All practice questions

Question 1441Question →

In a technology firm of $100$ software engineers, each engineer works on at least one of two projects: Project Alpha or Project Beta. If $70$ engineers work on Project Alpha, what is the average (arithmetic mean) monthly salary of all $100$ engineers at the firm?

(1) The average monthly salary of the engineers who work ONLY on Project Alpha is $\$6,000$ , and the average monthly salary of all engineers who work on Project Beta is $\$10,800$ .
(2) The average monthly salary of the engineers who work ONLY on Project Beta is $\$12,000$ , and the average monthly salary of the engineers who work on BOTH Project Alpha and Project Beta is $\$9,000$ .

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, but NEITHER statement ALONE is sufficient.

The option stating that BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient is correct. Evaluating either statement independently leaves the overlap count or salary undefined. However, combining both statements allows equating two expressions for the total salary of Project Beta, which uniquely yields

20

engineers working on both projects, leading to a single deterministic overall average salary of

\$8,400

.

Step-by-Step Solution

1

Define variables from the question stem.

Let

n_1

be the number of engineers working ONLY on Project Alpha,

n_2

be the number working ONLY on Project Beta, and

n_3

be the number working on BOTH. Total engineers =

n_1 + n_2 + n_3 = 100

. Given

70

work on Alpha (

n_1 + n_3 = 70

), we find

n_2 = 100 - 70 = 30

.

Establishing exact set counts reduces unknowns.

2

Evaluate Statement (1) alone.

Total salary =

(n_1 \times 6000) + ((n_2 + n_3) \times 10800) = (70 - n_3)(6000) + (30 + n_3)(10800) = 744,000 + 4800 n_3

. Since

n_3

is unknown, overall average cannot be determined.

Statement (1) depends on the unknown overlap

n_3

, so Statement (1) alone is NOT sufficient.

3

Evaluate Statement (2) alone.

We know the average salary for

n_2 = 30

is

\$12,000

and for

n_3

is

\$9,000

, but we have no salary information for the

n_1

engineers working ONLY on Project Alpha.

Without salary information for Project Alpha only, Statement (2) alone is NOT sufficient.

4

Evaluate Statements (1) and (2) together.

From Statement (2), total salary of Project Beta =

(30 \times 12000) + (n_3 \times 9000) = 360,000 + 9000 n_3

. From Statement (1), total salary of Project Beta =

(30 + n_3) \times 10800 = 324,000 + 10800 n_3

. Equating both:

360,000 + 9000 n_3 = 324,000 + 10800 n_3 \implies 1800 n_3 = 36,000 \implies n_3 = 20

.

Finding

n_3 = 20

gives

n_1 = 50

, allowing exact calculation of the overall average salary (

8,400

).

Key Concept

Weighted Averages and Overlapping Sets in Data Sufficiency

Question 1442Question →

Under maritime regulatory guidelines established in 2025, a commercial freight vessel qualifies for Tier-1 Eco-Status if and only if it operates using green hydrogen propulsion and maintains a certified bi-annual hull inspection log. Furthermore, every commercial freight vessel holding Tier-1 Eco-Status is fully exempt from regional carbon emissions surcharges. During fiscal year 2026, the commercial freight vessel Aura was required to pay regional carbon emissions surcharges.

Based on the information provided, evaluate the validity of the following statement: During fiscal year 2026, the Aura either did not operate using green hydrogen propulsion or did not maintain a certified bi-annual hull inspection log.

True

False

Show answer & explanation

Answer: True

Answer

The statement is True because the premises deductively guarantee that the vessel failed at least one of the two requirements for Tier-1 Eco-Status.

The conclusion is deductively forced by combining the contrapositive of the surcharge rule with the negation of a conjunction (De Morgan's Law). Because the vessel paid the surcharge, it did not have Tier-1 Eco-Status, which strictly means it lacked green hydrogen propulsion, lacked a certified log, or lacked both.

Step-by-Step Solution

1

Analyze the given conditional premises.

Premise 1 establishes a biconditional relationship: Tier-1 Eco-Status

\iff

(Green Hydrogen Propulsion

\land

Certified Bi-annual Inspection Log). Premise 2 establishes a standard conditional relationship: Tier-1 Eco-Status

\implies

Exempt from Carbon Surcharges.

Mapping the logical structure allows for formal deductive reasoning.

2

Apply the contrapositive to the factual premise regarding the vessel.

The vessel was required to pay carbon surcharges (Not Exempt). By contrapositive of Premise 2 (Not Exempt

\implies

Not Tier-1 Eco-Status), the vessel did not hold Tier-1 Eco-Status in 2026.

If a conditional statement

A \implies B

is true, its contrapositive

\neg B \implies \neg A

must also be true.

3

Deduce the status of the specific requirements using De Morgan's Law.

Since Tier-1 Eco-Status requires both Green Hydrogen Propulsion AND a Certified Log, the negation of Tier-1 Eco-Status (

\neg(\text{Green Hydrogen} \land \text{Certified Log})

) is equivalent to (Not Green Hydrogen OR Not Certified Log).

Failing a joint necessary condition means failing at least one of the individual components.

Key Concept

Identifying Must-Be-True Statements using Contrapositive Logic and De Morgan's Laws of Negation

Question 1443Question →

A municipal transit authority recently applied a new light-reflecting polymer coating to crosswalks across three high-density commercial districts. Over the subsequent twelve months, nighttime pedestrian accidents in these districts decreased by 25 percent compared to the previous year. The authority concluded that the enhanced visual contrast provided by the polymer coating was the primary factor in reducing nighttime pedestrian accidents. Which of the following would it be most useful to determine in evaluating the transit authority's conclusion?

Whether municipal authorities also increased traffic enforcement patrols and lowered speed limits in those three districts during the same twelve-month period

Whether the decrease in pedestrian accidents directly led municipal officials to lower the maintenance budget for standard asphalt pavements

Whether reflective polymer coatings create daytime surface glare that slightly increases driver discomfort during peak sunlight hours

Whether adjacent suburban residential areas experienced an increase in personal vehicle registrations over the same twelve-month span

Whether local civic groups had initially voiced objections regarding the high upfront procurement costs of the polymer coating

Show answer & explanation

Answer: Whether municipal authorities also increased traffic enforcement patrols and lowered speed limits in those three districts during the same twelve-month period

Answer

Determining whether municipal authorities also increased traffic enforcement patrols and lowered speed limits in those three districts during the same twelve-month period is the most critical factor for evaluating the argument.

