66 6.6%
= 0.375 = 37.5%
≈ 0.83 ≈ 8 83 3.3%
= 0.5 = 50%
= 0.25 = 25%
= 0.625 = 62.5%
= 0.5 = 50%
= 0.75 = 75%
= 0.75 = 75%
= 0.875 = 87.5%
- using % of total total in each set: MEDIAN: The number in the middle of a set, when the terms are put in order. If there is an even number of terms in the set, the median is the average of the two middle terms.
STANDARD DEVIATION: Measures how “spread out” the elements in a set are. {51, 50, 51, 51, 52} has a lower standard deviation than {2, 6, 24, 25, 34}. INTEREST FORMULAS: For initial investment of P dollars at an annual interest rate of r percent. percent. • Simple interest: Amount of money in an account making simple interest after t years: years: P + P rt
• Compound interest: Amount of money in an account where interest r nt is compounded n times per year, after t years: P (1 + n )
Common formulas • Distance = speed × time • Work = work rate × time
x + a )( x + b ) • Factoring: x 2 + Cx + D = ( means ab = D and a ( + b ) = C • FOIL: First, Outside, Inside, Last x + y • x ( + y )( ) = x 2 + 2 xy + y 2 x – • x ( – y )( – y ) = x 2 – 2 xy + y 2 • Difference of squares: ( + + y )( – y ) = x 2 – y 2 x x –
Systems with too few equations • Look for ways to solve for the value of an expression, or of a single variable. • Keep an eye out for dependent equations, especially on word problems.
NUMBERS AND ARITHMETIC “Percent of” translations:
MEAN: Average =
= 0.5 = 50%
8
8 = 64
2
DECIMAL/PERCENTAGE/FRACTION EQUIVALENCIES:
Polynomials:
• Profit = revenue – cost • Revenue = volume × unit cost
10 • 10% of x : ( 100 ) x x • n% of x : ( n ) x x 100
Combinatorics:
• General method: Draw blanks, fill in # of possibilities, multiply together. • Always begin with the blank with the most restrictions. • Combinations (order doesn’t matter): - Numbe Numberr of ways to choose choose k items items n ! from a set of n: k !( ! n ( − k )! - Number Number of ways to choose choose 2 items ( 1) from a set of n: n n2− - Numbe Numberr of ways to choose choose 1 item from a set of n: n • Permutations (order matters): - Numbe Numberr of ways to arrange arrange k items items n! from a set of n: n ( − k )!
• Factorials: n! = n × n n – 2) × ... ( – 1) × ( × 3 × 2 × 1 Common factorials - 0! = 1! = 1 - 4! = 24 24 - 2! = 2 - 5! = 12 120 0 - 3! = 6 - 6! = 720 720
KNEWTON KNOTES
GMAT Quantitative NUMBER PROPERTIES • Multiplication/division: - Pos (÷/ × ) Neg = Neg (÷/ × ) Pos = Neg (different signs gives a negative result) - Pos (÷/ × ) Pos = Pos (same sign gives a positive result) - Neg (÷/ × ) Neg = Pos (same sign gives a positive result)
Properties of 0:
• 0 = 0 × n = 0 n • n + 0 = n – 0 = n • n0 = 1 • 0 is even. • 0 is neither positive nor negative. Properties of 1:
Absolute value:
• n × 1 = n = n1 = n 1 • 1 n = 1
• Absolute value of a number is always greater than or equal to zero. • Equation with absolute value is really two equations: |x| = n x = n or x = – n.
Odd and even integers:
• Addition/subtraction: - Odd + Even = Odd - Even + Even = Even - Odd + Odd = Even • Multiplication: - Odd × Even = Even - Even × Even = Even - Odd × Odd = Odd
• Inequalities with absolute value:
|x| < n
− n < x < n
|x | > n
x < − n OR n < x
Place value: Know the names of thousands
through thousandths place. tenths hundredths thousandths
Positive and negative integers:
• Addition/subtraction: - Pos + Pos = Pos - Neg + Neg = Neg - Pos − Neg = Pos - Neg − Pos = Neg
1,234.567 thousands hundreds tens ones
Factors, multiples, divisibility:
Prime numbers:
• In a list of n consecutive integers, exactly one will be a multiple of n. • The sum or difference of two multiples of a number is also a multiple of that number. • All of the following phrases are equivalent: - a is evenly divisible by b - b evenly divides a - b divides a with no remainder - b is a factor of a - a is a multiple of b • Divisibility rules: - 3: digits add to a multiple of 3 (87 is divisible by 3 because 8 + 7 = 15 and 15 = 5 × 3) - 4: last 2 digits alone are divisible by 4 (24 ÷ 4 = 6, so 124 and 3,524 are divisible by 4) - 5: last digit is 5 or 0 - 6: divisible by 2 AND 3 - 9: digits add to a multiple of 9 60 - 12: divisible by 3 AND 4 • Use factor trees to find all the 12 prime factors of a number. 2 6 2
• A prime number is any number that only has 2 factors: itself and 1. • 1 is not prime. • 2 is the only even prime. • Prime numbers to know: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47. Integers:
• Integers are whole numbers, including negative whole numbers. • 0 is an integer. Use Venn diagrams to organize information in sets problems.
A
B x
y
A A
z
B = x + y + z B = y
The ratio of a to b can be written as or a : b. Direct and inverse proportions:
• If x and y are directly proportional, then x = cy , where c is a constant. • If x and y are inversely proportional, then xy = c, where c is a constant.
5
3
GEOMETRY
Geometry shortcuts
Quadrilaterals
• On xy -plane problems, if you have any one of these, you can find any of the other 4: - Any linear equation containing both x and y - Slope-intercept form of a line - 2 points on a line - The intercepts of a line - The slope and one point on a line
Squares:
w RECTANGLE
and perimeters by combining the areas and perimeters of these familiar shapes.
