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Question 1401Question →

If aa and bb are non-zero real numbers, what is the value of ab\frac{a}{b}?

(1) 5a−3b=05a - 3b = 0
(2) a+b=8a + b = 8

Show answer & explanation

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

Answer

Statement (1) ALONE is sufficient to determine the value of ab\frac{a}{b}, but statement (2) alone is not sufficient.
Statement (1) can be rearranged algebraically to 5a=3b5a = 3b, which directly gives ab=35\frac{a}{b} = \frac{3}{5}, providing a single definitive value. Statement (2), a+b=8a + b = 8, allows infinitely many pairs of (a,b)(a, b) yielding different ratios for ab\frac{a}{b}. Therefore, Statement (1) alone is sufficient, while Statement (2) alone is not.

Step-by-Step Solution

1
Rephrase the question stem target
The target is to find a numerical value for the ratio ab\frac{a}{b}.
Simplifying the target helps determine if an equation directly yields the ratio without needing individual variable values.
2
Evaluate Statement (1) independently: 5a−3b=05a - 3b = 0
Rearranging gives 5a=3b5a = 3b. Dividing both sides by 5b5b (since b≠0b \neq 0) yields ab=35\frac{a}{b} = \frac{3}{5}.
This yields a single, unique numerical value for the target expression. Thus, Statement (1) alone is sufficient.
3
Evaluate Statement (2) independently: a+b=8a + b = 8
If a=4a = 4 and b=4b = 4, then ab=1\frac{a}{b} = 1. If a=2a = 2 and b=6b = 6, then ab=13\frac{a}{b} = \frac{1}{3}.
Multiple values of the ratio ab\frac{a}{b} are possible. Thus, Statement (2) alone is not sufficient.

Key Concept

Determining ratios from homogeneous linear equations in Data Sufficiency
Question 1402Question →

A department has 4040 employees, and each employee speaks at least one of two languages: French or Spanish. How many employees in the department speak Spanish?

(1) 2525 employees speak French.
(2) 1010 employees speak both French and Spanish.

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 total number of employees who speak at least one language is given by Total=French+Spanish−Both\text{Total} = \text{French} + \text{Spanish} - \text{Both}. Neither statement alone provides both the number of French speakers and the number of employees who speak both languages. However, combining both statements gives 40=25+Spanish−1040 = 25 + \text{Spanish} - 10, which uniquely solves to Spanish=25\text{Spanish} = 25. Therefore, both statements together are sufficient, but neither alone is sufficient.

Step-by-Step Solution

1
Rephrase the question stem using the overlapping sets formula.
Since every employee speaks at least one language, Neither=0\text{Neither} = 0. The relationship is Total=French+Spanish−Both\text{Total} = \text{French} + \text{Spanish} - \text{Both}, which simplifies to 40=French+Spanish−Both40 = \text{French} + \text{Spanish} - \text{Both}.
Establishing the mathematical relationship before evaluating statements clarifies what data is missing.
2
Evaluate Statement (1) independently.
Statement (1) gives French=25\text{French} = 25. Substituting this gives 40=25+Spanish−Both40 = 25 + \text{Spanish} - \text{Both}, or Spanish−Both=15\text{Spanish} - \text{Both} = 15.
Since Both\text{Both} is unknown, Spanish\text{Spanish} could take multiple values. Statement (1) is NOT sufficient.
3
Evaluate Statement (2) independently.
Statement (2) gives Both=10\text{Both} = 10. Substituting this gives 40=French+Spanish−1040 = \text{French} + \text{Spanish} - 10, or French+Spanish=50\text{French} + \text{Spanish} = 50.
Since French\text{French} is unknown, Spanish\text{Spanish} cannot be uniquely determined. Statement (2) is NOT sufficient.
4
Evaluate Statement (1) and Statement (2) together.
Substitute both values into the equation: 40=25+Spanish−10  ⟹  40=15+Spanish  ⟹  Spanish=2540 = 25 + \text{Spanish} - 10 \implies 40 = 15 + \text{Spanish} \implies \text{Spanish} = 25.
The equation yields a single, unique value for the target variable. Both statements together are sufficient.

Key Concept

Overlapping Sets (Two Groups)
Estimated Time:1m 0s
Question 1403Question →

If mm and nn are non-zero real numbers, what is the value of m2+n2mn\frac{m^2 + n^2}{mn}?

(1) m2n+mn2=6(m+n)m^2 n + m n^2 = 6(m + n)
(2) m2−n2=3(m−n)m^2 - n^2 = 3(m - n)

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.
Evaluating each statement alone yields multiple possible numerical outcomes because factoring produces independent cases (m+n=0m+n=0 vs mn=6mn=6 for the first statement, and m−n=0m-n=0 vs m+n=3m+n=3 for the second). However, when combining both statements, three of the four sub-cases fail: two violate non-zero or consistency conditions, and the sub-case where mn=6mn=6 and m+n=3m+n=3 has a negative discriminant (32−4(1)(6)=−15<03^2 - 4(1)(6) = -15 < 0), yielding no real solutions. This leaves only the sub-case mn=6mn=6 and m−n=0m-n=0, which uniquely determines the value of the target expression to be 2. Therefore, both statements together are sufficient.