The correct answer provides information about potential confounding variables. In causal reasoning, establishing that event A caused event B requires verifying that no third factor (event C) occurred at the same time that could account for event B. If speed limits were lowered and police patrols were increased during the exact same period, those interventions offer a plausible alternative explanation for the drop in pedestrian accidents, directly challenging the conclusion that the polymer coating was the primary driver.

Step-by-Step Solution

1

Identify the argument structure and causal claim.

Premise: Crosswalks received reflective polymer coating; nighttime pedestrian accidents dropped by 25%. Conclusion: The polymer coating was the primary cause of the accident reduction.

Understanding the precise leap from observed correlation to claimed cause is essential for evaluating causal validity.

2

Identify the standard key vulnerabilities of causal arguments on the GMAT.

Causal claims can be evaluated by checking for alternative causes (confounding variables), reverse causality, or control group comparisons.

If another change occurred simultaneously, the observed effect might be attributable to that confounding factor rather than the polymer coating.

3

Evaluate the choices to find the factor that tests an alternative explanation.

Inquiring whether traffic police enforcement and speed limits were changed simultaneously tests whether an alternative cause produced the 25% decrease.

If enforcement increased and speed limits were lowered, those actions could account for the reduced accident rate, severely weakening the author's claim that the polymer coating was the primary cause.

Key Concept

Evaluating Causal Arguments: Identifying Confounding Variables and Alternative Explanations

Question 1444Question →

If $m$ and $n$ are non-zero real numbers such that $m^2 \neq n^2$ , is $\frac{m^3 - m^2 n + m n^2 - n^3}{m^2 - n^2} < 0$ ?

(1) $m + n = -5$
(2) $m - n = 3$

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.

Simplifying the target expression by factoring gives

\frac{(m - n)(m^2 + n^2)}{(m - n)(m + n)} = \frac{m^2 + n^2}{m + n}

. Because

m^2 + n^2 > 0

for all non-zero real numbers, the quotient is negative if and only if

m + n < 0

. Statement (1) gives

m + n = -5 < 0

, providing a definitive 'Yes' answer. Statement (2) gives

m - n = 3

, which allows

m + n

to be either positive or negative depending on the specific values of

m

and

n

. Thus, Statement (1) alone is sufficient.

Step-by-Step Solution

1

Rephrase the target question stem algebraically

The target expression

\frac{m^3 - m^2 n + m n^2 - n^3}{m^2 - n^2}

simplifies to

\frac{m^2 + n^2}{m + n}

.

Factor the numerator by grouping:

m^2(m - n) + n^2(m - n) = (m - n)(m^2 + n^2)

. Factor the denominator:

m^2 - n^2 = (m - n)(m + n)

. Since

m^2 \neq n^2

,

m - n \neq 0

, so cancel

(m - n)

.

2

Determine the condition for the simplified expression to be negative

The rephrased target question is 'Is

m + n < 0

?'

Since

m

and

n

are non-zero real numbers,

m^2 + n^2 > 0

always. Therefore, the sign of

\frac{m^2 + n^2}{m + n}

depends solely on the denominator

m + n

.

3

Evaluate Statement (1):

m + n = -5

Statement (1) is SUFFICIENT.

Statement (1) directly tells us

m + n = -5

, which is less than

0

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

m + n < 0

?'.

4

Evaluate Statement (2):

m - n = 3

Statement (2) is NOT SUFFICIENT.

Knowing

m - n = 3

gives

m = n + 3

, so

m + n = 2n + 3

. If

n = 0.5

,

m + n = 4 > 0

('No'). If

n = -5

,

m + n = -7 < 0

('Yes'). Since

m + n

can be positive or negative, Statement (2) is insufficient.

Key Concept

Question Stem Simplification in Data Sufficiency

Estimated Time:2m 0s

Question 1445Question →

Passage:
In an economic study examining thirty regional technology hubs, researchers observed that fifteen hubs introduced targeted tax incentives for green energy startups in 2023. Every hub that introduced these tax incentives experienced a measurable increase in foreign direct investment in 2024. However, none of the hubs that introduced the incentives reported an increase in local patent filings during that same year.

Statement: Based on the passage, the claim 'The targeted tax incentives were the primary driver of the increase in foreign direct investment in the fifteen technology hubs' must be true.

True

False

Show answer & explanation

Answer: False

Answer

False. The statement cannot be properly inferred as a guaranteed fact from the passage premises.

The evaluation of False is correct. On GMAT Critical Reasoning inference items, a statement must be logically guaranteed by the text to be considered a valid deduction. The passage establishes that fifteen tech hubs introduced tax incentives in 2023 and subsequently experienced an increase in foreign direct investment in 2024. However, correlation and temporal sequence do not logically establish causation or prove that the incentives were the primary driver. Because the claim introduces an unstated causal link, it represents a 'could be true' / out-of-scope trap rather than a necessary truth.

Step-by-Step Solution

1

Analyze the given premises in the passage

The passage establishes two factual relationships: 15 hubs introduced tax incentives in 2023, and 100% of those 15 hubs saw increased foreign direct investment in 2024.

Identify what is explicitly stated without adding outside assumptions.

2

Evaluate the logical connection required by the statement

The statement claims that the tax incentives were the 'primary driver' (causal mechanism) of the investment growth.

Determine whether causal mechanism is logically necessitated by temporal sequence.

3

Identify common inference traps

Mistaking correlation or sequence for causation is an out-of-scope extrapolation trap. The claim could be true, but it does not MUST be true.

On GMAT Critical Reasoning inference questions, claims that are merely plausible or 'could be true' are invalid deductions.

Key Concept

Avoiding Causal and Out-of-Scope Inference Traps

Question 1446Question →

If $u$ and $v$ are real numbers such that $u + v \neq 0$ , what is the value of $\frac{u - v}{u + v}$ ?

(1) $u^2 - 3uv + 2v^2 = 0$
(2) $2u^2 - 5uv + 2v^2 = 0$

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, but NEITHER statement ALONE is sufficient.

The correct answer is the option stating that both statements together are sufficient, but neither alone is sufficient. Statement (1) yields two possible ratios (

u=v

giving 0, or

u=2v

giving

1/3

), so it is not sufficient. Statement (2) also yields two possible ratios (

u=v/2

giving

-1/3

, or

u=2v

giving

1/3

), so it is not sufficient. When combined,

u=v

and

u=v/2

would require

u=v=0

, which is forbidden by the condition

u+v \neq 0

. Therefore,

u=2v

is the only valid relation, producing a unique value of

1/3

.

Step-by-Step Solution

1

Rephrase the target question stem.

