All practice questions

Question 1421Question →

In Metropolitan Area M, municipal agricultural policies dictate that every commercial urban farm utilizing hydroponic vertical farming methods receives a Tier-1 municipal water rate discount. Furthermore, any agricultural initiative in Metropolitan Area M receiving a Tier-1 municipal water rate discount is subject to an annual environmental audit conducted by the Department of Ecology. Finally, no agricultural initiative in Metropolitan Area M that is subject to an annual environmental audit conducted by the Department of Ecology is permitted to use synthetic chemical pesticides. During the last fiscal year, exactly 40 percent of all urban farms operating in Metropolitan Area M used synthetic chemical pesticides.

If the statements above are true, which of the following must also be true on the basis of them?

No urban farm in Metropolitan Area M that refrained from using synthetic chemical pesticides during the last fiscal year was subject to an annual environmental audit.

Every urban farm in Metropolitan Area M that was subject to an annual environmental audit during the last fiscal year utilized hydroponic vertical farming methods.

At least 40 percent of the urban farms operating in Metropolitan Area M during the last fiscal year did not receive a Tier-1 municipal water rate discount.

During the last fiscal year, more than half of all urban farms in Metropolitan Area M received a Tier-1 municipal water rate discount.

At least 40 percent of the urban farms operating in Metropolitan Area M during the last fiscal year were subject to an annual environmental audit.

Show answer & explanation

Answer: At least 40 percent of the urban farms operating in Metropolitan Area M during the last fiscal year did not receive a Tier-1 municipal water rate discount.

Answer

At least 40 percent of the urban farms operating in Metropolitan Area M during the last fiscal year did not receive a Tier-1 municipal water rate discount.

The stimulus establishes a conditional chain: Hydroponic vertical farming implies a Tier-1 water discount, which implies an environmental audit, which in turn implies that synthetic chemical pesticides are NOT used. Thus, any farm receiving a Tier-1 water discount cannot use synthetic chemical pesticides (Tier-1 Discount → No Synthetic Pesticides). Taking the contrapositive reveals that if a farm uses synthetic chemical pesticides, it CANNOT receive a Tier-1 water discount (Synthetic Pesticides → No Tier-1 Discount). Since exactly 40 percent of urban farms used synthetic chemical pesticides, those specific 40 percent must not have received a Tier-1 discount. Therefore, it must be true that at least 40 percent did not receive the discount.

Step-by-Step Solution

1

Formalize the conditional statements given in the premises.

Hydroponic (H) → Tier-1 Discount (T) → Environmental Audit (E) → NO Synthetic Pesticides (~P).

Establishing the chain of necessary conditions allows us to trace deductions across all premises.

2

Deduce the transitive relation between Tier-1 Discount and Synthetic Pesticides, then find its contrapositive.

Direct chain: T → ~P. Contrapositive: P → ~T (If a farm uses synthetic pesticides, it does NOT receive a Tier-1 discount).

Taking the contrapositive provides a valid deduction connecting pesticide use directly to discount status.

3

Apply the given numerical fact to the contrapositive deduction.

Since 40% of farms used synthetic pesticides (P), those exact 40% cannot receive a Tier-1 discount (~T).

Combining the contrapositive rule with the explicit percentage proves that at least 40% of farms must not receive the discount.

Key Concept

Conditional Logic Chains and Contrapositive Inferences

Question 1422Question →

A regional agricultural board plans to encourage commercial fruit orchard operators to replace conventional synthetic pesticide sprays with a newly developed biological control agent consisting of predatory mites. Because extensive field trials demonstrated that these predatory mites effectively eliminate target pest insects without producing any chemical runoff into adjacent waterways, board officials claim that widespread adoption of the mites will result in a significant net reduction of toxic chemical contamination in the region's river ecosystem. Which of the following would be most useful to investigate in order to evaluate the validity of the agricultural board's claim?

Whether fruit orchard operators who adopt the predatory mites will need to apply secondary chemical fungicides to protect the mite population from naturally occurring fungal pathogens.

Whether neighboring agricultural regions cultivating non-fruit crops plan to introduce biological pest controls within the next five years.

Whether the purchase price of the predatory mite agent per acre is higher than the historical retail cost of conventional synthetic pesticides.

Whether conventional synthetic pesticide sprays currently contribute to atmospheric emissions in addition to water contamination.

Whether the pest insects targeted by the predatory mites can also feed on wild flora growing along the riverbanks.

Show answer & explanation

Answer: Whether fruit orchard operators who adopt the predatory mites will need to apply secondary chemical fungicides to protect the mite population from naturally occurring fungal pathogens.

Answer

Determining whether orchard operators who adopt the predatory mites must apply secondary chemical fungicides to safeguard the mites from fungal pathogens is most useful to evaluate the argument.

The correct answer provides the key information required to assess whether replacing conventional pesticides with predatory mites truly achieves a net reduction in water contamination. Under the Test of Variance, if orchard operators must spray secondary chemical fungicides to protect the mites, those fungicides could introduce new toxic runoff into the river ecosystem, thereby undermining the conclusion. Conversely, if no secondary chemical applications are required, the adoption of predatory mites will successfully reduce overall chemical runoff into the river.

Step-by-Step Solution

1

Deconstruct the argument structure into premise and conclusion.

Premise: Predatory mites eliminate pest insects without generating chemical runoff. Conclusion: Replacing synthetic pesticides with predatory mites will yield a net reduction in toxic chemical contamination in the river ecosystem.

Identifying the author's logical leap reveals unstated assumptions regarding potential secondary chemical usage.

2

Identify the unstated assumption in the causal plan.

The argument assumes that adopting predatory mites will not necessitate any new chemical interventions that could introduce toxic runoff into the river.

If alternative chemical treatments are triggered by adopting the biological agent, total chemical runoff might not decrease.

3

Apply the Test of Variance to evaluate potential answers.

Testing extreme answers to the option regarding secondary chemical fungicides: If YES (fungicides are required), chemical runoff may remain high or increase (weakens conclusion). If NO (no fungicides required), chemical runoff will decrease as predicted (strengthens conclusion).

An option whose two opposite extreme responses yield opposite impacts on the conclusion is the decisive factor required to evaluate argument validity.

Key Concept

Test of Variance for Argument Evaluation

Question 1423Question →

If $x$ and $y$ are real numbers such that $x \neq 0$ , is $\frac{|x - y|}{x} > 1$ ?