Step-by-Step Solution

1
Rephrase the target expression.
The target expression m2+n2mn\frac{m^2 + n^2}{mn} can be rewritten as mn+nm\frac{m}{n} + \frac{n}{m}. We need a unique numerical value for this expression.
Simplifying the target helps evaluate what parameters or relationships are required.
2
Evaluate Statement (1) independently.
Rearrange Statement (1): mn(m+n)−6(m+n)=0  ⟹  (mn−6)(m+n)=0mn(m + n) - 6(m + n) = 0 \implies (mn - 6)(m + n) = 0. This gives two cases: Case 1: m+n=0  ⟹  n=−mm + n = 0 \implies n = -m. Since m,n≠0m, n \neq 0, m2+(−m)2m(−m)=2m2−m2=−2\frac{m^2 + (-m)^2}{m(-m)} = \frac{2m^2}{-m^2} = -2. Case 2: mn=6mn = 6. For instance, if m=2,n=3m = 2, n = 3, then 4+96=136\frac{4+9}{6} = \frac{13}{6}. Multiple values exist (−2-2 and 136\frac{13}{6}), so Statement (1) alone is NOT sufficient.
Factoring instead of dividing by (m+n)(m+n) preserves the root m+n=0m+n=0.
3
Evaluate Statement (2) independently.
Rearrange Statement (2): (m−n)(m+n)−3(m−n)=0  ⟹  (m−n)(m+n−3)=0(m - n)(m + n) - 3(m - n) = 0 \implies (m - n)(m + n - 3) = 0. This gives two cases: Case 1: m−n=0  ⟹  m=nm - n = 0 \implies m = n. Since m,n≠0m, n \neq 0, m2+m2m2=2\frac{m^2 + m^2}{m^2} = 2. Case 2: m+n=3m + n = 3. For instance, if m=1,n=2m = 1, n = 2, then 1+42=52\frac{1+4}{2} = \frac{5}{2}. Multiple values exist (22 and 52\frac{5}{2}), so Statement (2) alone is NOT sufficient.
Factoring preserves the root m−n=0m-n=0.
4
Combine Statement (1) and Statement (2).
We test the four combinations of cases:
- Case A (m+n=0m + n = 0) & Case X (m−n=0m - n = 0): System gives m=0,n=0m = 0, n = 0. Contradicts the condition that m,nm, n are non-zero.
- Case A (m+n=0m + n = 0) & Case Y (m+n=3m + n = 3): 0=30 = 3, impossible.
- Case B (mn=6mn = 6) & Case X (m−n=0m - n = 0): m=n  ⟹  m2=6  ⟹  m=n=±6m = n \implies m^2 = 6 \implies m = n = \pm\sqrt{6}. Here m2+n2mn=6+66=2\frac{m^2+n^2}{mn} = \frac{6+6}{6} = 2.
- Case B (mn=6mn = 6) & Case Y (m+n=3m + n = 3): n=3−m  ⟹  m(3−m)=6  ⟹  m2−3m+6=0n = 3 - m \implies m(3-m) = 6 \implies m^2 - 3m + 6 = 0. The discriminant is b2−4ac=9−24=−15<0b^2 - 4ac = 9 - 24 = -15 < 0, which yields no real solutions for mm and nn.

Only Case B & Case X yields valid real non-zero solutions, providing a unique value of 2.
Checking real constraints eliminates non-real systems and isolates a single valid numerical outcome.

Key Concept

Factoring non-linear algebraic systems without illegal variable division, and verifying real-number constraints using discriminants in Data Sufficiency.
Estimated Time:2m 30s
Question 1404Question →

Consider the following argument: "A recent industry report notes that successful corporate managers display high ambition when setting quarterly targets. However, psychologists define ambition as an excessive desire for personal dominance over others. Therefore, successful corporate managers must possess an excessive desire for personal dominance."

Match each component or statement from the argument on the left with its corresponding contextual meaning or logical role on the right.

Click a left item, then click its matching right item

Items

Usage of 'ambition' in the first premise (industry report)
Usage of 'ambition' in the second premise (psychological definition)
Logical flaw in drawing the final conclusion

Matches

Show answer & explanation

Answer

The first premise's usage of 'ambition' corresponds to 'Professional drive to set and attain high performance goals'; the second premise's usage corresponds to 'Clinical trait characterized by an excessive desire for dominance'; and the logical flaw corresponds to 'Equivocating by shifting the definition of a key term between premises'.
Matching each item correctly demonstrates an understanding of how the term 'ambition' undergoes a shift in meaning from a general business drive to a psychological trait, causing the fallacy of equivocation.

Step-by-Step Solution

1
Analyze the meaning of 'ambition' in the first premise
In the first premise, 'ambition' describes managers aiming for high quarterly targets, indicating professional drive.
Contextual clues link the term to goal-setting in a business setting.
2
Analyze the meaning of 'ambition' in the second premise
The second premise explicitly introduces a different, psychological definition centered on personal dominance.
The author shifts from a colloquial/business sense to a specialized psychological sense.
3
Identify the logical flaw connecting the premises to the conclusion
The conclusion relies on treating these two separate definitions as if they were identical, which is the logical flaw of equivocation.
An argument is fallacious if its validity depends on a single term carrying two different meanings.

Key Concept

Equivocation and Shift in Term Meaning
Question 1405Question →

To combat declining wild bee populations, environmental officials in Region X plan to restrict the agricultural use of neonicotinoid pesticides, citing a study showing that bee colonies in uncultivated nature reserves suffer far lower mortality rates than those near commercial farmland. The officials conclude that enforcing this restriction across all farms in Region X will restore local wild bee populations to healthy levels.

Which of the following is an assumption on which the environmental officials' argument depends?

Show answer & explanation

Answer: The lower mortality rate of bee colonies in nature reserves is not primarily attributable to environmental differences other than the absence of neonicotinoid pesticides.

Answer

The argument assumes that the lower mortality rate of bee colonies in nature reserves is not primarily attributable to environmental differences other than the absence of neonicotinoid pesticides.
The conclusion relies on a causal link between neonicotinoid exposure and high bee mortality. For the proposed restriction to restore bee populations, the observed survival advantage in nature reserves must actually stem from the absence of neonicotinoids rather than other confounding factors (such as superior forage or fewer parasites). Negating this choice reveals that if other factors are the primary cause, the plan will fail, proving it is a necessary unstated assumption.