Dividing the numerator and denominator of

\frac{u - v}{u + v}

by

v

(assuming

v \neq 0

) shows that knowing the ratio

\frac{u}{v}

uniquely determines the value of the expression.

Simplifying the target expression reduces the problem to determining whether a unique ratio

\frac{u}{v}

exists.

2

Evaluate Statement (1) independently.

Factor

u^2 - 3uv + 2v^2 = 0

as

(u - v)(u - 2v) = 0

. This implies

u = v

or

u = 2v

. If

u = v

, then

\frac{u - v}{u + v} = \frac{0}{2v} = 0

. If

u = 2v

, then

\frac{u - v}{u + v} = \frac{2v - v}{2v + v} = \frac{v}{3v} = \frac{1}{3}

.

Since Statement (1) allows two distinct numerical values (

0

and

\frac{1}{3}

), Statement (1) alone is NOT sufficient.

3

Evaluate Statement (2) independently.

Factor

2u^2 - 5uv + 2v^2 = 0

as

(2u - v)(u - 2v) = 0

. This implies

u = \frac{1}{2}v

or

u = 2v

. If

u = \frac{1}{2}v

, then

\frac{u - v}{u + v} = \frac{\frac{1}{2}v - v}{\frac{1}{2}v + v} = \frac{-\frac{1}{2}v}{\frac{3}{2}v} = -\frac{1}{3}

. If

u = 2v

, then

\frac{u - v}{u + v} = \frac{1}{3}

.

Since Statement (2) allows two distinct numerical values (

-\frac{1}{3}

and

\frac{1}{3}

), Statement (2) alone is NOT sufficient.

4

Evaluate Statement (1) and Statement (2) combined.

Combine the conditions: Statement (1) requires

u = v

or

u = 2v

; Statement (2) requires

u = \frac{1}{2}v

or

u = 2v

. Testing

u = v

in Statement (2) yields

2v^2 - 5v^2 + 2v^2 = -v^2 = 0 \implies v = 0

, which implies

u = 0

, but

u + v \neq 0

rules out

u = v = 0

. Testing

u = \frac{1}{2}v

in Statement (1) similarly leads to

u = v = 0

. Thus, the only non-zero solution satisfying both equations is

u = 2v

, which gives a unique value of

\frac{1}{3}

.

Combining both statements eliminates the ambiguous cases that violate the stem constraint, establishing a unique value.

Key Concept

Homogeneous Quadratic Factoring and Constraint Evaluation in Data Sufficiency

Question 1447Question →

Consider the following argument: A pharmaceutical research institute evaluated a novel antihypertensive drug by administering it to a cohort of patients over twelve weeks. The study recorded a statistically significant reduction in blood pressure in patients receiving the new drug compared to those receiving the standard treatment. Based on these findings, the lead researcher concluded that adopting the new drug in standard clinical practice will decrease the overall rate of major cardiovascular events among hypertensive patients. Evaluate whether the following claim represents an unstated assumption necessary to the researcher's argument:

'The researcher's conclusion relies on the unstated assumption that a statistically significant short-term reduction in blood pressure translates into a long-term reduction in overall major cardiovascular events.'

True

False

Show answer & explanation

Answer: True

Answer

The statement is True. The argument requires the unstated assumption that short-term blood pressure lowering correlates with long-term reductions in major cardiovascular events.

The evaluated statement correctly identifies a necessary assumption. The argument connects a short-term surrogate marker (twelve-week blood pressure drop) to a broader clinical outcome (reduction in major cardiovascular events). By negating the statement—positing that short-term blood pressure drops do not lead to fewer long-term cardiovascular events—the conclusion loses all logical support. Therefore, the assumption is necessary.

Step-by-Step Solution

1

Deconstruct the argument structure into premise and conclusion.

Premise: Patients taking the new drug experienced statistically significant blood pressure reductions over a 12-week trial compared to standard treatment. Conclusion: Replacing the standard treatment with the new drug will reduce overall rates of major cardiovascular events.

Identifying the gap between the evidence provided (12-week surrogate marker) and the final claim (long-term clinical outcome) is essential for locating unstated assumptions.

2

Identify the logical leap between premise and conclusion.

The argument assumes that achieving a short-term reduction in a biological metric (blood pressure) necessarily leads to a reduction in long-term health outcomes (major cardiovascular events).

An unstated assumption must bridge the gap between what is known from the premise and what is claimed in the conclusion.

3

Apply the Negation Test to the statement.

Negated statement: Short-term blood pressure reductions do NOT translate into long-term reductions in major cardiovascular events. If this negated claim is true, the 12-week study results provide no basis for concluding that major cardiovascular events will decrease.

If the negated assumption destroys the validity of the argument, the original assumption is strictly necessary.

Key Concept

Negation Test for Unstated Necessary Assumptions

Question 1448Question →

Passage:
A two-year study across thirty regional hospitals evaluated the implementation of an automated AI-driven triage system for emergency room admissions. Hospitals that adopted the system experienced a 25% reduction in average emergency room wait times compared to hospitals that retained manual triage protocols. Based on this correlation, health administrators concluded that the automated AI triage system was directly responsible for streamlining patient processing and reducing wait times.

Statement:
Evaluating whether the hospitals that adopted the AI triage system also increased their total emergency department nursing and administrative staffing during the study period addresses a potential confounding variable in the argument.

Is the statement above True or False?

True

False

Show answer & explanation

Answer: True

Answer

True

The statement is True. In Critical Reasoning, a confounding variable is an alternate factor that systematically differs between comparison groups and could account for the observed outcome. If hospitals that implemented the AI triage system also hired additional nurses and staff, that staffing increase serves as a plausible alternative explanation for the 25% reduction in wait times. Therefore, evaluating whether staffing levels changed directly tests for a confounding variable.

Step-by-Step Solution

1

Analyze the passage's core causal argument.

Premise: Hospitals adopting an AI triage system saw a 25% drop in ER wait times compared to non-adopting hospitals. Conclusion: The AI triage system directly caused the reduction in wait times.

Identifying the premise-to-conclusion structure isolates the underlying causal claim.

2

Identify potential causal flaws and confounding variables.

The argument assumes that the introduction of the AI system was the only relevant operational difference between the adopting and non-adopting hospitals during the study period.

In observational studies, a confounding variable (a third variable that correlates with the cause and influences the outcome) offers an alternative explanation for the observed effect.

3

Evaluate the statement's impact on the argument.

Determining whether AI-adopting hospitals simultaneously expanded staffing tests for an unmeasured third variable (increased human staffing). If staffing increased in those hospitals, it provides a competing cause for the reduced wait times.

