All practice questions

Question 401Question →

A city recently increased the number of traffic police officers deployed downtown by 50 percent in an effort to catch speeding drivers. Surprisingly, during the six months following this deployment, the total number of speeding tickets issued downtown per month decreased significantly.

Which of the following, if true, best helps to resolve the apparent discrepancy described above?

Drivers aware of the increased police presence exercised greater caution and drove within the speed limit far more consistently.

The newly deployed traffic police officers underwent advanced training on detecting speeding vehicles.

Neighboring municipalities that did not increase police patrols saw a minor increase in traffic violations during the same period.

Most traffic police officers prefer patrolling quiet residential avenues over heavily congested commercial streets.

The city recently increased the monetary fines associated with speeding tickets issued in commercial districts.

Show answer & explanation

Answer: Drivers aware of the increased police presence exercised greater caution and drove within the speed limit far more consistently.

Answer

The apparent discrepancy is best resolved by the fact that drivers, aware of the increased police presence, exercised greater caution and drove within the speed limit far more consistently.

The statement about drivers exercising greater caution reconciles both facts: although there were 50 percent more police officers looking for speeding infractions, the drivers altered their behavior to obey speed limits, resulting in fewer total speeding violations and thus fewer tickets issued.

Step-by-Step Solution

1

Identify the two contradictory facts in the stimulus.

Fact 1: Traffic police presence downtown increased by 50%. Fact 2: The monthly number of speeding tickets issued downtown decreased significantly.

Resolving a paradox requires finding an additional fact that allows both stated facts to be true simultaneously.

2

Evaluate how driver behavior affects the number of tickets issued.

Even with more officers looking for speeders, if drivers stop speeding, fewer tickets will be written.

A reduction in actual speeding incidents directly bridges the gap between increased enforcement effort and decreased ticket volume.

Key Concept

Resolving Paradoxes and Discrepancies

Question 402Question →

If $r$ and $s$ are integers, is $r + s$ an even integer?

(1) $r^2 - s^2$ is an odd integer.
(2) $r$ is an odd integer.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Show answer & explanation

Answer: Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Answer

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

The correct response indicates that Statement (1) alone is sufficient while Statement (2) alone is not. Statement (1) establishes algebraically that

r + s

must be odd because

(r - s)(r + s)

is an odd integer, yielding a definitive 'No' answer to whether

r + s

is even. Statement (2) leaves the value and parity of

s

completely unrestricted, allowing for both 'Yes' and 'No' outcomes.

Step-by-Step Solution

1

Rephrase the target question

The target is a Yes/No question: Is

r + s

even? A statement is sufficient if it yields a definitive 'Yes' or a definitive 'No'.

In Yes/No Data Sufficiency questions, proving that a condition is always false (a definitive 'No') is just as sufficient as proving it is always true.

2

Evaluate Statement (1):

r^2 - s^2

is an odd integer

Factor

r^2 - s^2 = (r - s)(r + s)

. Since the product of two integers is odd, both factors

(r - s)

and

(r + s)

must be odd integers. Thus,

r + s

is definitely odd, which means

r + s

is NOT even.

Since Statement (1) provides a definitive 'No' to the target question, Statement (1) alone is SUFFICIENT.

3

Evaluate Statement (2):

r

is an odd integer

If

r = 3

and

s = 1

, then

r + s = 4

(even

\rightarrow

Yes). If

r = 3

and

s = 2

, then

r + s = 5

(odd

\rightarrow

No).

Because Statement (2) allows both 'Yes' and 'No' answers depending on the value of

s

, Statement (2) alone is NOT sufficient.

Key Concept

Definitive Yes/No Decision Logic in Data Sufficiency

Question 403Question →

If $a$ and $b$ are real numbers, is $a^2 + b^2 < 4$ ?

(1) $(a - 2)^2 + (b - 2)^2 = 18$
(2) $a^2 - 6a + b^2 - 8b = -24$

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Show answer & explanation

Answer: Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

Answer

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

Statement (1) permits values of

a^2 + b^2

ranging from 2 to 50, which can be either less than 4 (yielding a 'Yes') or greater than or equal to 4 (yielding a 'No'), making Statement (1) insufficient. Statement (2) defines a geometric locus of points where

a^2 + b^2 \ge 16

, which means

a^2 + b^2 < 4

is conclusively false for all possible values. Because Statement (2) yields a definitive 'No' answer, it is sufficient on its own. Consequently, Statement (2) alone is sufficient, but Statement (1) alone is not sufficient.

Step-by-Step Solution

1

Understand the Data Sufficiency decision logic for Yes/No questions.

A statement is sufficient if it yields a definitive 'Yes' (always true) OR a definitive 'No' (always false). It is insufficient only if it allows both 'Yes' and 'No'.

Data Sufficiency requires determinism regarding the truth value of the target question.

2

Evaluate Statement (1):

(a - 2)^2 + (b - 2)^2 = 18

.

This equation represents a circle centered at

(2, 2)

with radius

r = \sqrt{18} = 3\sqrt{2}

. The distance from the origin to the center is

\sqrt{2^2 + 2^2} = 2\sqrt{2}

. The distance

d = \sqrt{a^2 + b^2}

of any point on the circle from the origin ranges from

|2\sqrt{2} - 3\sqrt{2}| = \sqrt{2}

to

2\sqrt{2} + 3\sqrt{2} = 5\sqrt{2}

. Thus,

a^2 + b^2 = d^2

ranges from

(\sqrt{2})^2 = 2

to

(5\sqrt{2})^2 = 50

.

Since

a^2 + b^2

can be

2

(which is

< 4

, giving 'Yes') or

50

(which is

\ge 4

, giving 'No'), Statement (1) cannot yield a single definitive Yes/No answer and is INSUFFICIENT.

