1. Verbal Reasoning
Overview and Strategies

2. Verbal Overview 3 types of questions: 40% sentence correction
30% reading comprehension
30% critical reasoning
The entire verbal section is about structure rather than content.

3. Verbal Overview Sentence correction content:
All basic grammar, but especially:
Parallel constructions / clauses
Verb tense or agreement errors
Misplaced / dangling modifiers
Inappropriate pronoun usage
Selection among prepositions

4. Verbal Overview Sentence correction questions:
A sentence will be given, with a section underlined; the answer choices will represent alternative renderings for the underlined section (including no chnage).

5. Verbal Overview Reading comprehension content:
Identify the “main topic” of a piece
Answer “according to…” questions
Make [very] simple inferences
Identify narrative structure
NOTHING ABOUT CONTENT

6. Verbal Overview Reading comprehension questions:
“The main topic of the passage is…”
“The passage suggests which of…”
“According to the passage, which…”
“The author considers which of…”
“The 2nd paragraph serves what role...”

7. Verbal Overview Critical reasoning content:
Distinguish between assumptions, premises/evidence, and conclusions
Analyze structure of an argument
Consider additional information
Identify analogous structures

8. Verbal Overview Critical reasoning questions:
“The passage assumes that…”
“The purpose of the passage is…”
“Which if true would most weaken / strengthen the above conclusion…”
“Which is best supported by…”

9. Verbal Overview Critical reasoning questions:
“Which must be true based on…”
“Which information would be most useful in evaluating the argument…”
“Which most closely resembles the argument / logic in the passage…”

10. Grammar Strategies
The GMAT is not a vocabulary test. The purpose of sentence correction questions is to assess whether you can construct a sentence following the standard procedural grammar rules, not whether you can write “at grade”.

11. Grammar Strategies
You are NOT looking for the best or most natural way to render a sentence. Your only goal is to identify the option that contains no errors—specifically, the option that is most unambiguously correct, even if it doesn’t “sound” right.

12. Grammar Strategies
“Unambiguously correct” sentences are usually those in which auxiliary words that are often omitted in speech are included consistently throughout. Look at the little words: is, are, has, have, do, does, that, those, which, etc

13. Grammar Strategies
For sentence correction questions, be sure to assess the correctness of each option within the entire sentence rather than as an isolated clause. The options may be long, but usually only one or two key words matter.

14. Grammar Strategies
Focus on structural words / sentence parts—identify all subjects, objects, verbs, and auxiliary words present. Ignore the content of the sentence, and check only that these components are used correctly and are “matched”.

15. Grammar Strategies
“Many political pundits believe that dissension over immigration priorities among an increasingly polarized U.S. congress decrease the likelihood of significant action being taken in 2016.” The choices here will obscure the error.

16. Grammar Strategies
“dissension over immigration priorities among an increasingly polarized U.S. congress decrease the likelihood of significant action being taken in 2016.” Answers might be “… likelihood that signification action will be”, and so on.

17. Grammar Strategies
“dissension over immigration priorities among an increasingly polarized U.S. congress decrease the likelihood of significant action being taken in 2016.” The only error is one of subject-verb agreement: “dissension”-“decrease”.

18. Grammar Strategies
“dissension over immigration priorities among an increasingly polarized U.S. congress decrease the likelihood of significant action being taken in 2016.” “over immigration priorities” is only an adjective—“priorities” is not a subject.

19. Grammar Strategies
“dissension over immigration priorities among an increasingly polarized U.S. congress decrease the likelihood of significant action being taken in 2016.” The most likely solution is “decreases”; an alternate could be “will decrease”.

20. Grammar Strategies
GRE vocabulary questions will offer multiple potential sentence renderings that make intuitive sense or parse easily. Your task is not to find a “best sounding” sentence, but to identify the two or three words in the sentence which are related most to the word you are asked to fill in.

21. Grammar Strategies
“Since she believed him to be both candid and trustworthy, she refused to consider the possibility that his statement to the board of directors should be construed as _________.” The majority of words are unnecessary.

22. Grammar Strategies
“Since she believed him to be both candid and trustworthy, she refused to consider the possibility that his statement to the board of directors should be construed as _________.” First find the “key” / “target” phrase.

23. Grammar Strategies
“Since she believed him to be both candid and trustworthy, she refused to consider the possibility …” Identify intended word relationships. Does the target phrase negate or affirm the word you are intended to fill in?

24. Grammar Strategies
“Since she believed him to be both candid and trustworthy, she refused to consider the possibility that his statement to the board of directors should be construed as insincere.” Choose based on target and relation.

25. Grammar Examples
“Stories are a haunted genre. Far from a _Blank 1_ kind of story, the ghost story is almost the paradigm of the form, and a feeling of _Blank 2_ was undoubtedly one effect that Poe had in mind when he wrote about how stories work.”
debased
pessimism
normative
goosebumps
meticulous
curiousity

26. Grammar Examples
“Stories are a haunted genre. Far from a _Blank 1_ kind of story, the ghost story is almost the paradigm of the form, and a feeling of _Blank 2_ was undoubtedly one effect that Poe had in mind when he wrote about how stories work.”
debased
pessimism
normative
goosebumps
meticulous
curiousity

27. Grammar Examples
“Dreams are _______ in and of themselves, but, when combined with other data, they can tell us much about the dreamer.” Select two of the six answer choices
astonishing
inscrutable
disordered
revealing
harmless
uninformative

28. Grammar Examples
“Dreams are _______ in and of themselves, but, when combined with other data, they can tell us much about the dreamer.” Select two of the six answer choices
astonishing
inscrutable
disordered
revealing
harmless
uninformative