Step-by-Step Solution

1
Deconstruct the argument structure
Premise: Bee colonies in nature reserves (where neonicotinoids are absent) have lower mortality than those near commercial farmland. Conclusion: Restricting neonicotinoids on farmland will restore wild bee populations.
Identifying the gap between the observed correlation (reserves vs. farmland) and the proposed causal intervention (restricting neonicotinoids) is essential.
2
Identify the logical gap
The author assumes that the absence of neonicotinoids is the primary cause of lower bee mortality in reserves, ruling out alternative confounding variables.
If other differences between reserves and farmland cause the mortality gap, removing neonicotinoids will fail to achieve the intended result.
3
Apply the Negation Test to verify the correct assumption
Negated claim: 'The lower mortality rate in nature reserves IS primarily attributable to environmental differences other than the absence of neonicotinoids.' If true, restricting neonicotinoids will not restore bee populations, completely invalidating the conclusion.
A statement is a required assumption if and only if its logical negation shatters the conclusion of the argument.

Key Concept

Identifying Unstated Assumptions via the Negation Test
Estimated Time:2m 0s
Question 1406Question →

If xx and yy are non-zero real numbers, is xy<1\frac{x}{y} < 1?

(1) ∣x+y∣<∣x−y∣|x + y| < |x - y|
(2) x>yx > y

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

Show answer & explanation

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

Answer

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
The correct option states that Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. Squaring both sides of Statement (1) gives x2+2xy+y2<x2−2xy+y2x^2 + 2xy + y^2 < x^2 - 2xy + y^2, which simplifies directly to 4xy<04xy < 0, meaning xy<0xy < 0. When two numbers have opposite signs, their quotient xy\frac{x}{y} must be negative, making it strictly less than 1. This provides a definitive 'Yes' answer. Statement (2) gives x>yx > y; if x=3x = 3 and y=2y = 2, 32>1\frac{3}{2} > 1 (No), but if x=1x = 1 and y=−2y = -2, 1−2=−0.5<1\frac{1}{-2} = -0.5 < 1 (Yes). Hence Statement (2) is not sufficient.

Step-by-Step Solution

1
Rephrase the target question
The target question asks if xy<1\frac{x}{y} < 1.
Understanding the algebraic condition required for a fraction to be less than 1 helps evaluate given statements.
2
Evaluate Statement (1): ∣x+y∣<∣x−y∣|x + y| < |x - y|
Squaring both sides yields (x+y)2<(x−y)2  ⟹  x2+2xy+y2<x2−2xy+y2  ⟹  4xy<0  ⟹  xy<0(x + y)^2 < (x - y)^2 \implies x^2 + 2xy + y^2 < x^2 - 2xy + y^2 \implies 4xy < 0 \implies xy < 0.
Since both sides of the inequality are non-negative distance expressions, squaring preserves the inequality sign.
3
Determine the sufficiency of Statement (1)
Since xy<0xy < 0, xx and yy must have opposite signs. Therefore, xy\frac{x}{y} is negative, which means xy<0<1\frac{x}{y} < 0 < 1. This gives a definitive 'Yes' to the question.
Any negative number is strictly less than 1, so Statement (1) alone is sufficient.
4
Evaluate Statement (2): x>yx > y
If x=3x = 3 and y=2y = 2, then x>yx > y and xy=1.5>1\frac{x}{y} = 1.5 > 1 (No). If x=1x = 1 and y=−2y = -2, then x>yx > y and xy=−0.5<1\frac{x}{y} = -0.5 < 1 (Yes).
Testing cases with positive vs. negative denominators shows that Statement (2) leads to both 'Yes' and 'No' answers.
5
Determine the sufficiency of Statement (2) and select the overall answer choice
Statement (2) alone is not sufficient. Therefore, Statement (1) alone is sufficient, but statement (2) alone is not sufficient.
Only Statement (1) provides a definitive answer to the question stem.

Key Concept

Distance interpretation of absolute values and algebraic rephrasing of inequalities involving quotient sign analysis.
Estimated Time:2m 0s
Question 1407Question →

A bio-pharmaceutical research consortium enforces the following regulatory guidelines for all clinical trials:

1. Every international multi-site trial conducted by the consortium requires central Ethics Board approval prior to the initiation of patient enrollment.
2. Any trial receiving central Ethics Board approval that includes pediatric participants must undergo an independent pediatric safety review.

During a recent compliance audit, investigators noted that Trial X was conducted as an international multi-site trial and enrolled pediatric participants, but never underwent an independent pediatric safety review.

Evaluate the validity of the following statement based on the information above: "Patient enrollment for Trial X was initiated without prior approval from the central Ethics Board."

Show answer & explanation

Answer: True

Answer

True. It logically follows from the premises that patient enrollment for Trial X began without approval from the central Ethics Board.
The deduction relies on formal contrapositive logic: Premise 2 states that receiving Ethics Board approval AND including pediatric participants guarantees a safety review. Trial X included pediatric participants but lacked the safety review. Therefore, Trial X could not have received Ethics Board approval. Since Trial X did enroll pediatric participants, patient enrollment was initiated without central Ethics Board approval.

Step-by-Step Solution

1
Formalize the second regulatory guideline into conditional logic.
(Ethics Board Approval AND Pediatric Participants) → Independent Pediatric Safety Review.
This establishes the necessary condition for any trial that has both approval and pediatric participants.
2
Apply the facts of Trial X to the contrapositive of the second guideline.
Since Trial X had pediatric participants but NO Pediatric Safety Review, Trial X did NOT have Ethics Board Approval.
The contrapositive states: NOT(Pediatric Safety Review) → NOT(Ethics Board Approval AND Pediatric Participants). Given Pediatric Participants is true, Ethics Board Approval must be false.
3
Combine the deduction with the fact that patient enrollment occurred.
Trial X enrolled pediatric participants (enrollment initiated) without Ethics Board Approval.
Because enrollment was initiated for Trial X and Ethics Board Approval was proven to be absent, enrollment was initiated without approval.