Investigating potential co-occurring factors directly evaluates the presence of a confounding variable, making the statement True.

Key Concept

Causal Flaws and Confounding Variables

Question 1449Question →

A major telecommunications company recently replaced traditional underground fiber-optic cables with low-Earth-orbit satellite connections for all its remote enterprise clients. Over the following year, customer satisfaction scores regarding connection stability among these remote clients rose by 40 percent. The company's executives concluded that satellite technology is inherently more reliable than underground physical infrastructure for serving remote geographic locations. Which of the following would it be most useful to establish in order to evaluate the executives' conclusion?

Whether the company simultaneously replaced the client-side network routers for all remote enterprise accounts at the time of the transition

Whether satellite transmission technology is generally less expensive to install and maintain per client than traditional underground fiber-optic cabling

Whether the higher customer satisfaction scores led the telecommunications company to expand its satellite service offerings to urban markets

Whether remote enterprise clients plan to expand their business operations into additional geographic regions in the next fiscal year

Whether satellite coverage extended to new remote areas where residents previously had no access to any telecommunications infrastructure

Show answer & explanation

Answer: Whether the company simultaneously replaced the client-side network routers for all remote enterprise accounts at the time of the transition

Answer

The argument is best evaluated by establishing whether the company simultaneously replaced the client-side network routers for all remote enterprise accounts at the time of the transition.

The correct answer introduces a potential confounding factor: simultaneous client-side router replacement. If new routers were introduced at the same time as the satellite system, the observed stability increase could be attributed to the upgraded hardware rather than the satellite technology itself. Conversely, if no hardware was replaced, the argument's causal claim is significantly strengthened.

Step-by-Step Solution

1

Identify the premises and conclusion of the causal argument.

Premise: Fiber-optic cables were replaced with satellite connections, and stability satisfaction rose by 40%. Conclusion: Satellite technology caused the increase in connection reliability.

Understanding the precise cause-and-effect claim is necessary before evaluating alternative explanations.

2

Formulate key evaluation criteria for causal reasoning.

To test whether satellite tech caused the improvement, check if another simultaneous change (a confounding variable) could explain the positive outcome instead.

Causal claims based on observed before-and-after correlations are vulnerable to third-variable confounders.

3

Assess the answer choices for a factor that confirms or rules out an alternative cause.

Knowing whether client-side routers were replaced at the same time reveals whether the stability gain was caused by satellite tech or by superior router hardware.

If hardware was updated concurrently, the satellite connection itself may not be the cause of the performance improvement.

Key Concept

Evaluating Causal Arguments and Alternative Explanations

Estimated Time:1m 30s

Question 1450Question →

If $u$ and $v$ are real numbers such that $u \neq v$ , what is the value of $\frac{u + v}{u - v}$ ?

(1) $u^2 - v^2 = 4(u - v)^2$
(2) $u^2 + v^2 = 5uv$

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.

Statement (1) allows factoring the difference of squares into

(u - v)(u + v) = 4(u - v)^2

. Because

u \neq v

, we can divide by

(u - v)

to determine directly that

\frac{u + v}{u - v} = 4

, which provides a unique value. Statement (2) gives

\left(\frac{u + v}{u - v}\right)^2 = \frac{7}{3}

, which yields two possible values (

\pm\sqrt{\frac{7}{3}}

) and is therefore not sufficient. Thus, Statement (1) alone is sufficient, but Statement (2) alone is not.

Step-by-Step Solution

1

Rephrase the target question and examine constraints.

The target expression is

\frac{u + v}{u - v}

. The stem specifies

u \neq v

, so

u - v \neq 0

, making the fraction well-defined.

Establishing non-zero denominator conditions allows division by

(u - v)

when evaluating statements.

2

Evaluate Statement (1) independently.

Rewrite the left side of

u^2 - v^2 = 4(u - v)^2

as

(u - v)(u + v) = 4(u - v)^2

. Since

u - v \neq 0

, divide both sides by

(u - v)

to get

u + v = 4(u - v)

. Divide both sides by

(u - v)

again to obtain

\frac{u + v}{u - v} = 4

.

This yields a single, definitive numerical value for the target expression. Statement (1) is sufficient.

3

Evaluate Statement (2) independently.

Expand

(u + v)^2 = u^2 + v^2 + 2uv

and

(u - v)^2 = u^2 + v^2 - 2uv

. Substituting

u^2 + v^2 = 5uv

gives

(u + v)^2 = 7uv

and

(u - v)^2 = 3uv

. Taking the ratio gives

\left(\frac{u + v}{u - v}\right)^2 = \frac{7uv}{3uv} = \frac{7}{3}

(since

uv \neq 0

). Taking the square root yields

\frac{u + v}{u - v} = \sqrt{\frac{7}{3}}

or

-\sqrt{\frac{7}{3}}

.

Because two distinct numerical values are possible, Statement (2) alone does not yield a unique answer. Statement (2) is not sufficient.

Key Concept

Factoring difference of squares and avoiding degree miscounts in quadratic ratios

Question 1451Question →

If $k$ is a real number such that $k \neq -1$ , is $\frac{|k - 3|}{k + 1} < 1$ ?

(1) $|k| > 1$
(2) $k^2 - 2k - 3 > 0$

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: EACH statement ALONE is sufficient.

Answer

EACH statement ALONE is sufficient.

Rephrasing the question stem shows that

\frac{|k - 3|}{k + 1} < 1

holds whenever

k < -1

or

k > 1

. Statement (1) gives

k < -1

or

k > 1

, yielding a definitive YES. Statement (2) gives

k < -1

or

k > 3

, which is a subset of

k < -1

or

k > 1

, also yielding a definitive YES. Thus, each statement alone is sufficient.

Step-by-Step Solution

1

Rephrase the question stem target inequality

The target inequality

\frac{|k - 3|}{k + 1} < 1

is satisfied if and only if

k < -1

or

k > 1

.

Analyze by cases depending on the sign of the denominator

k + 1

.
- Case 1:

k + 1 < 0 \implies k < -1

. The denominator is negative and the numerator

|k - 3| \ge 0

, so

\frac{|k - 3|}{k + 1} \le 0 < 1

is always true for all

k < -1

.
- Case 2:

k + 1 > 0 \implies k > -1

. Multiplying by

k + 1 > 0

yields

|k - 3| < k + 1

, which expands to

-(k + 1) < k - 3 < k + 1

. The left boundary

-k - 1 < k - 3

reduces to

2k > 2 \implies k > 1

. The right boundary

k - 3 < k + 1 \implies -3 < 1

is universally true. Thus for

k > -1

, the condition holds when

k > 1

.
Combining both cases, the inequality holds whenever

k < -1

or

k > 1

.