29. Grammar Examples
“Even in this business, where ______ is part of everyday life, a talent for lying is not something usually found on one’s resume.” Select two of the six answer choices
ambition
insensitivity
mendacity
prejudice
prevarication
avarice

30. Grammar Examples
“Even in this business, where ______ is part of everyday life, a talent for lying is not something usually found on one’s resume.” Select two of the six answer choices
ambition
insensitivity
mendacity
prejudice
prevarication
avarice

31. Grammar Examples
“While in many ways their personalities could not have been more different—she was ebullient where he was glum, relaxed where he was awkward, garrulous where he was________—they were surprisingly well suited.” Select two of the six answer choices
solicitous
laconic
munificent
fastidious
irresolute
taciturn

32. Grammar Examples
“While in many ways their personalities could not have been more different—she was ebullient where he was glum, relaxed where he was awkward, garrulous where he was________—they were surprisingly well suited.” Select two of the six answer choices
solicitous
laconic
munificent
fastidious
irresolute
taciturn

33. Grammar Overview
Misplaced modifiers occur when a phrase clarifies or describes another element in the sentence (subject or verb, usually), but the construction of the sentence positions the phrase too far from the element it describes.

34. Grammar Overview
In order for a phrase to be considered a misplaced modifier, see whether it contains both a subject (could be just a pronoun) and a verb; if the phrase (or part of the phrase) cannot stand on its own as a sentence, it is a modifier.

35. Grammar Overview
To identify whether a phrase could be a misplaced modifier, see whether it contains both a subject (could be just a pronoun) and a verb; if the phrase (or part of the phrase) cannot stand on its own as a sentence, it is a modifier.

36. Grammar Overview
“Although not as liquid an investment as a money-market account, financial experts often recommend a certificate of deposit for its high and stable yield.” The modifier here is misplaced next to “experts” (who are not investments)

37. Grammar Overview
“Although not as liquid an investment as a money-market account, financial experts often recommend a certificate of deposit for its high and stable yield.” We could say “money-market account, a certificate of deposit is often…”

38. Grammar Overview
“Although not as liquid an investment as a money-market account, financial experts often recommend a certificate of deposit for its high and stable yield.” But, the latter half of the sentence is not underlined, so we cannot fix it.

39. Grammar Overview
“Although not as liquid an investment as a money-market account, financial experts often recommend a certificate of deposit for its high and stable yield.” The only fix is to make the first half into a phrase that can stand alone.

40. Grammar Overview
“Although it is not as liquid an invest-ment as a money-market act., financial experts often recommend a certificate of deposit for its high and stable yield.” The only fix is to make the first half into a phrase that can stand alone.

41. Grammar Overview
“Upset by the recent downturn in productivity, the possibility of new pay incentives was raised by the board of directors in their annual meeting.” The modifier here (not underlined) is misplaced (a “possibility” is not upset).

42. Grammar Overview
“Upset by the recent downturn in productivity, the possibility of new pay incentives was raised by the board of directors in their annual meeting.” We can change the sentence to put the modified element next to the modifier.

43. Grammar Overview
“Upset by the recent downturn in productivity, the board of directors raised the possibility of new pay incentives at their annual meeting.” Who is upset? The “board of directors”, so that phrase must follow the comma.

44. Grammar Overview
Parallel construction errors occur when a sentence contains a compound subject or verb whose parts are not “matched”, when it contains two full clauses whose structures are not the same, or when it makes a comparison between elements of different types.

45. Grammar Overview
To identify whether a sentence might contain a poor parallel construction or comparison, see whether it contains a list of elements joined by “and” / “or”, whether it contains multiple clauses, or whether it makes a comparison or an analogy between two elements.

46. Grammar Overview
“The reasons cited for the governor’s decision not to seek re-election were the high cost of a campaign, the lack of support from his party, and desiring to spend more time with his family.” There’s a list; check for poor parallels.

47. Grammar Overview
“…the high cost of a campaign, the lack of support from his party, and desiring to spend more time with his family.” “… the high cost” “… the lack of support” “… desiring to spend more time”

48. Grammar Overview
“…the high cost of a campaign, the lack of support from his party, the desire to spend more time with his family.” “… the high cost” “… the lack of support” “… the desire to spend more time”

49. Grammar Overview
“To say that the songs of the common robin are less complex than those of the indigo bunting is doing a great disservice to both species of birds.” There are 2 clauses; are they parallel?

50. Grammar Overview
“To say that the songs of the common robin are less complex than those of the indigo bunting is doing a great disservice to both species of birds.” “To say that the songs…” “[is] doing a great disservice…”

51. Grammar Overview
“To say that the songs of the common robin are less complex than those of the indigo bunting is to do a great disservice to both species of birds.” “To say that the songs…” “[is] to do a great disservice…”

52. Grammar Overview
“Over the last 20 years, the growth of info. technology has been more rapid than any other business field.” The word “more” indicates that there is a comparison; are the elements that are compared actually comparable?

53. Grammar Overview
“Over the last 20 years, the growth of info. technology has been more rapid than any other business field.” “the growth of info. technology” “any other business field”

54. Grammar Overview
“Over the last 20 years, the growth of info. technology has been more rapid than any other business field.” “the growth of info. technology” “the growth of any other business field”

55. Grammar Overview
“Over the last 20 years, the growth of info. technology has been more rapid than that of any other business field.” “the growth of info. technology” “that of any other business field”

56. Grammar Overview
“In contrast to classical guitars, whose players prefer the rounded tones of nylon strings, folk guitar players prefer the bright sound only steel can create.” The sentence directly indicates that there is a comparison; is it parallel?