3

Evaluate Statement (2):

a^2 - 6a + b^2 - 8b = -24

.

Completing the square gives

(a - 3)^2 - 9 + (b - 4)^2 - 16 = -24 \implies (a - 3)^2 + (b - 4)^2 = 1

. This is a circle centered at

(3, 4)

with radius

1

. The distance from the origin to the center

(3,4)

is

\sqrt{3^2 + 4^2} = 5

. The minimum distance from the origin to any point on this circle is

5 - 1 = 4

, so

d \ge 4

. Therefore,

a^2 + b^2 = d^2 \ge 4^2 = 16

.

Because

a^2 + b^2 \ge 16

for all points satisfying Statement (2), the inequality

a^2 + b^2 < 4

is NEVER true. Statement (2) yields a definitive 'No' answer, which makes it SUFFICIENT.

4

Conclude the final Data Sufficiency choice.

Statement (1) alone is insufficient, while Statement (2) alone is sufficient.

A definitive 'No' from Statement (2) satisfies sufficiency under standard Data Sufficiency decision logic.

Key Concept

Value vs. Yes/No Data Sufficiency Decision Logic

Question 404Question →

Archaeologists analyzing high-resolution synthetic aperture radar imagery of the Eastern Desert observed subsurface linear anomalies near ancient copper mines. Carbon dating of organic residue in nearby soil strata dates to the 12th century BCE. The researchers concluded that these linear anomalies represent buried irrigation canals constructed specifically to support the agricultural needs of the mining settlements.

Which of the following, if true, provides the strongest support for the researchers' conclusion?

Soil samples extracted directly from inside the linear structures contain preserved pollen from cultivated grain crops that cannot survive in desert environments without artificial watering.

Geological surveys show that subterranean fault lines in the Eastern Desert naturally form linear patterns that trap organic sediment over centuries.

Synthetic aperture radar technology has been proven effective at detecting ancient subterranean trade routes in several other desert regions.

Copper smelting at the 12th-century BCE mining sites required substantial quantities of timber brought in from non-desert regions.

Metal production at the Eastern Desert mining settlements declined sharply towards the end of the 12th century BCE as local ore deposits became exhausted.

Show answer & explanation

Answer: Soil samples extracted directly from inside the linear structures contain preserved pollen from cultivated grain crops that cannot survive in desert environments without artificial watering.

Answer

The discovery of fossilized pollen from cultivated grain crops requiring artificial watering inside the linear structures provides direct empirical evidence that the anomalies were indeed irrigation canals built to support local agriculture.

The correct answer provides direct physical evidence connecting the subsurface linear structures to agricultural irrigation. Finding preserved pollen from crops that require artificial watering inside the features demonstrates that they functioned as water channels designed for crop cultivation, directly reinforcing the researchers' conclusion.

Step-by-Step Solution

1

Deconstruct the argument into premise and conclusion

Premise: Radar imagery shows subsurface linear anomalies near 12th-century BCE copper mines, with nearby organic soil strata dating to that period. Conclusion: The anomalies are buried irrigation canals built to supply agricultural needs for the mining settlements.

Strengthening questions require identifying the logical gap between the evidence (linear shapes near mines) and the conclusion (irrigation canals for farming).

2

Identify the key assumption underlying the conclusion

The argument assumes that the linear anomalies are artificial structures intended for carrying water to crops rather than natural geological formations or non-agricultural conduits.

Providing evidence that directly links the interior of the structures to crop cultivation requiring diverted water validates this core assumption.

3

Evaluate the answer options against the strengthening objective

The option establishing the presence of crop pollen requiring artificial watering inside the structures explicitly confirms their agricultural and hydrological function.

Direct evidence of agricultural water usage inside the features supplies the missing evidentiary link.

Key Concept

Strengthening Arguments by Providing Direct Evidence for Unstated Assumptions

Question 405Question →

During the annual performance review, every project manager who met their sales target received a performance bonus. Furthermore, any manager who received a performance bonus was granted extra vacation days.

Evaluate the following statement based strictly on the information above:
If a project manager did not receive extra vacation days following the annual performance review, that project manager did not meet their sales target.

True

False

Show answer & explanation

Answer: True

Answer

True

The statement represents a valid contrapositive deduction. By combining the two premises (Target Met → Bonus and Bonus → Extra Vacation), we deduce that meeting the target guarantees extra vacation days. The contrapositive (lacking extra vacation days guarantees the target was not met) must therefore be true.

Step-by-Step Solution

1

Identify and formalize the conditional statements given in the passage.

Premise 1: Target Met → Bonus. Premise 2: Bonus → Extra Vacation.

Converting natural language into formal conditional statements clarifies logical relationships.

2

Chain the conditional premises together.

Target Met → Extra Vacation.

If statement A implies statement B, and statement B implies statement C, then statement A logically guarantees statement C.

3

Form the contrapositive of the chained conditional statement.

NOT Extra Vacation → NOT Target Met.

The contrapositive of a valid conditional statement is always logically equivalent and must be true.

Key Concept

Identifying Must-Be-True Statements and Valid Deductions

Question 406Question →

A technology company recently introduced an asynchronous communication policy, replacing mandatory daily team meetings with written status updates. Over the subsequent twelve months, the average time required for the software development team to complete project features decreased by 35 percent. The director of operations concluded that eliminating daily meetings directly caused the reduction in feature completion time by minimizing workplace disruptions. Which of the following would be most useful to determine in evaluating the director of operations' conclusion?

Whether the company implemented automated code-testing software during the same twelve-month period that significantly reduced deployment delays.

Whether the reduction in feature completion time directly caused software engineers to report higher overall job satisfaction.

Whether software developers at competing firms that retained mandatory daily meetings expressed a preference for written status updates.

Whether the asynchronous communication policy was applied to non-technical departments within the company in addition to the software development team.