Key Concept

Conditional Logic Deductions and Contrapositive Chains
Question 1408Question →

Five years ago, a commercial airline began applying a newly developed synthetic hydrophobic coating to the exterior fuselages of its aircraft. Wind-tunnel tests and flight logs under controlled conditions confirmed that the coating reduced aerodynamic drag, allowing engines to burn 8 percent less fuel per mile flown compared to untreated aircraft. Surprisingly, however, over the five-year period following the fleet-wide application, the airline's average fuel consumption per flight hour across its standard domestic routes actually increased, even though flight speeds, flight altitudes, and fuel prices remained unchanged. Which of the following, if true, most helps to resolve the apparent discrepancy described above?

Show answer & explanation

Answer: Routine exposure to standard de-icing chemicals used during winter operations chemically altered the coating, causing its surface to become significantly rougher than standard untreated metal.

Answer

Routine exposure to standard de-icing chemicals used during winter operations chemically altered the coating, causing its surface to become significantly rougher than standard untreated metal.
The correct answer identifies a key difference between controlled testing conditions and actual routine operational conditions. While the coating successfully reduces drag under test conditions, exposure to de-icing fluids during regular operations causes the coating to degrade into a surface rougher than standard untreated metal. This accounts for both the initial test results and the observed increase in fuel burn per flight hour during actual service.

Step-by-Step Solution

1
Identify the two contradictory facts in the stimulus.
Fact 1: Controlled tests show the synthetic coating reduces drag and fuel burn by 8 percent per mile. Fact 2: In real-world domestic operations, average fuel consumption per flight hour increased.
Resolving a paradox requires finding an explanation that allows both statements to be completely true simultaneously.
2
Evaluate how real-world operational conditions differ from controlled testing environments.
Controlled tests evaluate pristine coatings under ideal conditions, whereas real-world domestic flights expose aircraft to operational elements like de-icing chemicals.
An operational difference that degrades performance below baseline level can account for the net increase in hourly fuel consumption.
3
Select the choice that reconciles both facts without denying either premise.
The option describing chemical alteration from de-icing explains why controlled tests showed savings while actual routine flights experienced increased drag and fuel burn.
This provides a mechanism for why the coating became rougher than untreated metal during real-world use.

Key Concept

Resolving Paradoxes and Discrepancies
Question 1409Question →

From 2015 to 2023, all municipal transit authorities in Region M derived their annual operational funding exclusively from passenger fares and regional tax subsidies. In 2019, a regional ordinance mandated that any transit authority receiving more than 40 percent of its operational funding from passenger fares must provide discounted passes to senior citizens. Furthermore, under the ordinance, no transit authority could receive regional tax subsidies if passenger fares accounted for more than 50 percent of its operational funding.

If the statements above are true, which of the following must also be true about municipal transit authorities in Region M after the 2019 ordinance took effect?

Show answer & explanation

Answer: Any municipal transit authority whose passenger fares accounted for more than 50 percent of its operational funding was required to provide discounted passes to senior citizens.

Answer

Any municipal transit authority whose passenger fares accounted for more than 50 percent of its operational funding was required to provide discounted passes to senior citizens.
The correct response logically follows by synthesizing the rules given in the stem. If passenger fares exceeded 50 percent of operational funding, the authority could receive no regional tax subsidies. Because operational funding came exclusively from fares and subsidies, an authority receiving no subsidies derived 100 percent of its funding from fares. Because 100 percent is greater than 40 percent, the authority was required under the 2019 ordinance to provide discounted passes to senior citizens.

Step-by-Step Solution

1
Identify the strict constraints provided in the premises.
Constraint 1: Total Funding = Passenger Fares + Tax Subsidies (exclusively).
Constraint 2: If Fares > 40%, Senior Discounts are mandatory.
Constraint 3: If Fares > 50%, Tax Subsidies = 0.
Establishing the exact formal logic parameters allows synthesizing multi-sentence premises.
2
Analyze the condition where Fares > 50%.
By Constraint 3, Tax Subsidies must equal 0%. Combined with Constraint 1, Fares must account for 100% of total operational funding.
Synthesizing the exclusive funding sources with the subsidy restriction reveals the precise percentage of funding coming from fares.
3
Apply Constraint 2 to the calculated result.
Since 100% is greater than 40%, the condition for mandatory senior discounts is satisfied.
Deducing that Fares = 100% guarantees that the > 40% threshold is met, making the requirement inescapable.

Key Concept

Drawing Deductive Inferences from Synthesized Constraints
Estimated Time:2m 0s
Question 1410Question →

A car dealership's inventory consists exclusively of sedans and SUVs. What is the ratio of the number of sedans to the number of SUVs in the inventory?

(1) The total number of sedans and SUVs in the inventory is 150150.
(2) The number of sedans in the inventory is 6060.

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.
Statement (1) gives S+U=150S + U = 150, which is insufficient by itself. Statement (2) gives S=60S = 60, which is also insufficient by itself. Together, we can deduce U=90U = 90, giving a unique ratio S:U=60:90=2:3S : U = 60 : 90 = 2 : 3. Therefore, both statements together are sufficient.

Step-by-Step Solution

1
Rephrase the target question.
Let SS be the number of sedans and UU be the number of SUVs. The target ratio is SU\frac{S}{U}.
Establishing explicit variable definitions simplifies statement analysis.
2
Evaluate Statement (1) independently.
S+U=150S + U = 150.
Knowing only the total does not fix the specific values of SS or UU, so the ratio SU\frac{S}{U} can take multiple values. Statement (1) is insufficient.
3
Evaluate Statement (2) independently.
S=60S = 60.
Without knowing UU, the ratio SU\frac{S}{U} cannot be evaluated. Statement (2) is insufficient.
4
Evaluate Statements (1) and (2) combined.
Substitute S=60S = 60 into S+U=150S + U = 150 to get 60+U=15060 + U = 150, so U=90U = 90. Thus, SU=6090=23\frac{S}{U} = \frac{60}{90} = \frac{2}{3}.
Combining both statements yields a single, definitive ratio value. Statements (1) and (2) together are sufficient.