57. Grammar Overview
“In contrast to classical guitars, whose players prefer the rounded tones of nylon strings, folk guitar players prefer the bright sound only steel can create.” “classical guitars, whose owners prefer” “folk guitar players prefer”

58. Grammar Overview
“In contrast to classical guitar players, who prefer the rounded tones of nylon strings, folk guitar players prefer the bright sound only steel can create.” This is one possible solution.

59. Grammar Overview
“In contrast to classical guitars, whose players prefer the rounded tones of nylon strings, folk guitars have steel strings which create a bright sound, which their players prefer.” This is another possible solution.

60. Grammar Overview
Tense/subj-verb agreement errors occur most frequently in sentences that contain complex structures with multiple subjects and multiple verbs simultaneously, especially when it has clauses that overlap or are separated by a lot of intervening material.

61. Grammar Examples
“Unlike Franklin D. Roosevelt’s bootstrap program that helped to restart economic growth in the 1930s through public works, Ronald Reagan proposed a program of trickle-down economics to restart the economy.”
Franklin D. Roosevelt’s bootstrap program that helped
Franklin D. Roosevelt, whose bootstrap program helped
Franklin D. Roosevelt and his bootstrap program, that helped
the bootstrap program of Franklin D. Roosevelt that helped
Franklin D. Roosevelt and his bootstrap program, helping

62. Grammar Examples
“As its reputation for making acquisitions of important masterpieces has grown, the museum has increasingly turned down gifts of lesser-known paintings that they would in the past have accepted gratefully.”
they would in the past have accepted gratefully
they would have accepted gratefully in the past
it would in the past have accepted gratefully
it previously would have accepted gratefully in the past
previously would have been accepted gratefully

63. Grammar Examples
“During the 1970s, it became evident that writing about someone else’s research was far easier for social scientists who wanted to make a name for themselves than it was to do their own research.”

64. Grammar Examples
that writing about someone else’s research was far easier for social scientists who wanted to make a name for themselves
that writing about someone else’s research was, for social scientists who wanted to make a name for themselves, far easier
to social scientists who wanted to make a name for themselves, that writing about someone else’s research was far easier
that for social scientists who wanted to make a name for them-selves, writing about someone else’s research was far easier
that for social scientists who wanted to make a name for them-selves, it was far easier to write about someone else’s research

65. Grammar Examples
that writing about someone else’s research was far easier for social scientists who wanted to make a name for themselves
that writing about someone else’s research was, for social scientists who wanted to make a name for themselves, far easier
to social scientists who wanted to make a name for themselves, that writing about someone else’s research was far easier
that for social scientists who wanted to make a name for them-selves, writing about someone else’s research was far easier
that for social scientists who wanted to make a name for them-selves, it was far easier to write about someone else’s research

66. Reading Strategies
The GMAT does not test your actual understanding of the content of any passage [not even in the essay].
The purpose of the verbal section is to identify if you know how verbal communication works as a process.

67. Reading Strategies Tips for finding the correct answer:
Focus on why the passage author said something rather than what was said.
What is being claimed vs. what is being used to support that claim?
What is missing from the passage?

68. Reading Strategies Tips for finding the correct answer:
Put aside your personal perspective.
Do not read a passage for content then develop counter-examples or weaknesses; “ours is not to make reply, ours is not to reason why”

69. Reading Strategies Tips for finding the correct answer:
A conclusion is not an outcome that may result if an argument is true; it is only that which is being argued.
Work within the narrow boundaries of the passage / argument as given.

70. Reading Examples
“A fashion designer’s fall line for women utilizing new softer fabrics broke all sales records last year. To capitalize on her new success, the designer plans to launch a line of clothing for men this year that makes use of the same new softer fabrics.”

71. Reading Examples The designer’s plan makes which assumption?
Other designers will not also introduce lines for men with the new softer fabrics.
The designer will have time enough to develop lines for both men and women.
Men will be as interested in the new softer fabrics as women were the year before.
The line for men will be considered to be innovative because of its use of new fabrics.

72. Reading Examples
“A newly discovered disease is thought to be caused by a certain bacterial strain. However, a recent study notes that this bacteria also thrives in the presence of a certain virus, implying that the virus is actually what causes the new disease.”

73. Reading Examples Which would most support the study’s finding?
In the absence of the virus, the disease has been observed to follow bacterial infection.
The virus has been observed alone, apart from the bacteria, in some disease cases.
In cases where the disease does not develop, bacterial infection is preceded by the virus.
The virus has been shown to aid the growth of the bacteria associated with the disease.

74. Reading Examples
“In contrast to the drivers who live in Mountainview, a greater proportion of the drivers who live in Oak Valley exceed the speed limit regularly. This explains why there are more accidents each year in Oak Valley than there are in Mountainview.”

75. Reading Examples
All of the following would weaken the argument, except which?
Per capita, there are fewer traffic officers in Oak Valley than there are in Mountainview.
There are a greater number of drivers in Oak Valley than there are in Mountainview.
The roads in Oak Valley are icy for a greater portion of the year than in Mountainview.
Oak Valley has a greater number of blind intersections than Mountainview has.

76. Reading Examples
“To reduce the deficit, the US could cut military and defense spending, the largest budget component. However, many critical industries depend on military contracts. If the government reduces military spending, it may have to provide these industries with other economic relief in peacetime.”

77. Reading Examples What is the main point of this passage?
The military-industrial complex’s size is a disincentive for cutting military spending.
If we cut US military spending, we will probably have to increase it again later.
We must maintain military spending or risk being unprepared in event of war.
Reducing military spending may result in an increase in other areas of spending.