(1) $x < 0$
(2) $y < 0$

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: 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) tells us

x < 0

. Since

|x - y| \ge 0

for all real numbers, dividing a non-negative numerator by a negative denominator guarantees that

\frac{|x - y|}{x} \le 0

. Consequently,

\frac{|x - y|}{x}

can never be greater than

1

, giving a definitive 'No' answer to the question. Because Statement (1) provides a single definitive answer, it is sufficient. Statement (2) tells us

y < 0

, but leaves the sign and value of

x

unconstrained. Choosing

x = -1

with

y = -2

gives a value of

-1

(answering 'No'), while choosing

x = 1

with

y = -2

gives a value of

3

(answering 'Yes'). Since Statement (2) allows both 'Yes' and 'No' outcomes, it is not sufficient. Thus, the option stating that Statement (1) alone is sufficient while Statement (2) alone is not sufficient is correct.

Step-by-Step Solution

1

Analyze the target question stem and algebraic constraints.

We are asked whether

\frac{|x - y|}{x} > 1

. Notice that for any real numbers

x

and

y

, the absolute value in the numerator

|x - y| \ge 0

. The sign of the denominator

x

determines the sign of the entire quotient.

Understanding the algebraic behavior of non-negative numerators over signed denominators simplifies statement evaluation.

2

Evaluate Statement (1):

x < 0

.

Since

|x - y| \ge 0

and

x < 0

, the ratio

\frac{|x - y|}{x}

is a non-negative number divided by a negative number. Thus,

\frac{|x - y|}{x} \le 0

. Since a number

\le 0

can never be greater than

1

, the answer to the question is a definitive 'No'. A definitive 'No' means the statement IS sufficient.

In Data Sufficiency, any statement that allows us to answer the question with a single, unambiguous 'Yes' or 'No' is sufficient.

3

Evaluate Statement (2):

y < 0

.

Test values for

x

while holding

y < 0

(e.g.,

y = -2

). If

x = -1

, then

\frac{|-1 - (-2)|}{-1} = \frac{1}{-1} = -1 \ngtr 1

(Answer: No). If

x = 1

, then

\frac{|1 - (-2)|}{1} = \frac{3}{1} = 3 > 1

(Answer: Yes). Because both 'Yes' and 'No' are possible, Statement (2) is not sufficient.

Getting conflicting answers from permissible test cases proves a statement is insufficient.

Key Concept

Definitive Yes/No logic and sign properties of absolute value quotients in Data Sufficiency

Question 1424Question →

Consider the following argument:

A regional agricultural authority plans to mitigate groundwater depletion by offering financial subsidies to farmers who replace water-intensive flood irrigation systems with high-efficiency drip irrigation. The authority argues that because drip irrigation reduces per-acre water consumption by 30 percent, implementing this subsidy program will lead to a net reduction in total agricultural groundwater extraction across the region.

Statement: The agricultural authority's argument relies on the unstated assumption that farmers receiving the subsidies will not expand their total irrigated acreage using the water volume saved through drip irrigation.

True

False

Show answer & explanation

Answer: True

Answer

The statement is True. The argument relies on the unstated assumption that farmers receiving subsidies will not expand their total irrigated acreage using the conserved water.

The statement accurately identifies a necessary unstated assumption. The author leaps from a per-acre efficiency gain (a 30% reduction per acre) to a regional aggregate reduction (net decrease in total groundwater extraction). For this claim to hold, total irrigated acreage must not expand to absorb the conserved water. Applying the Negation Test shows that if farmers do expand their acreage using the saved water, the conclusion collapses.

Step-by-Step Solution

1

Deconstruct the core argument into premises and conclusion

Premise: Drip irrigation reduces per-acre water consumption by 30 percent.
Premise: Subsidies will incentivize farmers to switch from flood to drip irrigation.
Conclusion: The subsidy program will lead to a net reduction in total agricultural groundwater extraction across the region.

Identifying the explicit claims reveals the structural gap between per-acre efficiency and total regional consumption.

2

Identify the logical gap between the premise and the conclusion

The premise concerns rate/efficiency (water consumption per acre), whereas the conclusion asserts an outcome regarding total volume (total regional extraction).

Total volume extracted is a function of both rate per acre and total acreage irrigated (

Total = Rate \times Acres

).

3

Apply the Negation Test to evaluate the assumption

Negated statement: Farmers receiving subsidies WILL expand their total irrigated acreage using the saved water volume.
Impact: If farmers increase their irrigated acreage using the 30% saved water, aggregate regional groundwater extraction will not drop, invalidating the conclusion.

A statement is a necessary unstated assumption if its logical negation invalidates the argument's conclusion.

Key Concept

Identifying Unstated Assumptions (Per-Unit Efficiency vs. Total Aggregate Volume Gap)

Question 1425Question →

To reduce long-term infrastructure expenses, the coastal transit authority of Metropolis plans to install continuous ultrasonic vibration sensors on the underwater concrete pillars of its main commuter bridges. Engineers argue that by detecting micro-fractures in real time, maintenance crews can perform minor localized patch repairs before major structural fatigue occurs, thereby significantly lowering total repair expenditures over the next decade.

Which of the following, if true, provides the strongest support for the engineers' argument?

The cost to purchase, install, and maintain the ultrasonic sensors is substantially lower than the net financial savings produced by preventing even a single major structural overhaul.

Ambient underwater noise from nearby commercial shipping channels routinely causes ultrasonic monitoring systems to log false readings, necessitating costly diver deployments to perform manual verifications.

Total vehicular traffic across the commuter bridges is expected to expand by nearly twenty percent over the coming decade due to regional population growth.

The transit authority has already successfully integrated ultrasonic monitoring technology to track track-bed shifts along its above-ground light rail network.

Most surface roadway damage on coastal bridges stems from chemical de-icing salts rather than structural issues originating in underwater support pillars.

Show answer & explanation

Answer: The cost to purchase, install, and maintain the ultrasonic sensors is substantially lower than the net financial savings produced by preventing even a single major structural overhaul.

Answer

The statement establishing that the cost of purchasing, installing, and maintaining the ultrasonic sensors is substantially lower than the net financial savings produced by preventing a major structural overhaul.