Circles
CIRCLE
• A tangent and a radius make a 90° angle. • Full angle of a circle is 360°. • Radius: any line from the center to the edge of the circle • Diameter = 2 r • Circumference = 2π r • Area of a circle = π r 2
Equilateral triangle: All sides equal, 60 ˚
60 ˚
x ˚
x ˚
EQUILATERAL
arc length circumference
=
sector area circle area
sector r
INSCRIBED and CENTRAL ANGLES
central angle 360°
=
x °
• Inscribed angle has half the measure of central angle with same endpoints.
ISOSCELES
with sides opposite those angles also equal.
y °
Angles
Similar triangles: If two triangles have all the
same angles, their side lengths will all be in the same ratio to one another, and vice versa.
x
• Supplementary angles add to 180º. • Complementary angles add to 90°.
z
than the sum of the other two sides, and longer than the difference of the other two sides.
| x − y | < z < (x + y )
y =
x
2
• Angles that make a full rotation around a point add to 360°.
y
Triangle Inequality: Any side must be shorter
a
TRIANGLE INEQUALITY
the longest side (hypotenuse).
b
RIGHT TRIANGLE
• Pythagorean theorem: a2 + b2 = c2
FULL ROTATION a°+b°+ c°+d°+ e° = 360°
• Parallel lines with transversals: - Big angles ( ) all equal y - Small angles ( ) all equal x - Big + Small = 180°
c
Right triangle: One 90° angle, across from
e°
y ˚ x ˚ x ˚ y ˚ y ˚ x ˚ x ˚ y ˚ CYLINDER
a° b°
°
d
PARALLEL LINES with TRANSVERSALS
c°
3D shapes
h
Cylinder: volume of cylinder = π r 2 h
• Special right triangles:
45 ˚
- 45-45-90 (with side lengths x , x , x √2 ) - 30-60-90 (with side lengths x ,
s
Combined figures are always combinations of familiar shapes. Find their areas
• Sector and arc:
Isosceles triangles: Two equal angles,
s
• Opposite sides equal • All angles 90° • Area of a rectangle = ℓw
60 ˚
all angles equal to 60°.
s
Rectangles:
Triangles
base × height 2 • Angles in any triangle add to 180°.
w
• All sides equal • All angles 90° • Area of a square = s2
• On rectangle/right triangle geometry problems, if you have any 2 of these, you can find any of the other 3: - Length of rectangle a ( ) - Width of rectangle b ( ) c a - Length of diagonal ( c ) - Area of rectangle b - Perimeter of rectangle • Area of a triangle =
s
60 ˚
x
, 2 x )
x
- Pythagorean triples: 3-4-5, 6-8-10, 9-12-15, 30-40-50, 5-12-13, 10-24-26, 8-15-17
r
• surface area of cylinder = Bases + Lateral Area = 2π r 2 + 2π r h .
x
45 ˚ x
SPECIAL RIGHT TRIANGLES
Rectangular solid: volume of box = ℓwh
2 x
CUBE
• surface area of box = 2 ℓw + 2wh + 2 ℓ h.
BOX
˚
30
s
Cube: volume of cube = s 3
• surface area of cube = 6 s 2
s s
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h w
KNEWTON KNOTES
GMAT Sentence Correction SC APPROACH - MOST/ALL OF SENTENCE UNDERLINED If most or all of the sentence is underlined, there are probably several errors among the answer choices. Likely, at least one answer choice contains a modifier or structure error. 1) Read through the entire sentence for meaning. 2) Use clues to determine which rule(s) may have been violated. If you find an error in the original, eliminate A. Note : awkwardness alone is NOT ENOUGH to eliminate an answer choice. The choice must violate a grammatical rule. 3) Read through choice B, and, by using the clues you have already found and by noting the differences between A and B, determine the type of error you’re looking for. If you spot an error, eliminate B.
4) Repeat the process for choices C-E. Continue to use differences between options to spot potential errors. 5) If more than one choice still remains, plug each underlined portion back into the original sentence; choose the option that clearly and unambiguously expresses the intended meaning of the sentence. 6) In a 50/50 guessing situation, favor the more concise option - the one without extra pronouns or prepositional phrases.
SUBJECT-VERB AGREEMENT
SC APPROACH - ONLY A FEW WORDS UNDERLINED If only a few words are underlined, the question is likely testing one very specific rule. 1. Read the entire sentence to determine what role the underlined portion plays. The underlined portion may connect two clauses, be part of an idiomatic expression, etc. Use clues to determine what the error may be. 2. If you’re having trouble figuring out what is being tested, use the differences between the answer choices to guide you; for example,
if one choice uses “as” but another uses “than,” look for a correctly formed comparison. 3. Choose the option that uses the correct word or phrase in the context of the sentence. 4. If you’re down to two options and one contains an extra pronoun or awkward prepositional phrase, opt for concision and clarity.
MODIFIERS RULE: An adjective or a phrase or clause that acts as an adjective must describe a noun. Adjectival modifiers follow strict placement rules. RULE: An adverb or a phrase or clause that acts as an adverb can describe a verb, adjective, or another adverb. The placement of adverbial modifiers is often somewhat flexible.
RULE: Clauses must each contain a subject-verb pair. A verb must agree with its subject in number.
ELIMINATE OPTIONS in which adjectival modifiers DO NOT describe a specific noun.
ELIMINATE OPTIONS in which the main subject-verb pair does not agree.
Adjective Clauses begin with relative pronouns: “which,” “that,” “who,” “whose,” “whom.”
ELIMINATE OPTIONS in which a subject-verb pair in an adjective clause or other dependent clause does not agree.
RULE: An adjective clause must describe the noun or noun idea immediately before it. When no new subject is introduced, the verb in the clause must agree with the noun described by the clause. If a “noun + prepositional phrase” makes up a “noun idea,” the clause can describe either the noun before the prepositional phrase or the object of the preopositional phrase. The verb in the clause must agree with the logically modified noun.