Key Concept

Data Sufficiency evaluation for linear systems involving ratios and sums.
Question 1411Question →

Passage:
In an international pharmaceutical trial evaluating a new anti-inflammatory drug, every participant was categorized as exhibiting either full remission, moderate symptom reduction, or no measurable improvement. Every participant who achieved full remission had completed a six-week physical therapy regimen prior to the trial. However, less than forty percent of all participants who completed the physical therapy regimen achieved full remission. Additionally, no participant enrolled at the trial's European testing centers achieved full remission.

Statement:
Based solely on the passage above, the statement 'At least one participant enrolled at the trial's European testing centers completed the six-week physical therapy regimen prior to the trial' MUST BE TRUE.

Show answer & explanation

Answer: False

Answer

The statement is False because the premise only establishes physical therapy as a necessary condition for full remission, not a sufficient one, making it possible that zero European participants took physical therapy.
The claim is evaluated as False because the premises do not guarantee that any European participant completed the physical therapy regimen. The passage establishes that physical therapy was necessary for full remission (Full Remission→Physical TherapyFull\ Remission \rightarrow Physical\ Therapy). Knowing that no European participant achieved full remission allows for the possibility that some European participants took physical therapy without achieving full remission, but it equally permits the possibility that zero European participants took physical therapy. Because the statement is not strictly required by the facts, classifying it as a 'must be true' statement is logically invalid.

Step-by-Step Solution

1
Analyze the formal logic of the conditional premise
Full Remission →\rightarrow Physical Therapy. This means completing physical therapy is a necessary condition for full remission, but not a sufficient condition.
Understanding the direction of conditional statements prevents confusing necessary conditions with sufficient outcomes.
2
Apply the evidence regarding European testing center participants
No European participant achieved full remission (¬Full Remission\neg Full\ Remission).
We must evaluate what can be deduced about European participants given their lack of full remission.
3
Test whether physical therapy completion is required for European participants
Denying the antecedent or negating the consequent of a necessary condition does not force the necessary condition to occur. European participants could have done physical therapy and achieved moderate/no improvement, OR they could have omitted physical therapy entirely.
Since the facts allow for zero European participants having taken physical therapy, asserting that at least one MUST have taken it is an unwarranted extrapolation.

Key Concept

Avoiding 'Could Be True' and Out-of-Scope Inference Traps in Conditional Premises
Estimated Time:1m 30s
Question 1412Question →

The chief economic adviser argued that to ensure the long-term viability of the national health infrastructure, the government must prioritize investments that maximize public interest. Because recent opinion polls demonstrate that over 80 percent of the public is intensely interested in cosmetic dermatology and anti-aging therapies, the adviser concluded that the ministry should reallocate substantial funding from primary preventative care toward subsidizing these elective treatments.

Which of the following best describes the logical flaw in the chief economic adviser's argument?

Show answer & explanation

Answer: It treats a term meaning the general welfare or benefit of society as though it meant whatever subjects currently find intriguing or engaging.

Answer

The argument is flawed because it equivocates on the phrase 'public interest,' shifting from its normative sense of societal health and welfare to a descriptive sense of what holds the public's curiosity.
The argument commits a classic equivocation flaw by conflating two distinct senses of the term 'interest.' In the initial policy objective, 'public interest' refers to societal well-being and benefit. However, the evidence presented relies on what the public is 'interested in'—that is, what captures public curiosity or consumer desire. Reallocating essential healthcare funds based solely on public fascination assumes that popular attention equals public welfare, which is an invalid shift in term meaning.

Step-by-Step Solution

1
Analyze the main premise and conclusion to identify key terms.
Premise goal: Maximize the 'public interest' (welfare/benefit of the population). Evidence cited: The public is 'interested in' (curious about) cosmetic procedures.
Tracking how key terms are deployed across premises and conclusions reveals conceptual shifts.
2
Compare the two uses of the root concept 'interest'.
The premise uses 'public interest' as a noun phrase denoting collective wellbeing, whereas the evidence uses 'interested in' as a verb phrase denoting popular attention or preference.
Equivocation occurs when an argument relies on two distinct meanings of the same word or phrase to draw an invalid conclusion.
3
Evaluate the answer options to select the choice describing this shift.
The choice identifying the shift from general welfare/benefit of society to what subjects find intriguing accurately highlights the flaw.
This choice directly pinpoints the semantic slide between normative welfare and descriptive curiosity.

Key Concept

Equivocation and Shift in Term Meaning
Question 1413Question →

If xx and yy are non-zero real numbers, is x3y+xy3<2x2y2x^3 y + x y^3 < 2 x^2 y^2?

(1) ∣x+y∣<∣x−y∣|x + y| < |x - y|
(2) x2y3>0x^2 y^3 > 0

Show answer & explanation

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

Answer

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
The correct response is the option stating that Statement (1) alone is sufficient, but Statement (2) alone is not sufficient. Rephrasing the stem x3y+xy3<2x2y2x^3 y + x y^3 < 2 x^2 y^2 by factoring yields xy(x−y)2<0xy(x - y)^2 < 0. Because (x−y)2>0(x - y)^2 > 0 whenever x≠yx \neq y, the inequality holds if and only if xy<0xy < 0. Statement (1) reduces to (x+y)2<(x−y)2(x+y)^2 < (x-y)^2, which simplifies directly to xy<0xy < 0, giving a definitive YES. Statement (2) reduces to y>0y > 0, which leaves the sign of xx (and thus xyxy) unknown.