78. Reading Examples
“To improve our overcrowded elementary schools, the town council has proposed new construction and smaller classes—a plan to be paid for with increased property tax for high-income homeowners. Though our schools need improving, the proposal should be rejected since the people who would pay for it receive no benefit from it.”

79. Reading Examples Which if true would most strengthen this claim?
Other nearby towns that reduced class size did not find an increase in education quality.
High-income residents already pay taxes for other unused services, like mass transit.
Tax records indicate many homeowners in high income brackets have no kids at home.
The higher tax is a disincentive to seek out profit and will reduce economic growth.

80. Reading Examples
“Professional athletes around the world pace themselves during practice, and do not exercise more than around 5 hours a day. If an athlete is exercising more than 5 hours a day, they are probably not a pro.”

81. Reading Examples Which argument most closely parallels this one?
If you sleep more than 5 hours a night, you are probably getting enough sleep; experts say an adult needs around 5-6 hours of sleep.
High-quality paint usually requires 2 coats. If you buy 1-coat paint, it is likely low quality.
Good runners run in the morning; if you run in the morning, you will be a better runner.
Healthy people don’t smoke, so you should quit.

82. Math / Quantitative
Algebra Content Review

83. Fundamentals of Algebra
Algebra is a process that has 2 parts:
Being able to translate a problem into mathematical expressions, functions, or algebraic equations
Knowing how to use or modify the resulting expressions or equations

84. Fundamentals of Algebra
Unknown, to-be-solved-for values are represented by letters in algebra: “The price of a pair of shoes is equal to 3 times the price of a pair of jeans.”
Unknown, to-be-solved-for values are represented by letters in algebra:
“The price of a pair of shoes is equal to 3 times the price of a pair of jeans.”
𝑠=3∗𝑗

85. Fundamentals of Algebra
You can perform the same operation, any operation, to both sides of an equation and it will remain equal. 𝑠=3∗𝑗 (𝑠) 40 +2= (3∗𝑗) 40 +2

86. Fundamentals of Algebra
Usually, you do this in order to simplify, not add to, an equation with the goal of a single letter (unknown value) on one side:
𝑥−5 = 2+3𝑥
−3𝑥 −3𝑥
−2𝑥 = 7

87. Fundamentals of Algebra
Usually, you do this in order to simplify, not add to, an equation with the goal of a single letter (unknown value) on one side:
−2𝑥 = 7
−2𝑥 − = 7 −2

88. Fundamentals of Algebra
Usually, you do this in order to simplify, not add to, an equation with the goal of a single letter (unknown value) on one side:
𝑥 =− =−3.5

89. Fundamentals of Algebra
“If x is equal to 1 more than the product of 3 and z, and y is equal to 1 less than the product of 2 and z, then how much greater is 2x than 3y when z has a value of 4?” 1 2 3 5 6
𝑥=3𝑧+1→3∗4+1=13→2𝑥=26
𝑥=3𝑧+1→3∗4+1=13→2𝑥=26
𝑥=3𝑧+1→3∗4+1=13→2𝑥=26
𝑦=2𝑧−1→2∗4−1=7→3𝑦=21
𝑦=2𝑧−1→2∗4−1=7→3𝑦=21
𝑦=2𝑧−1→2∗4−1=7→3𝑦=21

90. Systems of Equations
In general, to solve for a given number of unknown values (i.e. letters), you need as many equations as letters. 3𝑥+2𝑦=6 5𝑥 − 𝑦 =10

91. Systems of Equations 2 ways to solve a “system of equations”:
Solve one equation for a letter (y), then plug into the other equation.
Change one equation so that if you add it to the other equation, one letter (y) will be cancelled out.

92. Systems of Equations
1. Solve for y, plug in, then solve for x: 3𝑥+2𝑦=6 5𝑥 − 𝑦 =10 −5𝑥 −5𝑥 −𝑦=10−5𝑥 𝑦=−10+5𝑥
Solve for y →

93. Systems of Equations
1. Solve for y, plug in, then solve for x: 3𝑥+2𝑦=6 5𝑥 − 𝑦 =10
Solve for y →

94. Systems of Equations
1. Solve for y, plug in, then solve for x: 3𝑥+2𝑦=6 𝑦=−10+5𝑥
Plug in y →

95. Systems of Equations
1. Solve for y, plug in, then solve for x: 3𝑥+2(−10+5𝑥)=6 3𝑥+2(−10)+2(5𝑥)=6 3𝑥−20 +10𝑥 =6 13𝑥=26→𝑥= =2
Plug in y →

96. Systems of Equations
1. Solve for y, plug in, then solve for x: 3𝑥+2𝑦=6 𝑦=−10+5𝑥 𝑦=−10+10
Plug in y →
→ 𝑥=2
𝑦=−10+5(2)
→ 𝑦=0

97. Systems of Equations
2. Change equations so y cancels out: 3𝑥+2𝑦=6 5𝑥 − 𝑦 =10 2∗ 5𝑥−𝑦 =10∗2 2 5𝑥 +2 −𝑦 =20 10𝑥−2𝑦=20
Change →

98. Systems of Equations 2. Change equations so y cancels out: 3𝑥+2𝑦=6
10𝑥−2𝑦=20
13𝑥 =26
𝑥 =2
Change →
/ /13

99. Systems of Equations
If x and y satisfy the following system of equations, what is the value of x-y? 7𝑥+3𝑦=12 3𝑥+7𝑦 =6
2/3
3/2 2 4 6