2

Evaluate Statement (1)

Statement (1) alone is SUFFICIENT.

Statement (1) states

|k| > 1

, which unwraps to

k < -1

or

k > 1

. This matches the target condition exactly. For every value of

k

satisfying this statement, the answer to the stem question is a definitive YES.

3

Evaluate Statement (2)

Statement (2) alone is SUFFICIENT.

Statement (2) states

k^2 - 2k - 3 > 0

, which factors as

(k - 3)(k + 1) > 0

. Solving gives

k < -1

or

k > 3

.
- If

k < -1

, it falls into the left target region

k < -1

.
- If

k > 3

, it falls into the right target region

k > 1

.
In all cases allowed by Statement (2), the answer to the stem question is a definitive YES.

Key Concept

Inequalities with absolute values and variables in the denominator require case analysis based on the denominator's sign.

Question 1452Question →

Over the past three years, the transit authority of a major metropolitan area instituted a policy requiring daily mechanical inspections of its bus fleet, whereas previously inspections occurred on a weekly basis. During this same three-year period, the average number of unscheduled service interruptions per bus decreased by 40 percent. The head of the transit authority concluded that conducting daily mechanical inspections was the primary cause of the reduction in unscheduled service interruptions and recommended that all municipal transit agencies adopt daily inspection schedules. Which of the following highlights the primary vulnerability in the reasoning of the head of the transit authority?

It fails to consider that another concurrent development, such as the gradual replacement of older buses with newer models, could account for the decrease in unscheduled service interruptions.

It mistakes an effect for a cause by assuming that the reduction in unscheduled service interruptions led transit officials to institute daily mechanical inspections.

It presumes without entitlement that passenger satisfaction ratings are exclusively determined by the frequency of unscheduled service interruptions.

It rejects the possibility that weekly mechanical inspections could ever be executed without incurring prohibitive operational expenditure.

It treats a condition that is necessary for reducing unscheduled service interruptions as though it were sufficient to ensure that no breakdowns ever occur.

Show answer & explanation

Answer: It fails to consider that another concurrent development, such as the gradual replacement of older buses with newer models, could account for the decrease in unscheduled service interruptions.

Answer

The argument fails to consider that another concurrent development, such as the gradual replacement of older buses with newer models, could account for the decrease in unscheduled service interruptions.

The correct answer identifies the fundamental flaw in the argument's causal reasoning. The author observes that a policy change (daily inspections) coincided over a three-year window with a 40 percent reduction in unscheduled interruptions and concludes that the policy change caused the reduction. However, over a extended period of three years, other relevant factors—such as replacing old, failure-prone buses with new ones—could easily account for the improvement. By failing to control for or rule out such confounding variables, the author's causal conclusion remains unsubstantiated.

Step-by-Step Solution

1

Deconstruct the argument structure into premises and conclusion

Premise: Daily inspections replaced weekly inspections over three years. Premise: Unscheduled service interruptions fell by 40% during those three years. Conclusion: Daily inspections were the primary cause of the reduction.

Identifying the author's precise premise-to-conclusion link is necessary to isolate the logical leap.

2

Evaluate the causal jump in the author's reasoning

The argument observes a correlation over a three-year period between a policy change (daily inspections) and an outcome (fewer interruptions) and claims a direct cause-and-effect relationship.

Correlation between two events occurring in the same time frame does not automatically establish direct causation.

3

Identify potential logical vulnerabilities and alternative explanations

Over a multi-year period, confounding variables—such as fleet modernization, route changes, or driver retraining—could have caused the drop in breakdowns independent of inspection frequency.

A valid causal argument must rule out obvious confounding factors that could produce the observed effect.

Key Concept

Causal Fallacy / Confounding Variables

Question 1453Question →

To reduce municipal water treatment expenses, Metro City's utility department plans to install bio-filters containing specialized bacterial strains in the main reservoir. Laboratory tests demonstrated that these bacteria rapidly break down microplastics, which currently clog standard filtration membranes at downstream treatment facilities. Department officials conclude that implementing these bio-filters will significantly decrease overall water treatment operating costs across the city. Which of the following, if true, most seriously weakens the utility department's conclusion?

The bacterial strains release acidic organic compounds during microplastic digestion that severely corrode plant intake pumps, creating new repair costs that exceed membrane maintenance savings.

Standard filtration membrane replacement currently accounts for the single largest line-item maintenance expenditure across all downstream facilities.

Several neighboring municipalities have recently expressed interest in licensing bio-filter technology for their own industrial wastewater treatment units.

Downstream purification plants currently report higher overall water filtration efficiency during cold winter months than during summer months.

Microplastics are not the only type of particulate pollutant present in Metro City's central reservoir that requires filtration.

Show answer & explanation

Answer: The bacterial strains release acidic organic compounds during microplastic digestion that severely corrode plant intake pumps, creating new repair costs that exceed membrane maintenance savings.

Answer

The statement regarding the release of acidic organic compounds during microplastic digestion that corrode intake pumps and create costs exceeding membrane savings most seriously weakens the argument.

The correct answer weakens the argument by demonstrating an unintended negative consequence of the proposed solution. If the bacterial digestion process releases corrosive byproducts that damage intake pumps and incur replacement costs higher than the savings from reduced membrane clogging, the net financial impact will be negative, directly refuting the conclusion that overall expenses will decrease.

Step-by-Step Solution

1

Identify the premise and conclusion of the argument.

Premise: Lab tests show specialized bacteria digest microplastics that clog downstream membranes. Conclusion: Deploying these bio-filters will significantly decrease overall city water treatment operating costs.

Understanding the logic chain reveals the underlying assumption that microplastic removal will yield net financial savings.

2

Identify the unstated assumption in the plan.

The argument assumes that deploying the bacterial bio-filters will not introduce new hidden operational expenses or side effects that outweigh the cost savings from clogging reduction.

A plan-based argument is vulnerable to evidence showing unexpected negative side effects or implementation barriers.

3

Evaluate the choices to find new information that shatters this assumption.

The choice describing corrosive acidic compounds proves that implementing the bio-filters introduces a new repair expense that exceeds the savings gained from reduced membrane clogging.

This directly invalidates the conclusion that overall water treatment operating costs will decrease.

Key Concept

Weakening a Plan-to-Goal Argument via Implementation Side Effects

Question 1454Question →

An investor purchased a total of $N$ shares split between Stock $X$ and Stock $Y$ . What percentage of the total number of shares purchased were shares of Stock $X$ ?