Step-by-Step Solution

1
Rephrase the target question stem algebraically.
The target inequality x3y+xy3<2x2y2x^3 y + x y^3 < 2 x^2 y^2 can be rearranged as x3y+xy3−2x2y2<0x^3 y + x y^3 - 2 x^2 y^2 < 0. Factoring out xyxy yields xy(x2−2xy+y2)<0xy(x^2 - 2xy + y^2) < 0, which is xy(x−y)2<0xy(x - y)^2 < 0.
Simplifying the stem isolates the core algebraic condition needed to answer the question.
2
Analyze the conditions under which xy(x−y)2<0xy(x - y)^2 < 0 holds.
Since xx and yy are non-zero, if x=yx = y, xy(x−y)2=0xy(x-y)^2 = 0, so the inequality is false (NO). If x≠yx \neq y, (x−y)2>0(x - y)^2 > 0, so the sign of xy(x−y)2xy(x - y)^2 is determined entirely by xyxy. If xy<0xy < 0, then x≠yx \neq y and xy(x−y)2<0xy(x - y)^2 < 0 (YES). If xy>0xy > 0, then xy(x−y)2≥0xy(x - y)^2 \ge 0 (NO). Thus, the target question simplifies to: 'Is xy<0xy < 0?'
A perfect square of a real number is non-negative, so sign evaluation reduces to checking xy<0xy < 0.
3
Evaluate Statement (1): ∣x+y∣<∣x−y∣|x + y| < |x - y|.
Squaring both sides of ∣x+y∣<∣x−y∣|x + y| < |x - y| gives (x+y)2<(x−y)2  ⟹  x2+2xy+y2<x2−2xy+y2  ⟹  4xy<0  ⟹  xy<0(x + y)^2 < (x - y)^2 \implies x^2 + 2xy + y^2 < x^2 - 2xy + y^2 \implies 4xy < 0 \implies xy < 0.
Since Statement (1) directly proves xy<0xy < 0, it gives a definitive YES to the rephrased target question, making Statement (1) sufficient.
4
Evaluate Statement (2): x2y3>0x^2 y^3 > 0.
Since x≠0x \neq 0, x2>0x^2 > 0. Thus x2y3>0x^2 y^3 > 0 simplifies to y3>0y^3 > 0, which means y>0y > 0. However, xx can still be positive (yielding xy>0xy > 0) or negative (yielding xy<0xy < 0).
Knowing only that y>0y > 0 leaves the sign of xyxy undetermined, so Statement (2) is not sufficient.

Key Concept

Data Sufficiency Target Simplification and Algebraic Rephrasing
Question 1414Question →

A class of 50 students took tests in both Mathematics and Science. How many students passed both tests?

(1) 35 students passed Mathematics and 30 students passed Science.
(2) 10 students failed both tests.

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 option is the choice stating that both statements together are sufficient, but neither alone is sufficient. Statement (1) leaves the number of students failing both tests unknown, while Statement (2) leaves the individual subject pass numbers unknown. Combined, the standard set equation Total=Group A+Group B−Both+Neither\text{Total} = \text{Group A} + \text{Group B} - \text{Both} + \text{Neither} yields a unique value of 2525 for students passing both tests.

Step-by-Step Solution

1
Set up the overlapping sets formula for two groups
Total=Math+Science−Both+Neither\text{Total} = \text{Math} + \text{Science} - \text{Both} + \text{Neither}, which becomes 50=Math+Science−Both+Neither50 = \text{Math} + \text{Science} - \text{Both} + \text{Neither}.
This formula connects all four components of a two-group overlapping set.
2
Evaluate Statement (1) independently
Substituting Math=35\text{Math} = 35 and Science=30\text{Science} = 30 into the formula yields 50=35+30−Both+Neither50 = 35 + 30 - \text{Both} + \text{Neither}, or Both−Neither=15\text{Both} - \text{Neither} = 15.
Since Neither\text{Neither} is unknown, Both\text{Both} cannot be determined. Statement (1) alone is insufficient.
3
Evaluate Statement (2) independently
Substituting Neither=10\text{Neither} = 10 yields 50=Math+Science−Both+1050 = \text{Math} + \text{Science} - \text{Both} + 10.
Since Math\text{Math} and Science\text{Science} are unknown, Both\text{Both} cannot be determined. Statement (2) alone is insufficient.
4
Evaluate Statements (1) and (2) combined
Combining all given values gives 50=35+30−Both+10⇒50=75−Both⇒Both=2550 = 35 + 30 - \text{Both} + 10 \Rightarrow 50 = 75 - \text{Both} \Rightarrow \text{Both} = 25.
A single unique value of 25 is obtained for the number of students who passed both tests.

Key Concept

Overlapping Sets (Two-Group Venn Diagram Formula)
Question 1415Question →

In a technology consulting firm of 100 employees, every employee works in either the Analytics department, the Engineering department, or both. How many employees work in both departments?

(1) Exactly 70 employees work in the Analytics department, and 60 employees work in the Engineering department.
(2) Exactly 40 employees work ONLY in the Analytics department.

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 states that Statement (1) ALONE is sufficient, but Statement (2) ALONE is not sufficient. Using the standard formula for overlapping sets, Total = Group A + Group B - Both, Statement (1) provides values for Total (100), Group A (70), and Group B (60), allowing us to solve directly for Both = 30. Statement (2) only specifies the number of members belonging strictly to Group A, which leaves the overlap dependent on the unstated size of Group B.

Step-by-Step Solution

1
Set up the overlapping sets formula for two groups.
Total = N(Analytics) + N(Engineering) - N(Both)
Since every employee belongs to at least one of the two departments, the union of the two sets equals the total number of employees, 100.
2
Evaluate Statement (1) independently.
100 = 70 + 60 - N(Both) => N(Both) = 30.
Statement (1) provides N(Analytics) = 70 and N(Engineering) = 60. Substituting these values into the formula yields a unique value of 30 for N(Both). Thus, Statement (1) alone is sufficient.
3
Evaluate Statement (2) independently.
100 = N(Only Analytics) + N(Only Engineering) + N(Both) => 100 = 40 + N(Only Engineering) + N(Both).
We have two unknown variables: N(Only Engineering) and N(Both). Multiple non-negative integer pairs satisfy this equation (e.g., N(Both) could be 0, 10, 20, etc.). Thus, Statement (2) alone is not sufficient.

Key Concept

Overlapping Sets Formula for Two Groups
Estimated Time:1m 30s
Question 1416Question →

If xx is a real number, is ∣x+2∣+∣x−4∣≤8|x + 2| + |x - 4| \le 8?