100. Systems of Equations
“A symphony sells 3 kinds of tickets: box seats for $40, general admission for $20, and student admission for $10. On a recent night they sold 200 tickets, 40 to students, and made $4,000 in all. How many general admission tickets did the symphony sell?” 96
120
140
160
180

101. Systems of Equations
“15 years ago, Adam was 3 times as old as Bob. Today, Adam is twice as old as Bob. How old will Adam be 5 years from now?” 35 45 50 60 65
𝐴−15 =3∗(𝐵−15)
𝐴=2𝐵

102. Systems of Equations Cautions about systems of equations:
If the 2 equations are equivalent to each other, you cannot solve them.
3𝑥+2𝑦=6
6𝑥+4𝑦=12

103. Systems of Equations
On the other hand, sometimes you don’t need 2 equations; GMAT will put 2 variables in 1 equation, but do so such that 1 variable cancels out.
15 𝐴+𝐵 −6 𝐵 2 =30+3𝐵(5−2𝐵)
15𝐴+15𝐵−6 𝐵 2 =30+15𝐵−6 𝐵 2

104. Quadratic Equations
The GMAT tests your ability to handle quadratic equations (squared terms), in both expanded and factored forms. factored 𝑥+2 𝑥+5 = expanded 𝑥 2 +7𝑥+10

105. Quadratic Equations
To expand a quadratic equation, first multiply each part (taken separately) in the first factor by every part in the second factor, then combine like terms. 𝑥+4 𝑥+𝑦+3

106. Quadratic Equations
𝑥+4 𝑥+𝑦+3 1. 𝑥∗ 𝑥+𝑦 ∗ 𝑥+𝑦+3 𝑥 2 +𝑥𝑦+3𝑥 + 4𝑥+4𝑦+12 𝑥 2 +𝑥𝑦+7𝑥+4𝑦+12

107. Quadratic Equations
To factor a quadratic equation, first break apart the squared term, then the final constant into two factors that can be added to produce the middle term. 𝑥 2 −9𝑥+20

108. Quadratic Equations 𝑥 2 −9𝑥+20 Factor: 𝑥+__ ∗ 𝑥+__ 𝑥+ −4 ∗ 𝑥+ −5
[20 = (2 * 10), (4 * 5), etc]
𝑥+ −4 ∗ 𝑥+ −5
𝑥−4 ∗ 𝑥−5

109. Quadratic Equations
There are 3 special types of quadratic equations the GMAT likes to test: 𝑥+𝑦 2 = 𝑥 2 +2𝑥𝑦+ 𝑦 2 𝑥+𝑦 𝑥−𝑦 = 𝑥 2 − 𝑦 2 𝑥−𝑦 2 = 𝑥 2 −2𝑥𝑦+ 𝑦 2

110. Time, Rate, & Distance/Work
To solve a problem involving distance or work over a period of time, write out the formula and fill in what you know. 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑜𝑟 𝑤𝑜𝑟𝑘 =𝑟𝑎𝑡𝑒∗𝑡𝑖𝑚𝑒

111. Time, Rate, & Distance/Work
“Jim and Bob are standing 450 feet apart and start walking toward each other. If Jim walks 4 feet per second and Bob walks 5 feet per second, how far has Bob walked when they meet?”
200 ft.
225 ft.
250 ft.
270 ft.
300 ft.

112. Time, Rate, & Distance/Work
“Jim/Bob are 450 feet apart. Jim walks 4 feet/sec.; Bob walks 5 feet/sec. How far has Bob walked when they meet?” 𝑑𝑖𝑠𝑡 𝐵 =5 𝑓𝑡/𝑠𝑒𝑐∗ 𝑡𝑖𝑚𝑒 𝐵 A critical fact is their times are equal: 𝑑𝑖𝑠𝑡 𝑜𝑣𝑒𝑟𝑎𝑙𝑙 = 𝑟𝑎𝑡𝑒 𝑜𝑣𝑒𝑟𝑎𝑙𝑙 ∗ 𝑡𝑖𝑚𝑒 𝐽 &/𝑜𝑟 𝐵

113. Time, Rate, & Distance/Work
“Jim/Bob are 450 feet apart. Jim walks 4 feet/sec.; Bob walks 5 feet/sec. How far has Bob walked when they meet?” 𝑑𝑖𝑠𝑡 𝐵 =5 𝑓𝑡/𝑠𝑒𝑐∗ 𝑡𝑖𝑚𝑒 𝐵 Their overall rate is also a bit tricky: 450 𝑓𝑒𝑒𝑡= 𝑟𝑎𝑡𝑒 𝑜𝑣𝑒𝑟𝑎𝑙𝑙 ∗ 𝑡𝑖𝑚𝑒 𝐽 &/𝑜𝑟 𝐵

114. Time, Rate, & Distance/Work
“Jim/Bob are 450 feet apart. Jim walks 4 feet/sec.; Bob walks 5 feet/sec. How far has Bob walked when they meet?” 𝑑𝑖𝑠𝑡 𝐵 =5 𝑓𝑡/𝑠𝑒𝑐∗ 𝑡𝑖𝑚𝑒 𝐵 From this we can find the overall time: 450=(4+5 𝑓𝑡/𝑠𝑒𝑐 )∗ 𝑡𝑖𝑚𝑒 𝐽 &/𝑜𝑟 𝐵