RULE: The GMAT injects filler between a subject-verb pair. Ignore this filler. • The subject of a sentence will never be inside of a prepositional phrase.
- If the subject is plural, the GMAT often puts a singular noun next to the verb to make the error more difficult to identify. With singular subjects, the GMAT often puts a plural noun next to the verb.
prep. phrase
adjective clause
“trick” plural noun
8
WRONG: The start of the races that will be run by the best athletes are at 6 pm.
9
RIGHT: The start of the races that will be run by the best athletes is at 6 pm.
RULE: The GMAT uses particularly tricky nouns as subjects. Unusual nouns rules: • Collective nouns refer to a group as a unit, and, on the GMAT, they are almost always singular. Ex: Team, jury, committee, company, cluster, group • Indefinite Pronouns fall into three categories: - Singular: each, every, anyone, everyone, nobody - Plural: both, few - Depends on the noun to which it refers : all, some, most, majority • Compound Subjects connected by “and” are almost always plural. • Noun Clauses beginning with “what” (“what the doctors have found is startling”) are usually singular. • Gerunds, nouns that end in –ing (“collecting cards is my f avorite hobby”), are singular. • Inverted Sentences disguise agreement by placing the verb after the subject. When a verb is preceded only by modifiers (prepositional phrase, participial phrase, etc.), the subject must come after the verb, and the verb must agree with its subject.
prep. phrase
relative clause
8
WRONG: In my closet, which is in the front hall, is all of my toys.
9
RIGHT: In my closet, which is in t he front hall, are all of my toys.
SUMMARY: When you see THESE CLUES, check for S-V agreement: • Answer choices differ by the use of singular/plural verb. • Sentences are full of adjective clauses and prepositional phrases. • “Unusual nouns” are used as subjects.
PRONOUNS RULE: Every pronoun must have a specific antecedent. The GMAT doesn’t often use gendered pronouns, so look specifically for it/its/they/them/their. RULE: Every pronoun must agree in number with its antecedent. ELIMINATE OPTIONS that contain a pronoun but do not contain a noun that agrees in number for the pronoun to logically replace. ELIMINATE OPTIONS that do not CLEARLY use a pronoun to refer to one, specific noun. RULE: When the subject of a dependent clause is a pronoun, this pronoun’s antecedent should be the subject of the main clause. When a sentence contains two independent clauses and the subject of the second clause is a pronoun, this pronoun’s antecedent should be the subject of the first clause. 8
WRONG: Because it was robbed last month, the door to the house is locked. (“it” is the subject of the dependent clause, so it must be illogically referring to “door.”)
SUMMARY: When you see THESE CLUES, look for pronoun-antecedent agreement: • EVERY TIME you see a pronoun, find its antecedent • Unusual nouns (especially collective nouns): the GMAT uses the same unusual nouns to test S-V and P-A agreement
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ELIMINATE OPTIONS in which the placement of the modifier is incorrect.
9
RIGHT: The books on the desk, which was the most expensive item in the furniture store, are open. (“which” logically refers to the noun “desk.”)
9
RIGHT: The books on the desk, which include a history textbook and a fiction novel, are open. (“which” logically refers to the noun idea “the books on the desk.”)
ELIMINATE OPTIONS in which the adjective clause refers to an abstract idea. 8
WRONG: Taylor broke her leg, which kept her from competing in the race.
ELIMINATE OPTIONS in which the adjective clause does not describe the noun or noun idea immediately before it. Participial Phrases begin with present participles (-ing words) or past participles (usually –ed) words. They must describe a logical noun. RULE: A participial phrases at the beginning of a sentence must modify the first noun after the comma.
ELIMINATE OPTIONS in which the first noun after t he comma is not the noun described by the phrase. 8
WRONG: Wanting to finish its project by Monday, the team’s meetings were on Saturday and Sunday. (The team wanted to finish the “project,” not the “meetings.”)
9
RIGHT: Wanting to finish its project by Monday, the team held meetings on Saturday and Sunday. (The “team” is correctly modified by the participial phrase.) Note: A pronoun inside of a modifying phrase (seen above) must refer to the noun being modified. RULE: A participial phrase that is set off by a comma, but is not at the beginning of a sentence, is more flexible with its placement, as long as it is clear what is being modified. If the participial phrase is not set off by a comma, it must describe the noun before it. ELIMINATE OPTIONS in which the participial phrase within the sentence does not clearly describe the specific noun it should logically describe. 8 9 9
WRONG: The manager hired the new employee, excited for the future of the company. (Unclear as to who is excited, the “manager” or the “employee.”) RIGHT: The manager hired the new employee, who was excited for the future of the company. (Relative clause clarifies that the “employee” is being modified.) RIGHT: The manager hired the new employee, believing that the candidate’s prior work experiences were valuable. (Logically, the participle must describe the “manager.”)
Summative modifiers begin with “a” or “an” and a noun that re-names or summarizes the gist of the previous clause. Summative modifiers often fix other modifier errors because they don’t need to describe a specific noun. 8
WRONG: The researchers hypothesized that the old method is flawed, which is likely to cause a great deal of controversy.
9
RIGHT: The researchers hypothesized that the old method is flawed, a proposal that is likely to cause a great deal of controversy.
Prepositional Phrases begin with a preposition (of, to, for, in, with, etc.) and end with a noun. They can be adjectival or adverbial. RULE: A prepositional phrase that describes a noun follows the same rules as a participial phrase (see above). RULE: When a prepositional phrase describes actions, its placement in the sentence is flexible, as long it is clear what it describes. 9
RIGHT: Until last week, the professor had never missed a day of work.
9
RIGHT: The professor had never missed a day of work until last week.
9
RIGHT: The professor had, until last week, never missed a day of work.