(1) ∣x−1∣≤4|x - 1| \le 4
(2) x2−x−6≤0x^2 - x - 6 \le 0

Show answer & explanation

Answer: EACH statement ALONE is sufficient.

Answer

Each statement alone is sufficient.
Rephrasing the question stem shows that ∣x+2∣+∣x−4∣≤8|x + 2| + |x - 4| \le 8 is equivalent to −3≤x≤5-3 \le x \le 5. Statement (1) gives −3≤x≤5-3 \le x \le 5, which directly yields a definitive 'Yes'. Statement (2) gives −2≤x≤3-2 \le x \le 3, which is a subset of [−3,5][-3, 5], so any value of xx satisfying Statement (2) must also satisfy −3≤x≤5-3 \le x \le 5, also yielding a definitive 'Yes'. Thus, each statement alone is sufficient.

Step-by-Step Solution

1
Rephrase the question stem target by analyzing critical points x=−2x = -2 and x=4x = 4 for ∣x+2∣+∣x−4∣≤8|x + 2| + |x - 4| \le 8.
The target inequality ∣x+2∣+∣x−4∣≤8|x + 2| + |x - 4| \le 8 simplifies to the range −3≤x≤5-3 \le x \le 5.
For x≥4x \ge 4: (x+2)+(x−4)≤8  ⟹  2x−2≤8  ⟹  x≤5(x + 2) + (x - 4) \le 8 \implies 2x - 2 \le 8 \implies x \le 5, giving [4,5][4, 5]. For −2≤x<4-2 \le x < 4: (x+2)+(4−x)=6≤8(x + 2) + (4 - x) = 6 \le 8, which is true for all x∈[−2,4)x \in [-2, 4). For x<−2x < -2: (−x−2)+(4−x)≤8  ⟹  2−2x≤8  ⟹  −2x≤6  ⟹  x≥−3(-x - 2) + (4 - x) \le 8 \implies 2 - 2x \le 8 \implies -2x \le 6 \implies x \ge -3, giving [−3,−2)[-3, -2). Combining all intervals gives −3≤x≤5-3 \le x \le 5.
2
Evaluate Statement (1): ∣x−1∣≤4|x - 1| \le 4.
−4≤x−1≤4  ⟹  −3≤x≤5-4 \le x - 1 \le 4 \implies -3 \le x \le 5.
Statement (1) states that xx is precisely in the range [−3,5][-3, 5]. This provides a definitive 'Yes' answer to the question 'Is −3≤x≤5-3 \le x \le 5?'. Thus, Statement (1) alone is sufficient.
3
Evaluate Statement (2): x2−x−6≤0x^2 - x - 6 \le 0.
(x−3)(x+2)≤0  ⟹  −2≤x≤3(x - 3)(x + 2) \le 0 \implies -2 \le x \le 3.
Since [−2,3][-2, 3] is entirely contained within [−3,5][-3, 5], every value of xx satisfying Statement (2) automatically satisfies −3≤x≤5-3 \le x \le 5. This yields a definitive 'Yes' to the question. Thus, Statement (2) alone is sufficient.
4
Combine evaluations of Statement (1) and Statement (2).
Each statement alone is sufficient.
Because both statements independently provide enough information to give a definitive 'Yes' answer, the correct answer choice is that each statement alone is sufficient.

Key Concept

Absolute Value Distance Interpretation and Subset Range Sufficiency
Question 1417Question →

If mm and nn are real numbers, is m2<n2m^2 < n^2?

(1) ∣m∣+n<0|m| + n < 0
(2) m+n>0m + n > 0

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.
Rephrased, the question asks whether ∣m∣<∣n∣|m| < |n|. Evaluating the first condition gives n<−∣m∣n < -|m|, which shows that nn must be negative and its absolute value ∣n∣=−n|n| = -n must be strictly greater than ∣m∣|m|. Hence, m2<n2m^2 < n^2 is definitely true. Evaluating the second condition allows both m=2,n=5m=2, n=5 (where m2<n2m^2 < n^2) and m=5,n=2m=5, n=2 (where m2>n2m^2 > n^2), making it insufficient. Therefore, the first condition alone is sufficient while the second condition alone is not.

Step-by-Step Solution

1
Rephrase the target question stem m2<n2m^2 < n^2.
The target condition m2<n2m^2 < n^2 is equivalent to ∣m∣2<∣n∣2|m|^2 < |n|^2, which is true if and only if ∣m∣<∣n∣|m| < |n|.
Since both m2m^2 and n2n^2 are non-negative, taking the square root of both sides preserves the inequality order for non-negative magnitudes.
2
Evaluate Statement (1): ∣m∣+n<0|m| + n < 0.
Rearranging gives n<−∣m∣n < -|m|. Since ∣m∣≥0|m| \ge 0, this implies nn is strictly negative. Taking absolute values of both sides of n<−∣m∣n < -|m| gives ∣n∣=−n>∣m∣|n| = -n > |m|. Therefore, ∣n∣>∣m∣|n| > |m|, which means n2>m2n^2 > m^2 or m2<n2m^2 < n^2.
Statement (1) yields a definitive 'Yes' to the target question. Thus, Statement (1) alone is sufficient.
3
Evaluate Statement (2): m+n>0m + n > 0.
Case 1: Let m=2m = 2 and n=5n = 5. Then 2+5=7>02 + 5 = 7 > 0, and 22=4<25=522^2 = 4 < 25 = 5^2 (Yes). Case 2: Let m=5m = 5 and n=2n = 2. Then 5+2=7>05 + 2 = 7 > 0, but 52=25>4=225^2 = 25 > 4 = 2^2 (No).
Because Statement (2) allows both 'Yes' and 'No' outcomes, Statement (2) alone is not sufficient.

Key Concept

Data Sufficiency evaluation of absolute values and algebraic inequalities
Question 1418Question →

If xx and yy are real numbers such that x+y≠0x + y \neq 0, what is the value of x3+y3x+y\frac{x^3 + y^3}{x + y}?