Whether the director of operations had previously advocated for similar workplace policy changes at other software companies.

Show answer & explanation

Answer: Whether the company implemented automated code-testing software during the same twelve-month period that significantly reduced deployment delays.

Answer

Determining whether the company implemented automated code-testing software during the same twelve-month period that significantly reduced deployment delays is most useful in evaluating the conclusion.

The correct answer provides a critical test for an alternative explanation (confounding variable). In GMAT causal evaluation, an argument inferring causation from a temporal correlation can be evaluated by checking whether another major operational change occurred simultaneously. If automated testing software was introduced, that software could be the real reason completion times dropped, thereby weakening the director's claim. If no such software was introduced, that potential alternative cause is eliminated, strengthening the director's claim.

Step-by-Step Solution

1

Deconstruct the argument structure.

Premise: Meeting elimination was followed by a 35% decrease in feature completion time. Conclusion: Eliminating meetings directly caused the reduction in completion time by eliminating disruptions.

Identifying the premise and conclusion clarifies that the argument relies on a correlation-to-causation leap.

2

Identify the primary logical vulnerability in causal reasoning.

The argument assumes no external confounding factors or alternative explanations caused the productivity increase during the twelve-month period.

Causal claims based on chronological sequence (post hoc reasoning) are highly vulnerable to third-variable confounders.

3

Evaluate the answer choices against the Variance Test for causal evaluation.

If automated testing software was introduced (Yes), it offers an alternative cause for the faster completion times, weakening the conclusion. If no automated testing was introduced (No), an alternative cause is ruled out, strengthening the conclusion.

An option that yields opposite logical effects depending on a 'Yes' or 'No' answer is the correct evaluation factor.

Key Concept

Evaluating Causal Arguments by Rule-Out or Identification of Alternative Explanations

Question 407Question →

In a GMAT Data Sufficiency 'Yes/No' question asking whether $a^2 > b^2$ , a statement establishing that $a + b = 0$ (where $a \neq 0$ ) is sufficient to answer the question.

True

False

Show answer & explanation

Answer: True

Answer

The statement is True. In Yes/No Data Sufficiency, proving that a target condition is conclusively false constitutes a definitive 'No' answer, which makes the statement sufficient.

In Data Sufficiency Yes/No decision logic, sufficiency is attained whenever a statement leads to a single, unambiguous answer—either always 'Yes' or always 'No'. Here,

a + b = 0

forces

a^2 = b^2

, meaning

a^2 > b^2

is never true. Thus, the statement provides a definitive 'No' answer to the question, making it sufficient. Therefore, the assertion is correct (True).

Step-by-Step Solution

1

Identify the target question type and sufficiency condition.

The target is a Yes/No Data Sufficiency question: 'Is

a^2 > b^2

?' Sufficiency requires either a definitive 'Yes' in all cases or a definitive 'No' in all cases.

Establishing decision logic boundaries is required before evaluating statement sufficiency.

2

Rephrase the given statement algebraically.

The statement gives

a + b = 0 \implies a = -b

(with

a \neq 0

).

Expressing one variable in terms of another allows direct substitution into the target inequality.

3

Substitute the expression into the target inequality

a^2 > b^2

.

(-b)^2 > b^2 \implies b^2 > b^2

, which simplifies to

0 > 0

. This statement is universally false for any non-zero real number

b

.

Evaluating the target inequality under the given constraint tests whether the outcome is deterministic.

4

Apply Data Sufficiency decision logic to the result.

Since

a^2 > b^2

is false for all allowable values, the answer to the question 'Is

a^2 > b^2

?' is a definitive 'No'. A definitive 'No' answer means the statement is sufficient.

In GMAT DS logic, both a definitive 'Yes' and a definitive 'No' satisfy the requirement for sufficiency.

Key Concept

Definitive No Sufficiency Rule in Yes/No Data Sufficiency

Question 408Question →

If $x$ and $y$ are real numbers, what is the value of $x + y$ ?

(1) $3x + 3y = 12$
(2) $x - y = 2$

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Show answer & explanation

Answer: Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Answer

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

The correct option correctly identifies that Statement (1) alone is sufficient because dividing

3x + 3y = 12

by 3 yields

x + y = 4

, directly answering the question without needing to solve for

x

and

y

individually. Statement (2) alone leaves

x + y

undetermined.

Step-by-Step Solution

1

Analyze the target expression in the question stem.

The target to find is a single value for

x + y

.

Rephrasing the question stem clarifies what exact numerical value or algebraic combination is required.

2

Evaluate Statement (1) independently:

3x + 3y = 12

.

Factor out 3 to get

3(x + y) = 12

, then divide by 3 to obtain

x + y = 4

.

Statement (1) gives a unique value for the target expression

x + y

, so Statement (1) alone is sufficient.

3

Evaluate Statement (2) independently:

x - y = 2

.

If

x = 3

and

y = 1

, then

x - y = 2

and

x + y = 4

. If

x = 4

and

y = 2

, then

x - y = 2

and

x + y = 6

.

Multiple values for

x + y

are possible, so Statement (2) alone is not sufficient.

Key Concept

Question Stem Rephrasing and Linear Expression Combination

Estimated Time:1m 0s

Question 409Question →

Under a regulatory framework established in 2024, any commercial cargo vessel operating on Baltic maritime routes must reduce its sulfur oxide emissions by at least 75 percent relative to its 2020 baseline during any calendar quarter in which it transits within 100 nautical miles of a major port. In the first quarter of 2026, every vessel in Maritime Corp's fleet operated exclusively on Baltic maritime routes and completed multiple transits within 100 nautical miles of major ports. During this same quarter, every vessel in Maritime Corp's fleet maintained sulfur oxide emissions at least 80 percent below its 2020 baseline, even though Maritime Corp did not purchase any low-sulfur synthetic fuels.