SUMMARY: When you see THESE CLUES, look for logical modification: • The words “which” and “that” are at the beginnings of clauses • A describing phrase is at the beginning of a sentence • Sentences that have a lot underlined and jumble the order of phrases and clauses
KNEWTON KNOTES
GMAT Sentence Correction PARALLELISM/COMPARISONS RULE: Ideas within the same clause must be connected correctly. Connected items include those in lists, those linked by correlative conjunctions, and those in comparisons. ELIMINATE OPTIONS that do not form correlative conjunctions correctly. When items are linked by correlative conjunctions, eliminate options that do not pair the first word in the construction with the correct FANBOYS conjunction.
MEMORIZE THE CORRELATIVE CONJUNCTIONS:
8
9
Either…or
Between…and
Not only…but also
Neither…nor
At once…and
Just as…so
Both…and
Not…but
Whether…or
WRONG: In the final round, the judges had to choose between the smartest competitor or the most attractive one. RIGHT: In the final round the judges had to choose between the smartest competitor and the most attractive one.
ELIMINATE OPTIONS that do not properly separate items in a list using a conjunction. 8
WRONG: The new employee impressed the boss, a man named Jim, made an impression on the CEO.
9
RIGHT: The new employee impressed the boss, a man named Jim, and made an impression on the CEO.
RULE: Items connected by “and” must be the same part of speech. The word “and” is a huge clue that items are in a list. Look to t he word or phrase after the “and,” and match it to an earlier word or phrase that is the same part of speech. If no such word or phrase exists, then the option does not contain a parallel list. Make sure the list does in fact connect items that play the same role in the sentence. ELIMINATE OPTIONS that do not link items that are the same part of speech when a sentence contains a list, a pair of correlative conjunctions, or a comparison. 8
WRONG: The exam testing natural intelligence and that one has work ethic will be given tomorrow. (The item after “and” is a “that clause” – there must be another “that clause” earlier)
9
RIGHT: The exam testing natural intelligence and work ethic will be given tomorrow.
8
WRONG: The television program neither impressed viewers nor critics. (“impressed” and “critics” are not the same part of speech)
9
RIGHT: The television program impressed neither viewers nor critics.
ELIMINATE OPTIONS that do not form comparisons correctly. Commonly used comparison terms are: MEMORIZE THE COMPARISON FORMATIONS:
as…as more/less/greater/higher…than like/unlike (must compare two nouns) 8
WRONG: The new SUV uses gas more quickly as the small sedan does.
9
RIGHT: The new SUV uses gas more quickly than the small sedan does.
ELIMINATE OPTIONS that do not form logical comparisons. RULE: In comparisons, “that” can be used to replace a singular item compared, and “those”can replace a plural item. A possessive can be used if the item belonging to the possessive is the other item compared. 8
WRONG: The couches in the living room are smaller than that in the den. (“that” is singular, but it refers to the plural noun “couches”)
9
RIGHT: The couches in the living room are smaller than those in the den.
8
WRONG: Unlike the head chef, whose dishes are always healthy, the assistant chef’s are quite unhealthy. (the implied “assistant chef’s dishes” are being compared to the “head chef”)
9
RIGHT: Unlike the head chef’s dishes, which are always healthy, the assistant chef’s are quite unhealthy.
SUMMARY: When you see THESE CLUES, check for parallelism: • The word “and” • Any correlative conjunction pair • Comparison words: “as,” “than,” unlike,” “like”
SENTENCE STRUCTURE
VERB T ENSE
RULE: An independent clause contains the main subject-verb pair; a sentence is incomplete without at least one independent clause.
RULE: Events that take place at the same time should be in the same verb tense. When there is a time shift in a sentence, the verb tenses used should correctly reflect this shift.
ELIMINATE OPTIONS that are f ragments because all clauses are dependent or because they contain a clause in which a subject’s verb is “stolen” by an adjective clause.
RULE: Use the past perfect and present perfect tenses correctly; the GMAT commonly tests these tenses.
8
WRONG: The company’s business proposal is not ready because the manager, who
has not yet edited it. 9
RIGHT: The company’s business proposal is not ready because the manager has
not yet edited it. RULE: Clauses must be connected correctly. Two independent clauses must be connected using a semicolon or using a “comma + coordinating conjunction” (coordinating conjunctions are FANBOYS words: For, And, Nor, But, Or, Yet, So). Ways to correctly connect clauses:
INDEPENDENT CLAUSE + COMMA + FANBOYS + INDEPENDENT CLAUSE INDEPENDENT CLAUSE + SEMICOLON + INDEPENDENT CLAUSE DEPENDENT CLAUSE + COMMA + INDEPENDENT CLAUSE INDEPENDENT CLAUSE + COMMA + DEPENDENT CLAUSE (Sometimes there is no comma when the dependent clause comes second) ELIMINATE OPTIONS that use a comma only or a coordinating conjunction only to connect clauses that are independent. 8
WRONG: This winter has been unusually cold and last summer was unusually hot.
9
RIGHT: This winter has been unusually cold, and last summer was unusually hot.
ELIMINATE OPTIONS that contain an incomplete clause on one side of a semicolon or a “comma + FANBOYS” construction. 8
WRONG: New computers have been purchased for the entire office; although most employees can use only the old system.
9
RIGHT: New computers have been purchased for the entire office; however, most employees can use only the old system.
The Past Perfect tense , or “had + past participle,” can only be used when t he verb in this tense took place in the far past, before another event in the more recent past. The Present Perfect tense , or “has/have + past participle,” describes an event that began at a point in the past and may continue. The word “since” is a great clue that a sentence may need the present perfect. ELIMINATE OPTIONS that use different verb tenses to describe two events that take place during the same time period. The word “when” links events that take place at the same time, and verbs that are linked in lists or by correlative conjunctions must typically be in the same tense. ELIMINATE OPTIONS that use the past perfect when the event in this tense did not come before another past tense event. 8
WRONG: By the time the movie had started, we w ere waiting in line for an hour. (Logically, we waited before the movie started.)