115. Time, Rate, & Distance/Work
“Jim/Bob are 450 feet apart. Jim walks 4 feet/sec.; Bob walks 5 feet/sec. How far has Bob walked when they meet?” 𝑑𝑖𝑠𝑡 𝐵 =5 𝑓𝑡/𝑠𝑒𝑐∗ 𝑡𝑖𝑚𝑒 𝐵 And we can fill that in for Bob ↑ 450 𝑓𝑒𝑒𝑡=9 𝑓𝑡/𝑠𝑒𝑐∗50 𝑠𝑒𝑐𝑜𝑛𝑑𝑠

116. Time, Rate, & Distance/Work
“Jim/Bob are 450 feet apart. Jim walks 4 feet/sec.; Bob walks 5 feet/sec. How far has Bob walked when they meet?” 𝑑𝑖𝑠𝑡 𝐵 =5 𝑓𝑡/𝑠𝑒𝑐∗50 𝑠𝑒𝑐𝑠 𝑑𝑖𝑠𝑡 𝐵 =250 𝑓𝑒𝑒𝑡
200 ft.
225 ft.
250 ft.
270 ft.
300 ft.

117. Time, Rate, & Distance/Work
“Able can finish a job in 3 hours. Guy can finish the same job in 12 hours. How many hours will it take Able and Guy working together to finish a job?” A critical fact is each rate is: 𝟏 𝒋𝒐𝒃 __ ℎ𝑜𝑢𝑟𝑠
hrs.
hrs.
hrs.
hrs.
hrs.

118. Time, Rate, & Distance/Work
“Able can finish a job in 3 hours. Guy can finish the same job in 12 hours. How many hours will it take Able and Guy working together to finish a job?” 1 𝑗𝑜𝑏= 𝑟𝑎𝑡𝑒 𝑜𝑣𝑒𝑟𝑎𝑙𝑙 ∗ 𝑡𝑖𝑚𝑒 𝑡𝑜𝑔𝑒𝑡ℎ𝑒𝑟
hrs.
hrs.
hrs.
hrs.
hrs.

119. Time, Rate, & Distance/Work
“Able can finish a job in 3 hours. Guy can finish the same job in 12 hours. How many hours will it take Able and Guy working together to finish a job?” 1 𝑗𝑜𝑏= ∗ 𝑡𝑖𝑚𝑒 𝑡𝑜𝑔𝑒𝑡ℎ𝑒𝑟
hrs.
hrs.
hrs.
hrs.
hrs.

120. Time, Rate, & Distance/Work
“Able can finish a job in 3 hours. Guy can finish the same job in 12 hours. How many hours will it take Able and Guy working together to finish a job?” 1 𝑗𝑜𝑏= 5/12 ∗ 𝑡𝑖𝑚𝑒 𝑡𝑜𝑔𝑒𝑡ℎ𝑒𝑟
hrs.
hrs.
hrs.
hrs.
hrs.

121. Algebraic Functions
Some GMAT questions use strange symbols in algebraic equations to represent an arbitrary function: 𝛻𝑥=2𝑥+3 This means, given: 𝛻4+𝛻 −1 2∗ ∗ −1 +3

122. Math / Quantitative
Data Sufficiency Review

123. Data Sufficiency An example question:
“What is the value of x? 1) x2 = ) x is negative”
Statement 1 alone is sufficient; Statement 2 alone is not
Statement 2 alone is sufficient; Statement 1 alone is not
Both statements 1 and 2 together are sufficient; neither statement alone is sufficient to answer the question
Each statement alone is sufficient to answer the question
Statements 1 and 2 together are not sufficient

124. Data Sufficiency Solving data sufficiency problems:
What is being asked? + What do you know? What else is needed?
Evaluate Statement 1 on its own.
Evaluate Statement 2 on its own.
Evaluate Statements 1 & 2 together.

125. Data Sufficiency Answering data sufficiency problems:
If #1 is sufficient alone, choose A or D (based on step 3 from previous-skip 4)
If #1 is not sufficient, choose B, C, or E (based on steps 3 and 4 from previous)
Statement 1 alone is sufficient; Statement 2 alone is not
Statement 2 alone is sufficient; Statement 1 alone is not
Both statements 1 & 2 together are sufficient; neither alone is
Each statement alone is sufficient to answer the question
Statements 1 and 2 together are not sufficient

126. Data Sufficiency Strategies for data sufficiency:
You don’t need to actually answer the question asked, only be able to.
If you face a yes/no question, being able to answer in the negative is ok!
Ignore St. 1 when evaluating St. 2.

127. 𝑀𝑎𝑦, 𝐽𝑢𝑛𝑒, 𝑜𝑟 𝑒𝑙𝑠𝑒 𝑎𝑛𝑜𝑡ℎ𝑒𝑟 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛
Data Sufficiency
“A store sold 50 more TVs in June than in May. By what percent did sales rise?”
What is being asked?
What do you know?
What else is needed?
50 𝑀𝑎𝑦 = 𝑥? 100
𝐽𝑢𝑛𝑒−𝑀𝑎𝑦=50
𝑀𝑎𝑦, 𝐽𝑢𝑛𝑒, 𝑜𝑟 𝑒𝑙𝑠𝑒 𝑎𝑛𝑜𝑡ℎ𝑒𝑟 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛

128. Data Sufficiency
“A store sold 50 more TVs in June than in May. By what percent did sales rise?”
1) In June, the store sold 150 TVs
2) In May, the store sold 90% of June’s total
Statement 1 alone is sufficient; Statement 2 alone is not
Statement 2 alone is sufficient; Statement 1 alone is not
Both statements 1 & 2 together are sufficient; neither alone is
Each statement alone is sufficient to answer the question
Statements 1 and 2 together are not sufficient

129. Data Sufficiency “What is the value of 𝑎 𝑏 ?” What is being asked?
What else is needed?
𝐶𝐴𝑅𝐸𝐹𝑈𝐿! 𝑁𝑜𝑡 𝑎 𝑜𝑟 𝑏!
𝐵𝑜𝑡ℎ 𝑎 𝑎𝑛𝑑 𝑏
𝑜𝑟 𝑠𝑜𝑚𝑒𝑡ℎ𝑖𝑛𝑔 𝑡ℎ𝑎𝑡
𝑒𝑥𝑝𝑟𝑒𝑠𝑠𝑒𝑠 𝑜𝑟 𝑟𝑒𝑑𝑢𝑐𝑒𝑠
𝑡𝑜 𝑡ℎ𝑒 𝑟𝑎𝑡𝑖𝑜 𝑜𝑓 𝑎/𝑏

130. Data Sufficiency “What is the value of 𝑎 𝑏 ?” 1) 𝑏=25 2) 16𝑎+13𝑏=0
→16𝑎=−13𝑏
−13 16 ∗ ∗ −13 16
16𝑎 −13𝑏 = 1
→ 𝑎 𝑏 =− 13 16
Statement 1 alone is sufficient; Statement 2 alone is not
Statement 2 alone is sufficient; Statement 1 alone is not
Both statements 1 & 2 together are sufficient; neither alone is
Each statement alone is sufficient to answer the question
Statements 1 and 2 together are not sufficient

131. Data Sufficiency “Does the average of {a, b, c} equal 8?”
What is being asked?
What else is needed?
8= 𝑎+𝑏+𝑐 3
i𝑠 𝑎+𝑏+𝑐=24?
𝑎 𝑎𝑛𝑑 𝑏 𝑎𝑛𝑑 𝑐,
or just their sum

132. Data Sufficiency “Does the average of {a, b, c} equal 8?”
1) 3 𝑎+𝑏+𝑐 =72
2) 𝑐=25−𝑎−𝑏
→𝑎+𝑏+𝑐=72/3
→𝑎+𝑏+𝑐=25
𝐶𝐴𝑅𝐸𝐹𝑈𝐿!𝑇ℎ𝑖𝑠 𝐷𝑂𝐸𝑆 𝑎𝑛𝑠𝑤𝑒𝑟 𝑡ℎ𝑒 𝑞𝑢𝑒𝑠𝑡𝑖𝑜𝑛!
Statement 1 alone is sufficient; Statement 2 alone is not
Statement 2 alone is sufficient; Statement 1 alone is not
Both statements 1 & 2 together are sufficient; neither alone is
Each statement alone is sufficient to answer the question
Statements 1 and 2 together are not sufficient

133. Data Sufficiency
“If Alaina spent $700 in July on rent, how much in all did she earn in July?”
1) Alaina earned 10% more than in June
2) Alaina saved ¼ of her earnings and spent half of what was left on rent
Statement 1 alone is sufficient; Statement 2 alone is not
Statement 2 alone is sufficient; Statement 1 alone is not
Both statements 1 & 2 together are sufficient; neither alone is
Each statement alone is sufficient to answer the question
Statements 1 and 2 together are not sufficient

134. Data Sufficiency
“If Alaina spent $700 in July on rent, how much in all did she earn in July?”
1) 10% more than June
2) Rent is ½ of ¾ July
→𝐽𝑢𝑙𝑦=1.1∗𝐽𝑢𝑛𝑒
→700=3/8∗𝐽𝑢𝑙𝑦
Statement 1 alone is sufficient; Statement 2 alone is not
Statement 2 alone is sufficient; Statement 1 alone is not
Both statements 1 & 2 together are sufficient; neither alone is
Each statement alone is sufficient to answer the question
Statements 1 and 2 together are not sufficient

135. Data Sufficiency
“Chad can cut down 3 trees per hour. How long does David take to cut down 9 trees?” 1) David works 6 hours a day cutting trees 2) David cuts down half as many trees per day as what Chad cuts down in 4 hours
Statement 1 alone is sufficient; Statement 2 alone is not
Statement 2 alone is sufficient; Statement 1 alone is not
Both statements 1 & 2 together are sufficient; neither alone is
Each statement alone is sufficient to answer the question
Statements 1 and 2 together are not sufficient

136. Data Sufficiency
“Chad can cut down 3 trees per hour. How long does David take to cut down 9 trees?” 1) David works 6 hours a day cutting trees 2) 𝑟𝑎𝑡𝑒 𝐷 ∗6= 1 2 ∗ 𝑟𝑎𝑡𝑒 𝐶 ∗4
Statement 1 alone is sufficient; Statement 2 alone is not
Statement 2 alone is sufficient; Statement 1 alone is not
Both statements 1 & 2 together are sufficient; neither alone is
Each statement alone is sufficient to answer the question
Statements 1 and 2 together are not sufficient

137. Data Sufficiency
“Chad can cut down 3 trees per hour. How long does David take to cut down 9 trees?” 1) David works 6 hours a day cutting trees 2) 𝑟𝑎𝑡𝑒 𝐷 ∗6= 1 2 ∗3∗4
→ 𝑟 𝐷 = 3∗4 2∗6 =1 𝑡𝑟𝑒𝑒 ℎ𝑟.
Statement 1 alone is sufficient; Statement 2 alone is not
Statement 2 alone is sufficient; Statement 1 alone is not
Both statements 1 & 2 together are sufficient; neither alone is
Each statement alone is sufficient to answer the question
Statements 1 and 2 together are not sufficient