The correct answer provides the missing piece of economic logic necessary for the plan's success. The author claims that performing minor localized repairs guided by sensors will lower total repair expenditures over ten years. However, if buying and maintaining these continuous sensors costs more than the money saved by preventing major overhauls, overall expenditures would increase. Confirming that sensor acquisition and maintenance expenses are substantially lower than the savings from avoiding overhauls directly supports the conclusion.

Step-by-Step Solution

1

Analyze the core argument structure

Premise: Installing ultrasonic sensors allows detection of micro-fractures so minor patch repairs can be made early. Conclusion: This plan will significantly lower total repair expenditures over the next decade.

Identifying the premise-to-conclusion leap reveals the underlying assumption needed to reach the conclusion.

2

Identify logical gaps in the plan

The argument assumes that the financial savings from avoiding major overhauls will exceed the costs of buying, installing, and servicing the sensors.

If the technology itself is excessively expensive to run, overall expenditures could rise despite avoiding major overhauls.

3

Evaluate the choices to find the one that confirms the assumption

The option stating that sensor costs are substantially lower than savings from preventing a major overhaul directly bridges this financial gap and confirms net savings.

Providing evidence that net savings are positive strongly reinforces the conclusion that expenditures will decline.

Key Concept

Strengthening Arguments by Supplying Missing Financial Link / Cost-Benefit Viability

Estimated Time:1m 30s

Question 1426Question →

A boutique fitness studio offers only yoga and pilates classes. What is the ratio of the number of yoga classes to the number of pilates classes offered each week?

(1) Yoga classes account for 60% of the total number of classes offered each week.
(2) The total number of classes offered each week is 40.

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 answer identifies that Statement (1) alone is sufficient because knowing that yoga classes account for 60% of total classes implies pilates classes account for 40%, giving a fixed ratio of 60 to 40 (or 3 to 2). Statement (2) only provides the total number of classes without any relative distribution, making it insufficient on its own.

Step-by-Step Solution

1

Rephrase the target question

We need to find the ratio

Y : P

, where

Y

is the number of yoga classes and

P

is the number of pilates classes. Note that total classes

T = Y + P

.

Simplifying the target helps determine whether percentage data or numerical data is needed.

2

Evaluate Statement (1)

If yoga classes are 60% of the total

T

, then

Y = 0.60T

. Since the studio offers only yoga and pilates classes, pilates classes account for the remaining 40%, so

P = 0.40T

. The ratio

Y : P = \frac{0.60T}{0.40T} = \frac{60}{40} = \frac{3}{2}

.

Statement (1) yields a unique numerical ratio of

3:2

. Thus, Statement (1) alone is SUFFICIENT.

3

Evaluate Statement (2)

Statement (2) gives

T = 40

. We have

Y + P = 40

, but without knowing the proportion or count of either class type,

Y : P

can take multiple values (e.g.,

20:20 = 1:1

or

30:10 = 3:1

).

Statement (2) alone does not give a unique ratio. Thus, Statement (2) alone is INSUFFICIENT.

Key Concept

Data Sufficiency with Ratios and Percentages

Estimated Time:1m 0s

Question 1427Question →

Read the argument below regarding energy infrastructure and match each statement from the text to its corresponding logical role within the argument.

Argument:
Although utility companies contend that investing in municipal microgrids will fragment national energy markets and raise consumer rates, recent empirical data from regional pilots show that decentralized solar-plus-storage systems significantly reduce peak-demand spikes. Lowering peak-demand strain directly reduces the need for expensive fossil-fuel peaking plants. Consequently, municipal microgrid expansion actually enhances overall grid resilience and long-term cost stability. Therefore, federal energy regulators should mandate municipal microgrid integration standards immediately, despite short-term transition costs for legacy utilities.

Match each statement on the left with its precise logical role on the right.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The statement urging federal regulators to mandate integration standards matches the Main Conclusion; the statement asserting that microgrid expansion enhances resilience and stability matches the Intermediate Conclusion; the statement regarding reduction of peak-demand spikes matches the Supporting Premise; and the statement about market fragmentation and rate increases matches the Counterargument.

The call for immediate federal integration mandates represents the Main Conclusion because it is the ultimate policy recommendation that the entire argument builds toward. The claim regarding grid resilience and cost stability is an Intermediate Conclusion because it is inferred from empirical pilot data and directly provides the logical justification for the main policy mandate. The finding that solar-plus-storage systems reduce peak-demand spikes is a Supporting Premise containing raw empirical evidence. Finally, the utility companies' position regarding market fragmentation and rate increases is a Counterargument introduced as an opposing stance.

Step-by-Step Solution

1

Identify the primary thesis or ultimate recommendation of the argument.

The final sentence, introduced by the conclusion indicator 'Therefore,' states that federal regulators should mandate municipal microgrid integration standards immediately.

This prescriptive policy call is the ultimate claim that all other evidence and intermediate claims in the passage aim to prove.

2

Identify intermediate claims that link evidence to the main conclusion.

The sentence introduced by 'Consequently' claims that municipal microgrid expansion enhances overall grid resilience and long-term cost stability.

This claim is supported by the peak-demand evidence and in turn acts as the direct rationale for the final policy mandate, making it an intermediate conclusion.

3

Differentiate empirical evidence from inferred conclusions.

The assertion that decentralized solar-plus-storage systems significantly reduce peak-demand spikes is introduced as empirical data from regional pilots.

It functions as foundational factual evidence (a premise) rather than an inferred statement.

4

Classify opposing claims framed within concessions.

The utility companies' assertion that microgrids fragment markets and raise consumer rates is introduced using the concessionary framing 'Although utility companies contend...'.

This represents an opposing position (counterargument) introduced to establish the context that the author's argument seeks to overcome.

Key Concept

Structural Role Analysis and Main vs. Subsidiary Conclusions

Question 1428Question →

In 2025, an independent logistics firm evaluated all 80 regional distribution centers operated by a retail enterprise. The firm's audit reported that every distribution center that implemented an automated sorting system reduced its order processing error rate by at least 30 percent relative to 2024. Furthermore, the report noted that every distribution center that reduced its order processing error rate by at least 30 percent in 2025 was located in the Western Region.

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

Every distribution center located in the Western Region implemented an automated sorting system in 2025.

Distribution centers that did not implement an automated sorting system failed to achieve any reduction in their order processing error rates in 2025.

Every distribution center evaluated in 2025 that implemented an automated sorting system was located in the Western Region.

Automated sorting systems were the most cost-effective method for reducing order processing error rates among the retailer's distribution centers in 2025.