Which of the following can be properly inferred from the information provided above?

During the first quarter of 2026, every vessel in Maritime Corp's fleet satisfied the regulatory framework's sulfur oxide emission reduction requirement during its transits within 100 nautical miles of major ports on Baltic maritime routes.

Maritime Corp achieved its sulfur oxide reductions by retrofitting its fleet engines with alternative mechanical filtration systems.

In the first quarter of 2026, Maritime Corp incurred higher total fleet operating costs than competing shipping lines that relied on low-sulfur synthetic fuels.

No commercial cargo vessel operating on Baltic maritime routes in the first quarter of 2026 emitted a lower volume of sulfur oxide than Maritime Corp's vessels.

Maritime Corp produced a smaller total volume of sulfur oxide emissions in the first quarter of 2026 than any competitor operating on Baltic maritime routes.

Show answer & explanation

Answer: During the first quarter of 2026, every vessel in Maritime Corp's fleet satisfied the regulatory framework's sulfur oxide emission reduction requirement during its transits within 100 nautical miles of major ports on Baltic maritime routes.

Answer

During the first quarter of 2026, every vessel in Maritime Corp's fleet satisfied the regulatory framework's sulfur oxide emission reduction requirement during its transits within 100 nautical miles of major ports on Baltic maritime routes.

The correct option is logically guaranteed by the text. The regulation requires a minimum 75% emission reduction for ships transiting within 100 nautical miles of major ports on Baltic routes. The passage confirms that every vessel in Maritime Corp's fleet operated on these routes, made such transits in the first quarter of 2026, and maintained reductions of at least 80%. Because 80% is greater than or equal to 75%, every vessel in the fleet necessarily met the regulatory requirement.

Step-by-Step Solution

1

Analyze the explicit regulatory condition given in the stimulus.

Regulations mandate at least a 75 percent reduction in sulfur oxide emissions relative to a 2020 baseline for vessels operating on Baltic routes within 100 nautical miles of a major port.

Establishing the precise criteria for compliance is necessary to evaluate claims about meeting regulatory standards.

2

Synthesize the facts regarding Maritime Corp's fleet during the first quarter of 2026.

All Maritime Corp vessels operated on Baltic routes, completed transits within 100 nautical miles of major ports, and achieved emission levels at least 80 percent below their 2020 baseline.

Comparing the fleet's actual performance (80 percent reduction) to the required threshold (75 percent reduction) reveals that the compliance standard was met.

3

Evaluate answer choices to eliminate out-of-scope hypotheses and 'could be true' speculations.

Only the statement declaring complete fleet compliance is logically guaranteed by combining the facts provided.

GMAT Critical Reasoning inference questions require selecting an option that MUST be true based solely on the given premises without external assumptions.

Key Concept

Identifying Must-Be-True Deductions vs. Out-of-Scope Speculations

Question 410Question →

A regional agricultural board planned to reduce crop losses caused by the invasive stem borer beetle by introducing a parasitic wasp species known to prey exclusively on stem borer larvae. In field trial plots located in low-altitude valleys, the wasp was released, and stem borer larva populations dropped by $70\%$ within one season, leading to a substantial decrease in crop damage. Based on these results, the board concluded that releasing the wasp regionally will reliably protect crops from stem borer damage across all commercial farming zones in the region. Which of the following, if true, most seriously weakens the agricultural board's argument?

The parasitic wasp cannot survive winter temperatures characteristic of the region's primary commercial farming zones, which are located at much higher altitudes than the trial plots.

Farms in neighboring regions that did not participate in the field trials experienced no decrease in crop losses caused by stem borer beetles during the trial period.

Chemical pesticides currently utilized by farmers in the region have shown a

25\%

decline in effectiveness over the past decade due to developed pest resistance.

Introducing the parasitic wasp will require commercial farmers to invest a minor amount of time attending introductory workshops on insect release protocols.

The stem borer beetle's larval stage was identified by entomologists as the primary driver of agricultural damage prior to the initiation of the trial.

Show answer & explanation

Answer: The parasitic wasp cannot survive winter temperatures characteristic of the region's primary commercial farming zones, which are located at much higher altitudes than the trial plots.

Answer

The argument is most seriously weakened by the fact that the parasitic wasp cannot survive the winter temperatures in the region's primary commercial farming zones.

The correct answer weakens the argument by exposing a flaw in generalizability. The agricultural board assumes that success in low-altitude trial plots guarantees success across all commercial farming zones. If the main commercial farming zones experience winter temperatures too cold for the wasp to survive due to higher altitude, the wasp population will fail to establish itself and cannot provide reliable crop protection.

Step-by-Step Solution

1

Deconstruct the argument's premises and main conclusion.

Premise: Releasing wasps in low-altitude trial plots reduced stem borer larvae by 70% and significantly decreased crop damage. Conclusion: Regional release of the wasp will reliably protect crops across all commercial farming zones in the region.

Understanding the precise leap from trial plot results to region-wide application is essential to identifying logical vulnerabilities.

2

Identify the unstated assumption linking premise to conclusion.

The argument assumes that environmental and operational conditions in the trial plots are representative of all commercial farming zones in the region.

For trial results to predict regional outcomes, key conditions affecting wasp survival must be consistent across zones.

3

Evaluate answer choices for information that shatters this assumption.

Evidence that the wasp cannot survive winter temperatures in the main commercial farming zones (due to altitude differences) proves the trial results cannot be replicated regionally.

If the wasp dies off during winter in primary farming areas, it cannot provide sustained or reliable crop protection.

Key Concept

Generalizability of Trial Results and Environmental Constraints in Biological Plans

Question 411Question →

If $x$ and $y$ are real numbers, is $|x - y| < |x + y|$ ?

