Overview and Strategies
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.
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
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).
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
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...”
Verbal Overview
Critical reasoning content:
Distinguish between assumptions,
premises/evidence, and conclusions
Analyze structure of an argument
Consider additional information
Identify analogous structures
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…”
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…”
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.
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.
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)
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…”
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.
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.
Grammar Overview
“Although it is not as liquid an investment 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.
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).
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.
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.
Grammar Overview
“Upset by the recent downturn in
productivity, the possibility of worker
incentives was raised by the board of
directors in their annual meeting.”
Who is upset? The “board of directors”,
so that phrase must follow the comma.
Grammar Overview
Parallel construction errors can occur
when a sentence contains a compound
subject or verb that are not presented
consistently, when it contains two full
clauses whose structures are not the
same, or when it makes a comparison
between elements of different types.
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.
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.
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”
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”
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?
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…”
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…”
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?
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”
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 growthof any other business field”
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”
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?
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”
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.
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.
Verbal 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.
Verbal 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.
Verbal 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?
Verbal 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”
Verbal 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.
Verbal 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.”
Verbal Examples
The designer’s plan makes which assumption?
A. Other designers will not also introduce
lines for men with the new softer fabrics.
B. The designer will have time enough to
develop lines for both men and women.
C. Men will be as interested in the new softer
fabrics as women were the year before.
D. The line for men will be considered to be
innovative because of its use of new fabrics.
Verbal 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.”
Verbal Examples
Which would most support the study’s finding?
A. In the absence of the virus, the disease has
been observed to follow bacterial infection.
B. The virus has been observed alone, apart
from the bacteria, in some disease cases.
C. In cases where the disease does not develop,
bacterial infection is preceded by the virus.
D. The virus has been shown to aid the growth
of the bacteria associated with the disease.
Verbal Examples
“In contrast to the drivers who live in
Moutainview, 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.”
Verbal Examples
All of the following would weaken the argument, except which?
A. Per capita, there are fewer traffic officers in
Oak Valley than there are in Mountainview.
B. There are a greater number of drivers in
Oak Valley than there are in Mountainview.
C. The roads in Oak Valley are icy for a greater
portion of the year than in Mountainview.
D. Oak Valley has a greater number of blind
intersections than Mountainview has.
Verbal 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.”
Verbal Examples
What is the main point of this passage?
A. The military-industrial complex’s size is a
disincentive for cutting military spending.
B. If we cut US military spending, we will
probably have to increase it again later.
C. We must maintain military spending or
risk being unprepared in event of war.
D. Reducing military spending may result in
an increase in other areas of spending.
Verbal 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.”
Verbal Examples
Which if true would most strengthen this claim?
A. Other nearby towns that reduced class size
did not find an increase in education quality.
B. High-income residents already pay taxes for
other unused services, like mass transit.
C. Tax records indicate many homeowners in
high income brackets have no kids at home.
D. The higher tax is a disincentive to seek out
profit and will reduce economic growth.
Verbal 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.”
Verbal Examples
Which argument most closely parallels this one?
A. 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.
B. High-quality paint usually requires 2 coats.
If you buy 1-coat paint, it is likely low quality.
C. Good runners run in the morning; if you run
in the morning, you will be a better runner.
D. Healthy people don’t smoke, so you should quit.
Algebra Content Review
Fundamentals of Algebra
Algebra is a process that has 2 parts:
1. Being able to translate a problem
into mathematical expressions,
functions, or algebraic equations
2. Knowing how to use or modify the
resulting expressions or equations
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.”
𝑠 =3∗𝑗
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)
(3 ∗ 𝑗)
+2=
+2
40
40
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 + 5
+ 5 − 3𝑥
−2𝑥
= 7
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
−2
7
=
−2
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
7
=−
2
= −3.5
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
Systems of Equations
2 ways to solve a “system of equations”:
1. Solve one equation for a letter (y),
then plug into the other equation.
2. Change one equation so that if you
add it to the other equation, one
letter (y) will be cancelled out.