A majority of the retail enterprise's 80 regional distribution centers were located in the Western Region in 2025.

Show answer & explanation

Answer: Every distribution center evaluated in 2025 that implemented an automated sorting system was located in the Western Region.

Answer

Every distribution center evaluated in 2025 that implemented an automated sorting system was located in the Western Region.

The correct answer directly combines the two premises given in the stimulus via formal transitive deduction: implementing an automated sorting system guarantees an error reduction of at least 30 percent, and achieving that reduction guarantees location in the Western Region. Therefore, any facility with an automated sorting system must be located in the Western Region.

Step-by-Step Solution

1

Analyze the formal logic premises provided in the stimulus.

Premise 1: Automated Sorting System

\rightarrow

Error Reduction

\ge 30\%

. Premise 2: Error Reduction

\ge 30\% \rightarrow

Western Region Location.

Mapping the premises using conditional logic establishes the formal relationships between the variables.

2

Apply the transitive property of conditional logic to combine the premises.

Automated Sorting System

\rightarrow

Western Region Location.

Since statement A leads to statement B, and statement B leads to statement C, statement A must logically lead to statement C.

3

Evaluate the options to identify the choice that represents a strict hypothetical syllogism without introducing outside concepts.

The statement asserting that any evaluated facility implementing an automated sorting system was located in the Western Region directly expresses this deduction.

A valid inference on the GMAT Critical Reasoning section must be strictly proven by the premises without relying on unstated assumptions.

Key Concept

Hypothetical Syllogism and Avoiding Out-of-Scope Inferences

Question 1429Question →

To prevent soil degradation in arid farmland, agricultural researchers propose planting salt-tolerant halophytes alongside commercial crops. Halophytes absorb excess sodium ions from topsoil, which the researchers claim will allow commercial crops to absorb soil nitrogen more efficiently and thereby increase overall crop yields within two growing seasons.

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

Halophytes consume substantial quantities of soil nitrogen during their growth cycle, reducing the net nitrogen available in the topsoil.

Commercial crops grown in non-arid climate zones typically produce significantly higher yields per hectare than those grown in arid regions.

High concentrations of sodium ions in topsoil chemically inhibit the molecular channels that commercial crop roots use to absorb nitrogen.

Chemical soil-desalination treatments cost far more to manufacture and apply than purchasing and planting halophyte seeds.

Halophytes harvested from coastal agricultural plots can be processed into commercial animal feed and biofuel products.

Show answer & explanation

Answer: High concentrations of sodium ions in topsoil chemically inhibit the molecular channels that commercial crop roots use to absorb nitrogen.

Answer

High concentrations of sodium ions in topsoil chemically inhibit the molecular channels that commercial crop roots use to absorb nitrogen.

The correct answer supplies the underlying physiological mechanism required for the argument to hold. If topsoil sodium ions actively inhibit root channels responsible for nitrogen uptake, then reducing sodium via halophyte absorption removes this inhibition, directly enabling commercial crops to absorb nitrogen more effectively and boost yields.

Step-by-Step Solution

1

Isolate the premises and conclusion of the argument.

Premise: Halophytes absorb excess sodium ions from topsoil. Conclusion: Halophytes will enable commercial crops to absorb nitrogen more efficiently and increase overall yields.

Strengthening an argument requires identifying the logical gap between the evidence provided and the claim being made.

2

Identify the unstated assumption linking topsoil sodium to nitrogen absorption.

The argument assumes that sodium ions currently hinder nitrogen uptake, and that removing sodium will resolve this bottleneck.

Without a causal connection between sodium levels and nitrogen absorption efficiency, the conclusion remains unproven.

3

Evaluate the options for a premise that establishes this causal connection.

Confirming that sodium ions directly block root transport channels for nitrogen proves that removing sodium enables increased nitrogen absorption.

Supplying the missing biological mechanism directly validates the core premise-to-conclusion transition.

Key Concept

Strengthening Causal Arguments via Underlying Mechanism

Question 1430Question →

If $m$ is a real number, is $|m^2 - 2m - 3| < 5$ ?

(1) $|m - 2| < 3$
(2) $|m + 1| < 4$

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.

Rephrasing the question stem requires solving

|m^2 - 2m - 3| < 5

. This breaks down into

-5 < m^2 - 2m - 3 < 5

. The left inequality

m^2 - 2m + 2 > 0

holds for all real numbers because its discriminant is negative and the quadratic opens upward. The right inequality

m^2 - 2m - 8 < 0

factors as

(m - 4)(m + 2) < 0

, giving the target range

-2 < m < 4

.

Evaluating the first statement gives

|m - 2| < 3 \implies -1 < m < 5

. This interval extends past 4 (e.g.,

m = 4.5

yields a 'No', while

m = 1

yields a 'Yes'), so the first statement alone is not sufficient.

Evaluating the second statement gives

|m + 1| < 4 \implies -5 < m < 3

. This interval extends past

-2

(e.g.,

m = -4

yields a 'No', while

m = 1

yields a 'Yes'), so the second statement alone is not sufficient.

Combining both statements requires taking the intersection of

-1 < m < 5

and

-5 < m < 3

, which gives

-1 < m < 3

. Since every number in the range

(-1, 3)

lies strictly between

-2

and

4

, the target condition is guaranteed to be true. Therefore, both statements together are sufficient, but neither statement alone is sufficient.

Step-by-Step Solution

1

Rephrase the target question stem algebraically.

The target inequality

|m^2 - 2m - 3| < 5

is equivalent to

-5 < m^2 - 2m - 3 < 5

. Solving the upper bound gives

m^2 - 2m - 8 < 0 \implies (m - 4)(m + 2) < 0 \implies -2 < m < 4

. The lower bound

m^2 - 2m - 3 > -5 \implies m^2 - 2m + 2 > 0 \implies (m - 1)^2 + 1 > 0

is true for all real

m

. Thus, the question simplifies to: 'Is

-2 < m < 4

?'

Simplifying the target stem establishes the exact numerical interval required to yield a definitive 'Yes' or 'No' response.

2

Evaluate Statement (1) independently.

Statement (1) states

|m - 2| < 3 \implies -3 < m - 2 < 3 \implies -1 < m < 5

. If

m = 1

,

-2 < 1 < 4

(Yes). However, if

m = 4.5

,

m

is not in

(-2, 4)

(No). Since Statement (1) allows both 'Yes' and 'No' outcomes, it is NOT sufficient.

