All practice questions

Question 1541Question →

If $n$ is a real number, is $n$ an integer?

(1) $n^2 + 5n$ is an integer.

(2) $n^2 - n$ is an integer.

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

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

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

EACH statement ALONE is sufficient.

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

Show answer & explanation

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

Answer

Both statements together are sufficient, but neither statement alone is sufficient.

Evaluating each statement alone shows that non-integer real numbers can produce integer values for

n^2 + 5n

or

n^2 - n

. However, combining both statements gives

6n = (n^2 + 5n) - (n^2 - n)

, proving

n = \frac{k}{6}

for some integer

k

. Substituting

n = \frac{k}{6}

back into

n^2 - n

requires

\frac{k^2 - 6k}{36}

to be an integer, which forces

k(k-6)

to be divisible by 36. Examining remainders modulo 6 shows

k

must be a multiple of 6, ensuring

n

is an integer.

Step-by-Step Solution

1

Evaluate Statement (1) independently.

Statement (1) is NOT sufficient.

If

n^2 + 5n = p

where

p

is an integer,

n

can be an integer (e.g.,

n=1 \implies n^2+5n=6

) or a non-integer real number (e.g.,

n^2+5n-1=0 \implies n = \frac{-5 + \sqrt{29}}{2}

, which gives

n^2+5n=1

).

2

Evaluate Statement (2) independently.

Statement (2) is NOT sufficient.

If

n^2 - n = q

where

q

is an integer,

n

can be an integer (e.g.,

n=2 \implies n^2-n=2

) or a non-integer real number (e.g.,

n^2-n-1=0 \implies n = \frac{1 + \sqrt{5}}{2}

, which gives

n^2-n=1

).

3

Combine Statement (1) and Statement (2).

6n

is equal to an integer

k = p - q

, so

n = \frac{k}{6}

.

Subtracting

(n^2 - n = q)

from

(n^2 + 5n = p)

yields

6n = p - q

. Since

p

and

q

are integers,

k = p - q

must be an integer.

4

Substitute

n = \frac{k}{6}

back into Statement (2) to check integer constraints on

k

.

k

must be a multiple of 6, which implies

n = \frac{k}{6}

is an integer.

Substituting

n = \frac{k}{6}

into

n^2 - n = q

gives

\left(\frac{k}{6}\right)^2 - \frac{k}{6} = \frac{k^2 - 6k}{36} = q

. Thus,

36

must divide

k(k-6)

. Consequently,

6

divides

k(k-6)

. Since

k

and

k-6

have the same remainder

r

modulo 6,

k(k-6) \equiv r^2 \pmod 6

. For

r^2

to be divisible by 6 where

r \in \{0, 1, 2, 3, 4, 5\}

,

r

must be

0

. Hence,

k

is a multiple of 6, making

n

an integer.

Key Concept

Number properties of non-integer real variables vs. integer constraints when combining polynomial equations.

Question 1542Question →

In 2024, a financial services company replaced its entire fleet of gasoline-powered corporate vehicles with high-efficiency electric vehicles, expecting to lower its annual vehicle energy expenditures. Over the subsequent twelve months, the electricity cost per mile driven for the electric vehicles was 40 percent lower than the gasoline cost per mile driven for the former fleet, and the total annual mileage driven by employees remained constant. Nevertheless, the company's total annual expenditure on vehicle fuel and charging increased significantly over that period.

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

Unlike gasoline stations, which charged only for fuel, public fast-charging stations used by the fleet levied heavy hourly parking fees and peak-demand surcharges during every charging session.

Average gasoline prices in the metropolitan area where the fleet operates fell by 15 percent over the course of the twelve-month period.

The company qualified for federal clean-energy tax rebates that offset a portion of the initial purchase price of the new electric fleet.

Routine maintenance and battery inspection costs for electric vehicles are expected to increase as the vehicle fleet ages over the next five years.

Many employees reported that driving the electric vehicles was more convenient than driving the former gasoline-powered vehicles.

Show answer & explanation

Answer: Unlike gasoline stations, which charged only for fuel, public fast-charging stations used by the fleet levied heavy hourly parking fees and peak-demand surcharges during every charging session.

Answer

The apparent discrepancy is resolved by the option stating that public fast-charging stations used by the fleet levied heavy hourly parking fees and peak-demand surcharges during every charging session.

The correct answer accounts for both premises simultaneously. While the raw electricity consumed per mile was 40% cheaper than gasoline, the total cost of using charging stations included substantial ancillary charges (hourly parking and peak-demand surcharges) that were not present at gasoline stations. These extra fees raised the total charging expenditure above the previous gasoline spending even though total mileage remained identical.

Step-by-Step Solution

1

Identify the two contradictory facts in the stimulus.

Fact 1: The per-mile electricity cost of the new electric fleet was 40% lower than the per-mile gasoline cost of the old fleet, and total miles driven stayed constant.
Fact 2: Total annual expenditure on fleet fuel and charging increased significantly.

Resolving a paradox requires understanding why lower per-mile energy costs and constant distance traveled did not yield overall spending reductions.

2

Evaluate the answer choices to find a factor that explains how total charging costs could rise despite lower per-mile electricity rates.

The correct explanation introduces additional fees—such as hourly parking fees and demand surcharges per charging session—that are incurred during charging but are not captured in the per-mile electricity cost rate.

This additional fee structure reconciles both premises: per-mile energy consumption cost was lower, but total spending on charging sessions was higher due to auxiliary fees.

Key Concept

Resolving Paradoxes: Confounding Cost Factors and Total vs. Unit Cost Discrepancies

Question 1543Question →

Under the revised regulations of the Global Mineral Oversight Commission, any rare-earth mining facility operating in Zone 4 that extracts dysprosium is required to undergo a quarterly heavy-metals soil audit. Furthermore, every mining facility that undergoes a quarterly heavy-metals soil audit is automatically awarded an Eco-Compliance Certificate by the commission. Facility Alpha operates in Zone 4, but it has not been awarded an Eco-Compliance Certificate by the commission.

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

Facility Alpha does not extract dysprosium.

Facility Alpha underwent a quarterly heavy-metals soil audit but failed to meet the required environmental safety standards.

If a mining facility has been awarded an Eco-Compliance Certificate, that facility must extract dysprosium.

Facility Alpha extracts rare-earth elements other than dysprosium at its Zone 4 location.

No mining facility operating outside of Zone 4 is required to undergo a quarterly heavy-metals soil audit.

Show answer & explanation

Answer: Facility Alpha does not extract dysprosium.

Answer

Facility Alpha does not extract dysprosium.

Combining the premises yields the conditional rule: if a facility operates in Zone 4 and extracts dysprosium, it receives an Eco-Compliance Certificate. The contrapositive dictates that if a facility lacks an Eco-Compliance Certificate, it either does not operate in Zone 4 or does not extract dysprosium. Because the stimulus explicitly states that Facility Alpha does operate in Zone 4, it must logically follow that Facility Alpha does not extract dysprosium.

Step-by-Step Solution

1

Analyze the conditional premises provided in the stimulus.

Premise 1: (Operates in Zone 4 AND Extracts Dysprosium) → Undergoes quarterly soil audit. Premise 2: Undergoes quarterly soil audit → Awarded Eco-Compliance Certificate.

Establishing conditional relationships allows for logical chaining.

2

Chain the premises together to form a combined conditional statement.

(Operates in Zone 4 AND Extracts Dysprosium) → Awarded Eco-Compliance Certificate.

If A leads to B, and B leads to C, then A directly leads to C.

3

Formulate the contrapositive of the combined conditional statement.

NOT Awarded Eco-Compliance Certificate → NOT (Operates in Zone 4 AND Extracts Dysprosium) = (Does NOT operate in Zone 4) OR (Does NOT extract Dysprosium).

The contrapositive of any valid conditional statement is logically equivalent and must also be true.

4

Apply the facts about Facility Alpha to the contrapositive.

Facility Alpha has NOT been awarded an Eco-Compliance Certificate. Since Facility Alpha IS operating in Zone 4, the second condition of the OR clause must hold: Facility Alpha does NOT extract dysprosium.

Since the first disjunct (does not operate in Zone 4) is false, the second disjunct must be true for the contrapositive to hold.

Key Concept

Identifying Must-Be-True Statements and Valid Deductions

Question 1544Question →

If $x$ is a real number, is $x$ an integer?

(1) $x^2$ is an integer.
(2) $3x$ is an integer.

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

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

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

EACH statement ALONE is sufficient.

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

Show answer & explanation

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

Answer

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

Evaluating Statement (1) alone allows

x = \sqrt{2}

(not an integer) or

x = 2

(an integer), so Statement (1) is insufficient. Evaluating Statement (2) alone allows

x = \frac{1}{3}

(not an integer) or

x = 1

(an integer), so Statement (2) is insufficient. Combining both statements establishes that

x = \frac{m}{3}

for an integer

m

, and

x^2 = \frac{m^2}{9} = k

for an integer

k

. This requires

m^2 = 9k

, which implies

m

must be a multiple of 3. Hence

x

must be an integer, giving a definitive 'Yes' answer. Thus, both statements together are sufficient.

Step-by-Step Solution

1

Analyze the question stem constraint

x

is constrained to be a real number (

x \in \mathbb{R}

). The target question is a Yes/No question: 'Is

x

an integer?'

Without an explicit integer constraint in the stem, non-integer real numbers must be tested as counterexamples.

2

Evaluate Statement (1) alone

If

x = \sqrt{2}

, then

x^2 = 2

(an integer), but

x

is not an integer (Answer: No). If

x = 2

, then

x^2 = 4

(an integer), and

x

is an integer (Answer: Yes). Statement (1) is NOT sufficient.

Statement (1) allows both integer and non-integer values for

x

.

3

Evaluate Statement (2) alone

If

x = \frac{1}{3}

, then

3x = 1

(an integer), but

x

is not an integer (Answer: No). If

x = 1

, then

3x = 3

(an integer), and

x

is an integer (Answer: Yes). Statement (2) is NOT sufficient.

Statement (2) allows fractional values with a denominator of 3 as well as integers.

4

Evaluate Statements (1) and (2) together

From Statement (2),

x = \frac{m}{3}

for some integer

m

. Substituting into Statement (1) yields

x^2 = \left(\frac{m}{3}\right)^2 = \frac{m^2}{9} = k

, where

k

is an integer. Thus,

m^2 = 9k

. Since

9k

is a multiple of 9,

m^2

is divisible by 9, which means

m

must be a multiple of 3. Let

m = 3p

for some integer

p

. Then

x = \frac{3p}{3} = p

, which guarantees that

x

is an integer. The answer is a definitive 'Yes'. Statements (1) and (2) together are SUFFICIENT.

Combining both conditions restricts

x

to rational numbers whose square is an integer, forcing

x

to be an integer.

Key Concept

Number Properties and Integer Constraints in Data Sufficiency

Estimated Time:1m 30s

Question 1545Question →

If $m$ and $n$ are positive integers such that $m \neq n$ , is $\frac{m^2 + mn - 2n^2}{m - n}$ an even integer?

(1) $m + 3$ is an odd integer.
(2) $n$ is an odd integer.

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

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

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

EACH statement ALONE is sufficient.

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

Show answer & explanation

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

Answer

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

By factoring the numerator of the expression in the question stem,

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

. Since

2n

is always even,

m + 2n

is even if and only if

m

is even. Rephrasing the question target yields: "Is

m

an even integer?" Statement (1) tells us

m + 3

is odd, which means

m

must be even (giving a definitive "Yes" answer). Statement (2) provides information about

n

, which is irrelevant to whether

m

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

Step-by-Step Solution

1

Simplify the target expression in the question stem

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

Factoring the numerator simplifies the target expression for

m \neq n

.

2

Rephrase the target question using parity properties

Since

2n

is an even integer for any integer

n

,

m + 2n

is even if and only if

m

is an even integer. The rephrased question target is: "Is

m

an even integer?"

Simplifying the target question eliminates the variable

n

from the requirement.

3

Evaluate Statement (1):

m + 3

is an odd integer

If

m + 3

is odd, then

m = \text{odd} - 3 = \text{even}

. Thus,

m

is definitively even. Statement (1) alone is SUFFICIENT.

Subtracting an odd integer from an odd integer yields an even integer.

4

Evaluate Statement (2):

n

is an odd integer

Statement (2) gives information about

n

, but gives no information about

m

. Since the rephrased question depends solely on

m

, Statement (2) alone is INSUFFICIENT.

Knowing

n

does not determine whether

m

is even.

Key Concept

Algebraic factoring and target rephrasing in Data Sufficiency questions

Question 1546Question →

Under the guidelines of the Global Logistics Safety Board, any cargo vessel operating along Route K that carries hazardous liquid chemicals must be equipped with an automated leak-detection telemetry system. Furthermore, any cargo vessel equipped with an automated leak-detection telemetry system is subject to mandatory unannounced monthly inspections by maritime safety authorities. During the previous calendar year, the commercial vessel Vanguard operated along Route K and carried hazardous liquid chemicals.

Based on the information provided above, evaluate the truth value of the following statement: The Vanguard was subject to mandatory unannounced monthly inspections by maritime safety authorities.

True

False

Show answer & explanation

Answer: True

Answer

The statement is True.

Combining the premises yields a valid chain of logic: (Route K + Hazardous Chemicals) → Leak-Detection Telemetry System → Mandatory Unannounced Monthly Inspections. Since the Vanguard satisfies the initial conditions, the final outcome must logically hold true.

Step-by-Step Solution

1

Identify the initial factual premises regarding the specific vessel.

The vessel Vanguard operated on Route K and carried hazardous liquid chemicals.

This establishes the antecedent conditions for the first rule.

2

Apply the first conditional rule to the established premises.

The Vanguard was required to be equipped with an automated leak-detection telemetry system.

Rule 1 dictates that operating on Route K with hazardous liquid chemicals necessitates the leak-detection system.

3

Apply the second conditional rule to the result from Step 2.

The Vanguard was subject to mandatory unannounced monthly inspections by maritime safety authorities.

Rule 2 dictates that any vessel equipped with the leak-detection telemetry system is subject to these inspections.

Key Concept

Identifying Must-Be-True Statements and Valid Deductions via Conditional Logic Chains

Question 1547Question →

A regional logistics firm operates four distribution centers: Facility Alpha, Facility Beta, Facility Gamma, and Facility Delta. The table below displays the total tonnage of cargo processed and the average processing cost per ton (in dollars) at each facility during the last quarter:

Distribution Center	Total Tonnage Processed (tons)	Average Cost per Ton ($)
Facility Alpha	2,400	50
Facility Beta	1,600	40
Facility Gamma	$T$	80
Facility Delta	3,000	$C$

Is the overall average processing cost per ton across all four distribution centers combined less than $60?

(1) Facility Gamma processed 2,000 tons of cargo ( $T = 2,000$ ).
(2) Facility Delta's average cost per ton was $55 ($ C = 55$).

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

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

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

EACH statement ALONE is sufficient.

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

Show answer & explanation

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

Answer

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

The correct answer is the option stating that both statements together are sufficient, but neither alone is sufficient. By setting up the total cost and total tonnage expressions from the table, the question 'Is the overall average cost less than

60?' simplifies to the inequality 'Is

T + 150C < 11,800

?'. Neither statement alone supplies both variables (

T

and

C

), but together they specify

T = 2,000

and

C = 55

, giving

2,000 + 8,250 = 10,250 < 11,800$, which answers the question with a definitive 'Yes'.

Step-by-Step Solution

1

Set up the algebraic expression for the overall combined average processing cost.

Total Tonnage =

2,400 + 1,600 + T + 3,000 = 7,000 + T

.
Total Cost (

) =

(2,400 \times 50) + (1,600 \times 40) + (T \times 80) + (3,000 \times C) = 120,000 + 64,000 + 80T + 3,000C = 184,000 + 80T + 3,000C$.

Weighted average requires dividing total dollar expenditure across all facilities by total tonnage processed.

2

Rephrase the target question inequality.

Is

\frac{184,000 + 80T + 3,000C}{7,000 + T} < 60

?
Multiply by

(7,000 + T)

:

184,000 + 80T + 3,000C < 60(7,000 + T)

184,000 + 80T + 3,000C < 420,000 + 60T

20T + 3,000C < 236,000

Divide by 20:
Is

T + 150C < 11,800

?

Simplifying the target question establishes a direct numerical threshold to test against both statements.

3

Evaluate Statement (1) alone.

Given

T = 2,000

, the target inequality becomes: Is

2,000 + 150C < 11,800 \implies 150C < 9,800 \implies C < 65.33

?
Since

C

is unknown, if

C = 50

, the answer is Yes (

9,500 < 11,800

). If

C = 70

, the answer is No (

12,500 \not< 11,800

).

Statement (1) leaves

C

undetermined, allowing both 'Yes' and 'No' outcomes.

4

Evaluate Statement (2) alone.

Given

C = 55

, the target inequality becomes: Is

T + 150(55) < 11,800 \implies T + 8,250 < 11,800 \implies T < 3,550

?
Since

T

is unknown, if

T = 2,000

, the answer is Yes (

10,250 < 11,800

). If

T = 4,000

, the answer is No (

12,250 \not< 11,800

).

Statement (2) leaves

T

undetermined, allowing both 'Yes' and 'No' outcomes.

5

Evaluate Statements (1) and (2) together.

Combining

T = 2,000

and

C = 55

into the rephrased inequality:

T + 150C = 2,000 + 150(55) = 2,000 + 8,250 = 10,250

.
Since

10,250 < 11,800

, we obtain a definitive 'Yes'.

Both variables are uniquely determined, resolving the target inequality definitively.

Key Concept

Weighted Average Calculation and Data Sufficiency Inequality Rephrasing

Estimated Time:2m 0s

Question 1548Question →

Tab 1 (Strategy Memo):
The Operations Director claims that launching a automated customer chat assistant will increase overall customer satisfaction by resolving basic inquiries instantly.

Tab 2 (Pilot Program Results):
During a 3-month pilot of the automated chat assistant, 70% of users reported frustration due to unresolved inquiries, resulting in a 15% decrease in overall satisfaction ratings among participating customers.

Statement: The pilot program results in Tab 2 weaken the Operations Director's claim in Tab 1 regarding customer satisfaction.

True

False

Show answer & explanation

Answer: True

Answer

The statement is True. The empirical data in Tab 2 demonstrates a decline in customer satisfaction, which directly weakens the optimistic claim made in Tab 1.

The claim in Tab 1 predicts an increase in customer satisfaction. The pilot program data in Tab 2 shows an actual 15% decrease in satisfaction. Because real-world outcomes contradicting a prediction weaken the premise of that prediction, the statement correctly asserts that Tab 2 weakens Tab 1.

Step-by-Step Solution

1

Identify the core claim in Tab 1

The Operations Director claims that the automated chat assistant will increase customer satisfaction.

Understanding the baseline premise is necessary to determine what evidence supports or weakens it.

2

Analyze the evidence provided in Tab 2

Tab 2 reports that customer satisfaction decreased by 15% during the pilot program.

Evaluating actual performance data reveals whether the real-world outcome aligns with or contradicts the claim.

3

Synthesize across tabs to evaluate argument impact

A 15% drop in satisfaction directly contradicts the claim of increased satisfaction, thereby weakening the claim.

Cross-tab synthesis determines the logical relationship (strengthen vs. weaken) between source documents.

Key Concept

Evaluating Argument Support and Weakening Across Tabs

Question 1549Question →

In a GMAT Data Sufficiency Value question asking for the unique numerical value of a variable $x$ , if Statement (1) alone restricts $x$ to a set of two distinct real numbers and Statement (2) alone also restricts $x$ to a set of two distinct real numbers, then combining Statement (1) and Statement (2) is guaranteed to determine a unique value for $x$ .

True

False

Show answer & explanation

Answer: False

Answer

The statement is false because the intersection of the solution sets of two statements can contain multiple values, failing to yield a unique numerical answer.

The statement is false. Combining two Data Sufficiency statements requires finding the values common to both solution sets. If both statements individually narrow the variable to the exact same pair of values, combining them does not eliminate either value. Consequently, the combined statements still leave multiple possibilities and remain insufficient.

Step-by-Step Solution

1

Formulate the mathematical condition for combining Data Sufficiency statements

Evaluating Statement (1) and Statement (2) together requires finding the set of values that satisfy both statements simultaneously, which is the set intersection

S_1 \cap S_2

of their individual solution sets

S_1

and

S_2

.

Both statements are taken to be true at the same time when analyzing their combined sufficiency.

2

Analyze whether the intersection of two 2-element sets necessarily contains exactly one element

If

S_1 = \{2, 3\}

and

S_2 = \{2, 3\}

, then

S_1 \cap S_2 = \{2, 3\}

, which contains two elements rather than one.

Two independent DS statements can be algebraically redundant or share identical candidate roots without contradicting one another.

3

Determine the sufficiency outcome of the combined statements

Since

x

can still equal either

2

or

3

, the combined statements do not yield a single, unique value for

x

.

A Value-type Data Sufficiency question requires a single, unambiguous value for a condition to be deemed sufficient.

Key Concept

Statement Combination and Solution Set Intersection

Question 1550Question →

A medical research trial evaluated $100$ patients, each of whom experienced side effect A, side effect B, or both. Exactly $65$ patients experienced side effect A, and exactly $55$ patients experienced side effect B. What was the average (arithmetic mean) duration of side effect A, in days, among all patients who experienced side effect A?

(1) The average duration of side effect A for patients who experienced ONLY side effect A was $8$ days.
(2) The average duration of side effect A for patients who experienced BOTH side effects was $12$ days.

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

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

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

EACH statement ALONE is sufficient.

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

Show answer & explanation

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

Answer

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

The correct choice indicates that both statements combined provide sufficient data while neither is sufficient on its own. Using the standard inclusion-exclusion principle for two sets,

100 = 65 + 55 - N(\text{Both})

, giving exactly

20

patients with both side effects and

45

patients with only side effect A. Statement (1) provides the sum of durations for the

45

patients (

45 \times 8 = 360

), and Statement (2) provides the sum of durations for the

20

patients (

20 \times 12 = 240

). Neither statement alone gives both sums, but together they yield a total sum of

600

days for

65

patients, resulting in a single definitive mean of

\frac{600}{65}

days.

Step-by-Step Solution

1

Rephrase the question stem using overlapping sets principles.

Let

N(A) = 65

,

N(B) = 55

, and total patients

N(A \cup B) = 100

. Using the set formula

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

, we get

100 = 65 + 55 - N(A \cap B)

, so

N(A \cap B) = 20

.

Determining the exact count of patients in the overlap region (

A \cap B

) and the exclusive region (

A \text{ only}

) is necessary to set up the weighted average formula.

2

Calculate the subgroup sizes for patients experiencing side effect A.

Patients with ONLY side effect A =

N(A) - N(A \cap B) = 65 - 20 = 45

. Patients with BOTH side effects =

20

. Total patients with side effect A =

65

.

The total duration of side effect A across all

65

patients is the sum of durations from the

45

'only A' patients and the

20

'both' patients.

3

Evaluate Statement (1) independently.

Statement (1) states that the mean for the

45

'only A' patients is

8

days. The total duration for this group is

45 \times 8 = 360

days. However, the duration for the

20

'both' patients is unknown.

Without the total duration for the overlap group, the overall mean cannot be determined. Thus, Statement (1) alone is NOT sufficient.

4

Evaluate Statement (2) independently.

Statement (2) states that the mean for the

20

'both' patients is

12

days. The total duration for this group is

20 \times 12 = 240

days. However, the duration for the

45

'only A' patients is unknown.

Without the total duration for the exclusive A group, the overall mean cannot be determined. Thus, Statement (2) alone is NOT sufficient.

5

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

Combining both statements, the total duration for all

65

patients with side effect A is

360 + 240 = 600

days. The overall average duration is

\frac{600}{65} = \frac{120}{13}

days.

A unique numerical value is obtained for the target question, so both statements together are sufficient.

Key Concept

Combining overlapping set subgroup sizes with weighted arithmetic mean formulas to evaluate Data Sufficiency statements.

Estimated Time:2m 0s

Question 1551Question →

Corporate Strategist: Our firm's new expansion plan is guaranteed to achieve unprecedented commercial success over the next decade. We know this because any strategic initiative that establishes absolute market dominance in high-growth sectors inevitably delivers exceptional long-run profitability. Furthermore, achieving unprecedented commercial success is itself the essential defining feature of establishing absolute market dominance in these sectors. Therefore, the plan's financial objectives will certainly be realized.

Which of the following best describes the flaw in the corporate strategist's reasoning?

It treats a claim that merely restates the desired outcome in different terms as an independent premise supporting that outcome.

It confuses an intermediate goal of the expansion plan with its ultimate long-term strategic objective.

It assumes without evidence that broader macroeconomic conditions will remain stable throughout the ten-year period.

It mistakes a condition that is necessary for achieving market dominance for one that is sufficient to guarantee it.

It draws an ungrounded comparison between the current expansion plan and successful strategies implemented by competing firms.

Show answer & explanation

Answer: It treats a claim that merely restates the desired outcome in different terms as an independent premise supporting that outcome.

Answer

The argument commits circular reasoning by assuming the truth of its conclusion within its premises, rephrasing 'unprecedented commercial success' as 'establishing absolute market dominance' to justify its claim.

The strategist attempts to prove that the plan will achieve 'unprecedented commercial success' by asserting that market dominance leads to profitability, and then defining 'market dominance' as 'unprecedented commercial success'. Because the premise simply restates the conclusion in synonymous terms, the argument assumes the very point in question without offering independent justification.

Step-by-Step Solution

1

Identify the conclusion and premises of the argument.

Conclusion: The expansion plan is guaranteed to achieve unprecedented commercial success. Premise 1: Market dominance leads to long-run profitability. Premise 2: Unprecedented commercial success is the defining feature of market dominance.

Deconstructing the argument structure isolates the logical link between evidence and conclusion.

2

Analyze the relationship between Premise 2 and the Conclusion.

Premise 2 equates 'establishing absolute market dominance' directly with 'achieving unprecedented commercial success'.

By defining the supporting condition using the exact outcome it seeks to prove, the argument offers no independent evidence.

3

Match the logical vulnerability to the correct structural flaw description.

The argument assumes what it purports to prove (begging the question / circular reasoning).

Circular reasoning occurs when a premise relies on the validity of the conclusion it is intended to support.

Key Concept

Circular Reasoning (Begging the Question)

Question 1552Question →

If $a$ and $b$ are non-zero real numbers, what is the value of $\frac{a^2 + 9b^2}{ab}$ ?

(1) $a^2 - 3ab - 18b^2 = 0$
(2) $a > 0$ and $b > 0$

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

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

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

EACH statement ALONE is sufficient.

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

Show answer & explanation

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

Answer

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

The correct option is the choice stating that both statements together are sufficient, but neither statement alone is sufficient. Statement (1) produces two valid roots for the ratio of the variables, which lead to two different target values. Statement (2) specifies that both variables are positive, which eliminates the negative root and uniquely identifies the ratio and target value when combined with Statement (1).

Step-by-Step Solution

1

Rephrase the target expression in terms of the ratio

k = \frac{a}{b}

.

\frac{a^2 + 9b^2}{ab} = \frac{a^2}{ab} + \frac{9b^2}{ab} = \frac{a}{b} + 9\left(\frac{b}{a}\right) = k + \frac{9}{k}

. Finding the ratio

k = \frac{a}{b}

is sufficient to answer the question.

Dividing the numerator by the denominator simplifies the expression into a function of a single ratio variable.

2

Evaluate Statement (1) independently.

Divide

a^2 - 3ab - 18b^2 = 0

by

b^2

to get

\left(\frac{a}{b}\right)^2 - 3\left(\frac{a}{b}\right) - 18 = 0

, or

k^2 - 3k - 18 = 0

. Factoring yields

(k - 6)(k + 3) = 0

, so

k = 6

or

k = -3

. If

k = 6

, target value is

6 + \frac{9}{6} = 7.5

. If

k = -3

, target value is

-3 + \frac{9}{-3} = -6

.

Since two distinct numerical outcomes are possible for the target expression, Statement (1) alone is NOT sufficient.

3

Evaluate Statement (2) independently.

a > 0

and

b > 0

implies

k = \frac{a}{b} > 0

, but gives no numerical constraint on

k

.

Without an equation linking

a

and

b

, Statement (2) alone is NOT sufficient.

4

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

Statement (1) gives

k = 6

or

k = -3

. Statement (2) requires

k > 0

, eliminating

k = -3

and establishing

k = 6

uniquely. Thus, the target value is uniquely

7.5

.

Combining both statements uniquely determines the ratio

k

and therefore the target expression.

Key Concept

Rephrasing homogeneous algebraic expressions into single ratio variables and analyzing degree constraints in quadratic systems.

Question 1553Question →

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

(1) $x^2 + y^2 = 25 - 2xy$
(2) $x^3 + y^3 = 125 - 3xy(x+y)$

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

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

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

EACH statement ALONE is sufficient.

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

Show answer & explanation

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

Answer

Statement (2) ALONE is sufficient to determine that

x + y = 5

, but Statement (1) ALONE is not sufficient because it allows

x + y = 5

or

x + y = -5

.

Statement (2) alone allows us to factor the expression into

(x+y)^3 = 125

. Because real numbers have a unique real cube root,

x+y

must equal

5

. Thus, Statement (2) alone provides a single, definitive answer to the question stem. Statement (1) yields

(x+y)^2 = 25

, which leads to

x+y = 5

or

x+y = -5

, making it insufficient on its own.

Step-by-Step Solution

1

Rephrase the target question

The target is to find a unique numerical value for the expression

(x + y)

. We do not need individual values for

x

and

y

.

Data Sufficiency targets involving expressions often do not require solving for individual variables.

2

Evaluate Statement (1):

x^2 + y^2 = 25 - 2xy

Rearranging terms gives

x^2 + 2xy + y^2 = 25

, which factors as

(x + y)^2 = 25

. Taking the square root gives

x + y = 5

or

x + y = -5

.

Since there are two distinct real values for

x + y

, Statement (1) alone is insufficient.

3

Evaluate Statement (2):

x^3 + y^3 = 125 - 3xy(x+y)

Rearranging terms gives

x^3 + 3xy(x+y) + y^3 = 125

, which is the expanded form of

(x + y)^3 = 125

. Taking the cube root gives

x + y = 5

.

For real numbers, every real number has exactly one real cube root. Thus,

x + y = 5

uniquely. Statement (2) alone is sufficient.

Key Concept

Algebraic Rephrasing and Degree of Real Polynomial Identities

Estimated Time:2m 0s

Question 1554Question →

A corporate strategy consultant presented the following argument to executive leadership: 'Over the past two years, every department within our software development division that adopted flexible core working hours recorded a 25 percent increase in annual project delivery rate. By contrast, departments in the same division that maintained rigid, fixed-office schedules demonstrated no increase in delivery rate. Therefore, mandating flexible core working hours across our risk management division will guarantee a similar increase in productivity.'

Statement: The consultant's reasoning is vulnerable to criticism on the grounds that it assumes a policy correlated with productivity gains in one specialized operational environment will automatically produce identical results in a fundamentally different operational setting.

True

False

Show answer & explanation

Answer: True

Answer

True. The argument commits a flaw of unwarranted extrapolation across different operational contexts.

The statement accurately pinpoints the argument's flaw. The author takes evidence from software development and applies it directly to risk management without establishing that the two divisions function similarly enough for the policy to have the same effect.

Step-by-Step Solution

1

Deconstruct the argument premises and conclusion.

Premise: Flexible hours correlated with +25% delivery rate in software development. Conclusion: Mandating flexible hours in risk management will guarantee a similar increase.

Understanding the gap between the evidence domain and the target domain is essential for identifying reasoning vulnerabilities.

2

Analyze the logical leap between premise and conclusion.

The author assumes that because flexible hours worked in software development, they are sufficient to produce the same result in risk management.

Extrapolating results from one group or setting to a non-equivalent group without justification constitutes a scope shift flaw.

3

Evaluate the statement against the identified flaw.

The statement asserts that the argument relies on the unwarranted assumption that a policy successful in one specialized environment will yield identical results in a different setting, which matches the core flaw.

Determining whether the description of the flaw is accurate.

Key Concept

Extrapolation and Scope Shift Flaws

Question 1555Question →

To reduce patient wait times for critical care, a hospital network plans to replace its manual paper triage system with an automated algorithmic triage program. In a preliminary trial, the software analyzed patient vital signs and symptom checklists in under 15 seconds, compared to the 8 minutes required by triage nurses. Hospital management concludes that implementing this software across all its emergency departments will significantly decrease the average time critical-care patients wait before receiving treatment from an emergency physician. Which of the following, if true, most seriously weakens the hospital management's argument?

The automated triage software requires a high-speed server connection that occasionally experiences minor network latency during peak operating hours.

Emergency physicians reported that patients triaged by the software had more accurate preliminary diagnostic profiles than those triaged manually by nurses.

Hospital safety rules require an attending emergency physician to manually verify and sign off on any algorithm-generated triage score before a patient can be transferred to a critical care bed.

The introduction of the automated triage system will not reduce the average waiting time for non-urgent patients presenting with minor injuries.

Several triage nurses voiced concerns that automated software reduces human interaction during the initial intake consultation.

Show answer & explanation

Answer: Hospital safety rules require an attending emergency physician to manually verify and sign off on any algorithm-generated triage score before a patient can be transferred to a critical care bed.

Answer

The argument is most seriously weakened by the statement that hospital safety rules mandate an attending emergency physician to manually verify and sign off on algorithm-generated triage scores before transferring patients to critical care.

The argument assumes that reducing triage time from 8 minutes to 15 seconds will decrease overall wait times for critical-care patients. The option pointing out mandatory physician sign-off before transfer introduces a procedural dependency on physician availability. Because physicians must clear the triage result before transfer occurs, the delay is simply shifted to physician sign-off availability, directly undermining the conclusion that overall wait times will decrease.

Step-by-Step Solution

1

Identify the argument's premises and conclusion

Premise: Automated software triages patients in 15 seconds vs. 8 minutes for nurses. Conclusion: Implementing software will reduce average wait times for critical-care patients before seeing a physician.

The conclusion relies on the unstated assumption that saving time at the initial triage intake stage directly translates into faster overall access to a physician.

2

Formulate the central assumption

The argument assumes no downstream procedural bottlenecks will offset or eliminate the time saved during the 15-second triage phase.

To weaken a plan-to-goal argument, an option must show that an unstated obstacle or side effect prevents the plan from achieving its stated goal.

3

Evaluate the choices for a weakening factor

Mandating physician sign-off on the automated score prior to transfer introduces a bottleneck tied directly to physician availability, undermining the time savings.

If patients cannot proceed until a busy physician signs off, the total wait time to receive critical care remains constrained by physician availability rather than initial triage speed.

Key Concept

Weakening Arguments - Uncovering Downstream Bottlenecks

Question 1556Question →

The table below shows performance and operational metrics for six regional fulfillment hubs operated by a global logistics network:

Hub ID	Region	Automation Level	On-Time Delivery Rate (%)	Daily Order Volume (thousands)
H-101	North America	High	94.5%	120
H-102	Europe	Medium	91.0%	85
H-103	Asia-Pacific	High	96.2%	150
H-104	North America	High	89.5%	110
H-105	Europe	High	95.0%	95
H-106	Asia-Pacific	Medium	92.8%	140

A fulfillment hub qualifies for an operational excellence award if it meets all of the following three conditions:
- Its Automation Level is 'High'.
- Its On-Time Delivery Rate is at least 93.0%.
- Its Daily Order Volume is greater than 100 thousand orders.

What is the total combined Daily Order Volume (in thousands) of the fulfillment hubs that qualify for the operational excellence award?

270

365

380

475

615

Show answer & explanation

Answer: 270

Answer

270 thousand orders

To qualify for the award, a hub must satisfy three conditions simultaneously: Automation Level must be 'High', On-Time Delivery Rate must be at least 93.0%, and Daily Order Volume must exceed 100 thousand. Evaluating each hub row by row:
- H-101 meets all three criteria (High, 94.5% ≥ 93.0%, 120 > 100).
- H-102 fails on all criteria.
- H-103 meets all three criteria (High, 96.2% ≥ 93.0%, 150 > 100).
- H-104 fails the delivery rate condition (89.5% < 93.0%).
- H-105 fails the order volume condition (95 ≤ 100).
- H-106 fails the automation and delivery rate conditions.

The only qualifying hubs are H-101 and H-103. Summing their order volumes gives 120 + 150 = 270 thousand.

Step-by-Step Solution

1

Identify the compound filter criteria

Hubs must simultaneously satisfy: (1) Automation Level = 'High', (2) On-Time Delivery Rate ≥ 93.0%, and (3) Daily Order Volume > 100.

The stem specifies that a hub must meet all three conditions to qualify.

2

Evaluate each hub against all three criteria

H-101: High (Yes), 94.5% ≥ 93.0% (Yes), 120 > 100 (Yes) -> Qualifies.
H-102: Medium (No), 91.0% < 93.0% (No), 85 ≤ 100 (No) -> Does not qualify.
H-103: High (Yes), 96.2% ≥ 93.0% (Yes), 150 > 100 (Yes) -> Qualifies.
H-104: High (Yes), 89.5% < 93.0% (No), 110 > 100 (Yes) -> Does not qualify.
H-105: High (Yes), 95.0% ≥ 93.0% (Yes), 95 ≤ 100 (No) -> Does not qualify.
H-106: Medium (No), 92.8% < 93.0% (No), 140 > 100 (Yes) -> Does not qualify.

A row is excluded if even one conjunctive condition is violated.

3

Aggregate the target attribute for qualifying rows

Qualifying Hubs: H-101 (120 thousand) and H-103 (150 thousand).
Total Daily Order Volume = 120 + 150 = 270 thousand.

The question asks for the sum of Daily Order Volume for the qualifying subset.

Key Concept

Multi-Column Conjunctive Table Filtering

Estimated Time:1m 30s

Question 1557Question →

If $x$ is a positive real number, is $x$ an integer?

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

(2) $x^2 + \frac{1}{x}$ is an integer.

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

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

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

EACH statement ALONE is sufficient.

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

Show answer & explanation

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

Answer

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

The option stating that both statements together are sufficient is correct. Evaluating Statement (1) alone allows irrational solutions like

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

. Evaluating Statement (2) alone allows irrational roots of

x^3 - 3x + 1 = 0

. However, combining both statements reveals that

x^2 - x

is an integer. Subtracting the resulting linear system forces

x

to be a rational number. For any positive rational

x = \frac{p}{q}

in simplest form,

x + \frac{1}{x}

being an integer requires

p = 1

and

q = 1

, proving that

x = 1

, which is an integer.

Step-by-Step Solution

1

Evaluate Statement (1) independently.

Statement (1) is NOT sufficient.

If

x = 1

, then

x + \frac{1}{x} = 2

(an integer), and

x

is an integer. However, if

x = \frac{3 + \sqrt{5}}{2} \approx 2.618

(a positive real non-integer), then

x + \frac{1}{x} = \frac{3 + \sqrt{5}}{2} + \frac{3 - \sqrt{5}}{2} = 3

(an integer). Since

x

can be an integer or a non-integer, Statement (1) alone does not uniquely answer the question.

2

Evaluate Statement (2) independently.

Statement (2) is NOT sufficient.

If

x = 1

, then

x^2 + \frac{1}{x} = 2

(an integer), and

x

is an integer. If

x^2 + \frac{1}{x} = 3

, multiplying by

x

yields

x^3 - 3x + 1 = 0

. Evaluating

f(x) = x^3 - 3x + 1

shows

f(1) = -1 < 0

and

f(2) = 3 > 0

, so by the Intermediate Value Theorem, there exists a real root

x \in (1, 2)

, which is positive but not an integer. Hence, Statement (2) alone is insufficient.

3

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

Both statements together are SUFFICIENT.

Let

x + \frac{1}{x} = a

and

x^2 + \frac{1}{x} = b

, where

a

and

b

are integers. Subtracting Statement (1) from Statement (2) gives

\left(x^2 + \frac{1}{x}\right) - \left(x + \frac{1}{x}\right) = b - a \implies x^2 - x = k

, where

k = b - a

is an integer. From Statement (1),

x^2 - ax + 1 = 0

, and from the difference,

x^2 - x - k = 0

. Subtracting these two quadratic equations gives

(1 - a)x + (1 + k) = 0

. For any positive real

x

,

a = x + \frac{1}{x} \ge 2

, so

1 - a \neq 0

. Thus

x = \frac{1 + k}{a - 1}

. Because

k

and

a

are integers,

x

must be a rational number. Let

x = \frac{p}{q}

in lowest terms, where

p

and

q

are positive integers with

\gcd(p, q) = 1

. Then

x + \frac{1}{x} = \frac{p}{q} + \frac{q}{p} = \frac{p^2 + q^2}{pq} = a \implies p^2 + q^2 = a p q

. Since

p^2 = q(ap - q)

,

q

must divide

p^2

. But

\gcd(p, q) = 1

, so

q = 1

. Similarly,

p

must divide

q^2

, so

p = 1

. Thus

x = 1

, which is an integer. Both statements together definitively answer YES.

Key Concept

Deduce integer constraints and rationality by combining non-linear algebraic expressions for real variables in Data Sufficiency.

Question 1558Question →

If $p$ and $q$ are positive real numbers, is $p$ an integer?

(1) $p^2 + pq$ is an integer.
(2) $q^2 + pq$ is an integer.

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

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

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

EACH statement ALONE is sufficient.

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

Show answer & explanation

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

Answer

Statements (1) and (2) TOGETHER are NOT sufficient to determine if

p

is an integer.

The correct answer states that statements (1) and (2) together are not sufficient. Statement (1) alone allows

p = 0.5, q = 1.5

(giving

p^2+pq=1

) as well as

p = 1, q = 1

(giving

p^2+pq=2

). Statement (2) alone similarly allows

p = 0.5, q = 1.5

(giving

q^2+pq=3

) as well as

p = 1, q = 1

(giving

q^2+pq=2

). When both statements are evaluated together, adding the two equations reveals

p+q = \sqrt{(p^2+pq) + (q^2+pq)}

. Choosing

p^2+pq = 1

and

q^2+pq = 3

yields

p+q = 2

and

p = 0.5

, demonstrating that

p

does not have to be an integer even when both statements are satisfied.

Step-by-Step Solution

1

Analyze the Question Stem and Constraints

Variables

p

and

q

are positive real numbers (not constrained to integers). The goal is to determine if

p

must be an integer.

Recognizing that variables are real numbers prevents premature integer assumptions.

2

Evaluate Statement (1) independently:

p^2 + pq = p(p+q)

is an integer

Case 1: If

p = 1

and

q = 1

, then

p(p+q) = 1(2) = 2

(integer), so

p

IS an integer.
Case 2: If

p = 0.5

and

q = 1.5

, then

p(p+q) = 0.5(2) = 1

(integer), so

p

IS NOT an integer.
Statement (1) is INSUFFICIENT.

Testing non-integer real values tests whether the condition forces

p

to be an integer.

3

Evaluate Statement (2) independently:

q^2 + pq = q(p+q)

is an integer

Case 1: If

p = 1

and

q = 1

, then

q(p+q) = 1(2) = 2

(integer), so

p

IS an integer.
Case 2: If

p = 0.5

and

q = 1.5

, then

q(p+q) = 1.5(2) = 3

(integer), so

p

IS NOT an integer.
Statement (2) is INSUFFICIENT.

Statement (2) primarily constrains

q(p+q)

, leaving

p

unconstrained.

4

Evaluate Statements (1) and (2) Combined

Let

p(p+q) = k_1

and

q(p+q) = k_2

where

k_1, k_2

are positive integers.
Adding gives

(p+q)^2 = k_1 + k_2 \implies p+q = \sqrt{k_1 + k_2}

.
Thus,

p = \frac{k_1}{\sqrt{k_1 + k_2}}

.
Case 1: Let

k_1 = 1

and

k_2 = 3

. Then

p+q = \sqrt{4} = 2 \implies p = \frac{1}{2} = 0.5

(not an integer).
Case 2: Let

k_1 = 2

and

k_2 = 2

. Then

p+q = \sqrt{4} = 2 \implies p = \frac{2}{2} = 1

(an integer).
Since

p

can still be either an integer or a non-integer, both statements combined are INSUFFICIENT.

Algebraic combination yields

p = \frac{k_1}{\sqrt{k_1 + k_2}}

, which produces non-integers for appropriate choices of integer constants

k_1

and

k_2

.

Key Concept

Avoiding Implicit Integer Assumptions in Real-Valued Data Sufficiency

Estimated Time:2m 0s

Question 1559Question →

The table below displays the financial metrics for six companies operating across two industry sectors:

Company	Sector	Revenue ($M)	R&D Spend ($M)
Alpha Therapeutics	Biotech	120	30
Beta Bio	Biotech	200	50
Gamma Tech	IT	450	40
Delta Pharma	Biotech	160	40
Epsilon Health	Biotech	320	70
Zeta Systems	IT	500	60

Statement: For the subset of companies operating in the Biotech sector, the median annual revenue is $180 million.

True

False

Show answer & explanation

Answer: True

Answer

True

Filtering for Biotech companies gives revenues of

120M,

160M,

200M, and

320M. With 4 items, the median is the average of the 2nd and 3rd terms:

(160 + 200) / 2 = 180

million dollars.

Step-by-Step Solution

1

Filter the dataset by Sector

Identified the four Biotech companies: Alpha Therapeutics (

120M), Beta Bio (

200M), Delta Pharma (

160M), and Epsilon Health (

320M).

The statement specifically concerns companies in the Biotech sector.

2

Sort the filtered revenues in ascending order

Ordered revenues:

120M,

160M,

200M,

320M.

Determining the median requires data elements to be ordered sequentially.

3

Calculate the median for an even-count dataset

Middle two values are

160M and

200M. Median = (

160M +

200M) / 2 = $180M.

For an even number of observations

N = 4

, the median is the arithmetic mean of the

(N/2)^{\text{th}}

and

(N/2 + 1)^{\text{th}}

elements.

Key Concept

Calculating the median of a filtered table subset with an even number of items

Estimated Time:45s

Question 1560Question →

A tech consulting firm billed a client for a project completed by senior developers and junior developers. Senior developers were billed at a constant hourly rate of $x$ , and junior developers were billed at a constant hourly rate of $y$ . What was the average (arithmetic mean) hourly rate billed across all developer hours on the project?

(1) The total amount billed for senior developer hours was $50\%$ greater than the total amount billed for junior developer hours.
(2) Senior developers worked $40\%$ fewer total hours on the project than junior developers did.

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

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

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

EACH statement ALONE is sufficient.

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

Show answer & explanation

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

Answer

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

The correct choice is the option stating that both statements together are not sufficient. Statement (1) provides the ratio of total revenues between the two developer types, which is insufficient to find the overall average rate. Statement (2) provides the ratio of hours worked between senior and junior developers, which is also insufficient on its own. Combining both statements allows us to find that senior developers earn

2.5

times the hourly rate of junior developers and to express the average hourly rate as

1.5625 y

. However, because neither statement provides an absolute dollar amount for

x

,

y

, or total revenue, the numerical average hourly rate cannot be calculated.

Step-by-Step Solution

1

Rephrase the target question mathematically.

Let

h_s

be the senior hours and

h_j

be the junior hours. Total billing is

x h_s + y h_j

, and total hours is

h_s + h_j

. The average hourly rate is

A = \frac{x h_s + y h_j}{h_s + h_j}

. A specific numerical value for

A

is required.

Clarifying the target prevents confusing relative ratios with absolute dollar values.

2

Evaluate Statement (1) independently.

Statement (1) states that

x h_s = 1.5 y h_j

. Total billing equals

2.5 y h_j

, so

A = \frac{2.5 y h_j}{h_s + h_j}

.

Without knowing the ratio of hours

\frac{h_s}{h_j}

or specific dollar values for

x

or

y

,

A

cannot be determined. Statement (1) alone is insufficient.

3

Evaluate Statement (2) independently.

Statement (2) states that

h_s = 0.6 h_j

. Thus

A = \frac{0.6 x h_j + y h_j}{1.6 h_j} = \frac{0.6 x + y}{1.6}

.

Without the specific values of

x

and

y

,

A

cannot be determined. Statement (2) alone is insufficient.

4

Evaluate Statements (1) and (2) together.

From (2), substitute

h_s = 0.6 h_j

into (1):

x (0.6 h_j) = 1.5 y h_j \implies 0.6 x = 1.5 y \implies x = 2.5 y

. Substituting

x = 2.5 y

and

h_s = 0.6 h_j

into

A

yields

A = \frac{(2.5 y)(0.6 h_j) + y h_j}{1.6 h_j} = \frac{1.5 y h_j + y h_j}{1.6 h_j} = \frac{2.5 y}{1.6} = 1.5625 y

.

Combining both statements allows us to express the average rate as a multiple of

y

(

1.5625 y

), but since no concrete dollar value for

y

or

x

is provided in either statement, the exact average dollar rate cannot be calculated. Thus, both statements together are insufficient.

Key Concept

Distinguishing relative ratios from absolute numerical values in Data Sufficiency

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