(1) $x^3 y > 0$
(2) $|x| < y$

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Show answer & explanation

Answer: Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Answer

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Rephrasing the stem by squaring both non-negative sides reveals that

|x - y| < |x + y|

is strictly equivalent to

xy > 0

. Statement (1) specifies

x^3 y > 0

. Because

x^3

and

x

always have the same sign,

x^3 y > 0

means

x

and

y

share the same sign, which guarantees

xy > 0

. Thus, Statement (1) alone provides a definitive 'Yes'. Statement (2) states

|x| < y

, which forces

y > 0

but allows

x

to be either positive or negative, so

xy

can be either positive or negative. Therefore, Statement (1) alone is sufficient, but Statement (2) alone is not sufficient.

Step-by-Step Solution

1

Rephrase the question stem by squaring both sides of the inequality.

Since both

|x - y|

and

|x + y|

are non-negative,

|x - y| < |x + y| \iff (x - y)^2 < (x + y)^2 \iff x^2 - 2xy + y^2 < x^2 + 2xy + y^2 \iff 0 < 4xy \iff xy > 0

. The question asks whether

x

and

y

have the same sign (and neither is zero).

Simplifying absolute value inequalities before analyzing statements prevents misinterpretation.

2

Evaluate Statement (1):

x^3 y > 0

.

Since

x^3

always has the exact same algebraic sign as

x

for all real numbers,

x^3 y > 0 \implies xy > 0

. This yields a definitive 'Yes' to the rephrased question.

Statement (1) alone provides enough information to answer the target question conclusively.

3

Evaluate Statement (2):

|x| < y

.

Since

|x| \ge 0

, this implies

y > 0

. However,

x

can be positive (e.g.,

x=2, y=3 \implies xy=6 > 0

, Yes) or negative (e.g.,

x=-2, y=3 \implies xy=-6 < 0

, No).

Because Statement (2) allows both 'Yes' and 'No' answers, it is not sufficient.

Key Concept

Rephrasing absolute value inequalities and analyzing product sign constraints in Data Sufficiency

Question 412Question →

In a GMAT Data Sufficiency 'Yes/No' question regarding whether a real variable $x$ is positive, a statement establishing that $x^2 - 4x + 3 = 0$ is sufficient, whereas in a 'Value' question asking for the exact numerical value of $x$ , the exact same statement is insufficient.

True

False

Show answer & explanation

Answer: True

Answer

The statement is True.

The statement accurately reflects GMAT Data Sufficiency decision rules: a 'Yes/No' question requires only that all possible cases produce a uniform 'Yes' or uniform 'No' answer (which occurs here since both

x=1

and

x=3

are positive), while a 'Value' question strictly requires a single unique value (which fails here due to having two solutions).

Step-by-Step Solution

1

Analyze the mathematical implications of the given statement.

Solving

x^2 - 4x + 3 = 0

yields

(x-1)(x-3) = 0

, so

x = 1

or

x = 3

.

Determine the exact solution set allowed by the statement.

2

Evaluate sufficiency for a 'Yes/No' Data Sufficiency question stem.

Since both

x = 1

and

x = 3

are strictly greater than

0

, the answer to 'Is

x > 0

?' is unconditionally 'Yes'. Thus, the statement is sufficient.

In 'Yes/No' DS logic, any statement that yields a definitive, consistent 'Yes' (or a definitive, consistent 'No') is sufficient.

3

Evaluate sufficiency for a 'Value' Data Sufficiency question stem.

Since

x

can be either

1

or

3

, a single unique value for

x

cannot be determined. Thus, the statement is insufficient.

In 'Value' DS logic, a statement is sufficient if and only if it leads to exactly one numerical value.

4

Compare the conclusions with the given assertion.

The assertion correctly identifies that the statement is sufficient for 'Yes/No' logic and insufficient for 'Value' logic.

Confirm total alignment between the assertion and GMAT DS decision logic.

Key Concept

Value vs. Yes/No Data Sufficiency Decision Logic

Question 413Question →

Consider the following argument:

'Pharmaceutical executives contend that extending patent exclusivity periods is essential to funding high-risk biomedical research. However, public health economists observe that protracted exclusivity inflates healthcare expenditures without demonstrably increasing the rate of novel drug discovery. Although industry advocates argue that reduced patent lifespans would deter capital investment in critical therapeutic areas, comparative cross-national data demonstrate that robust public-venture funding effectively offsets private investment declines. Therefore, lengthening patent monopolies is an inefficient strategy for fostering pharmaceutical innovation.'

Which of the following correctly matches each claim from the passage with its structural role in the overall argument?

Click a left item, then click its matching right item

Show answer & explanation

Answer

The correct pairings align each statement from the passage to its precise structural function: the executive assertion matches the primary opposing position; the economist observation matches the empirical counter-evidence; the advocate warning matches the secondary defensive counter-claim; and the final sentence matches the author's main conclusion.

Each statement is correctly mapped based on its logical relationship within the text: the executive assertion serves as the primary opposing position; the economist observation supplies empirical counter-evidence against it; the advocate assertion introduces a defensive counter-claim; and the final sentence expresses the author's main conclusion.

Step-by-Step Solution

1

Analyze the passage structure and identify key transition words.

Pivot words such as 'However', 'Although', and 'Therefore' signal shifts between opposing positions, counterarguments, defensive claims, and the main conclusion.

Recognizing structural pivots is essential for accurately mapping the logical relationships between distinct claims in complex Critical Reasoning passages.

2

Identify the main target position and the author's main conclusion.

The executive contention introduces the primary opposing position, while the final sentence preceded by 'Therefore' states the author's ultimate main conclusion.

The passage opens by framing the industry stance it intends to evaluate and ends by declaring the author's final judgment.

3

Distinguish between the two distinct counter-claims presented in the middle of the argument.

The economist observation provides evidence contradicting the executive claim, whereas the advocate warning serves as a defensive counterargument to preserve exclusivity against economic criticisms.