9
RIGHT: By the time the movie started, we had been waiting in line for an hour.
ELIMINATE OPTIONS that use the present perfect to describe an event that t ook place at a specific point in time; events in the present perfect span a non-specific period of time that began in the past. 8
WRONG: In 1960, the family has moved to Canada. (The action happened at a specific point in history.)
9
RIGHT: In 1960, the family moved to Canada.
SUMMARY: When you see THESE CLUES, look for tense agreement: • Prepositional phrases that describe a time: “since 1960,” or “in 1960” for example. • “Had/Has/Have + past participle”
WORDY AND AWKWARD CONSTRUCTIONS RULE: When you’re choosing between options that do not contain concrete violations of grammatical rules, choose options that do not contain the following:
SUMMARY: When you see THESE CLUES, make sure t hat clauses are connected correctly: • “Comma + FANBOYS” or (; ) • Some options contain adjective clauses while others do not
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being
having been
“preposition + the + -ing of”
extra or unnecessary pronouns
8
Awkward: With the finishing of the construction of the building, the architects went out to celebrate it.
9
Better: Because they finished the construction of the building, the architects went out to celebrate.
KNEWTON KNOTES
GMAT Reading Comprehension RC BASICS 1. RC tests your ability to locate, understand and paraphrase ideas in a passage.
3. Four passages appear on every test, each with 3-4 questions.
2. Passages: 250-400 words. The passage appears on the left; questions, one at a time, on the right.
4. Bring no prior knowledge to this section; all the information that is needed to answer the questions is contained in or implied by the passage.
KNEWTON STRATEGIES • Spend 2-3 minutes actively reading and • Don’t spend time re-reading a complex MAPping the passage, and 4 to 5 minutes idea or sentence; most details in a pasanswering the questions. An entire passage sage will not be tested, and if a confusshould take 6-8 minutes. ing detail is tested, revisit and decipher as needed. • Read the entire passage only once, and then re-visit sections of the passage as the questions dictate. • Read actively: focus on t he main idea and structure, not on complex details.
• Passages often include lists, long sentences, complex and convoluted phrasing, or scientific or academic jargon. Take note of where these
Passage Categories/Additional Reading
Points of View
The Ideas That Matter:
BUSINESS: Wall Street Journal, New York
• Passages with more than one POV: keep track of who agrees with which theory.
Take note of t he “conventional wisdom” (“Many say…,” “most agree”). Note comparisons or similarities between two ideas ( “Unlike,” “Alternatively,” “Another proposed solution”).
Times, Economist, Businessweek, US News and World Report SCIENCE: New Scientist, Science, Popular
Science, Wired HUMANITIES: New Yorker, Atlantic Monthly,
New Republic
Use MAPS to Understand the Passage MAIN IDEA: What is being said. Often called
the “primary concern” ATTITUDE: How it is said. The feeling or
• Keep in mind: the author may have a different POV than do the sources in the passage.
PAY ATTENTION TO:
DON’T WORRY ABOUT:
• Passages rarely use the first-person; POV must be spotted through context.
Relative numbers or dates (“before 1920…”)
Specific numbers or dates (“On July 14, 1882…”)
Unknown words or terms that are repeated or defined in the passage
Unknown words or terms that appear only once
Conflicting opinions, and why they conflict
Determining which opinion is “right” or which provides better reasoning
Problems and their proposed solutions
Whether a given solution seems effective or logical
Divisions between entities (“While all corporations must… only industries with strict hierarchies will profit from…”)
Lists of entities (“Corporations, public utilities, consumer agencies and political organizations all…”)
Policies, theories or processes and the author’s opinion about them
Complex details of an economic policy, theory, or scientific process
Any proper names and what they believe (“According to Harris,…”)
Committing names or theories to memory.
Use TONE words:
- Negative Words: unfortunately, problematic, failure, complicating, shortcomings, unrealistic, unresolved, etc.
PURPOSE: Why it’s being said. Purpose is
- Positive Words: practical, fortunately, thoughtful, clever, set a new standard, useful, effective, successful, etc.
usually expressed as a verb on the GMAT, • “defend,” “compare,” or “illustrate”
- Words of Emphasis: Important, Significant, Best, etc.
STRUCTURE: How the passage is organized.
- Words of Continuity or Causation: Moreover, Because, Therefore, Thus, etc.
opinion of the author, often called the “tone”
Briefly note a description of each paragraph. • Write down as much or as little as is needed in order to understand the main ideas and structure of the passage.
- Words of Contrast: However, Although, Despite, etc.
READING COMPREHENSION QUESTION TYPES Global Purpose Questions • Refer to the entire passage; info should come from your MAPS. • Main Idea Questions: Choose the answer that describes the entire passage. Avoid choices that refer to one detail of the passage and neglect bigger issues. • Primary Purpose Questions: Use the lead verb in the answer choices to eliminate incorrect answer choices. - e.g.: if purpose is positive, eliminate answers that say “dispute” or “recommend against.”
Detail Questions • Ask you to locate, understand and paraphrase a specific detail from the passage. • Use your MAPS to locate the relevant section, and then re-read. • Take the time to understand context when answering detail questions; read the sentence before and after the sentence referred to in the question. • Avoid answer choices that distort details from the passage or create paraphrases that exaggerate claims from the passage.
Inference Questions • Ask you to determine which answer choice is most strongly implied, but is not explicitly stated, by the passage. • Do not stray from passage logic; an inference must be true according to the passage.
READING IN REGULAR LIFE
READING ON THE GMAT
Every sent ence is import ant.
Many sentences are unimportant.
We stop to look up unknown words.
Gather the meaning of words through context.
• Re-read the entire paragraph in which the detail is included.
We compare what we’re learning while reading to what we already know.
Bring NO outside information to bear on the passage.
• Difficult inference questions combine several details; you may need to reread all relevant sections that refer t o the topic at hand.