(1) The number of shares of Stock $X$ purchased was $40\%$ of the number of shares of Stock $Y$ purchased.
(2) The investor purchased $120$ shares of Stock $Y$ .

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.

Statement (1) establishes a direct proportional relationship between the number of shares of Stock X and Stock Y (

X = 0.40Y

). Substituting this into the part-to-whole expression

\frac{X}{X+Y}

yields

\frac{0.40Y}{1.40Y} = \frac{2}{7}

, which provides a unique percentage without requiring the actual number of shares. Statement (2) provides only the absolute quantity of Stock Y (

120

), leaving the quantity of Stock X completely unknown and preventing the calculation of the percentage. Therefore, Statement (1) alone is sufficient, but Statement (2) alone is not sufficient.

Step-by-Step Solution

1

Rephrase the target question algebraically.

The target is to find

\frac{X}{X + Y} \times 100\%

.

Determining the ratio

\frac{X}{Y}

is sufficient to find

\frac{X}{X + Y}

.

2

Evaluate Statement (1).

Statement (1) gives

X = 0.40Y

, which means

\frac{X}{Y} = \frac{2}{5}

. Therefore,

\frac{X}{X + Y} = \frac{0.40Y}{1.40Y} = \frac{2}{7}

, giving a unique percentage of

\frac{200}{7}\%

.

Since a single unique value is obtained, Statement (1) is sufficient.

3

Evaluate Statement (2).

Statement (2) gives

Y = 120

, but gives no information regarding

X

.

Without knowing

X

, the total shares

X + Y

and the fraction

\frac{X}{X + Y}

cannot be determined. Thus, Statement (2) is insufficient.

Key Concept

Part-to-whole percentages can be determined solely from part-to-part ratios without knowing absolute quantities.

Estimated Time:45s

Question 1455Question →

If $x$ and $y$ are positive real numbers, is $x$ an integer?

(1) $x + y$ is an integer and $xy = 12$ .
(2) $x - y$ is an integer and $x^2 + y^2 = 25$ .

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 determine that

x

must be an integer, but neither statement alone is sufficient.

Evaluating both statements together establishes that

(x+y)^2 = x^2 + y^2 + 2xy = 25 + 2(12) = 49

, meaning

x+y=7

. The system

x+y=7

and

xy=12

yields solutions

x=3, y=4

or

x=4, y=3

. In both outcomes,

x

is guaranteed to be an integer, yielding a definitive 'Yes' answer.

Step-by-Step Solution

1

Evaluate Statement (1) independently.

Statement (1) gives

x + y = k

for some integer

k

, and

xy = 12

. If

k = 7

, the quadratic equation

t^2 - 7t + 12 = 0

gives roots

t = 3

and

t = 4

, which are integers. However, if

k = 9

, the quadratic equation

t^2 - 9t + 12 = 0

gives roots

t = \frac{9 \pm \sqrt{33}}{2}

. Here

x = \frac{9 + \sqrt{33}}{2}

is a positive real number,

y = \frac{9 - \sqrt{33}}{2} > 0

,

x+y = 9

(integer), and

xy = 12

, but

x

is NOT an integer. Thus,

x

can be an integer or a non-integer.

Demonstrate that positive real numbers whose sum and product are integers do not necessarily have to be integers themselves.

2

Evaluate Statement (2) independently.

Statement (2) gives

x - y = m

for some integer

m

, and

x^2 + y^2 = 25

. If

m = 1

, we can have

x = 4

and

y = 3

, so

x

is an integer. If

m = 2

, then

(x-y)^2 = x^2 - 2xy + y^2 \implies 4 = 25 - 2xy \implies 2xy = 21

. Then

(x+y)^2 = x^2 + y^2 + 2xy = 25 + 21 = 46 \implies x+y = \sqrt{46}

. Thus

x = \frac{\sqrt{46} + 2}{2}

, which is positive and real, but NOT an integer. Thus, Statement (2) alone is not sufficient.

Test whether integer difference and fixed sum of squares guarantees integer values.

3

Evaluate Statements (1) and (2) together.

From Statement (1),

xy = 12

and

x + y = k

(where

k

is an integer). From Statement (2),

x^2 + y^2 = 25

. Using the algebraic identity

x^2 + y^2 = (x+y)^2 - 2xy

, we substitute the known values:

25 = k^2 - 2(12) \implies 25 = k^2 - 24 \implies k^2 = 49

. Since

x > 0

and

y > 0

,

x + y = k > 0

, so

k = 7

. Now we have

x + y = 7

and

xy = 12

. The possible values for

x

and

y

are the solutions to

t^2 - 7t + 12 = 0 \implies (t-3)(t-4) = 0 \implies t \in \{3, 4\}

. Therefore,

x

must be either 3 or 4. In both cases,

x

is definitively an integer.

Combine the conditions to solve for

x+y

and prove

x

must be an integer.

Key Concept

Integer constraints vs. real number assumptions in Data Sufficiency algebraic systems

Question 1456Question →

For all positive real numbers $p$ and $q$ , the Data Sufficiency Yes/No target question "Is $\frac{p^2 + q^2}{2pq} > 1$ ?" is algebraically equivalent to the simplified target question "Is $p \neq q$ ?"

True

False

Show answer & explanation

Answer: True

Answer

True

The statement is true because clearing the positive denominator

2pq

and rearranging terms converts the target inequality

\frac{p^2 + q^2}{2pq} > 1

into

(p - q)^2 > 0

. A real squared quantity is strictly positive if and only if its base is non-zero, making the target equivalent to "Is

p \neq q

?"

Step-by-Step Solution

1

Multiply both sides of the target inequality

\frac{p^2 + q^2}{2pq} > 1

by

2pq

.

p^2 + q^2 > 2pq

, with the direction of the inequality preserved since

p > 0

and

q > 0

implies

2pq > 0

.

Clearing positive denominators simplifies fractional inequalities without changing the inequality sign.

2

Subtract

2pq

from both sides of the inequality.

p^2 - 2pq + q^2 > 0

.

Grouping all non-zero terms on one side sets up a recognizable quadratic form.

3

Factor the quadratic expression as a perfect square.

(p - q)^2 > 0

.

Recognizing

p^2 - 2pq + q^2

as the expansion of

(p - q)^2

isolates the core variable relationship.

4

Analyze the conditions required for

(p - q)^2 > 0

to hold for real numbers.

The square of any real number is always non-negative, and it is strictly greater than zero if and only if the base is non-zero (

p - q \neq 0

, which means

p \neq q

).

Simplifying the logical condition reveals the exact target question needed for Data Sufficiency evaluation.