A statement is sufficient only if every value in its allowed range produces the same answer to the rephrased question.

3

Evaluate Statement (2) independently.

Statement (2) states

|m + 1| < 4 \implies -4 < m + 1 < 4 \implies -5 < m < 3

. If

m = 1

,

-2 < 1 < 4

(Yes). However, if

m = -4

,

m

is not in

(-2, 4)

(No). Thus, Statement (2) is NOT sufficient.

Checking boundary values reveals that Statement (2) allows values outside the required interval.

4

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

Combining Statement (1) range

(-1 < m < 5)

and Statement (2) range

(-5 < m < 3)

requires taking their intersection:

-1 < m < 3

. Since every value in

(-1, 3)

satisfies

-2 < m < 4

, the answer to the question is a definitive 'Yes'.

The intersection of the two ranges falls strictly inside the target range, guaranteeing sufficiency.

Key Concept

Data Sufficiency range evaluation for quadratic absolute value inequalities

Estimated Time:2m 0s

Question 1431Question →

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

(1) $x^2 y + xy^2 = 30$
(2) $x^2 + y^2 = 19 - xy$

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 to determine a unique value for

x + y

.

Evaluating both statements together produces the polynomial relation

(x + y)^3 - 19(x + y) - 30 = 0

. Factoring gives

(x + y - 5)(x + y + 2)(x + y + 3) = 0

. Each of the three roots (

5

,

-2

,

-3

) produces real numbers

x

and

y

satisfying both original equations. Because

x + y

can take three different values, the two statements together are not sufficient to determine a unique value.

Step-by-Step Solution

1

Analyze Statement (1) independently

Factor Statement (1):

x^2 y + xy^2 = xy(x + y) = 30

.

If

(x, y) = (2, 3)

, then

xy(x + y) = 6(5) = 30

, giving

x + y = 5

. If

(x, y) = (1, 5)

, then

xy(x + y) = 5(6) = 30

, giving

x + y = 6

. Multiple values exist, so Statement (1) alone is insufficient.

2

Analyze Statement (2) independently

Rearrange Statement (2):

x^2 + y^2 + xy = 19

, which equals

(x + y)^2 - xy = 19

.

If

(x, y) = (3, 2)

, then

3^2 + 2^2 + 6 = 19

, giving

x + y = 5

. If

(x, y) = (-5, 3)

, then

(-5)^2 + 3^2 + (-15) = 19

, giving

x + y = -2

. Multiple values exist, so Statement (2) alone is insufficient.

3

Evaluate Statements (1) and (2) together

Express

xy

in terms of

S = x + y

: from Statement (1),

xy = \frac{30}{S}

; from Statement (2),

xy = S^2 - 19

.

Equating the two expressions for

xy

yields

\frac{30}{S} = S^2 - 19 \implies S^3 - 19S - 30 = 0

.

4

Solve the cubic polynomial for

S = x + y

and check for real roots

(x, y)

S^3 - 19S - 30 = (S - 5)(S + 2)(S + 3) = 0

, giving

S = 5

,

S = -2

, or

S = -3

.

For

S = 5

,

xy = 6

, giving real solutions

\{x, y\} = \{2, 3\}

. For

S = -2

,

xy = -15

, giving real solutions

\{x, y\} = \{-5, 3\}

. For

S = -3

,

xy = -10

, giving real solutions

\{x, y\} = \{-5, 2\}

. Since three distinct valid real sums exist, the combined statements are not sufficient.

Key Concept

Non-linear systems of equations may yield multiple distinct real solutions. Two equations in two variables do not guarantee a unique solution in Data Sufficiency unless the system is linear or constrained to a single root.

Question 1432Question →

A community library cataloged a collection of $120$ historical manuscripts. Each manuscript is written in either Latin, Ancient Greek, or both. How many of the manuscripts are written in both Latin and Ancient Greek?

(1) Exactly $80$ manuscripts are written in Latin.
(2) The number of manuscripts written only in Ancient Greek is twice the number of manuscripts written in both languages.

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.

Neither statement alone provides enough independent relationships to solve for the intersection. Statement (1) determines the count of manuscripts written exclusively in Ancient Greek (

120 - 80 = 40

), while Statement (2) establishes a proportional link between manuscripts exclusively in Ancient Greek and those in both languages (

2x

). Combining both facts produces the linear equation

2x = 40

, which uniquely solves to

x = 20

.

Step-by-Step Solution

1

Rephrase the question stem using overlapping set principles.

Let

L

be the set of Latin manuscripts,

G

be the set of Ancient Greek manuscripts, and

x

be the number of manuscripts in both (

L \cap G

). Since every manuscript is in at least one set, Total =

(\text{Latin only}) + (\text{Greek only}) + x = 120

. Also, Total =

|L| + (\text{Greek only}) = 120

. We need to find the unique value of

x

.

Formulating the set relationships establishes the exact algebraic system needed to evaluate sufficiency.

2

Evaluate Statement (1) independently.

Statement (1) gives

|L| = 80

. Substituting into the total formula gives

80 + (\text{Greek only}) = 120 \implies \text{Greek only} = 40

. However,

|L| = (\text{Latin only}) + x = 80

. The overlap

x

can range anywhere from

0

to

80

.

Statement (1) does not provide enough information to isolate

x

from Latin only.

3

Evaluate Statement (2) independently.

Statement (2) states that

\text{Greek only} = 2x

. Substituting into the total equation yields

(\text{Latin only}) + 2x + x = (\text{Latin only}) + 3x = 120

. Multiple integer solutions exist for

x

(e.g., if

\text{Latin only} = 90

,

x = 10

; if

\text{Latin only} = 60

,

x = 20

).

Statement (2) presents one equation with two unknown variables.

4

Evaluate Statements (1) and (2) together.

From Statement (1),

\text{Greek only} = 120 - 80 = 40

. From Statement (2),

\text{Greek only} = 2x

. Therefore,

2x = 40 \implies x = 20

. This gives a single, unique answer.

Combining both statements provides two independent linear equations, uniquely solving for

x

.

Key Concept

Overlapping Sets (Two-Group Venn Diagram) Data Sufficiency

Estimated Time:2m 0s

Question 1433Question →

A three-year study of two hundred software development companies observed that firms instituting mandatory daily meditation sessions for their engineering teams reported a 35 percent reduction in post-release software bugs compared to firms without such programs. The researchers concluded that daily meditation directly enhances cognitive focus, thereby reducing coding errors.