Differentiating an empirical counter-finding from a defensive counter-claim is critical when evaluating multi-viewpoint arguments.

Key Concept

Identifying Counterarguments and Opposing Positions

Question 414Question →

In Data Sufficiency, simplifying the question target "Is $x^2 < x$ ?" yields the algebraically equivalent question target "Is $0 < x < 1$ ?" for all real numbers $x$ .

True

False

Show answer & explanation

Answer: True

Answer

The statement is true because solving the inequality

x^2 < x

yields the exact range

0 < x < 1

.

Rephrasing the question stem target

x^2 < x

algebraically leads directly to

x(x - 1) < 0

, which holds true if and only if

x

is strictly between

0

and

1

. Therefore, the simplified target 'Is

0 < x < 1

?' is completely equivalent.

Step-by-Step Solution

1

Rearrange the inequality to set one side to zero.

x^2 - x < 0

Standard algebraic procedure for solving non-linear inequalities requires comparing a factored expression to zero.

2

Factor the quadratic expression.

x(x - 1) < 0

Factoring isolates the roots (

x = 0

and

x = 1

) that define the boundary intervals on the real number line.

3

Determine the interval where the product of the factors is negative.

The product is negative between the roots, which corresponds to

0 < x < 1

.

When

x < 0

, both factors are negative (product is positive). When

x > 1

, both factors are positive (product is positive). Only when

0 < x < 1

is

x

positive and

(x - 1)

negative.

Key Concept

Simplifying Data Sufficiency question stems by finding equivalent inequality ranges reduces complex targets to direct boundary checks.

Estimated Time:45s

Question 415Question →

Three months ago, a citrus orchard manager introduced a species of non-stinging predatory mite to combat Asian citrus psyllids, insects that transmit a destructive bacterial disease to orange trees. Over the subsequent three-month period, the population of Asian citrus psyllids in the orchard decreased by 60 percent. Based on this outcome, the manager concluded that the introduction of the predatory mites was responsible for the decline in the psyllid population.

Which of the following, if true, provides the strongest support for the manager's conclusion?

Right before introducing the predatory mites, the orchard manager applied a broad-spectrum chemical insecticide aimed at reducing insect pest populations.

Asian citrus psyllids cause significantly more harm to sweet orange trees than to lemon trees grown in adjacent fields.

In nearby citrus orchards with identical environmental conditions where predatory mites were not introduced, the Asian citrus psyllid population did not decrease over the same three-month period.

Deploying predatory mites requires specialized equipment that increases overall pest management costs compared to traditional chemical spraying.

The bacterial disease spread by Asian citrus psyllids leads to premature fruit drop and severely reduced harvest yields.

Show answer & explanation

Answer: In nearby citrus orchards with identical environmental conditions where predatory mites were not introduced, the Asian citrus psyllid population did not decrease over the same three-month period.

Answer

The argument is most strongly supported by the statement indicating that in nearby orchards under identical conditions where predatory mites were not introduced, the psyllid population did not decrease.

The conclusion asserts a causal relationship between introducing predatory mites and the 60 percent decline in psyllids. The correct choice provides a control comparison: in neighboring orchards where environmental conditions were identical but mites were absent, psyllid populations remained stable. This rules out seasonal changes, weather events, or natural population cycles as the cause of the decline, significantly strengthening the manager's claim that the mites were responsible.

Step-by-Step Solution

1

Identify the argument's premises and conclusion.

Premise: Mites were introduced, and psyllid numbers dropped by 60% over 3 months. Conclusion: Introducing the mites caused the drop in psyllids.

Understanding the precise cause-and-effect claim is necessary before evaluating strengthening evidence.

2

Analyze what evidence would strengthen a causal claim.

A causal argument is strengthened by ruling out alternative explanations (such as weather or pesticides) or showing a control scenario where the cause is absent and the effect does not occur.

Standard GMAT strengthening mechanisms for causal arguments rely on establishing correlation/control or eliminating competing causes.

3

Evaluate the option presenting a control group.

Showing that nearby orchards without mites experienced no decrease in psyllids under identical weather conditions confirms that the decline was specific to the orchard with mites.

This effectively rules out external weather/seasonal factors as the true cause of the population decline.

Key Concept

Strengthening Causal Arguments via Control Groups and Alternative Cause Elimination

Estimated Time:1m 15s

Question 416Question →

Following the introduction of an AI-driven automated diagnostic system at a major hospital network, the average triage evaluation time that human emergency room nurses spent per patient increased by 30 percent. Nevertheless, with the total number of emergency room nurses and overall patient volume remaining constant, the average waiting time for patients before receiving treatment decreased significantly across the hospital network.

Which of the following, if true, most helps to resolve the apparent discrepancy described above?

The automated system independently processed and cleared routine, minor cases without nurse involvement, leaving human nurses to handle only severe cases that naturally require longer evaluations.

Human emergency room nurses spent additional time entering medical history data into the new automated system during each patient evaluation.

The hospital network expanded its emergency room waiting facilities, allowing a greater number of non-emergency patients to wait simultaneously.

Emergency room visits during peak evening hours typically involve more complex medical conditions than visits occurring during morning hours.

The automated system significantly reduced the administrative processing time required for post-treatment hospital discharge paperwork.

Show answer & explanation

Answer: The automated system independently processed and cleared routine, minor cases without nurse involvement, leaving human nurses to handle only severe cases that naturally require longer evaluations.

Answer

The discrepancy is resolved by the fact that the automated system independently processed and cleared routine, minor cases without nurse involvement, leaving human nurses to handle only severe cases that naturally require longer evaluations.