Key Concept

Target Rephrasing via Perfect Square Inequalities

Question 1457Question →

Tab 1: Engineering Telemetry Summary
During Quarter 3, the facility's grid-scale battery energy storage system logged a roundtrip charge-discharge efficiency of $94\%$ . Telemetry sensors recorded energy input and output strictly across the battery cell terminals during active storage cycles.

Tab 2: Financial Utility Billing Report
Energy billing records for Quarter 3 indicate that total kilowatt-hours ( $\text{kWh}$ ) purchased from the regional utility grid exceeded baseline operating projections by $25\%$ . Financial auditors concluded that net facility energy losses were far higher than the $6\%$ loss implied by battery telemetry.

Tab 3: Operations Maintenance Log
All auxiliary HVAC cooling units and thermal control systems operate on dedicated grid sub-circuits that bypass battery cell meters. Operational policy mandates continuous thermal conditioning whenever ambient site temperatures exceed $30^\circ\text{C}$ . During Quarter 3, site temperatures exceeded $30^\circ\text{C}$ for $65\%$ of operational hours.

Based on the information provided across the three tabs, which of the following statements, if true, help resolve the apparent discrepancy between the $94\%$ battery efficiency reported in Tab 1 and the $25\%$ excess energy consumption reported in Tab 2? Select all that apply.

Select all that apply

The battery telemetry in Tab 1 records power passing directly through cell terminals during active cycles, omitting energy consumed by auxiliary cooling units running on separate grid sub-circuits.

Prolonged ambient temperatures above

30^\circ\text{C}

in Quarter 3 triggered continuous operation of auxiliary cooling units, drawing substantial unmetered grid electricity.

Regional electricity supply tariffs increased by

25\%

per kilowatt-hour during peak demand hours throughout Quarter 3.

The internal chemical energy storage capacity of the battery cells degraded severely during Quarter 3 due to high operating temperatures.

Battery charging cycles took place predominantly during off-peak night hours when overall regional grid demand was low.

Show answer & explanation

Answer: The battery telemetry in Tab 1 records power passing directly through cell terminals during active cycles, omitting energy consumed by auxiliary cooling units running on separate grid sub-circuits.; Prolonged ambient temperatures above $30^\circ\text{C}$ in Quarter 3 triggered continuous operation of auxiliary cooling units, drawing substantial unmetered grid electricity.

Answer

The statements resolving the discrepancy are: (1) battery telemetry measuring only terminal power while omitting auxiliary cooling power on separate sub-circuits, and (2) high ambient temperatures causing continuous operation of auxiliary cooling units that drew unmetered grid electricity.

The correct options work together to resolve the conflict by clarifying measurement scope. The statement noting that telemetry tracks only cell terminals explains that auxiliary HVAC power was excluded from Tab 1's efficiency calculation. The statement regarding high ambient temperatures triggering continuous auxiliary cooling provides the empirical cause for why Tab 2's grid purchases were 25% above projections.

Step-by-Step Solution

1

Identify the conflict between sources

Tab 1 reports high battery roundtrip efficiency (94%, or 6% loss), while Tab 2 shows facility grid energy draw exceeding projections by 25% (implying large net energy losses).

Resolving a discrepancy requires finding factors that make both reported figures factually true within their respective measurement boundaries.

2

Cross-reference operational boundaries and exceptions across tabs

Tab 1 specifies that telemetry measures energy strictly across battery cell terminals. Tab 3 notes that auxiliary HVAC systems run on dedicated grid sub-circuits bypassing cell meters and operate continuously when ambient temperature exceeds 30°C (which occurred 65% of the time in Q3).

Connecting the unmetered auxiliary HVAC draw to high Q3 temperatures explains why utility grid energy purchases (Tab 2) were 25% higher even though the battery cells themselves operated at 94% efficiency (Tab 1).

3

Evaluate candidate statements for valid reconciliation

The statement highlighting separate sub-circuit metering boundaries and the statement identifying high temperature HVAC activation together explain the missing energy draw without contradicting Tab 1.

Both valid hypotheses work together to explain why total grid draw exceeded battery-only telemetry projections.

Key Concept

Discrepancy and Conflict Resolution Between Sources

Question 1458Question →

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

(1) $x^2 + 2xy + y^2 - 5x - 5y + 6 = 0$
(2) $xy = 2$

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

Answer

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

The correct response identifies that neither statement alone establishes a single value for

x + y

, but combining them eliminates

x + y = 2

because no real numbers

x

and

y

have a sum of

2

and a product of

2

(discriminant

2^2 - 4(2) = -4 < 0

). Only

x + y = 3

yields real values for

x

and

y

(discriminant

3^2 - 4(2) = 1 \ge 0

). Thus, both statements together are sufficient to determine that

x + y = 3

.

Step-by-Step Solution

1

Analyze Statement (1) by grouping terms in terms of

(x + y)

.

Rewrite

x^2 + 2xy + y^2 - 5(x + y) + 6 = 0

as

(x + y)^2 - 5(x + y) + 6 = 0

, which factors into

((x + y) - 2)((x + y) - 3) = 0

. Hence,

x + y = 2

or

x + y = 3

. Since there are two possible values, Statement (1) alone is NOT sufficient.

Determining whether Statement (1) provides a unique value for the target expression.

2

Analyze Statement (2) independently.

Statement (2) gives

xy = 2

. Multiple real pairs satisfy this (e.g.,

(2, 1) \Rightarrow x + y = 3

or

(1, 2) \Rightarrow x + y = 3

vs.

(\sqrt{2}, \sqrt{2}) \Rightarrow x + y = 2\sqrt{2}

). Thus, Statement (2) alone is NOT sufficient.

Testing whether

xy = 2

alone restricts

x + y

to a single numerical value.

3

Combine Statement (1) and Statement (2) under the real number constraint.

Let

S = x + y

and

P = xy = 2

. For

x

and

y

to be real numbers, they must be the real roots of

t^2 - St + P = 0

, requiring discriminant

\Delta = S^2 - 4P \ge 0

.
- Case 1: If

S = 2

, then

\Delta = 2^2 - 4(2) = -4 < 0

. No real solutions exist for

x

and

y

.
- Case 2: If

S = 3

, then

\Delta = 3^2 - 4(2) = 1 \ge 0

. Real solutions exist (e.g.,

x = 2, y = 1

).

Applying the real-number constraint to eliminate algebraically impossible cases.

4

Conclude total sufficiency.

Only

x + y = 3

produces real values for

x

and

y

. Therefore, both statements together uniquely determine

x + y = 3

.