Firms that instituted mandatory meditation sessions were also significantly more likely to have simultaneously adopted automated continuous-integration testing protocols during the study period.

Decreasing post-release software bugs generally leads to higher client satisfaction scores and increased long-term contract renewal rates for software development firms.

The study tracked software bugs over a three-year period during which major cloud hosting providers introduced upgraded server infrastructure across the entire software industry.

Engineers at firms with mandatory meditation sessions reported higher overall job satisfaction than engineers who did not participate in daily mindfulness exercises.

Frequent post-release software bugs often cause severe operational strain that prevents engineering managers from scheduling non-essential workplace wellness activities.

Show answer & explanation

Answer: Firms that instituted mandatory meditation sessions were also significantly more likely to have simultaneously adopted automated continuous-integration testing protocols during the study period.

Answer

The argument assumes that because mandatory daily meditation correlated with a reduction in software bugs, the meditation directly caused the reduction. The correct answer weakens this causal claim by introducing a confounding variable: firms adopting meditation were also far more likely to implement automated continuous-integration testing protocols, which provides an alternative explanation for the decline in software bugs.

The correct answer identifies a major confounding variable. If companies that instituted daily meditation were also significantly more likely to adopt automated continuous-integration testing, the reduction in post-release bugs could easily be attributed to the automated testing tools rather than to meditation. This severely weakens the author's conclusion that meditation directly caused the improvement.

Step-by-Step Solution

1

Identify the premise and conclusion of the argument

Premise: Companies with mandatory meditation had 35% fewer post-release software bugs. Conclusion: Daily meditation directly causes the reduction in coding errors.

Understanding the precise logical transition from correlation to causation reveals the vulnerability of the argument.

2

Analyze the causal vulnerability

The author assumes no third factor (confounding variable) caused the decrease in software bugs in the meditation group.

Causal arguments on the GMAT are frequently weakened by demonstrating that an unmentioned third variable is responsible for the observed effect.

3

Evaluate the answer choices to locate a confounding variable or alternative cause

The option identifying simultaneous adoption of automated continuous-integration testing introduces a third variable that directly causes bug reductions.

If the firms with meditation also introduced automated testing, the testing—not the meditation—could be the true driver of the 35% reduction in bugs.

Key Concept

Causal Reasoning and Confounding Variables

Question 1434Question →

A commercial airline plans to replace its current regional fleet with newer aircraft models that consume 25 percent less fuel per seat-mile, maintaining that this fleet transition will lower the airline's overall annual operating expenses. Consider the following assertion: 'Determining whether global jet fuel prices will rise or fall over the next five years represents a valid application of the Test of Variance to evaluate the argument's conclusion.' Is this assertion true or false?

True

False

Show answer & explanation

Answer: False

Answer

The assertion is False. Evaluating jet fuel price trends fails the Test of Variance because opposite extreme outcomes (fuel prices rising vs. fuel prices falling) both result in lower fuel expenditures for the new fleet compared to the current fleet, rather than producing opposing impacts on the conclusion's validity.

The statement is false because applying opposite extremes to global fuel price movements (prices rising vs. prices falling) fails to produce opposite logical effects on the conclusion. In both extreme scenarios, an aircraft fleet that uses 25% less fuel per seat-mile will incur lower fuel costs than the existing fleet, thereby failing to test whether the cost-reduction plan succeeds or fails.

Step-by-Step Solution

1

Analyze the argument's premise and conclusion

Premise: The new aircraft consume 25% less fuel per seat-mile than the current fleet. Conclusion: Replacing the current fleet with the new aircraft will lower overall annual operating expenses relative to keeping the current fleet.

Evaluating argument validity requires isolating the comparative claim being made between the proposed plan and the status quo.

2

Define the standard for the Test of Variance

Under the Test of Variance, a question validly evaluates an argument if and only if one extreme response strongly strengthens the conclusion and the opposite extreme response strongly weakens it.

The Test of Variance is the formal method used in Critical Reasoning to test whether an information gap is decisive to an argument's structural validity.

3

Test opposite extreme outcomes for global fuel price trends

Extreme 1 (Fuel prices double): The 25% fuel efficiency yields even larger absolute dollar savings for the new fleet over the old fleet (strengthens). Extreme 2 (Fuel prices drop by 80%): The new fleet still spends 25% less on fuel than the old fleet, maintaining lower operating costs relative to keeping the old fleet (still strengthens).

Evaluating extreme values demonstrates whether the variable flips the logical standing of the conclusion.

4

Determine the truth value of the assertion

Because both extreme answers lead to the same logical outcome (the new fleet remains cheaper to operate than the old fleet), testing fuel price trends does not produce opposite impacts. Consequently, claiming that this variable represents a valid application of the Test of Variance is False.

Failing to produce opposing logical effects violates the core requirement of the Test of Variance.

Key Concept

The Test of Variance requires that opposite extreme answers to an evaluative question yield opposite logical impacts on an argument's conclusion. Variables that shift baseline and proposed options proportionally fail this test because they do not alter the relative validity of the argument.

Question 1435Question →

If $x$ and $y$ are real numbers with $x \neq 0$ , is $\frac{x^4 - y^4}{x^3 + x y^2} > x - y$ ?

(1) $y = 2x$
(2) $xy > 0$

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

Answer

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

Simplifying the numerator

(x^4 - y^4) = (x^2 - y^2)(x^2 + y^2)

and denominator

x^3 + xy^2 = x(x^2 + y^2)

reduces the left side of the inequality to

\frac{x^2 - y^2}{x}

. Rephrasing the question

\frac{x^2 - y^2}{x} > x - y

shows that it is equivalent to asking whether

\frac{y(x - y)}{x} > 0

.

Under Statement (1), substituting

y = 2x

transforms the target into asking 'Is

x < 0

?'. Because Statement (1) gives no information about the sign of

x

, it is insufficient.

Under Statement (2),

xy > 0

means

x

and

y

share the same sign, but testing values shows both 'Yes' and 'No' are possible (

x=2, y=1

yields Yes, whereas

x=1, y=2

yields No). Thus Statement (2) is insufficient.

Combining both statements gives

y = 2x

and

xy = 2x^2 > 0

, which holds true for all non-zero real numbers regardless of sign. Because

x

can still be positive or negative, the question 'Is

x < 0

?' cannot be answered. Hence, both statements together are NOT sufficient, making the option stating that statements (1) and (2) together are not sufficient the correct choice.