The correct answer reconciles both facts through a subgroup composition shift. When the AI system automatically handles routine minor cases, these patients bypass human triage nurses entirely, significantly shortening the overall patient queue and reducing wait times. Concurrently, human nurses are left evaluating only the severe, complex cases, which naturally take longer to assess. Thus, the average evaluation time per patient for human nurses rises even as overall patient waiting times fall.

Step-by-Step Solution

1

Identify the two apparently contradictory facts in the passage.

Fact 1: Average triage time per patient spent by human nurses increased by 30%. Fact 2: Total nurses and patient volume remained constant, yet average pre-treatment patient wait times decreased significantly.

Resolving a paradox requires finding an explanation that allows both facts to be true simultaneously.

2

Analyze the mathematical and logical relationship between average time per unit and total pool composition.

If the automated system removes a large volume of fast, routine cases from the human queue, the denominator of human-evaluated patients shrinks, consisting now only of complex cases.

A shift in subgroup composition (a weighted average effect) can cause the average time spent on remaining human cases to rise while overall patient throughput speeds up.

3

Evaluate the choices to find the one that addresses both the increase in per-patient nurse time and the decrease in overall wait times.

The option stating that minor cases were handled entirely by the automated system reconciles both phenomena perfectly without contradicting any premise.

By filtering out minor cases, human queues shrink (reducing wait times), while the average time per patient for nurses increases because they only evaluate complex cases.

Key Concept

Resolving Paradoxes: Subgroup Composition Shift (Weighted Average Paradox)

Estimated Time:2m 0s

Question 417Question →

If $x$ is a non-zero real number, is $x$ an integer?

(1) $x + \frac{6}{x}$ is an integer.

(2) $x^2 + \frac{36}{x^2}$ is an integer.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Show answer & explanation

Answer: Statements (1) and (2) TOGETHER are NOT sufficient.

Answer

Statements (1) and (2) TOGETHER are NOT sufficient.

The correct response establishes that neither statement individually nor both statements combined are sufficient to determine whether the real number is an integer. Testing specific real numbers demonstrates that integer values such as 2 and irrational values such as 3 plus the square root of 3 satisfy both algebraic conditions, producing affirmative and negative answers to the question.

Step-by-Step Solution

1

Analyze Statement (1):

x + \frac{6}{x} = k

for some integer

k

.

If

x = 2

, then

2 + \frac{6}{2} = 5

(an integer), and

x

IS an integer. If

x = 3 + \sqrt{3}

, then

x + \frac{6}{x} = 3 + \sqrt{3} + \frac{6(3 - \sqrt{3})}{6} = 6

(an integer), but

x

is NOT an integer.

Since Statement (1) yields both a 'Yes' and a 'No' answer, it is not sufficient.

2

Analyze Statement (2):

x^2 + \frac{36}{x^2} = m

for some integer

m

.

If

x = 2

, then

2^2 + \frac{36}{2^2} = 4 + 9 = 13

(an integer), so

x

IS an integer. If

x = \sqrt{2}

, then

(\sqrt{2})^2 + \frac{36}{(\sqrt{2})^2} = 2 + 18 = 20

(an integer), but

x

is NOT an integer.

Since Statement (2) yields both a 'Yes' and a 'No' answer, it is not sufficient.

3

Analyze Statements (1) and (2) combined.

Note that

\left(x + \frac{6}{x}\right)^2 = x^2 + 12 + \frac{36}{x^2}

. Thus

x^2 + \frac{36}{x^2} = \left(x + \frac{6}{x}\right)^2 - 12

. Whenever

x + \frac{6}{x}

is an integer

k

,

x^2 + \frac{36}{x^2} = k^2 - 12

is automatically an integer. Therefore, Statement (2) provides no extra constraints.

Both

x = 2

(integer) and

x = 3 + \sqrt{3}

(non-integer) satisfy both statements simultaneously, so the combined statements remain insufficient.

Key Concept

Evaluating algebraic expressions for non-integer real number solutions in Data Sufficiency

Question 418Question →

For all real numbers $x$ such that $x \neq 0$ and $x \neq 1$ , the Data Sufficiency Yes/No target question "Is $\frac{x^2 - x}{|x - 1|} > 0$ ?" is algebraically equivalent to asking "Is $x > 0$ ?".

True

False

Show answer & explanation

Answer: False

Answer

The statement is False.

The statement is False because simplifying the target inequality

\frac{x(x - 1)}{|x - 1|} > 0

yields

x < 0

or

x > 1

, which is not equivalent to

x > 0

.

Step-by-Step Solution

1

Factor the numerator of the expression in the target inequality.

\frac{x^2 - x}{|x - 1|} = \frac{x(x - 1)}{|x - 1|}

.

Factoring allows for analyzing the individual signs of the linear factors.

2

Analyze the denominator to simplify the inequality.

Since

|x - 1| > 0

for all

x \neq 1

, multiplying both sides of

\frac{x(x - 1)}{|x - 1|} > 0

by

|x - 1|

gives the equivalent inequality

x(x - 1) > 0

.

Multiplying an inequality by a strictly positive quantity preserves the direction of the inequality sign.

3

Solve the quadratic inequality

x(x - 1) > 0

.

The product

x(x - 1)

is positive when both factors have the same sign, yielding the solution set

x < 0

or

x > 1

.

A product of two real terms is positive when both terms are positive or both terms are negative.

4

Compare the rephrased target (

x < 0

or

x > 1

) with the proposed target (

x > 0

).

The range

0 < x < 1

makes

x > 0

true but makes

x(x - 1) > 0

false. Thus, the targets are not equivalent.

Two target questions are algebraically equivalent if and only if they yield identical truth values for all values in the domain.

Key Concept

Question Stem Simplification and Target Rephrasing

Question 419Question →

If $k$ is a non-zero real number, is $k$ an integer?

(1) $\frac{12}{k}$ is an integer.