Systems of Equations
1. Solve for y, plug in, then solve for x:
3𝑥 + 2𝑦 = 6
Solve for y → 5𝑥 − 𝑦 = 10
−5𝑥
− 5𝑥
−𝑦 = 10 − 5𝑥
𝑦 = −10 + 5𝑥
Systems of Equations
1. Solve for y, plug in, then solve for x:
3𝑥 + 2𝑦 = 6
Solve for y → 5𝑥 − 𝑦 = 10
Systems of Equations
1. Solve for y, plug in, then solve for x:
Plug in y →
3𝑥 + 2𝑦 = 6
𝑦 = −10 + 5𝑥
Systems of Equations
1. Solve for y, plug in, then solve for x:
Plug in y →
3𝑥 + 2(−10 + 5𝑥) = 6
3𝑥 + 2(−10) + 2(5𝑥) = 6
3𝑥 − 20
+ 10𝑥
=6
13𝑥 = 26 → 𝑥 = 13 26 = 2
Systems of Equations
1. Solve for y, plug in, then solve for x:
Plug in y →
3𝑥 + 2𝑦 = 6
→ 𝑥=2
𝑦 = −10 + 5𝑥
5(2)
𝑦 = −10 + 10 → 𝑦 = 0
Systems of Equations
2. Change equations so y cancels out:
Change →
3𝑥 + 2𝑦 = 6
5𝑥 − 𝑦 = 10
2 ∗ 5𝑥 − 𝑦 = 10 ∗ 2
2 5𝑥 + 2 −𝑦 = 20
10𝑥 − 2𝑦 = 20
Systems of Equations
2. Change equations so y cancels out:
Change →
3𝑥 + 2𝑦 = 6
10𝑥 − 2𝑦 = 20
13𝑥 /13 = 26 /13
𝑥
=2
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?”
A. 96
B. 120 C. 140 D. 160 E. 180
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?”
A. 35 B. 45 C. 50 D. 60 E. 65
𝐴 − 15 = 3 ∗ (𝐵 − 15)
𝐴 = 2𝐵
Systems of Equations
Cautions about systems of equations:
1. If the 2 equations are equivalent to
each other, you cannot solve them.
3𝑥 + 2𝑦 = 6
6𝑥 + 4𝑦 = 12
Systems of Equations
2. 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.
2
15 𝐴 + 𝐵 − 6𝐵 = 30 + 3𝐵(5 − 2𝐵)
2
15𝐴 + 15𝐵 − 6𝐵 = 30 + 15𝐵 − 6𝐵
2
Quadratic Equations
The GMAT tests your ability to handle
quadratic equations (squared terms),
in both expanded and factored forms.
factored
expanded
𝑥+2 𝑥+5 =
2
𝑥 + 7𝑥 + 10
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
Quadratic Equations
𝑥+4 𝑥+𝑦+3
1. 𝑥 ∗ 𝑥 + 𝑦 + 3
2. 4 ∗ 𝑥 + 𝑦 + 3
𝑥 2 + 𝑥𝑦 + 3𝑥 + 4𝑥 + 4𝑦 + 12
𝑥 2 + 𝑥𝑦 + 7𝑥 + 4𝑦 + 12
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.
2
𝑥 − 9𝑥 + 20
Quadratic Equations
𝑥 2 − 9𝑥 + 20
Factor: 𝑥 + __ ∗ 𝑥 + __
[20 = (2 * 10), (4 * 5), etc]
𝑥 + −4
∗ 𝑥 + −5
𝑥−4 ∗ 𝑥−5
Quadratic Equations
There are 3 special types of quadratic
equations the GMAT likes to test:
𝑥+𝑦
2
2
= 𝑥 + 2𝑥𝑦 + 𝑦
2
𝑥+𝑦 𝑥−𝑦 =𝑥 −𝑦
𝑥−𝑦
2
2
2
2
= 𝑥 − 2𝑥𝑦 + 𝑦
2
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
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?”
A. 200 ft. B. 225 ft. C. 250 ft. D. 270 ft. E. 300 ft.
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?”
𝑑𝑖𝑠𝑡𝐵 = 5 𝑓𝑡/𝑠𝑒𝑐 ∗ 𝑡𝑖𝑚𝑒𝐵
A critical fact is their times are equal:
𝑑𝑖𝑠𝑡𝑜𝑣𝑒𝑟𝑎𝑙𝑙 = 𝑟𝑎𝑡𝑒𝑜𝑣𝑒𝑟𝑎𝑙𝑙 ∗ 𝑡𝑖𝑚𝑒𝐽 &/𝑜𝑟 𝐵
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?”
𝑑𝑖𝑠𝑡𝐵 = 5 𝑓𝑡/𝑠𝑒𝑐 ∗ 𝑡𝑖𝑚𝑒𝐵
Their overall rate is also a bit tricky:
450 𝑓𝑒𝑒𝑡 = 𝑟𝑎𝑡𝑒𝑜𝑣𝑒𝑟𝑎𝑙𝑙 ∗ 𝑡𝑖𝑚𝑒𝐽 &/𝑜𝑟 𝐵
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?”
𝑑𝑖𝑠𝑡𝐵 = 5 𝑓𝑡/𝑠𝑒𝑐 ∗ 𝑡𝑖𝑚𝑒𝐵
From this we can find the overall time:
450 = (4 + 5 𝑓𝑡/𝑠𝑒𝑐 ) ∗ 𝑡𝑖𝑚𝑒𝐽 &/𝑜𝑟 𝐵
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?”
𝑑𝑖𝑠𝑡𝐵 = 5 𝑓𝑡/𝑠𝑒𝑐 ∗ 𝑡𝑖𝑚𝑒𝐵
And we can fill that in for Bob ↑
450 𝑓𝑒𝑒𝑡 = 9 𝑓𝑡/𝑠𝑒𝑐 ∗ 50 𝑠𝑒𝑐𝑜𝑛𝑑𝑠
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?”
𝑑𝑖𝑠𝑡𝐵 = 5 𝑓𝑡/𝑠𝑒𝑐 ∗ 50 𝑠𝑒𝑐𝑠
𝑑𝑖𝑠𝑡𝐵 = 250 𝑓𝑒𝑒𝑡
A. 200 ft. B. 225 ft. C. 250 ft. D. 270 ft. E. 300 ft.
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
4
3
4
1
4
2
5
5
8
A critical fact is each rate is:
𝟏 𝒋𝒐𝒃
__ ℎ𝑜𝑢𝑟𝑠
A. 1 hrs. B. 1 hrs. C. 2 hrs. D. 2 hrs. E. 2 hrs.
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 ∗ 4 + 3 + 2 ∗ −1 + 3
Data Sufficiency Review
Data Sufficiency
An example question:
“What is the value of x?
2
1) x = 9
A.
B.
C.
D.
E.
2) 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
Data Sufficiency
Solving data sufficiency problems:
1. What is being asked? + What do
you know? What else is needed?
2. Evaluate Statement 1 on its own.
3. Evaluate Statement 2 on its own.
4. Evaluate Statements 1 & 2 together.
Data Sufficiency
Answering data sufficiency problems:
If #1 is sufficient alone, choose A or D
(based on step 3 from previous-skip 4)
2. If #1 is not sufficient, choose B, C, or E
(based on steps 3 and 4 from previous)
1.
A.
B.
C.
D.
E.
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
Data Sufficiency
Strategies for data sufficiency:
1. You don’t need to actually answer
the question asked, only be able to.
2. If you face a yes/no question, being
able to answer in the negative is ok!
3. Ignore St. 1 when evaluating St. 2.
Data Sufficiency
“A store sold 50 more TVs in June than
in May. By what percent did sales rise?”
What is being asked?
50
𝑥?
=
𝑀𝑎𝑦 100
What do you know?
𝐽𝑢𝑛𝑒 − 𝑀𝑎𝑦 = 50
What else is needed?
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?”
1) In June, the store sold 150 TVs
2) In May, the store sold 90% of June’s total
A.
B.
C.
D.
E.
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
Data Sufficiency
“What is the value of
What is being asked?
𝑎
𝑏
?”
𝐶𝐴𝑅𝐸𝐹𝑈𝐿!
𝑁𝑜𝑡 𝑎 𝑜𝑟 𝑏!
𝐵𝑜𝑡ℎ 𝑎 𝑎𝑛𝑑 𝑏
𝑜𝑟 𝑠𝑜𝑚𝑒𝑡ℎ𝑖𝑛𝑔 𝑡ℎ𝑎𝑡
𝑒𝑥𝑝𝑟𝑒𝑠𝑠𝑒𝑠 𝑜𝑟 𝑟𝑒𝑑𝑢𝑐𝑒𝑠
𝑡𝑜 𝑡ℎ𝑒 𝑟𝑎𝑡𝑖𝑜 𝑜𝑓 𝑎/𝑏
What else is needed?
Data Sufficiency
“What is the value of
𝑎
𝑏
?”
1) 𝑏 = 25
2) 16𝑎 + 13𝑏 = 0 → 16𝑎 = −13𝑏
−13
16
A.
B.
C.
D.
E.
∗
16𝑎
−13𝑏
𝑎
13
−13
→ =−
=1∗
𝑏
16
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
Data Sufficiency
“Does the average of {a, b, c} equal 8?”
What is being asked?
What else is needed?
𝑎+𝑏+𝑐
8=
3
i𝑠 𝑎 + 𝑏 + 𝑐 = 24?
𝑎 𝑎𝑛𝑑 𝑏 𝑎𝑛𝑑 𝑐 ,
or just their sum
Data Sufficiency
“Does the average of {a, b, c} equal 8?”
1) 3 𝑎 + 𝑏 + 𝑐 = 72 → 𝑎 + 𝑏 + 𝑐 = 72/3
2) 𝑐 = 25 − 𝑎 − 𝑏 → 𝑎 + 𝑏 + 𝑐 = 25
𝐶𝐴𝑅𝐸𝐹𝑈𝐿! 𝑇ℎ𝑖𝑠 𝐷𝑂𝐸𝑆
𝑎𝑛𝑠𝑤𝑒𝑟 𝑡ℎ𝑒 𝑞𝑢𝑒𝑠𝑡𝑖𝑜𝑛!
A.
B.
C.
D.
E.
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
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
A.
B.
C.
D.
E.
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
Data Sufficiency
“If Alaina spent $700 in July on rent,
how much in all did she earn in July?”
1) 10% more than June → 𝐽𝑢𝑙𝑦 = 1.1 ∗ 𝐽𝑢𝑛𝑒
2) Rent is ½ of ¾ July → 700 = 3/8 ∗ 𝐽𝑢𝑙𝑦
A.
B.
C.
D.
E.
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
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
A.
B.
C.
D.
E.
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
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
1
2) 𝑟𝑎𝑡𝑒𝐷 ∗ 6 = ∗ 𝑟𝑎𝑡𝑒𝐶 ∗ 4
2
A.
B.
C.
D.
E.
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
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
3∗4
𝑡𝑟𝑒𝑒
1
2) 𝑟𝑎𝑡𝑒𝐷 ∗ 6 = ∗ 3 ∗ 4 → 𝑟𝐷 = 2∗6 = 1 ℎ𝑟.
2
A.
B.
C.
D.
E.
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
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 300 miles
2) Joann drove the first 160 miles at 80 mph
A.
B.
C.
D.
E.
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
Data Sufficiency
“Joann drove from home to the beach
in less than 4 hours. Was her average
speed greater than 70 miles per hour?”
300
4
1) 300 miles → 300 = 𝑟 ∗ 4 → 𝑟 =
= 75
2) Joann drove the first 160 miles at 80 mph
A.
B.
C.
D.
E.
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
Data Sufficiency
“What is the value of Z in {X, Y, Z} if X = 5
and all of the values are positive integers?”
1
1) The mean of {X, Y, Z} - 6 is equal to 3 𝑌
2) Y is equal to 7 and the set’s range is 8
A.
B.
C.
D.
E.
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
Data Sufficiency
“What is the value of Z in {X, Y, Z} if X = 5
and all of the values are positive integers?”
5+𝑌+𝑍
1
1) 3 − 6 = 3 𝑌
2) Y is equal to 7 and the set’s range is 8
A.
B.
C.
D.
E.
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
Data Sufficiency
“What is the value of Z in {X, Y, Z} if X = 5
and all of the values are positive integers?”
5+𝑌+𝑍 −18
1
1)
= 𝑌 → 𝑌 + 𝑍 − 13 = 𝑌 → 𝑍 = 13
3
3
2) Y is equal to 7 and the set’s range is 8
A.
B.
C.
D.
E.
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
Data Sufficiency
“What is the value of Z in {X, Y, Z} if X = 5
and all of the values are positive integers?”
5+𝑌+𝑍 −18
1
1)
= 𝑌 → 𝑌 + 𝑍 − 13 = 𝑌 → 𝑍 = 13
3
3
2) {5, 7, Z}, max – min = 8 → is 7 the max? is 5 the min?
7 − 𝑍 = 8?
A.
B.
C.
D.
E.
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
Data Sufficiency
“Is the value of ∴(𝑥) < 0 if 𝑥 = 4 ?”
1) ∴ 𝑥 = 𝑥 − 𝛻𝑥
2
2) 𝛻 𝑥 = 15 − 𝑥
A.
B.
C.
D.
E.
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
Data Sufficiency
“Is the value of ∴(𝑥) < 0 if 𝑥 = 4 ?”
1) ∴ 𝑥 = 𝑥 − 𝛻𝑥 → = 16 − 𝛻𝑥
2) 𝛻 𝑥 = 15 − 𝑥 → = 15 − 2 𝑜𝑟 15 + 2
= 13
𝑜𝑟 17
1) = 3
𝑜𝑟 − 1
2
A.
B.
C.
D.
E.
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