Step-by-Step Solution

1

Simplify the algebraic expression on the left-hand side of the target inequality.

\frac{x^4 - y^4}{x^3 + xy^2} = \frac{(x^2 - y^2)(x^2 + y^2)}{x(x^2 + y^2)} = \frac{x^2 - y^2}{x}

since

x^2 + y^2 > 0

for all non-zero real

x

.

Factoring the numerator as a difference of squares and factoring out

x

in the denominator allows canceling the non-zero common factor

(x^2 + y^2)

.

2

Rephrase the target inequality in terms of simplified components.

The target question 'Is

\frac{x^2 - y^2}{x} > x - y

?' subtracts to

\frac{x^2 - y^2 - x(x - y)}{x} > 0 \implies \frac{y(x - y)}{x} > 0

.

Simplifying the target question establishes the precise algebraic condition that must be evaluated by the statements.

3

Evaluate Statement (1):

y = 2x

.

Substituting

y = 2x

into the simplified target yields

\frac{2x(x - 2x)}{x} > 0 \implies \frac{-2x^2}{x} > 0 \implies -2x > 0 \implies x < 0

.

Statement (1) reduces the question to 'Is

x < 0

?'. Without information about the sign of

x

, this statement alone is NOT sufficient.

4

Evaluate Statement (2):

xy > 0

.

If

x = 2, y = 1

,

\frac{1(2-1)}{2} = 0.5 > 0

(Yes). If

x = 1, y = 2

,

\frac{2(1-2)}{1} = -2 \ngtr 0

(No).

Knowing that

x

and

y

have the same sign does not determine relative magnitudes, so Statement (2) alone is NOT sufficient.

5

Evaluate Statements (1) and (2) together.

Combining

y = 2x

and

xy > 0

gives

x(2x) = 2x^2 > 0

, which is true for all non-zero real

x

. If

x = 1, y = 2

, the target gives 'No'; if

x = -1, y = -2

, the target gives 'Yes'.

Combining both statements still does not determine whether

x < 0

or

x > 0

, so both statements together are NOT sufficient.

Key Concept

Question Stem Simplification and Target Rephrasing

Question 1436Question →

If $x$ is a real number, is $|x - 5| < x + 1$ ?

(1) $x^2 - 6x + 8 < 0$
(2) $|x - 3| < 2$

Which of the following choices correctly describes the sufficiency of the statements?

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 question target

|x - 5| < x + 1

simplifies directly to

x > 2

. Statement (1) bounds

x

to

2 < x < 4

, which guarantees that

x > 2

is always true, providing a definitive 'Yes'. Statement (2) bounds

x

to

1 < x < 5

, allowing values such as

1.5

(where

x > 2

is false) and

3

(where

x > 2

is true), which is insufficient. Therefore, Statement (1) alone is sufficient, but Statement (2) alone is not sufficient.

Step-by-Step Solution

1

Rephrase the question stem by simplifying the absolute value inequality

|x - 5| < x + 1

.

The inequality holds if and only if

-(x + 1) < x - 5 < x + 1

. Solving

-x - 1 < x - 5

yields

2x > 4

, or

x > 2

. Solving

x - 5 < x + 1

yields

-5 < 1

, which is always true. Thus, the question rephrases to: 'Is

x > 2

?'

Simplifying absolute value conditions in the stem reduces complex expressions to a single target range.

2

Evaluate Statement (1):

x^2 - 6x + 8 < 0

.

Factoring gives

(x - 2)(x - 4) < 0

, which yields the range

2 < x < 4

. For any value of

x

in

(2, 4)

,

x

is strictly greater than

2

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

Since every value satisfying Statement (1) satisfies

x > 2

, Statement (1) alone is sufficient.

3

Evaluate Statement (2):

|x - 3| < 2

.

Unfolding the absolute value gives

-2 < x - 3 < 2

, or

1 < x < 5

. If

x = 1.5

, then

x > 2

is false ('No'). If

x = 3

, then

x > 2

is true ('Yes'). Because both 'Yes' and 'No' answers are possible, Statement (2) is not sufficient.

Statement (2) permits values both above and below the threshold of 2.

Key Concept

Question Stem Rephrasing for Absolute Value Inequalities

Estimated Time:2m 0s

Question 1437Question →

A major software enterprise recently introduced a specialized cybersecurity training program for its third-party software vendors. The company's executive board claims that because vendors who voluntarily completed the training during its pilot phase experienced a 40 percent drop in data breach incidents compared to their pre-training levels, mandating the training for all remaining vendors will significantly reduce the enterprise's overall exposure to third-party data breaches.

Which of the following is an assumption on which the executive board's claim depends?

The vendors who voluntarily completed the training during the pilot phase were not inherently better equipped to implement security improvements than the remaining vendors.

Data breach incidents caused by third-party vendors represent the single largest operational risk to the software enterprise's annual revenue.

The financial cost to third-party vendors of completing the cybersecurity training is lower than the expected cost of remediating a data breach.

Mandating compliance with standardized cybersecurity training typically causes vendors to focus on passing audits rather than addressing novel security threats.

Completing the cybersecurity training program is sufficient to eliminate all third-party data breaches for any vendor that undergoes the instruction.

Show answer & explanation

Answer: The vendors who voluntarily completed the training during the pilot phase were not inherently better equipped to implement security improvements than the remaining vendors.

Answer

The argument depends on assuming that the vendors who voluntarily participated in the pilot program were not fundamentally better suited to benefit from the training than the non-pilot vendors.

The conclusion relies on generalizing the success of a voluntary pilot program to all remaining vendors. For this generalization to hold, the pilot participants must be representative of the whole population in their ability to translate the training into breach reductions. The correct choice addresses this self-selection gap: if the pilot vendors were inherently more capable of improving security, the remaining vendors would not achieve similar results, invalidating the executive board's expectation.

Step-by-Step Solution

1

Deconstruct the argument structure

Premise: Vendors who voluntarily took pilot training saw a 40% breach reduction. Conclusion: Mandating training for all remaining vendors will significantly reduce overall third-party breach exposure.

Identifying the jump from a voluntary pilot sample to a mandatory universal rule highlights potential sampling and self-selection gaps.

2

Formulate the required assumption (gap identification)

The author assumes that the success observed in the voluntary pilot group will replicate across the remaining non-pilot group.