(1) x2−xy+y2=12x^2 - xy + y^2 = 12
(2) x2+y2=20x^2 + y^2 = 20 and xy=8xy = 8

Show answer & explanation

Answer: EACH statement ALONE is sufficient.

Answer

EACH statement ALONE is sufficient.
Factoring the numerator using the sum of cubes identity x3+y3=(x+y)(x2−xy+y2)x^3 + y^3 = (x + y)(x^2 - xy + y^2) allows canceling (x+y)(x + y), since x+y≠0x + y \neq 0. Thus, the question asks for the value of x2−xy+y2x^2 - xy + y^2. Statement (1) directly states that x2−xy+y2=12x^2 - xy + y^2 = 12, which is sufficient. Statement (2) gives x2+y2=20x^2 + y^2 = 20 and xy=8xy = 8, so x2−xy+y2=(x2+y2)−xy=20−8=12x^2 - xy + y^2 = (x^2 + y^2) - xy = 20 - 8 = 12, which is also sufficient. Since each statement independently yields a unique value, the correct choice states that each statement alone is sufficient.

Step-by-Step Solution

1
Simplify and rephrase the target expression in the question stem.
Since x+y≠0x + y \neq 0, factor the numerator using the sum of cubes formula x3+y3=(x+y)(x2−xy+y2)x^3 + y^3 = (x + y)(x^2 - xy + y^2). The expression simplifies to (x+y)(x2−xy+y2)x+y=x2−xy+y2\frac{(x + y)(x^2 - xy + y^2)}{x + y} = x^2 - xy + y^2. The question target is equivalent to finding the value of x2−xy+y2x^2 - xy + y^2.
Simplifying the target expression before analyzing the statements eliminates unnecessary variables and reveals the exact algebraic value needed.
2
Evaluate Statement (1) independently.
Statement (1) gives x2−xy+y2=12x^2 - xy + y^2 = 12. Since this matches the simplified question target directly, the value is uniquely determined as 12.
Statement (1) provides the exact value of the rephrased target expression.
3
Evaluate Statement (2) independently.
Statement (2) provides x2+y2=20x^2 + y^2 = 20 and xy=8xy = 8. Substituting these into the target expression x2−xy+y2=(x2+y2)−xyx^2 - xy + y^2 = (x^2 + y^2) - xy gives 20−8=1220 - 8 = 12. The value is uniquely determined as 12.
Statement (2) supplies component values that combine to form the target expression uniquely.
4
Determine the final Data Sufficiency decision.
Because Statement (1) alone is sufficient and Statement (2) alone is sufficient, the correct choice is that EACH statement ALONE is sufficient.
Both statements independently yield a single, consistent answer to the rephrased question.

Key Concept

Question Stem Rephrasing and Algebraic Identity Simplification in Data Sufficiency
Question 1419Question →

In GMAT Critical Reasoning, evaluating the validity of an argument using the Test of Variance requires that applying two opposite extreme answers to a evaluation question (such as 'yes' and 'no') produces opposite impacts on the strength of the conclusion.

Show answer & explanation

Answer: True

Answer

True. The Test of Variance dictates that a key evaluation question must yield opposing logical effects (strengthening vs. weakening) when tested with opposite extreme responses.
The statement accurately summarizes the fundamental rule of the Test of Variance, where opposing extreme outcomes must exert opposite logical forces on the conclusion.

Step-by-Step Solution

1
Understand the core mechanism of evaluating argument validity.
To evaluate whether an argument's conclusion holds, one must identify missing information or assumptions that determine the argument's success.
The validity of an argument depends on whether key underlying conditions are met.
2
Apply the Test of Variance method.
Test both extreme logical answers (e.g., 100% vs 0%, or Yes vs No) to the proposed question.
If one extreme strengthens the conclusion and the other extreme weakens it, the question directly tests the argument's validity.

Key Concept

Evaluating Argument Validity and Test of Variance
Question 1420Question →

Despite widespread assertions by venture capitalists that quantum computing start-ups will achieve commercial viability within the next decade, commercial adoption of quantum processors will remain economically infeasible for the foreseeable future. Proponents often highlight recent laboratory breakthroughs in quantum error-correction protocols to support their optimistic timeline; however, translating these delicate laboratory demonstrations into mass-produced industrial hardware demands synthetic material purities currently impossible to achieve at scale. Therefore, although ongoing research in quantum information theory will surely yield profound insights for theoretical physics, it cannot overcome the overwhelming financial and physical barriers to near-term market viability.

Which of the following best expresses the main conclusion of the argument above?

Show answer & explanation

Answer: Widespread commercial adoption of quantum processing technology is unlikely to become economically viable in the near term.

Answer

The main conclusion of the argument is that widespread commercial adoption of quantum processing technology is unlikely to become economically viable in the near term.
The author's primary objective is to refute the optimistic timeline of venture capitalists and establish that commercial adoption of quantum processors will remain economically infeasible for the foreseeable future. The rest of the passage provides technical justification (material purity requirements) and addresses potential counterarguments or minor side benefits (theoretical physics insights) in service of this main claim.

Step-by-Step Solution

1
Analyze the argument structure and locate structural pivot words.
Identified the opening main claim ('commercial adoption... will remain economically infeasible'), followed by proponents' counter-evidence, a structural pivot ('however'), a supporting physical premise, and a concluding synthesis ('it cannot overcome...').
Understanding structural roles distinguishes the author's main assertion from evidence and concessions.
2
Evaluate the functional role of each sentence.
The statement about material purities serves as evidence explaining the physical barrier, while the statement about theoretical physics is a concession.
Subordinate claims and evidence exist to justify the primary thesis regarding economic infeasibility.
3
Match the primary thesis to the correct option.
The statement expressing that commercial adoption of quantum processing technology is unlikely to become economically viable in the near term accurately paraphrases the author's primary claim.
The correct option must state the ultimate assertion that all other premises and rebuttals serve to prove.

Key Concept

Identifying Main Conclusions in Complex Arguments
Estimated Time:2m 0s
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