We take time to stop, reread, and unravel complex details.
Skim complex details and descriptions, and only return to them if they are asked about.
We know who wrote what we’re reading and can take that into account while reading.
Determine the author’s purpose, tone and perspective without knowing who he or she is.
We trust the author to present points clearly, because it is his or her responsibility to be understood.
Actively MAP the purpose and structure of the passage to understand the overall ideas.
• Some ask why the author included a certain detail; these questions are almost always used either to exemplify one of the author’s points or to provide a counterargument.
Argument/Application Questions • Ask you to understand an argument and to strengthen, weaken, or apply it to a specific case. • Ask you to apply passage ideas to hypothetical situations not described in the passage. • Do not overextend the analogy; the correct answer will not require a large logical leap. • Often require you to determine a rule, restriction or category described in the passage. - e.g. If a passage states that “Sung’s 1972 study claims that when investments surpass personal savings, economic recessions are likely,” a question might ask for a situation most likely to engender an economic recession; the correct answer would provide these exact required conditions.
EXCEPT questions, begin with the answer choices and find the corresponding • Avoid extreme answer choices or answer section of the passage, one by one, eliminating choices that could be true according to them as you go. Watch out for distorted details the passage but are not definitely true. and extreme answer choices.
WRONG ANSWER CHOICES Some types of incorrect answer choices appear many times on the Reading Comprehension section:
STANDARD WACs
DIFFICULT WACs
Too Broad in Scope
Goes beyond subject of passage
Incorrect Comparisons
Erroneously compares two entities
Too Narrow in Scope
Addresses overly specific subject
Distorted Detail
Uses passage wording incorrectly
Opposite
States reversal of correct fact
Could be True
Possibly true but not inferable
Extreme
Overshoots answer/ extreme language
Irrelevant Topic
Addresses new subject erroneously
Note: On
Many answer choices pull eye-catching details from the wrong section of the passage; use your MAPS to eliminate these answers immediately.
KNEWTON KNOTES
GMAT Reading Comprehension Common wisdom in finance for the last three decades states that small commercial banks have little influence on the status of a national economy. 1 Fiscal policy is usually written to favor large commercial banks, based on the assumption that large banks play a critical role in most aspects of national economic health, including consumer lending rates, government interest rates, home mortgage rates, and the extension of credit. 2 However, evidence has recently emerged suggesting that some small commercial banks outperform their larger counterparts, especially during unusual economic periods. 3 Perhaps the best example of this is 4 during recessions, when some small commercial banks, which have limited assets and thus more stringent credit requirements, incur a far smaller debt burden than do dangerously over-leveraged large banks. Although no research has proven definitively that small commercial banks are more active providers of loans and credit during recessions than are large commercial banks, some economists 5 are claiming that small banks step into leading roles when necessary to keep national economies afloat.
It can be inferred
1
commercial banks
2
from the passage that proponents of the “alternative view” about small would be most likely to agree with which of the following statements?
The conflicting definitions of “small bank” disguise the limited economic role played by such institutions. 3 Small, local banks are more likely than are large commercial banks to support legislation that offers aid to small businesses. 4 Some geographic and financial limitations on small commercial banks become advantageous during economic recessions. 5 Fiscal policy should be overhauled to favor
6
the economic needs of small commercial banks.
7
Economic recessions would be avoided if only a larger number of commercial bank employees knew their clientele personally 8 .
Thus, an alternative view has emerged, claiming that small commercial banks are a necessary hedge against depressions. 6 First, 7 small banks tend to be restricted to a single locale; bankers are more likely to know their clientele personally and to provide emergency small-business assistance without bureaucratic delays. Small banks also pump money back into local communities at times when their residents face high levels of unemployment and malaise. Lastly, small banks must always ensure adequate reserves of capital in preparation for a bank run, which inherently limits their ability to acquire heavy risk. However, 8 such claims about the role played by small banks are based primarily on economic theory sometimes lacking solid empirical support. The issue is complicated by the fact that 9 policy makers utilize conflicting definitions of a “small bank.” The US Federal Reserve, for example, 10 states that any domestically chartered bank not among the 25 largest nationwide can be considered a “small bank”- this group includes federal banks that are far larger than the local, personal banks cited by economists.
MENTAL NOTES
PHYSICAL NOTES
(Inner monologue while reading the passage)
(Sample scratch work of notes about the passage)
1
The “common wisdom” is almost always going to be contradicted at some point in the passage.
2
Lists of entities are rarely important on their own; skim them and go back only if asked.
3
The word of contrast at the beginning of this sentence indicates a reversal of the conventional wisdom.
4
Take note of examples that support certain points in the passage. Some will be explicitly highlighted with the word “example” or “exemplify.”
5
This sentence starts with a contrast word, and states the notable point of view of “some economists.” This sentence plays the role of the “thesis” of this passage.
6
The word of continuity in this sentence indicates that the second paragraph is a continuation and extension of the first.
7
The words “first” and “lastly” indicate an enumerated list of reasons; thus, the second paragraph provides evidentiary examples to illustrate and expand upon claim made in the first sentence.
8
The third paragraph has a final reversal, indicated by a word of contrast.
9
A “complication” is likely to introduce a new fact not yet described.
10
This is an implicit comparison, as opposed to many explicit comparisons made in this passage (small vs. big banks, common wisdom vs. new theory). The Federal Reserve uses one definition, and those that subscribe to the “alternative view” use another.
M: Small banks- important for an economy? A: Even-handed, sees both sides P: Present “alternative view” about small c. banks S: P 1- Big banks more important? Maybe not P2-Small banks might be better, esp. in recession P3-Not enough evidence, definition problems
HOW TO READ A QUESTION (Question stems and wrong answer choices contain valuable clues) 1
The word “inferred” means that the correct answer will not be explicitly stated, but must be true according to the passage.
2
The proponents of the “alternative view” are discussed in the second paragraph, as per the MAP.
3
This choice states the opposite of what the proponents of the “alternative view” believe: that the economic role played by small banks is large, not “limited.”
4
“Legislation that offers aid to small businesses” is an irrelevant topic. It is similar, but not equivalent to emergency small-business assistance without bureaucratic delays.
5
This choice paraphrases the second paragraph and is correct. Small banks are “local” (a geographical limitation) and “must keep adequate reserves” (a financial limitation) and these qualities help in a recession economy .
6
This choice is extreme; the words “overhauled” and “favor” are unsupported by the passage.
7
The word “avoided” is extreme here; although small banks may be a necessary hedge against depression, the personal connection may not, in itself, help economies avoid recessions.
8
This use of direct passage wording is intended to trap test-takers. Notice that the correct answer primarily paraphrases the passage and does not quote directly.
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KNEWTON KNOTES
GMAT Critical Reasoning CRITICAL REASONING APPROACH 1. Read the question stem to determine question type. Doing so tells you what you’ll be asked to do.
2. Identify the conclusion of the argument and the evidence presented in the argument.
3. Identify any assumptions clearly present in the argument.
4. Make a categorical pre-phrase to determine the qualities of a correct answer.
DEFINITION OF TERM Argument – Two to five sentences that lay out a set of logical premises and usually draw a conclusion based on these premises. Premises – The claims upon which a conclusion is based; these consist of evidence and assumptions.
Assumption – Unstated, but necessary, premises of an argument. Assumptions often bridge a gap in reasoning between evidence and conclusions. Every GMAT argument contains multiple assumptions. Inference – A logical conclusion that must be true based on the premises of an argument but is not explicitly stated in an argument.
Conclusions – Opinions, predictions, recommendations, or general principles that are drawn based on the premises of an argument. Conclusions can be found anywhere in an argument. Conclusion keywords: thus, therefore, so, due to this, for this reason, will result in, should, this change will/would, apparently, clearly.
Entity – A noun used in an argument; a person, place, group, rate, rise in profit, etc. Evidence – Statements that are explicitly stated in an argument. Evidence is used to draw a conclusion in an argument. Evidence keywords: because, given that, since, due to the fact that, in t he past year, last month.
CRITICAL REASONING QUESTION TYPES The three most common CR question types (constituting over 60% of all test questions, on average) require test-takers to identify the evidence, conclusion, and assumptions provided in an argument. • WEAKEN (~30% of all CR questions)
• DETERMINING RELEVANT INFORMATION (~8%)
Many CR questions ask test-takers to weaken an argument, or to find evidence that undermines a prediction or recommendation made in the argument. Weaken questions are the most common CR question type. Test-takers can weaken an argument by: - Invalidating an assumption on which the conclusion depends. - Identifying an additional piece of evidence that makes the conclusion itself less likely to be valid. - Remembering that the correct answer does not have to invalidate or disprove the argument to be an effective weakener. - Keeping in mind that there are several ways to weaken a causal argument (an argument which concludes that a certain cause has a certain effect. (X Y): • Identify an alternative cause for the observed effect (Z
Y)
• Suggest reverse causation; that the effect in fact caused the proposed cause (Y X) • Strengthen the chance that the cause and effect are correlated but not causally related (X and Y, not X Y) • Negate the existence of the cause or effect (~X, ~Y) • STRENGTHEN (~20%)
Many CR questions ask test-takers to strengthen an argument, or to find evidence that supports a prediction or recommendation made in the argument. A strengthener does not have to prove the argument; it simply must make the conclusion more likely to be valid. A strengthener may provide a detail that is directly relevant to the situation described in the argument or a general principle that applies to the argument. • ASSUMPTION (~15%)
Some CR questions ask test-takers to identify an assumption upon which an argument depends. Assumptions either: 1. Fill a logical gap in the argument: If you have trouble coming up with assumptions, try creating a sentence that links the evidence with the conclusion. 2. Negate a potential argument weakener: An answer choice may introduce and immediately refute a piece of additional evidence that, if true, would weaken the argument. Refuting a weakener shows that the original argument continues to be valid. The Negation Test can be used to identify or check the answer on Assumption questions (see next page for Negation Test strategy). • INFERENCE (~10%)
- Some CR questions ask test-takers to infer a statement based on a series of pieces of evidence in an argument. - On Inference questions, the correct answer must be true. Wrong answers are either entirely false or could be, but are not necessarily, true according to the argument. - There are two ways to form a valid inference: • Paraphrase: In a long, complicated inference question, the correct answer will often paraphrase one idea in the prompt. • Logical Bridge: Most inference questions ask test-takers to connect two pieces of evidence in the argument.
- Some questions ask test-takers to determine which answer choice provides relevant information with which to evaluate the conclusion of an argument. - The answer choices in this question type sometimes assume unusual forms; for example, they may be questions or may begin with the word “whether,” as in “Whether the new strategy will cost more than the previous one did.” - Locate the key feature that connects the evidence and the conclusion of an argument to locate the correct answer. • EXPLAINING OBSERVED EVENTS/PARADOX (~8%)
Some CR questions present an observation and then ask test-takers to identify the piece of evidence that would best explain it. In a Paradox question, the observed event seems to conflict with the evidence. That is, the evidence leads away from the conclusion. To resolve a paradox, the answer will be either a connecting piece of evidence or a piece of evidence that explains both evidence and conclusion. • ROLE OF STATEMENT (~5%)
At times, a CR question will include two boldfaced statements and ask test-takers to identify the role that the statements play in the argument. Identify the conclusion of the argument and any evidence provided to support it, as well as any counterarguments. In most cases, the statements are either evidence or conclusions. • FLAW (<5%)
Flaw questions ask test-takers to identify a logical flaw. Common Flaws include: - Unrepresentative samples - Confusing correlation and causation - Overlooked possibilities - Confusing sets and their members - Confusing necessity and sufficiency • METHOD OF REASONING/PARALLEL REASONING (<5%)
On rare occasions, test-takers will be asked to explicitly describe how an argument is constructed, or to find an analogous situation that uses the same logical construction as the original. IF YOU GET STUCK...
1) Take Your Time Critical Reasoning prompts contain logical traps and demand close reading. CR questions often take far longer than SC or RC questions do - two and half minutes or more is not too long for a tough CR question. 2) Draw a Diagram Some students find it helpful to draw diagrams of the argument, using letters or pictures to represent entities and their relationships. Doing so can help untangle complicated arguments. 3) Eliminate Wrong Answers for a Reason When stuck between two answer choices, identify the exact reason why either answer choice is incorrect. Look for distortions and shifts in terms, irrelevant comparisons, extreme language, and other common traps.
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KNEWTON KNOTES
GMAT
Critical Reasoning KNEWTON CR STRATEGIES
Use the Negation Test
Paraphrase
Create a Categorical Prephrase
Notice Multiple Points of View
• Negating a valid assumption will invalidate an argument. Negate each answer choice in turn: if negating an answer choice invalidates the argument, that answer choice is correct.
• Critical reasoning questions often use overly complex or convoluted language just to confuse test-takers.
• A categorical prephrase is a general expectation of the logical features a correct answer must contain.
• Translate complicated arguments into simpler language as much as possible, and make mental abbreviations of long entities.
• Some prephrases can be specific (“If the alarm didn’t deter criminals, it would not make the neighborhood safer.”) but most are abstract definitions of the missing piece of the argument (“I need something that would lead to a safer neighborhood.”)
• Some CR arguments express a point of view other than that of the author. Take notice of who is citing each piece of evidence or drawing the conclusion in an argument.
• The Negation Test can be used on Assumption questions, but not on Strengthen questions.
• Many arguments and answer choices contain double and triple negatives. Translate these into simpler sentences.
• Categorical prephrases are useful for weaken, strengthen and assumption question types.
WRONG ANSWER CHOICES • Opposite - Answer choices that have the opposite effect of the correct answer, including “strengthen” answers on “weaken” questions and vice versa.
- if one answer is the opposite of another, chances are good that one of them is correct. • Scope Errors - Answer choices that are too specific or too broad to address the issue at hand, or that slightly shift the focus of the question away from what is being asked • Extreme Errors - Answer choices that overstate claims, often through the use of extreme language (only, never, must, cannot, all) when there are no matching claims in the argument.
- Extreme answer choices are almost never correct on Assumption Questions, but extreme terms can be found in the correct answer of many “Which of the following, if true...” questions.
• Role of Statement questions often express two opposing arguments simultaneously in one CR stem.
COMMON CR LOGICAL PATTERNS
• Distorted Details - Answer choices that repeat argument wording, but introduce errors, including false or irrelevant comparisons, chronological errors, or references to the wrong group or idea • Could be true - Answer choices that could possibly be true according to the argument but, that cannot be inferred as definitely true.
- Some wrong answer choices are factually accurate, but if an answer choice is not directly supported by the prompt, it is wrong. • Irrelevant Topic - Answer choices that refer to subjects that are tangentially related to the topic of the argument but are irrelevant to the question at hand. • Irrelevant Comparisons and Distinctions Answer choices that compare entities in a way that does not address the argument correctly. (this includes comparing a quantity to “the average” when such a comparison does not have an effect on the argument.)
• Comparisons
- Many critical reasoning arguments and answer choices use complex comparisons. - Make sure that both the entities and the criterion on which the entities are being compared are clear and consistent. • Causation or Correlation
- CR arguments often cite one or two pieces of correlated factual evidence and draw a causal conclusion. - Test-takers are often asked to either strengthen or weaken this causal conclusion with additional evidence. • Similar Conditions do not imply Similar Results
- Many CR arguments draw conclusions about one situation based on the results of another situation.
- In these cases, never assume that the similar conditions necessarily imply similar results; the correct answer often explains how the situations are less similar than is implied in the argument. - The citation of a similar situation in an answer choice is usually incorrect. • Absolute vs. Relative Quantities/ Groups vs. Members of Groups
- Some arguments confuse absolute and relative quantities, or confuse percentages with real numbers. - Arguments may state that because a number of groups is rising, the number of total members within those groups is also rising. This is not necessarily true, and the correct answer may point out the fact that this only occurs if the num ber of members per group decreases.
SAMPLE QUESTION STEMS • WEAKEN
• PARADOX
‘Which of the following, if true, most seriously weakens (undermines, calls into question) the argument presented above? • STRENGTHEN Which of the following, if true, would provide the most support for (strengthen, provides the strongest grounds for, etc.) the lawmaker’s prediction?
Which of the following, if true, most helps to explain this surprising finding? • ROLE OF STATEMENT
In the argument given, the two portions in boldface play which of the following roles? • FLAW
The economist’s argument is flawed because it fails to consider
• INFER CONCLUSION
Which of the following must be true on the basis of the statements presented above? Which of the following can properly be inferred regarding tax rates from the statement above? • DETERMINING RELEVANT INFO
This argument is most vulnerable to the objection that it fails to • METHOD OF ARGUMENT
The researcher replies to the CEO’s argument by • BLANK STEMS (~5%)
In evaluating the argument, it would be most useful to determine (or “most important to know”) • ASSUMPTION
Which of the following is an assumption made in drawing the conclusion above?
Some arguments have no question stem, but end with a blank. These tend to be either “explain events/paradox” questions (“However, these facts do not prove that the virus causes infection, because ______” ) or “infer conclusion” questions (“Thus, it can be expected that __________”). • EXCEPT STEMS
• EXPLAINING OBSERVED EVENTS
Which of the following hypotheses best accounts for the finding of the experiment?
Any of the following, if true, would strengthen the manager’s conclusion EXCEPT
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