138. Data Sufficiency
“Joann drove from home to the beach in less than 4 hours. Was her average speed greater than 70 miles per hour?”
1) Joann drove a total of 260 miles
2) Joann drove the first 160 miles at 80 mph
Statement 1 alone is sufficient; Statement 2 alone is not
Statement 2 alone is sufficient; Statement 1 alone is not
Both statements 1 & 2 together are sufficient; neither alone is
Each statement alone is sufficient to answer the question
Statements 1 and 2 together are not sufficient

139. Data Sufficiency
“Joann drove from home to the beach in less than 4 hours. Was her average speed greater than 70 miles per hour?”
1) 260 miles
2) Joann drove the first 160 miles at 80 mph
→𝑟= =65
→260=𝑟∗4
Statement 1 alone is sufficient; Statement 2 alone is not
Statement 2 alone is sufficient; Statement 1 alone is not
Both statements 1 & 2 together are sufficient; neither alone is
Each statement alone is sufficient to answer the question
Statements 1 and 2 together are not sufficient

140. Data Sufficiency
“What is the value of Z in {X, Y, Z} if X = 5 and all of the values are positive integers?” 1) The mean of {X, Y, Z} - 6 is equal to 1 3 𝑌 2) Y is equal to 7 and the set’s range is 8
Statement 1 alone is sufficient; Statement 2 alone is not
Statement 2 alone is sufficient; Statement 1 alone is not
Both statements 1 & 2 together are sufficient; neither alone is
Each statement alone is sufficient to answer the question
Statements 1 and 2 together are not sufficient

141. Data Sufficiency
“What is the value of Z in {X, Y, Z} if X = 5 and all of the values are positive integers?” 1) 5+𝑌+𝑍 3 −6= 1 3 𝑌 2) Y is equal to 7 and the set’s range is 8
Statement 1 alone is sufficient; Statement 2 alone is not
Statement 2 alone is sufficient; Statement 1 alone is not
Both statements 1 & 2 together are sufficient; neither alone is
Each statement alone is sufficient to answer the question
Statements 1 and 2 together are not sufficient

142. Data Sufficiency
“What is the value of Z in {X, Y, Z} if X = 5 and all of the values are positive integers?” 1) 5+𝑌+𝑍 −18 3 = 1 3 𝑌 2) Y is equal to 7 and the set’s range is 8
→ 𝑌+𝑍−13=𝑌
→ 𝑍=13
Statement 1 alone is sufficient; Statement 2 alone is not
Statement 2 alone is sufficient; Statement 1 alone is not
Both statements 1 & 2 together are sufficient; neither alone is
Each statement alone is sufficient to answer the question
Statements 1 and 2 together are not sufficient

143. Data Sufficiency
“What is the value of Z in {X, Y, Z} if X = 5 and all of the values are positive integers?” 1) 5+𝑌+𝑍 −18 3 = 1 3 𝑌 2) {5, 7, Z}, max – min = 8
→ 𝑌+𝑍−13=𝑌
→ 𝑍=13
→is 7 the max? is 5 the min?
7−𝑍=8?
𝑍−5=8?
Statement 1 alone is sufficient; Statement 2 alone is not
Statement 2 alone is sufficient; Statement 1 alone is not
Both statements 1 & 2 together are sufficient; neither alone is
Each statement alone is sufficient to answer the question
Statements 1 and 2 together are not sufficient

144. Data Sufficiency
“A dealership sells only sedans and coupes, in two colors each: red or blue. Last year, it sold 9,000 vehicles, half of which were red. How many coupes did the dealership sell last year?”
1) The dealership sold three times as many blue coupes as red sedans last year.
2) The dealership sold half as many blue sedans as blue coupes last year
Statement 1 alone is sufficient; Statement 2 alone is not
Statement 2 alone is sufficient; Statement 1 alone is not
Both statements 1 & 2 together are sufficient; neither alone is
Each statement alone is sufficient to answer the question
Statements 1 and 2 together are not sufficient

145. Data Sufficiency
“A dealership sells only sedans and coupes, in two colors each: red or blue. Last year, it sold 9,000 vehicles, half of which were red. How many coupes did the dealership sell last year?”
1) BC = 3 * RS
2) BC = 2 * BS
Statement 1 alone is sufficient; Statement 2 alone is not
Statement 2 alone is sufficient; Statement 1 alone is not
Both statements 1 & 2 together are sufficient; neither alone is
Each statement alone is sufficient to answer the question
Statements 1 and 2 together are not sufficient

146. Data Sufficiency “Is the value of ∴(𝑥)<0 if 𝑥=4 ?”
1) ∴ 𝑥 = 𝑥 2 − 𝛻𝑥
2) 𝛻 𝑥 =15− 𝑥
Statement 1 alone is sufficient; Statement 2 alone is not
Statement 2 alone is sufficient; Statement 1 alone is not
Both statements 1 & 2 together are sufficient; neither alone is
Each statement alone is sufficient to answer the question
Statements 1 and 2 together are not sufficient

147. Data Sufficiency “Is the value of ∴(𝑥)<0 if 𝑥=4 ?”
1) ∴ 𝑥 = 𝑥 2 − 𝛻𝑥
2) 𝛻 𝑥 =15− 𝑥
→ =16− 𝛻𝑥
→ =15−2 𝑜𝑟 15+2
= 𝑜𝑟 17
1) = 𝑜𝑟 −1
Statement 1 alone is sufficient; Statement 2 alone is not
Statement 2 alone is sufficient; Statement 1 alone is not
Both statements 1 & 2 together are sufficient; neither alone is
Each statement alone is sufficient to answer the question
Statements 1 and 2 together are not sufficient