If the pilot group possessed unique baseline advantages, the remaining vendors will not achieve similar improvements upon mandate.

3

Apply the Negation Test to the candidate assumption

Negated statement: The voluntary pilot vendors WERE inherently better equipped to implement security improvements than the remaining vendors.

If the remaining vendors lack the capacity to implement these improvements, mandating the course will fail to significantly reduce overall breach exposure, completely shattering the conclusion.

Key Concept

Identifying Unstated Assumptions (Selection Bias & Sample Representativeness)

Question 1438Question →

Among a panel of $150$ medical research trials, each trial evaluates at least one of three experimental drugs: Drug $X$ , Drug $Y$ , or Drug $Z$ . If $80$ trials evaluate Drug $X$ and $70$ trials evaluate Drug $Y$ , how many trials evaluate Drug $Z$ only?

(1) Exactly $25$ trials evaluate both Drug $X$ and Drug $Y$ .
(2) Exactly $40$ trials evaluate at least two of the three drugs, and no trial evaluates all three drugs.

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 choice is the option stating that Statement (1) alone is sufficient, but Statement (2) alone is not sufficient. By rephrasing the question, the number of trials evaluating Drug Z only equals the total number of trials (

150

) minus the number of trials evaluating Drug X or Drug Y (

|X \cup Y|

). Statement (1) gives the intersection

|X \cap Y| = 25

, which immediately yields

|X \cup Y| = 80 + 70 - 25 = 125

, so Drug Z only

= 150 - 125 = 25

. Statement (2) only establishes that the double overlap equals

40

, leaving the specific overlap between X and Y unknown.

Step-by-Step Solution

1

Rephrase the target question using set notation.

Let

U

be the total set of trials (

|U| = 150

). Since every trial evaluates at least one drug,

|X \cup Y \cup Z| = 150

. The number of trials evaluating Drug

Z

only is given by

|Z \text{ only}| = |X \cup Y \cup Z| - |X \cup Y| = 150 - |X \cup Y|

.

Simplifying the target target shows that finding

|X \cup Y|

is both necessary and sufficient to answer the question.

2

Evaluate Statement (1) independently.

Statement (1) gives

|X \cap Y| = 25

. Using the standard principle of inclusion-exclusion for two sets:

|X \cup Y| = |X| + |Y| - |X \cap Y| = 80 + 70 - 25 = 125

. Then

|Z \text{ only}| = 150 - 125 = 25

.

Statement (1) yields a single, unique numerical answer (

25

), so Statement (1) alone is sufficient.

3

Evaluate Statement (2) independently.

Let

a, b, c

be the double overlaps

|X \cap Y|

,

|Y \cap Z|

, and

|X \cap Z|

, and

d = |X \cap Y \cap Z| = 0

. Statement (2) states

a + b + c = 40

. Total set formula gives

150 = 80 + 70 + |Z| - 40 \implies |Z| = 40

. However,

|Z \text{ only}| = |Z| - (b + c) = 40 - (40 - a) = a = |X \cap Y|

, which is unknown.

Since the value of

|X \cap Y|

can vary, Statement (2) alone does not yield a unique numerical value and is insufficient.

Key Concept

Overlapping Sets and Data Sufficiency Target Rephrasing

Question 1439Question →

In GMAT Data Sufficiency 'Value' decision logic, if a question asks for the numerical value of a target expression $f(x)$ , a statement that yields more than one possible value for the variable $x$ is automatically insufficient.

True

False

Show answer & explanation

Answer: False

Answer

The statement is False.

The assertion is false because sufficiency in 'Value' Data Sufficiency questions is determined by whether the overall target expression resolves to one specific numerical value, not whether each constituent variable is uniquely determined.

Step-by-Step Solution

1

Identify the fundamental requirement for sufficiency in a GMAT Data Sufficiency 'Value' question.

A statement is sufficient if and only if it leads to a single, unique numerical value for the requested target expression.

The focus of a Value question is the target expression as a whole, not the isolated variables contained within it.

2

Test whether multiple possible values for a variable

x

can still yield a unique value for target expression

f(x)

.

Consider

f(x) = x^2

with

x = 2

or

x = -2

. In both cases,

f(x) = 4

.

Because all permissible values of

x

map to the exact same output for

f(x)

, the target value is uniquely determined.

3

Evaluate the claim made in the statement.

The claim asserts that having multiple values for

x

makes a statement 'automatically insufficient', which is disproven by counterexample.

Since a statement can yield multiple values for

x

and still be sufficient for

f(x)

, the assertion is false.

Key Concept

Value Data Sufficiency Decision Logic & Target Expression Sufficiency

Question 1440Question →

A corporate cafeteria serves only two types of lunches: hot meals and cold salads. On Tuesday, what was the ratio of the number of hot meals served to the number of cold salads served?

(1) The total number of lunches served on Tuesday was 200.
(2) On Tuesday, the number of hot meals served was 60% of the total number of lunches served.

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.

The correct choice is the option stating Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. Statement (1) gives only the total volume of lunches served (200), which allows for multiple possible combinations of hot meals and cold salads, resulting in different ratios. Statement (2) establishes that hot meals comprise 60% of the total, meaning cold salads comprise the remaining 40%. The ratio of hot meals to cold salads is therefore fixed at 60% to 40%, which simplifies to 3:2. Thus, Statement (2) alone provides sufficient information to answer the question uniquely.

Step-by-Step Solution

1

Define variables and target expression

Let

H

be the number of hot meals and

C

be the number of cold salads. Total lunches

T = H + C

. Target is to find the ratio

\frac{H}{C}

.

Rephrasing the question stem algebraically clarifies what information is needed.

2

Evaluate Statement (1)

Statement (1) states

T = H + C = 200

. Without knowing the proportion of either

H

or

C

,

\frac{H}{C}

could be

\frac{190}{10} = 19

or

\frac{100}{100} = 1

.

A single total count does not fix the internal ratio of the two components.

3

Evaluate Statement (2)

Statement (2) states

H = 0.60T

. Since

T = H + C

, then

C = T - 0.60T = 0.40T

. The ratio

\frac{H}{C} = \frac{0.60T}{0.40T} = \frac{0.60}{0.40} = \frac{3}{2}

.

Statement (2) yields a single, definitive numerical value for the ratio.

Key Concept

Data Sufficiency logic for percentage components and ratio determination

Answer

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