(2) $k^2 - 3k$ is an integer.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Show answer & explanation

Answer: BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Answer

Both statements together are sufficient to answer the question, but neither statement alone is sufficient.

The correct response demonstrates that neither statement individually restricts

k

exclusively to integer values, but combining them enforces that

k

is rational (from the reciprocal condition) and has a denominator of 1 (from the quadratic condition), proving conclusively that

k

must be an integer.

Step-by-Step Solution

1

Evaluate Statement (1) independently.

Statement (1) states that

\frac{12}{k} = m

for some non-zero integer

m

, so

k = \frac{12}{m}

. If

m = 1

,

k = 12

(an integer). If

m = 24

,

k = 0.5

(not an integer). Since

k

can be either an integer or a non-integer, Statement (1) is NOT sufficient.

Testing specific values reveals that

k

can be fractional while fulfilling the condition.

2

Evaluate Statement (2) independently.

Statement (2) states that

k^2 - 3k = n

for some integer

n

. If

k = 4

,

k^2 - 3k = 4

(an integer). However, if

k = \frac{3 + \sqrt{13}}{2}

,

k^2 - 3k = 1

(an integer), but

k

is irrational. Since

k

can be an integer or irrational, Statement (2) is NOT sufficient.

Quadratic expressions with integer outputs can have irrational inputs.

3

Evaluate Statements (1) and (2) together.

From Statement (1),

k

must be rational. Let

k = \frac{p}{q}

in lowest terms, where

p

and

q

are coprimes (

\gcd(p, q) = 1

) and

q \ge 1

. Substitute

k = \frac{p}{q}

into Statement (2):

k^2 - 3k = \frac{p^2 - 3pq}{q^2}

. For this expression to be an integer,

q^2

must divide

p(p - 3q)

. Since

\gcd(p, q) = 1

,

q

shares no common prime factors with

p

or

p - 3q

. Thus

\gcd(p^2 - 3pq, q^2) = 1

, which means

q^2

can divide

p^2 - 3pq

if and only if

q^2 = 1

, so

q = 1

. Since

q = 1

,

k = p

, which guarantees

k

is an integer. Thus, the combined statements yield a definitive 'Yes'.

Combining rationality from the first condition with the algebraic divisibility constraint of the second condition eliminates all non-integer possibilities.

Key Concept

Data Sufficiency logic with integer constraints and rational number representations

Estimated Time:2m 0s

Question 420Question →

At a technology firm, $60$ software engineers work on Project Alpha, Project Beta, or both. Exactly $40$ engineers work on Project Alpha, and exactly $35$ engineers work on Project Beta. Is the average (arithmetic mean) years of experience of all $60$ engineers combined greater than $7$ years?

(1) The average years of experience of the engineers who work ONLY on Project Alpha is $8$ years, and the average years of experience of the engineers who work ONLY on Project Beta is $5$ years.
(2) The average years of experience of all $40$ engineers on Project Alpha is $8.75$ years.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement alone is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Show answer & explanation

Answer: BOTH statements TOGETHER are sufficient, but NEITHER statement alone is sufficient.

Answer

Both statements together are sufficient to answer the question definitively with a 'Yes', but neither statement alone is sufficient.

Combining both statements provides the exact sum of values across all three mutually exclusive sub-groups (Only Alpha, Only Beta, and Both), yielding a unique overall average of 7.5 years, which yields a definitive 'Yes' answer.

Step-by-Step Solution

1

Determine the sizes of the non-overlapping and overlapping groups.

Group sizes: Only Alpha = 25, Only Beta = 20, Both = 15.

Using the overlapping sets formula

N(A \cup B) = N(A) + N(B) - N(A \cap B)

, we get

60 = 40 + 35 - N(A \cap B)

, which yields

N(A \cap B) = 15

. Thus,

N(\text{Only Alpha}) = 40 - 15 = 25

and

N(\text{Only Beta}) = 35 - 15 = 20

.

2

Evaluate Statement (1) alone.

Statement (1) is NOT sufficient.

Statement (1) gives

S_{\text{Only A}} = 25 \times 8 = 200

and

S_{\text{Only B}} = 20 \times 5 = 100

. The sum of experience for the

15

overlap engineers (

S_{\text{Both}}

) remains unknown. Total mean

= \frac{300 + S_{\text{Both}}}{60}

. If

S_{\text{Both}} = 150

(mean

10

), total mean

= 7.5 > 7

(Yes). If

S_{\text{Both}} = 30

(mean

2

), total mean

= 5.5 \le 7

(No). Hence, Statement (1) alone is insufficient.

3

Evaluate Statement (2) alone.

Statement (2) is NOT sufficient.

Statement (2) gives the total experience of all

40

engineers on Project Alpha:

S_{\text{Alpha}} = 40 \times 8.75 = 350

. This means

S_{\text{Only A}} + S_{\text{Both}} = 350

. Total mean

= \frac{350 + S_{\text{Only B}}}{60}

. Since

S_{\text{Only B}}

is unknown, the total mean could be

7.5

(if

S_{\text{Only B}} = 100

) or

6.17

(if

S_{\text{Only B}} = 20

). Hence, Statement (2) alone is insufficient.

4

Evaluate Statements (1) and (2) together.

Statements (1) and (2) together are SUFFICIENT.

From Statement (1),

S_{\text{Only A}} = 200

and

S_{\text{Only B}} = 100

. From Statement (2),

S_{\text{Only A}} + S_{\text{Both}} = 350

, which implies

200 + S_{\text{Both}} = 350 \implies S_{\text{Both}} = 150

. The combined total experience is

S_{\text{Total}} = 200 + 100 + 150 = 450

. The overall mean is

\frac{450}{60} = 7.5

years. Since

7.5 > 7

, we get a definitive 'Yes'.

Key Concept

Data Sufficiency evaluation for combined weighted averages and overlapping set partitions

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