Completing the Data Sufficiency evaluation.

Key Concept

Quadratic expression rephrasing and discriminant constraints for real variable existence in Data Sufficiency

Estimated Time:2m 0s

Question 1459Question →

A university research institute evaluated $N$ technology projects completed last year. Each project received grant funding from at least one of two foundations: Foundation $X$ or Foundation $Y$ . Exactly $30\%$ of the $N$ projects received funding from both foundations. If the average (arithmetic mean) grant amount per project among projects funded by Foundation $X$ was $\$50,000$ and the average grant amount per project among projects funded by Foundation $Y$ was $\$60,000$ , what was the average total grant funding per project across all $N$ projects?

(1) The number of projects that received funding from Foundation $X$ only was equal to the number of projects that received funding from Foundation $Y$ only.
(2) The total dollar amount disbursed by Foundation $Y$ exceeded the total dollar amount disbursed by Foundation $X$ by $\$1,800,000$ .

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 to answer the question, but statement (2) alone is not sufficient.

Statement (1) alone is sufficient because equating the counts of projects funded exclusively by Foundation X and exclusively by Foundation Y implies that the total number of projects funded by Foundation X equals the total number funded by Foundation Y. Combined with the principle of overlapping sets where thirty percent of projects received funding from both, this uniquely determines that sixty-five percent of projects were funded by Foundation X and sixty-five percent were funded by Foundation Y, yielding a unique overall average grant of $71,500. Statement (2) alone is insufficient because the equation relating total disbursements depends on the unknown total number of projects N.

Step-by-Step Solution

1

Express the total funding and set proportions in terms of the total number of projects

N

.

Let

n_X

be the number of projects funded by Foundation

X

and

n_Y

be the number of projects funded by Foundation

Y

. The number of projects funded by both is

0.30N

. Since every project is funded by at least one foundation,

N = n_X + n_Y - 0.30N \implies n_X + n_Y = 1.30N

. Dividing by

N

gives the ratio sum

\frac{n_X}{N} + \frac{n_Y}{N} = 1.30

.

Establishing the relationship between the subset counts and total projects simplifies the target average formula.

2

Formulate the expression for the average total grant funding per project.

Total Funding

= 50,000 n_X + 60,000 n_Y

. Therefore, the average funding per project is

\text{Average} = \frac{50,000 n_X + 60,000 n_Y}{N} = 50,000\left(\frac{n_X}{N}\right) + 60,000\left(\frac{n_Y}{N}\right)

.

Since total grant funding across all projects is the sum of all money disbursed by Foundation

X

and Foundation

Y

, the overall average depends strictly on the ratios

\frac{n_X}{N}

and

\frac{n_Y}{N}

.

3

Evaluate Statement (1) independently.

Statement (1) states that the number of projects funded by

X

only equals the number funded by

Y

only:

n_X - 0.30N = n_Y - 0.30N \implies n_X = n_Y

. Since

\frac{n_X}{N} + \frac{n_Y}{N} = 1.30

and

n_X = n_Y

, we get

2\left(\frac{n_X}{N}\right) = 1.30 \implies \frac{n_X}{N} = 0.65

and

\frac{n_Y}{N} = 0.65

. Substituting these into the average formula yields

\text{Average} = 50,000(0.65) + 60,000(0.65) = 71,500

. Statement (1) ALONE is sufficient.

Knowing that the two single-foundation set sizes are equal determines the exact proportions of

N

funded by each foundation.

4

Evaluate Statement (2) independently.

Statement (2) states that

60,000 n_Y - 50,000 n_X = 1,800,000

. Dividing by

N

gives

60,000\left(\frac{n_Y}{N}\right) - 50,000\left(\frac{n_X}{N}\right) = \frac{1,800,000}{N}

. Because

N

is unknown, the right-hand side is not fixed, so

\frac{n_X}{N}

and

\frac{n_Y}{N}

cannot be uniquely determined. Statement (2) ALONE is not sufficient.

An absolute dollar equation introduces a dependency on the total count

N

, preventing a unique calculation of the relative proportions.

Key Concept

Overlapping Sets and Weighted Averages in Data Sufficiency

Estimated Time:2m 0s

Question 1460Question →

Ecologists studying endangered forest elephants in a central African preserve recently introduced an automated acoustic monitoring network to track herd movements via low-frequency infrasound calls. The researchers hypothesized that installing these acoustic sensors at key waterholes would allow them to accurately estimate total elephant population trends across the preserve over time. However, skeptics argued that because elephants frequently travel between waterholes, acoustic detections at waterholes would lead to double-counting individual animals, thereby rendering population trend estimates unreliable.

Which of the following, if true, most strengthens the ecologists' argument?

Each elephant's low-frequency call possesses a unique vocal signature that allows the automated software to identify and distinguish individual animals across different sensor locations.

Dense forest canopy weakens low-frequency sound waves during the rainy season, reducing the effective detection radius of the acoustic sensors.

Forest elephants consume significantly greater volumes of water per day during dry seasons than savanna elephants do.

Acoustic monitoring networks are less expensive to maintain over long periods than satellite tracking collars.

Automated acoustic networks have been successfully deployed to track bird migration patterns in temperate coastal ecosystems.

Show answer & explanation

Answer: Each elephant's low-frequency call possesses a unique vocal signature that allows the automated software to identify and distinguish individual animals across different sensor locations.

Answer

The statement showing that each elephant possesses a unique vocal signature allowing the software to distinguish individual animals across locations provides the strongest support.

The core objection raised against the ecologists' hypothesis is that elephants traveling between waterholes will be counted multiple times, distorting the overall population estimates. The statement providing that each elephant has a unique vocal signature that automated software uses to track individuals directly resolves this issue. By demonstrating that double-counting can be avoided, the finding defends the validity of the data and strongly supports the conclusion that population trends can be estimated accurately.

Step-by-Step Solution

1

Identify the conclusion and premises of the argument.

Conclusion: Acoustic sensors at waterholes will accurately estimate overall elephant population trends over time. Vulnerability/Objection: Elephants visit multiple waterholes, causing double-counting.

To strengthen an argument facing a specific counterargument, we must find evidence that overcomes or neutralizes that specific weakness.

2

Evaluate the impact of demonstrating individual vocal signatures.

If the software identifies unique vocal signatures, moving between waterholes will not lead to double-counting because the system recognizes repeat individuals.

Eliminating the potential cause of data distortion directly protects the premise-to-conclusion logical path.

Key Concept

Strengthening an Argument by Neutralizing a Key Objection / Ruling Out Data Distortion

Answer

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Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept